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b(in)g(all)h(that)e(follo)o(ws\),)g(and)h(whic)o(h)g (are)f(lo)q(cated)h(at)f(p)q(oin)o(ts)h Fi(k)h Fn(=)f(\()p Fi(k)1349 2431 y Fv(1)1368 2424 y Fi(;)8 b(k)1413 2431 y Fv(2)1432 2424 y Fi(;)g(k)1477 2431 y Fv(3)1495 2424 y Fn(\))k(of)g(in)o(tegral)h(co)q(ordi-)37 2481 y(nates)i(in)h Fj(R)248 2463 y Fv(3)268 2481 y Fn(;)e(eac)o(h)h(n)o(ucleus)i (therefore)d(lies)j(at)d(the)h(cen)o(ter)g(of)f(a)h(cubic)h(unit)g (cell)h Fi(Q)1515 2488 y Fh(k)1549 2481 y Fn(=)12 b Fg(f)p Fn(\()p Fi(x)1663 2488 y Fv(1)1683 2481 y Fi(;)c(x)1730 2488 y Fv(2)1748 2481 y Fi(;)g(x)1795 2488 y Fv(3)1814 2481 y Fn(\))13 b Fg(2)37 2537 y Fj(R)76 2521 y Fv(3)96 2537 y Fn(;)8 b Fg(\000)157 2519 y Fv(1)p 157 2526 18 2 v 157 2553 a(2)194 2537 y Fi(<)15 b(x)270 2544 y Fh(i)295 2537 y Fg(\000)c Fi(k)365 2544 y Fh(i)393 2537 y Fg(\024)448 2519 y Fv(1)p 448 2526 V 448 2553 a(2)471 2537 y Fi(;)31 b(i)14 b Fn(=)h(1)p Fi(;)8 b Fn(2)p Fi(;)g Fn(3)p Fg(g)13 b Fn(\(with)j(the)h(con)o(v)o(en)o(tion)f(that)g Fi(Q)1308 2544 y Fv(0)1344 2537 y Fn(will)i(b)q(e)f(henceforth)f(denoted)37 2594 y(b)o(y)f Fi(Q)p Fn(\).)k(The)14 b(set)g(of)f(the)i(p)q(ositions)g (of)e(these)i(n)o(uclei)h(is)e(then)h(a)e(\014nite)i(subset)g(\003)e (of)h(the)g(set)g(of)g(all)h(p)q(oin)o(ts)37 2650 y(of)h(in)o(tegral)h (co)q(ordinates)f(that)g(is)h Fj(Z)678 2634 y Fv(3)712 2650 y Fg(\032)d Fj(R)800 2634 y Fv(3)820 2650 y Fn(.)23 b(The)17 b(union)g(of)f(all)h(cubic)h(cells)f(whose)g(cen)o(ter)f(is)h (a)e(p)q(oin)o(t)37 2706 y(of)i(\003)g(is)g(denoted)h(b)o(y)e (\000\(\003\);)h(its)g(v)o(olume)h(is)f(denoted)h(b)o(y)f Fg(j)p Fn(\003)p Fg(j)p Fn(.)24 b(Since)18 b(eac)o(h)f(cell)i(has)e (unit)g(v)o(olume)h(and)37 2763 y(eac)o(h)f(n)o(ucleus)g(is)g(of)f (unit)h(c)o(harge,)f Fg(j)p Fn(\003)p Fg(j)f Fn(is)h(also)h(the)f(n)o (um)o(b)q(er)h(of)e(n)o(uclei)j(and)f(the)f(total)g(n)o(uclear)h(c)o (harge.)p eop %%Page: 3 3 3 2 bop 37 50 a Fm(1.)24 b Fl(Intr)o(oduction)1470 b Fn(3)37 199 y(It)16 b(is)f(imp)q(ortan)o(t)g(to)g(note)g(that,)f(in)i 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Fn(ha)o(v)o(e)f(the)g(same)g(p)q (erio)q(dicit)o(y)j(as)d(the)g(assumed)g(p)q(erio)q(dicit)o(y)j(of)d (the)151 1101 y(n)o(uclei)h(?)37 1195 y(Let)h(us)f(precise)h(no)o(w)e (the)h(scop)q(e)h(of)e(this)i(article.)26 b(W)l(e)17 b(shall)h(not)e(deal)i(here)f(with)h(the)f(ph)o(ysical)h(bac)o(k-)37 1252 y(ground)h(of)g(this)g(theoretical)h(problem,)g(and)f(w)o(e)g(w)o (ould)g(rather)f(refer)h(the)g(reader)g(to)f(the)h(textb)q(o)q(oks)37 1308 y([6,)14 b(55])g(and)h(the)g(articles)g([27)o(,)f(28].)19 b(The)c(questions)g(w)o(e)g(tac)o(kle)g(here)g(are)f(indeed)j(close)e (to)f(questions)h(of)37 1365 y(in)o(terest)i(in)f(Solid)i(State)d(Ph)o (ysics,)i(b)q(oth)f(for)f(theoretical)i(and)f(n)o(umerical)h(purp)q (oses.)23 b(F)l(or)16 b(the)g(sak)o(e)f(of)37 1421 y(brevit)o(y)l(,)i (w)o(e)g(shall)g(not)g(detail)g(here)g(the)g(relationship)h(b)q(et)o(w) o(een)f(our)f(w)o(ork)g(and)h(Solid)h(State)e(Ph)o(ysics.)37 1478 y(W)l(e)g(only)g(men)o(tion)f(some)g(references)h(here,)f(namely)h ([23)o(,)f(42)o(],)g(and)g(also)g([2)o(,)g(6,)g(9)o(,)g(40,)f(43,)h(48) o(,)g(49)o(,)g(57].)108 1534 y(The)k(purp)q(ose)g(of)f(our)g(study)h (is)g(t)o(w)o(ofold:)26 b(\014rst,)19 b(w)o(e)f(w)o(an)o(t)f(to)h(c)o (hec)o(k)h(that)f(the)g(molecular)i(mo)q(del)37 1590 y(under)d(consideration)g(do)q(es)f(ha)o(v)o(e)g(the)g(exp)q(ected)h(b) q(eha)o(viour)g(in)g(the)f(limit)h(of)f(large)g(v)o(olumes;)g(second,) 37 1647 y(w)o(e)f(wish)g(to)e(set)i(a)f(limit)i(problem)f(that)e(is)i (w)o(ell-p)q(osed)h(mathematically)g(and)e(that)g(can)h(b)q(e)g (justi\014ed)g(in)37 1703 y(the)e(most)e(p)q(ossible)i(rigorous)f(w)o (a)o(y)f(\(in)h(particular)h(with)f(a)g(view)g(to)f(giv)o(e)i(a)e (sound)h(ground)g(for)g(n)o(umerical)37 1760 y(sim)o(ulations)17 b(of)d(the)i(condensed)g(phase\).)108 1816 y(The)j(mo)q(dels)g(w)o(e)f (shall)i(consider)f(here,)g(and)g(whic)o(h)g(are)g(describ)q(ed)h(in)f (Section)h(2)e(b)q(elo)o(w,)h(are)f(is-)37 1873 y(sued)i(from)e(Quan)o (tum)g(Chemistry)l(,)i(and)e(therefore,)h(they)g(are)f(mo)q(dels)i (that)d(are)i(only)g(v)m(alid)h(at)e(zero)37 1929 y(temp)q(erature.)i (F)l(rom)14 b(the)g(mathematical)h(viewp)q(oin)o(t,)h(the)e(thermo)q (dynamic)h(limit)h(problem)f(has)g(b)q(een)37 1986 y(extensiv)o(ely)k (studied,)e(in)h(the)f(zero)f(temp)q(erature)h(setting)g(as)f(w)o(ell)i (as)e(in)i(the)e(setting)h(of)f(strictly)i(p)q(os-)37 2042 y(itiv)o(e)g(temp)q(eratures)f(\(see)g([11)o(])g(or)f([13)o(])h (for)f(a)h(brief)g(historical)h(surv)o(ey\).)25 b(W)l(e)17 b(shall)h(only)g(men)o(tion)f(the)37 2099 y(ground-breaking)f(w)o(ork)e ([32)o(])g(b)o(y)g(Lieb)i(and)f(Simon)h(on)e(the)h(thermo)q(dynamic)g (limit)h(in)g(the)e(framew)o(ork)37 2155 y(of)k(the)f(Thomas-F)l(ermi)g (theory)g(\(TF)g(Theory)g(for)g(short\).)26 b(Indeed,)19 b(this)f(w)o(ork)e(w)o(as)h(at)g(the)g(origin)h(of)37 2211 y(our)c(o)o(wn)g(study)g([11)o(])g(on)g(the)g(Thomas-F)l(ermi-v)o (on)g(W)l(eizs\177)-23 b(ac)o(k)o(er)14 b(mo)q(del)h(\(TFW)e(mo)q(del)i (for)f(short\),)f(and)37 2268 y(has)j(largely)f(in\015uenced)j(our)d(w) o(ork.)108 2324 y(In)23 b([11)o(],)g(w)o(e)e(ha)o(v)o(e)h(pro)o(v)o(ed) f(that)h(the)g(three)g(questions)g Fj(\(i\){\(ii\){)q(\(ii)q(i\))k Fn(of)21 b(the)h(thermo)q(dynamic)37 2381 y(limit)g(problem)g(that)d(w) o(e)i(ha)o(v)o(e)f(raised)h(ab)q(o)o(v)o(e)f(can)h(b)q(e)g(answ)o(ered) f(p)q(ositiv)o(ely)i(in)g(the)e(setting)h(of)f(the)37 2437 y(TFW)c(theory)l(.)23 b(W)l(e)16 b(\014nd)h(it)g(useful)g(to)f (brie\015y)i(emphasize)f(the)g(fact)e(that)h(man)o(y)g(of)g(the)g (concepts)h(and)37 2494 y(tec)o(hniques)k(that)d(w)o(e)g(ha)o(v)o(e)h (used)g(in)h([11)o(])e(\(some)g(of)h(them)g(b)q(eing)h(inherited)g (from)e(Lieb)j(and)e(Simon,)37 2550 y(some)i(others)f(b)q(eing)i(in)o (tro)q(duced)g(b)o(y)e(us)h(in)g(order)f(to)g(treat)g(the)h(TFW)e (case\))i(will)h(b)q(e)f(useful)h(here.)37 2607 y(T)l(aking)c(b)q (ene\014t)g(from)f(the)g(w)o(ork)f(b)o(y)h(Lieb)i(and)e(Simon)h(who)f 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y(of)g(the)g(energy)g(p)q(er)h(unit)f(v)o(olume)h(in) f(the)g(thermo)q(dynamic)h(limit.)42 b(W)l(e)22 b(shall)h(nev)o (ertheless)g(pro)o(v)o(e)37 1441 y(the)18 b(con)o(v)o(ergence)h(of)e (the)h(energy)g(p)q(er)g(unit)h(v)o(olume)f(in)h(the)f(thermo)q (dynamic)g(limit)i(for)d(a)g(simpli\014ed)37 1498 y(Hartree-F)l(o)q(c)o (k)j(mo)q(del,)h(whose)f(energy)g(functional)h(is)f(con)o(v)o(ex)g (\(namely)g(the)g(reduced)h(Hartree-F)l(o)q(c)o(k)37 1554 y(mo)q(del,)16 b(treated)f(in)h(Section)g(4\).)108 1611 y(As)d(far)g(as)g(the)g(Hartree-F)l(o)q(c)o(k)g(mo)q(del)h(is)g (concerned,)g(w)o(e)f(shall)i(suggest)d(a)h(p)q(erio)q(dic)j(problem)e (candi-)37 1667 y(date)e(to)f(b)q(e)i(the)f(thermo)q(dynamic)g(limit)i (\(see)e(Section)g(5\).)18 b(W)l(e)12 b(shall)h(pro)o(v)o(e)f(that)f (this)h(p)q(erio)q(dic)i(problem)37 1724 y(is)k(mathematically)h(w)o (ell-p)q(osed.)28 b(By)17 b(the)h(w)o(a)o(y)l(,)f(it)g(is)h(w)o(orth)f (emphasizing)i(the)e(fact)g(that)g(the)g(Euler-)37 1780 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(osition)g(to)f(in)o(tro)q(duce)h(the)f(molecular)h(mo)q(dels)g(w)o(e)f (shall)i(deal)f(with.)108 2596 y(The)e(Hartree-F)l(o)q(c)o(k)e(mo)q (del,)i(whic)o(h)h(is)e(the)h(most)e(commonly)i(used)g(mo)q(del)g(in)g (Quan)o(tum)f(Molecular)37 2652 y(Chemistry)j([41)o(])f(can)g(b)q(e)h (written)f(as)g(follo)o(ws)561 2765 y Fi(I)585 2746 y Fh(H)s(F)581 2776 y Fv(\003)658 2765 y Fn(=)e(inf)5 b Fg(f)p Fi(E)822 2746 y Fh(H)s(F)819 2776 y Fv(\003)881 2765 y Fn(\()p Fi(K)s Fn(\))10 b(+)1019 2734 y(1)p 1019 2755 23 2 v 1019 2796 a(2)1047 2765 y Fi(U)1078 2772 y Fv(\003)1117 2765 y Fn(;)20 b Fi(K)15 b Fg(2)e(K)1282 2772 y Fv(\003)1308 2765 y Fg(g)p Fi(;)438 b Fn(\(2.3\))p eop %%Page: 6 6 6 5 bop 53 50 a Fm(2.)23 b Fl(General)17 b(setting)e(of)h(the)g(models) g(and)h(main)f(resul)m(ts)699 b Fn(6)37 199 y(where)16 b(the)f(set)g(of)g(minimization)i Fg(K)680 206 y Fv(\003)722 199 y Fn(consists)e(of)g(self-adjoin)o(t)h(op)q(erators)e Fi(K)k Fn(on)d Fi(L)1527 183 y Fv(2)1547 199 y Fn(\()p Fj(R)1604 183 y Fv(3)1623 199 y Fn(\))g(suc)o(h)h(that)372 298 y Fg(K)407 305 y Fv(\003)446 298 y Fn(=)d Fg(f)p Fn(0)f Fg(\024)h Fi(K)i Fg(\024)e Fj(1)8 b Fi(;)15 b Fn(T)l(r)g Fi(K)g Fn(=)e Fg(j)p Fn(\003)p Fg(j)p Fi(;)h Fn(T)l(r)1059 261 y Fe(\002)1078 298 y Fn(\()p Fg(\000)p Fn(\001)c Fg(\000)h Fi(V)1251 305 y Fv(\003)1278 298 y Fn(\))p Fi(K)1338 261 y Fe(\003)1369 298 y Fi(<)i Fn(+)p Fg(1g)p Fi(;)249 b Fn(\(2.4\))37 397 y(with)16 b Fj(1)f Fn(denoting)h(the)f(iden)o(tit)o(y)h(on)g Fi(L)710 380 y Fv(2)729 397 y Fn(\()p Fj(R)786 380 y Fv(3)806 397 y Fn(\).)j(The)d(energy)f(functional)h Fi(E)1345 380 y Fh(H)s(F)1342 410 y Fv(\003)1421 397 y Fn(in)g(\(2.3\))d(is)j(giv)o (en)g(b)o(y)72 525 y Fi(E)109 506 y Fh(H)s(F)106 536 y Fv(\003)169 525 y Fn(\()p Fi(K)s Fn(\))c(=)h(T)l(r)354 488 y Fe(\002)373 525 y Fn(\()p Fg(\000)p Fn(\001)d Fg(\000)g Fi(V)545 532 y Fv(\003)572 525 y Fn(\))p Fi(K)632 488 y Fe(\003)661 525 y Fn(+)711 494 y(1)p 711 515 23 2 v 711 556 a(2)759 463 y Fe(Z)5 b(Z)724 597 y Fd(R)754 588 y Ft(3)772 597 y Fu(\002)p Fd(R)829 588 y Ft(3)859 494 y Fi(\032)p Fn(\()p Fi(x;)j(x)p Fn(\))g Fi(\032)p Fn(\()p Fi(y)r(;)g(y)r Fn(\))p 859 515 266 2 v 927 556 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1137 525 y Fi(dxdy)h Fg(\000)1295 494 y Fn(1)p 1295 515 23 2 v 1295 556 a(2)1343 463 y Fe(Z)5 b(Z)1308 597 y Fd(R)1338 588 y Ft(3)1355 597 y Fu(\002)p Fd(R)1412 588 y Ft(3)1443 494 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1599 478 y Fv(2)p 1443 515 174 2 v 1465 556 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1629 525 y Fi(dxdy)r(;)42 b Fn(\(2.5\))37 686 y(with)20 b Fi(\032)p Fn(\()p Fg(\001)p Fi(;)8 b Fg(\001)p Fn(\))17 b(denoting)j(the)g(k)o(ernel)g(of)e(the)i(Hilb)q(ert-Sc)o(hmidt)h(op)q (erator)e Fi(K)s Fn(.)31 b(Let)20 b(us)f(no)o(w)g(de\014ne)h(the)37 742 y(v)m(arious)c(quan)o(tities)g(that)f(app)q(ear)g(in)h(the)f(ab)q (o)o(v)o(e)g(de\014nition)i(of)e(the)g(Hartree-F)l(o)q(c)o(k)g(mo)q (del.)108 799 y(The)i(op)q(erator)f Fi(K)k Fn(is)d(the)g(so-called)h Ff(\(r)n(e)n(duc)n(e)n(d\))f(one-p)n(article)g(density)g(matrix)p Fn(.)26 b(F)l(rom)16 b(the)g(general)37 855 y(theory)21 b(of)f(trace-class)g(op)q(erators)g(on)g Fi(L)774 839 y Fv(2)794 855 y Fn(\()p Fj(R)851 839 y Fv(3)870 855 y Fn(\))g(\(see,)i(for)e(example,)i([44)o(]\),)f(an)o(y)f(op)q(erator)f Fi(K)24 b Fn(in)d Fg(K)1849 862 y Fv(\003)37 912 y Fn(admits)h(a)f (complete)h(set)f(of)g(eigenfunctions)i(\()p Fi(')921 919 y Fh(n)943 912 y Fn(\))961 919 y Fh(n)p Fu(\025)p Fv(1)1051 912 y Fn(in)f Fi(H)1152 895 y Fv(1)1171 912 y Fn(\()p Fj(R)1228 895 y Fv(3)1248 912 y Fn(\))f(asso)q(ciated)g(to)g (the)g(eigen)o(v)m(alues)37 968 y(0)13 b Fg(\024)g Fi(\025)147 975 y Fh(n)183 968 y Fg(\024)g Fn(1)h(\(coun)o(ted)g(with)g(m)o (ultiplicit)o(y\).)22 b(Th)o(us)14 b(w)o(e)f(ma)o(y)h(decomp)q(ose)g Fi(K)j Fn(along)d(suc)o(h)g(an)g(eigen)o(basis)37 1024 y(of)h Fi(L)120 1008 y Fv(2)140 1024 y Fn(\()p Fj(R)197 1008 y Fv(3)217 1024 y Fn(\),)f(in)i(suc)o(h)g(a)e(w)o(a)o(y)h(that)f (its)h(Hilb)q(ert-Sc)o(hmidt)j(k)o(ernel)e(ma)o(y)f(b)q(e)g(written)h (as)672 1130 y Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))k(=)862 1087 y Fe(X)862 1185 y Fh(n)p Fu(\025)p Fv(1)936 1130 y Fi(\025)962 1137 y Fh(n)993 1130 y Fi(')1023 1137 y Fh(n)1046 1130 y Fn(\()p Fi(x)p Fn(\))c Fi(')1146 1111 y Fu(\003)1146 1141 y Fh(n)1169 1130 y Fn(\()p Fi(y)r Fn(\))p Fi(:)37 1273 y Fn(Owing)19 b(to)d(the)i(fact)e(that)h Fi(K)j Fn(is)e(trace-class,)f(the)h(corresp)q(onding)g(densit)o(y)g(is) g(w)o(ell-de\014ned)i(as)d(a)g(non-)37 1329 y(negativ)o(e)f(function)g (in)g Fi(L)479 1313 y Fv(1)499 1329 y Fn(\()p Fj(R)556 1313 y Fv(3)575 1329 y Fn(\))f(through)709 1435 y Fi(\032)p Fn(\()p Fi(x;)8 b(x)p Fn(\))j(=)901 1392 y Fe(X)900 1490 y Fh(n)p Fu(\025)p Fv(1)974 1435 y Fi(\025)1000 1442 y Fh(n)1032 1435 y Fg(j)p Fi(')1075 1442 y Fh(n)1098 1435 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1173 1416 y Fv(2)1191 1435 y Fi(;)37 1583 y Fn(and)16 b(T)l(r)f Fi(K)i Fn(=)d Fg(j)p Fn(\003)p Fg(j)e Fn(=)412 1546 y Fe(R)433 1599 y Fd(R)463 1589 y Ft(3)491 1583 y Fi(\032)p Fn(\()p Fi(x;)c(x)p Fn(\))g Fi(dx)j Fn(=)741 1549 y Fe(P)789 1596 y Fh(n)p Fu(\025)p Fv(1)865 1583 y Fi(\025)891 1590 y Fh(n)915 1583 y Fn(.)22 b(Moreo)o(v)o(er,)14 b(according)i(to)f(this)h(sp)q (ectral)g(decomp)q(o-)37 1639 y(sition)g(of)f Fi(K)s Fn(,)g(w)o(e)g(ma)o(y)f(giv)o(e)i(a)f(sense)g(to)580 1756 y(T)l(r[)p Fg(\000)p Fn(\001)8 b Fi(K)s Fn(])k(=)836 1713 y Fe(X)836 1811 y Fh(n)p Fu(\025)p Fv(1)910 1756 y Fi(\025)936 1763 y Fh(n)967 1694 y Fe(Z)993 1797 y Fd(R)1023 1788 y Ft(3)1057 1756 y Fg(jr)p Fi(')1138 1763 y Fh(n)1161 1756 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1236 1737 y Fv(2)1262 1756 y Fi(dx;)457 b Fn(\(2.6\))37 1905 y(while)17 b(T)l(r[)p Fi(V)242 1912 y Fv(\003)276 1905 y Fi(K)s Fn(])12 b(=)391 1871 y Fe(P)439 1919 y Fh(n)p Fu(\025)p Fv(1)516 1905 y Fi(\025)542 1912 y Fh(n)573 1869 y Fe(R)595 1921 y Fd(R)625 1912 y Ft(3)659 1905 y Fi(V)685 1912 y Fv(\003)713 1905 y Fn(\()p Fi(x)p Fn(\))c Fg(j)p Fi(')826 1912 y Fh(n)847 1905 y Fn(\()p Fi(x)p Fn(\))p Fg(j)922 1889 y Fv(2)949 1905 y Fi(dx)k Fn(=)1059 1869 y Fe(R)1080 1921 y Fd(R)1110 1912 y Ft(3)1145 1905 y Fi(V)1171 1912 y Fv(\003)1198 1905 y Fn(\()p Fi(x)p Fn(\))c Fi(\032)p Fn(\()p Fi(x;)g(x)p Fn(\))g Fi(dx)p Fn(.)108 1987 y(It)16 b(is)g(a)g(standard)f(fact)h([30)o(])f(that)g(this)i(form)o(ulation)f (of)f(the)h(Hartree-F)l(o)q(c)o(k)f(problem)i(is)f(equiv)m(alen)o(t)37 2043 y(to)f(the)g(follo)o(wing)h(one,)f(whic)o(h)h(migh)o(t)f(b)q(e)h (more)f(familiar)h(to)f(the)g(reader)77 2164 y Fi(I)101 2146 y Fh(H)s(F)97 2176 y Fv(\003)174 2164 y Fn(=)e(inf)277 2100 y Fe(\032)311 2164 y Fi(E)348 2146 y Fh(H)s(F)345 2176 y Fv(\003)408 2164 y Fn(\()p Fi(')456 2171 y Fv(1)475 2164 y Fn(;)8 b Fi(:::)p Fn(;)g Fi(')586 2174 y Fu(j)p Fv(\003)p Fu(j)629 2164 y Fn(\))i(+)708 2134 y(1)p 708 2154 23 2 v 708 2195 a(2)735 2164 y Fi(U)766 2171 y Fv(\003)805 2164 y Fn(;)20 b Fi(')868 2171 y Fh(i)895 2164 y Fg(2)13 b Fi(H)980 2146 y Fv(1)999 2164 y Fn(\()p Fj(R)1056 2146 y Fv(3)1075 2164 y Fn(\))p Fi(;)1114 2102 y Fe(Z)1139 2206 y Fd(R)1169 2196 y Ft(3)1196 2164 y Fi(')1226 2171 y Fh(i)1240 2164 y Fi(')1270 2146 y Fu(\003)1270 2176 y Fh(j)1302 2164 y Fn(=)g Fi(\016)1370 2171 y Fh(i;j)1410 2164 y Fi(;)8 b Fn(1)k Fg(\024)h Fi(i;)8 b(j)13 b Fg(\024)g(j)p Fn(\003)p Fg(j)1689 2100 y Fe(\033)1722 2164 y Fi(;)47 b Fn(\(2.7\))79 2399 y Fi(E)116 2380 y Fh(H)s(F)113 2410 y Fv(\003)176 2399 y Fn(\()p Fi(')224 2406 y Fv(1)243 2399 y Fn(;)8 b Fi(:::)p Fn(;)g Fi(')354 2408 y Fu(j)p Fv(\003)p Fu(j)397 2399 y Fn(\))13 b(=)486 2338 y Fu(j)p Fv(\003)p Fu(j)476 2356 y Fe(X)480 2454 y Fh(i)p Fv(=1)561 2337 y Fe(Z)587 2440 y Fd(R)617 2431 y Ft(3)644 2399 y Fg(jr)p Fi(')725 2406 y Fh(i)738 2399 y Fg(j)751 2380 y Fv(2)781 2399 y Fg(\000)826 2337 y Fe(Z)852 2440 y Fd(R)882 2431 y Ft(3)909 2399 y Fi(V)935 2406 y Fv(\003)962 2399 y Fn(\()p Fi(x)p Fn(\))8 b Fi(\032)p Fn(\()p Fi(x)p Fn(\))g Fi(dx)543 2543 y Fn(+)594 2513 y(1)p 594 2533 V 594 2575 a(2)629 2481 y Fe(Z)d(Z)705 2585 y Fd(R)735 2575 y Ft(3)752 2585 y Fu(\002)p Fd(R)809 2575 y Ft(3)842 2513 y Fi(\032)p Fn(\()p Fi(x)p Fn(\))j Fi(\032)p Fn(\()p Fi(y)r Fn(\))p 842 2533 176 2 v 864 2575 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1030 2543 y Fi(dxdy)g Fg(\000)1188 2513 y Fn(1)p 1188 2533 23 2 v 1188 2575 a(2)1223 2481 y Fe(Z)5 b(Z)1299 2585 y Fd(R)1329 2575 y Ft(3)1346 2585 y Fu(\002)p Fd(R)1403 2575 y Ft(3)1436 2513 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1592 2496 y Fv(2)p 1436 2533 174 2 v 1458 2575 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1623 2543 y Fi(dxdy)r(;)48 b Fn(\(2.8\))37 2683 y(where)18 b Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))16 b(=)369 2649 y Fe(P)417 2659 y Fu(j)p Fv(\003)p Fu(j)417 2697 y Fh(i)p Fv(=1)484 2683 y Fi(')514 2690 y Fh(i)528 2683 y Fn(\()p Fi(x)p Fn(\))p Fi(')620 2690 y Fh(i)633 2683 y Fn(\()p Fi(y)r Fn(\))693 2667 y Fu(\003)712 2683 y Fn(,)i Fi(\032)p Fn(\()p Fi(x)p Fn(\))f(=)g Fi(\032)p Fn(\()p Fi(x;)8 b(x)p Fn(\))16 b(=)1097 2649 y Fe(P)1145 2659 y Fu(j)p Fv(\003)p Fu(j)1145 2697 y Fh(i)p Fv(=1)1212 2683 y Fg(j)p Fi(')1255 2690 y Fh(i)1268 2683 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1343 2667 y Fv(2)1362 2683 y Fn(.)27 b(This)19 b(equiv)m(alence)h(means)37 2740 y(that)11 b(ev)o(ery)f(minimizer)j(of)e(\(2.3\){\(2.5)o(\))d(is)j(a)g(pro)s (jector)e(with)i(\014nite)h(rank)f Fg(j)p Fn(\003)p Fg(j)p Fn(.)17 b(In)11 b(this)h(latter)e(form)o(ulation)37 2796 y(the)15 b Fi(')145 2803 y Fh(i)159 2796 y Fn('s)e(are)h(in)o (terpreted)h(as)f(the)g(electronic)i(w)o(a)o(v)o(e-functions.)j(Let)c (us)f(observ)o(e)g(that)g(the)g(form)o(ulation)p eop %%Page: 7 7 7 6 bop 53 50 a Fm(2.)23 b Fl(General)17 b(setting)e(of)h(the)g(models) g(and)h(main)f(resul)m(ts)699 b Fn(7)37 199 y(in)14 b(terms)e(of)g (densit)o(y)h(matrices)f(is)h(more)f(in)o(trinsic,)j(and)d(therefore)g (sometimes)h(more)f(con)o(v)o(enien)o(t)h(to)f(use)37 256 y(than)g(the)f(second)h(one.)19 b(Indeed,)13 b(for)e(ev)o(ery)g (unitary)h(transform)e Fi(U)16 b Fn(in)d Fj(C)1292 239 y Fu(j)p Fv(\003)p Fu(j)1338 256 y Fn(,)f(and)g(for)f(ev)o(ery)g (orthonormal)37 312 y(family)16 b(\()p Fi(')224 319 y Fh(i)237 312 y Fn(\))255 321 y Fv(1)p Fu(\024)p Fh(i)p Fu(\024j)p Fv(\003)p Fu(j)400 312 y Fn(in)g Fi(H)495 296 y Fv(1)514 312 y Fn(\()p Fj(R)571 296 y Fv(3)590 312 y Fn(\))614 296 y Fu(j)p Fv(\003)p Fu(j)660 312 y Fn(,)e(w)o(e)h(ob)o(viously)g(ha)o(v)o(e)f Fi(E)1095 296 y Fh(H)s(F)1092 326 y Fv(\003)1155 312 y Fn(\()p Fi(U)5 b Fn(\()p Fi(')1257 319 y Fv(1)1276 312 y Fn(;)j Fi(:::)p Fn(;)g Fi(')1387 321 y Fu(j)p Fv(\003)p Fu(j)1430 312 y Fn(\)\))k(=)h Fi(E)1563 296 y Fh(H)s(F)1560 326 y Fv(\003)1623 312 y Fn(\()p Fi(')1671 319 y Fv(1)1691 312 y Fn(;)8 b Fi(:::)p Fn(;)g Fi(')1802 321 y Fu(j)o Fv(\003)p Fu(j)1845 312 y Fn(\),)37 369 y(while)16 b(the)f(densit)o(y)g (matrices)f(that)f(are)h(resp)q(ectiv)o(ely)i(asso)q(ciated)f(to)e Fi(U)5 b Fn(\()p Fi(')1343 376 y Fv(1)1362 369 y Fn(;)j Fi(:::)p Fn(;)g Fi(')1473 378 y Fu(j)p Fv(\003)p Fu(j)1517 369 y Fn(\))14 b(and)g(\()p Fi(')1684 376 y Fv(1)1703 369 y Fn(;)8 b Fi(:::)p Fn(;)g Fi(')1814 378 y Fu(j)p Fv(\003)p Fu(j)1857 369 y Fn(\))37 425 y(are)15 b(the)h(same.)108 482 y(It)22 b(is)g(of)f(course)g(straigh)o(tforw)o(ard)f(to)h(deduce)h (from)f(the)h(p)q(oin)o(t)g(n)o(uclei)h(setting)f(\(2.3\){\(2.5\))c (the)37 538 y(analogous)d(smeared)g(n)o(uclei)j(setting)d(for)f(the)i (HF)f(problem;)g(namely)79 674 y Fi(I)103 652 y Fh(m;H)s(F)99 688 y Fv(\003)217 674 y Fn(=)e(inf)320 610 y Fe(\032)354 674 y Fn(T)l(r)401 637 y Fe(\002)420 674 y Fg(\000)p Fn(\001)p Fi(K)535 637 y Fe(\003)564 674 y Fn(+)615 643 y(1)p 615 664 23 2 v 615 705 a(2)642 674 y Fi(D)q Fn(\()p Fi(\032)e Fg(\000)f Fi(m)818 681 y Fv(\003)845 674 y Fi(;)e(\032)h Fg(\000)i Fi(m)984 681 y Fv(\003)1010 674 y Fn(\))f Fg(\000)1088 643 y(j)p Fn(\003)p Fg(j)p 1088 664 57 2 v 1106 705 a Fn(2)1150 674 y Fi(D)q Fn(\()p Fi(m;)e(m)p Fn(\))874 803 y Fg(\000)924 772 y Fn(1)p 924 792 23 2 v 924 834 a(2)960 741 y Fe(Z)d(Z)1035 844 y Fd(R)1065 834 y Ft(3)1083 844 y Fu(\002)p Fd(R)1140 834 y Ft(3)1173 772 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1329 755 y Fv(2)p 1173 792 174 2 v 1194 834 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1359 803 y Fi(dxdy)j Fn(;)53 b Fi(K)15 b Fg(2)e(K)1667 810 y Fv(\003)1693 738 y Fe(\033)1727 803 y Fi(:)42 b Fn(\(2.9\))37 930 y(W)l(e)17 b(also)f(remark)g(that)f(the)h(equiv)m(alence)j(with)e(a)f (standard)g(form)f(of)h(the)g(t)o(yp)q(e)g(\(2.7\){\(2.8\))d(ob)o (viously)37 987 y(holds)j(true.)108 1068 y(As)g(announced)g(ab)q(o)o(v) o(e,)f(w)o(e)g(shall)i(also)e(consider)h(in)h(the)e(sequel)i(the)e (follo)o(wing)i(simpli\014ed)h(form)c(of)37 1125 y(the)i(Hartree-F)l(o) q(c)o(k)e(mo)q(del;)i(namely)g(the)f(reduced)i(Hartree-F)l(o)q(c)o(k)d (mo)q(del:)523 1248 y Fi(I)547 1230 y Fh(RH)s(F)543 1260 y Fv(\003)647 1248 y Fn(=)f(inf)750 1184 y Fe(\032)784 1248 y Fi(E)821 1230 y Fh(RH)s(F)818 1260 y Fv(\003)908 1248 y Fn(\()p Fi(K)s Fn(\))c(+)1046 1218 y(1)p 1046 1238 23 2 v 1046 1279 a(2)1074 1248 y Fi(U)1105 1255 y Fv(\003)1144 1248 y Fn(;)20 b Fi(K)15 b Fg(2)e(K)1309 1255 y Fv(\003)1335 1184 y Fe(\033)1369 1248 y Fi(;)377 b Fn(\(2.10\))529 1445 y Fi(E)566 1426 y Fh(RH)s(F)563 1456 y Fv(\003)652 1445 y Fn(\()p Fi(K)s Fn(\))12 b(=)h(T)l(r)837 1408 y Fe(\002)856 1445 y Fn(\()p Fg(\000)p Fn(\001)d Fg(\000)g Fi(V)1028 1452 y Fv(\003)1056 1445 y Fn(\))p Fi(K)1116 1408 y Fe(\003)1144 1445 y Fn(+)1195 1415 y(1)p 1195 1435 V 1195 1476 a(2)1222 1445 y Fi(D)q Fn(\()p Fi(\032;)e(\032)p Fn(\))p Fi(;)382 b Fn(\(2.11\))37 1559 y(\(where)16 b Fg(K)222 1566 y Fv(\003)263 1559 y Fn(is)g(still)g (de\014ned)h(b)o(y)e(\(2.4\)\),)e(and)i(resp)q(ectiv)o(ely)i(its)e (analogous)g(smeared)g(n)o(uclei)j(mo)q(del)114 1686 y Fi(I)138 1664 y Fh(m;RH)s(F)134 1700 y Fv(\003)279 1686 y Fn(=)13 b(inf)390 1622 y Fe(\032)424 1686 y Fn(T)l(r)471 1649 y Fe(\002)489 1686 y Fg(\000)p Fn(\001)p Fi(K)604 1649 y Fe(\003)634 1686 y Fn(+)684 1655 y(1)p 684 1675 V 684 1717 a(2)712 1686 y Fi(D)q Fn(\()p Fi(\032)d Fg(\000)g Fi(m)887 1693 y Fv(\003)914 1686 y Fi(;)e(\032)i Fg(\000)g Fi(m)1053 1693 y Fv(\003)1080 1686 y Fn(\))f Fg(\000)1158 1655 y Fn(1)p 1158 1675 V 1158 1717 a(2)1185 1686 y Fg(j)p Fn(\003)p Fg(j)f Fi(D)q Fn(\()p Fi(m;)g(m)p Fn(\))i(;)20 b Fi(K)15 b Fg(2)e(K)1602 1693 y Fv(\003)1628 1622 y Fe(\033)1670 1686 y Fi(:)76 b Fn(\(2.12\))108 1851 y(In)17 b(order)f(to)f(turn)h(to)g(the)g(thermo)q(dynamic)g(limit)i(problem)f Ff(p)n(er)g(se)p Fn(,)e(it)i(is)f(no)o(w)g(time)g(to)g(recall)h(the)37 1907 y(prop)q(erties)g(of)e(the)g(sequence)i(of)d(sets)i(\003)e(that)h (w)o(e)g(shall)i(consider.)k(F)l(or)15 b(the)g(sak)o(e)g(of)g (completeness,)h(w)o(e)37 1964 y(recall)h(here)e(the)h(follo)o(wing)g (de\014nition)h(tak)o(en)d(from)h([11)o(])g(and)g([32)o(].)37 2062 y Fj(De\014nition)21 b(1)c Ff(We)h(shal)r(l)g(say)f(that)i(a)f(se) n(quenc)n(e)e Fn(\(\003)972 2069 y Fh(i)985 2062 y Fn(\))1003 2069 y Fh(i)p Fu(\025)p Fv(1)1080 2062 y Ff(of)i(\014nite)e(subsets)h (of)h Fj(Z)1495 2045 y Fv(3)1533 2062 y Ff(go)n(es)f(to)h(in\014nity)e (if)37 2118 y(the)h(fol)r(lowing)f(two)h(c)n(onditions)e(hold)h(:)108 2175 y(\(a\))g(F)m(or)g(any)g(\014nite)f(subset)h Fi(A)c Fg(\032)h Fj(Z)735 2158 y Fv(3)755 2175 y Ff(,)j(ther)n(e)h(exists)e Fi(i)d Fg(2)h Fj(N)j Ff(such)h(that)688 2276 y Fg(8)p Fi(j)e Fg(\025)e Fi(i;)243 b(A)13 b Fg(\032)g Fn(\003)1194 2283 y Fh(j)1212 2276 y Fi(:)108 2402 y Ff(\(b\))i(If)i Fn(\003)262 2386 y Fh(h)300 2402 y Ff(is)f(the)h(set)f(of)g(p)n(oints)g (in)f Fj(R)777 2386 y Fv(3)813 2402 y Ff(whose)i(distanc)n(e)e(to)i Fi(@)s Fn(\000\(\003\))e Ff(is)h(less)f(than)h(h,)h(then)692 2533 y Fn(lim)682 2562 y Fh(i)p Fu(!1)777 2502 y Fg(j)p Fn(\003)822 2486 y Fh(h)822 2515 y(i)844 2502 y Fg(j)p 777 2522 80 2 v 781 2564 a(j)p Fn(\003)826 2571 y Fh(i)840 2564 y Fg(j)874 2533 y Fn(=)c(0)p Fi(;)125 b Fg(8)p Fi(h)13 b(>)g Fn(0)p Fi(:)37 2658 y Ff(Condition)k(\(b\))e(wil)r(l)h(b)n(e)g (her)n(e)n(after)g(r)n(eferr)n(e)n(d)g(to)h(as)f(the)g(V)m(an)g(Hove)g (c)n(ondition)p Fn(.)108 2740 y(Brie\015y)k(sp)q(eaking,)h(a)d (sequence)i(satisfying)g(the)f(V)l(an)g(Ho)o(v)o(e)f(condition)j(is)e (a)g(sequence)h(for)e(whic)o(h)37 2796 y(the)j(`b)q(oundary')g(is)g (negligible)i(in)f(fron)o(t)d(of)i(the)f(`in)o(terior'.)36 b(A)21 b(sequence)h(of)e(large)h(cub)q(es)h(t)o(ypically)p eop %%Page: 8 8 8 7 bop 53 50 a Fm(2.)23 b Fl(General)17 b(setting)e(of)h(the)g(models) g(and)h(main)f(resul)m(ts)699 b Fn(8)37 199 y(satis\014es)22 b(the)f(conditions)i(of)e(De\014nition)h(1.)38 b(W)l(e)21 b(shall)i(only)f(consider)g(henceforth)f(so-called)45 b Ff(V)m(an)37 256 y(Hove)16 b(se)n(quenc)n(es)g Fn(whic)o(h)g(are)e (going)h(to)f(in\014nit)o(y)i(in)g(the)e(sense)h(of)g(the)f(ab)q(o)o(v) o(e)h(de\014nition.)21 b(Occasionally)l(,)37 312 y(some)14 b(additional)h(conditions)g(will)g(also)f(b)q(e)h(required)f(\(see)g (Theorem)g(2.2\).)k(F)l(ollo)o(wing)c(the)g(notation)f(of)37 369 y([32,)h(11],)g(w)o(e)h(shall)h(write)g(henceforth)f(lim)785 376 y Fv(\003)p Fu(!1)890 369 y Fi(f)5 b Fn(\(\003\))14 b(instead)i(of)f(lim)1271 376 y Fh(i)p Fu(!1)1363 369 y Fi(f)5 b Fn(\(\003)1440 376 y Fh(i)1454 369 y Fn(\).)108 450 y(Before)23 b(in)o(tro)q(ducing)h(the)f(Hartree-F)l(o)q(c)o(k)f(t)o (yp)q(e)g(p)q(erio)q(dic)j(mo)q(dels,)g(it)e(is)g(to)f(b)q(e)i(noticed) f(that)f(a)37 506 y(k)o(ey-p)q(oin)o(t)14 b(for)f(their)h(de\014nition) h(is)f(the)f(understanding)i(of)e(la)o(ws)g(of)g(in)o(teractions)g(b)q (et)o(w)o(een)h(p)q(erio)q(dically)37 563 y(arranged)d(particles.)20 b(Indeed,)13 b(o)o(wing)e(to)f(the)h(long-range)g(of)g(the)g(Coulom)o (b)g(p)q(oten)o(tial,)i(the)e(electrostatic)37 619 y(p)q(oten)o(tial)h (created)f(b)o(y)g(the)g(in\014nite)i(lattice)e(of)g(n)o(uclei)h (cannot)f(b)q(e)h(simply)1314 585 y Fe(P)1362 633 y Fh(k)q Fu(2)p Fd(Z)1430 623 y Ft(3)1496 601 y Fv(1)p 1461 608 87 2 v 1461 636 a Fu(j)p Fh(x)p Fu(\000)p Fh(k)q Fu(j)1553 619 y Fn(,)f(since)h(this)f(series)37 676 y(ob)o(viously)17 b(do)q(es)e(not)g(mak)o(e)g(sense.)108 732 y(W)l(e)g(\014rst)g(of)g (all)h(in)o(tro)q(duce)g(the)f(p)q(erio)q(dic)j(p)q(oten)o(tial)e Fi(G)f Fn(that)f(is)i(uniquely)h(de\014ned)g(b)o(y)637 839 y Fg(\0004)p Fi(G)12 b Fn(=)h(4)p Fi(\031)860 802 y Fe(\000)880 839 y Fg(\000)p Fn(1)d(+)1003 795 y Fe(X)994 897 y Fh(y)q Fu(2)p Fd(Z)1061 887 y Ft(3)1085 839 y Fi(\016)r Fn(\()p Fg(\001)g(\000)g Fi(y)r Fn(\))1235 802 y Fe(\001)1256 839 y Fi(;)490 b Fn(\(2.13\))37 989 y(and)855 1039 y Fe(Z)881 1142 y Fh(Q)918 1101 y Fi(G)12 b Fn(=)h(0)p Fi(;)709 b Fn(\(2.14\))37 1232 y(with)20 b Fi(\016)g Fn(b)q(eing)g(the)f(Dirac)g(measure.)30 b(Due)19 b(to)f(our)g(c)o (hoice)i(of)e(normalization)i(\(2.14\))d(for)h Fi(G)p Fn(,)h(w)o(e)f(also)37 1289 y(need)f(to)d(de\014ne)i(the)g(constan)o(t) 720 1405 y Fi(M)i Fn(=)g(lim)830 1434 y Fh(x)p Fu(!)p Fv(0)910 1340 y Fe(\024)934 1405 y Fi(G)p Fn(\()p Fi(x)p Fn(\))9 b Fg(\000)1106 1374 y Fn(1)p 1092 1394 52 2 v 1092 1436 a Fg(j)p Fi(x)p Fg(j)1148 1340 y Fe(\025)1180 1405 y Fi(:)566 b Fn(\(2.15\))37 1532 y(W)l(e)17 b(shall)g(see)g(in)g (the)f(sequel)i(that)e(this)g(p)q(erio)q(dic)j(p)q(oten)o(tial)e Fi(G)f Fn(\(whic)o(h)h(is)f(also)h(the)f(Green's)g(function)37 1588 y(of)21 b(the)g(Laplacian)i(with)e(p)q(erio)q(dic)i(conditions)f (on)f(the)h(unit)f(cell\))i(is)e(the)g(in)o(teraction)h(electrostatic) 37 1645 y(p)q(oten)o(tial)16 b(created)g(b)o(y)f(the)g(p)q(erio)q(dic)i (distribution)g(of)e(c)o(harges)g(of)g(n)o(uclei.)21 b(W)l(e)16 b(denote)520 1765 y Fi(D)558 1772 y Fh(G)587 1765 y Fn(\()p Fi(f)r(;)8 b(f)d Fn(\))12 b(=)755 1703 y Fe(Z)6 b(Z)831 1806 y Fh(Q)p Fu(\002)p Fh(Q)924 1765 y Fi(f)f Fn(\()p Fi(x)p Fn(\))p Fi(G)p Fn(\()p Fi(x)k Fg(\000)h Fi(y)r Fn(\))p Fi(f)5 b Fn(\()p Fi(y)r Fn(\))j Fi(dxdy)r(:)37 1894 y Fn(W)l(e)16 b(also)f(de\014ne)715 1961 y Fi(f)5 b Fn(\()p Fi(x)p Fn(\))12 b(=)884 1930 y(1)p 869 1950 V 869 1992 a Fg(j)p Fi(x)p Fg(j)936 1961 y(\000)981 1899 y Fe(Z)1006 2002 y Fh(Q)1090 1930 y Fi(dy)p 1049 1950 131 2 v 1049 1992 a Fg(j)p Fi(x)e Fg(\000)g Fi(y)r Fg(j)1185 1961 y Fi(;)37 2071 y Fn(and)16 b(then)557 2187 y Fi(f)579 2194 y Fv(\003)606 2187 y Fn(\()p Fi(x)p Fn(\))c(=)728 2143 y Fe(X)728 2242 y Fh(k)q Fu(2)p Fv(\003)795 2123 y Fe(\022)888 2156 y Fn(1)p 833 2176 132 2 v 833 2218 a Fg(j)p Fi(x)e Fg(\000)g Fi(k)q Fg(j)980 2187 y(\000)1026 2125 y Fe(Z)1051 2228 y Fh(Q)1175 2156 y Fi(dy)p 1093 2176 212 2 v 1093 2218 a Fg(j)p Fi(x)g Fg(\000)g Fi(k)i Fg(\000)e Fi(y)r Fg(j)1310 2123 y Fe(\023)1343 2187 y Fi(:)37 2330 y Fn(It)16 b(is)f(con)o(v)o(enien)o(t)h(to)f(rewrite)g Fi(f)589 2337 y Fv(\003)631 2330 y Fn(as)737 2449 y Fi(f)759 2456 y Fv(\003)798 2449 y Fn(=)e Fi(V)872 2456 y Fv(\003)909 2449 y Fg(\000)e Fi(\037)983 2458 y Fv(\000\(\003\))1069 2449 y Fi(?)1121 2418 y Fn(1)p 1107 2439 52 2 v 1107 2480 a Fg(j)p Fi(x)p Fg(j)1163 2449 y Fi(;)37 2573 y Fn(where,)20 b(more)e(generally)l(,)j(w)o(e)d(shall)i(denote)f(b)o(y)f Fi(\037)934 2580 y Fv(\012)981 2573 y Fn(the)h(c)o(haracteristic)g (function)g(of)g(the)f(domain)i(\012.)37 2629 y(Besides,)d(it)e(is)h (pro)o(v)o(ed)f(in)h([32)o(],)e(and)i(recalled)g(in)g([11)o(],)f(that,) f(when)i Fi(Q)f Fn(is)g(a)g(cub)q(e,)829 2752 y Fg(j)p Fi(f)5 b Fn(\()p Fi(x)p Fn(\))p Fg(j)11 b(\024)1026 2722 y Fi(C)p 1008 2742 71 2 v 1008 2784 a Fg(j)p Fi(x)p Fg(j)1060 2770 y Fv(4)p eop %%Page: 9 9 9 8 bop 53 50 a Fm(2.)23 b Fl(General)17 b(setting)e(of)h(the)g(models) g(and)h(main)f(resul)m(ts)699 b Fn(9)37 199 y(almost)13 b(ev)o(erywhere)h(on)f Fj(R)517 183 y Fv(3)537 199 y Fn(,)g(for)g(some)g(p)q(ositiv)o(e)h(constan)o(t)e Fi(C)17 b Fn(and)d(that)e Fi(f)1344 206 y Fv(\003)1384 199 y Fn(con)o(v)o(erges)h(to)f(the)i(p)q(erio)q(dic)37 256 y(p)q(oten)o(tial)20 b Fi(G)12 b Fn(+)h Fi(d)p Fn(,)19 b(for)g(some)f(real)i(constan)o(t)e Fi(d)g Fn(that)g(is)i(indep)q (enden)o(t)h(of)e(\003,)g(uniformly)g(on)g(compact)37 312 y(subsets)14 b(of)g Fj(R)283 296 y Fv(3)310 312 y Fg(n)7 b Fj(Z)372 296 y Fv(3)392 312 y Fn(.)19 b(Moreo)o(v)o(er,)12 b(for)h(an)o(y)h(compact)f(subset)h Fi(K)j Fn(of)c Fj(R)1251 296 y Fv(3)1271 312 y Fn(,)h Fi(f)1320 319 y Fv(\003)1354 312 y Fg(\000)1396 278 y Fe(P)1444 326 y Fh(k)q Fu(2)p Fv(\003)p Fu(\\)p Fh(K)1616 294 y Fv(1)p 1582 301 87 2 v 1582 328 a Fu(j)p Fh(x)p Fu(\000)p Fh(k)q Fu(j)1686 312 y Fn(con)o(v)o(erges)37 380 y(uniformly)i(on)f Fi(K)j Fn(to)c Fi(G)9 b Fn(+)g Fi(d)g Fg(\000)586 346 y Fe(P)634 394 y Fh(k)q Fu(2)p Fd(Z)702 384 y Ft(3)719 394 y Fu(\\)p Fh(K)823 362 y Fv(1)p 789 369 V 789 396 a Fu(j)p Fh(x)p Fu(\000)p Fh(k)q Fu(j)895 380 y Fn(\(see)15 b([32)o(]\).)k(Therefore,) 14 b(w)o(e)h(ma)o(y)f(notew)o(orth)o(y)f(observ)o(e)37 443 y(that)i(the)g(p)q(erio)q(dic)i(p)q(oten)o(tial)f Fi(G)f Fn(whic)o(h)g(w)o(as)g(previously)h(de\014ned)g(b)o(y)f (\(2.13\))f(and)h(\(2.14\))e(is)i(also)g(giv)o(en)37 499 y(b)o(y)487 624 y Fi(G)p Fn(\()p Fi(x)p Fn(\))d(=)655 581 y Fe(X)645 682 y Fh(k)q Fu(2)p Fd(Z)713 673 y Ft(3)730 560 y Fe(\022)823 593 y Fn(1)p 768 614 132 2 v 768 655 a Fg(j)p Fi(x)e Fg(\000)h Fi(k)q Fg(j)915 624 y(\000)961 562 y Fe(Z)986 665 y Fh(Q)1123 593 y Fn(1)p 1028 614 212 2 v 1028 655 a Fg(j)p Fi(x)f Fg(\000)h Fi(y)h Fg(\000)e Fi(k)q Fg(j)1252 624 y Fi(dy)1300 560 y Fe(\023)1343 624 y Fg(\000)h Fi(d)p Fn(;)37 770 y(that)17 b(is,)h(the)f(sum)h(o)o(v) o(er)e(the)h(lattice)h(p)q(oin)o(ts)g(of)f(the)g(Coulom)o(b)g(p)q(oten) o(tial)h(created)f(b)o(y)h(a)f(p)q(oin)o(t)g(c)o(harge)37 827 y(placed)23 b(at)e(the)g(cen)o(ter)g(of)g(the)g(unit)h(cub)q(e,)i (and)d(whic)o(h)h(is)g(screened,)h(on)e(eac)o(h)h(cell,)i(b)o(y)d(a)g (uniform)37 883 y(bac)o(kground)f(of)g(negativ)o(e)f(unit)i(c)o(harge.) 33 b(This)21 b(screening)g(e\013ect)f(whic)o(h)h(is)f(commonly)g (observ)o(ed)g(in)37 939 y(thermo)q(dynamic)h(limit)g(issues)g(\(see)e ([25)o(,)h(26)o(,)f(27,)g(32,)g(11)o(,)h(13)o(]\))f(is)h(a)g (consequence)h(of)e(the)h(electrical)37 996 y(neutralit)o(y)c(of)f(the) g(molecular)h(systems)f(under)h(consideration.)108 1052 y(Let)d(us)h(no)o(w)e(turn)h(to)f(the)i(p)q(erio)q(dic)h(problems)e(w)o (e)g(w)o(an)o(t)f(to)g(de\014ne.)21 b(W)l(e)13 b(shall)h(detail)g(in)g (Section)g(3.2)37 1109 y(b)q(elo)o(w)j(the)f(reasons)g(wh)o(y)g(w)o(e)f (need)i(to)f(in)o(tro)q(duce)h(the)f(follo)o(wing)g(set)g(of)g(op)q (erators,)f(whic)o(h)i(are)e(aimed)37 1165 y(to)g(b)q(ecome)h(the)f (analogues)h(of)e(the)i(usual)g(densit)o(y)f(matrices)h(in)g(the)f(p)q (erio)q(dic)i(case.)37 1263 y Fj(De\014nition)j(2)15 b Ff(L)n(et)h Fi(Q)437 1247 y Fh(?)470 1263 y Fn(=)d([)p Fg(\000)p Fi(\031)r Fn(;)8 b(+)p Fi(\031)r Fn([)691 1247 y Fv(3)709 1263 y Ff(,)16 b(and,)g(for)h(every)f Fi(\030)i Ff(in)e Fi(Q)1164 1247 y Fh(?)1183 1263 y Ff(,)460 1365 y Fi(L)491 1347 y Fv(2)491 1377 y Fh(\030)511 1365 y Fn(\()p Fi(Q)p Fn(\))c(=)h Fg(f)p Fi(u)f Fg(2)h Fi(L)778 1347 y Fv(2)778 1377 y(lo)q(c)824 1365 y Fn(\()p Fj(R)881 1347 y Fv(3)901 1365 y Fn(\))f(;)20 b Fi(e)985 1347 y Fu(\000)p Fh(i\030)q Fu(\001)p Fh(x)1081 1365 y Fi(u)c Ff(is)g Fi(Q)10 b Fg(\000)g Ff(p)n(erio)n(dic)p Fg(g)p Fi(:)37 1468 y Ff(We)20 b(now)f(c)n(onsider)f(families)h(of)g(op)n(er)n (ators)g Fi(K)871 1475 y Fh(\030)909 1468 y Ff(\()p Fi(\030)26 b Fg(2)18 b Fi(Q)1058 1451 y Fh(?)1078 1468 y Ff(\),)h(which)h(ar)n(e)f (self-adjoint)g(on)f Fi(L)1678 1451 y Fv(2)1678 1481 y Fh(\030)1698 1468 y Fn(\()p Fi(Q)p Fn(\))p Ff(,)h(and)37 1524 y(which)e(enjoy)f(the)h(fol)r(lowing)f(pr)n(op)n(erties,)g(for)g (almost)g(every)h Fi(\030)d Fg(2)f Fi(Q)1230 1507 y Fh(?)1250 1524 y Ff(.)37 1618 y Fj(\(H2\)')24 b Fn(0)13 b Fg(\024)g Fi(K)305 1625 y Fh(\030)336 1618 y Fg(\024)g Fj(1)p Ff(,)j(with)h Fj(1)f Ff(b)n(eing)g(the)g(identity)g(on)g Fi(L)1035 1601 y Fv(2)1035 1632 y Fh(\030)1055 1618 y Fn(\()p Fi(Q)p Fn(\))p Ff(;)37 1742 y Fj(\(H3\))24 b Ff(the)16 b(op)n(er)n(ators)h Fi(K)480 1749 y Fh(\030)515 1742 y Ff(have)f(\014nite)g(tr)n(ac)n(es,)f (and)h(satisfy)1110 1680 y Fe(Z)1135 1783 y Fh(Q)1163 1774 y Fc(?)1191 1742 y Fn(T)l(r)1237 1754 y Fh(L)1261 1742 y Ft(2)1261 1766 y Fc(\030)1279 1754 y Fv(\()p Fh(Q)p Fv(\))1336 1742 y Fi(K)1375 1749 y Fh(\030)1441 1711 y Fi(d\030)p 1411 1731 106 2 v 1411 1773 a Fn(\(2)p Fi(\031)r Fn(\))1498 1760 y Fv(3)1534 1742 y Fn(=)d(1)p Ff(;)37 1870 y Fj(\(H4\))24 b Fn(T)l(r)215 1882 y Fh(L)239 1870 y Ft(2)239 1893 y Fc(\030)256 1882 y Fv(\()p Fh(Q)p Fv(\))313 1870 y Fn([)p Fg(\000)p Fn(\001)399 1877 y Fh(\030)418 1870 y Fi(K)457 1877 y Fh(\030)476 1870 y Fn(])12 b Fi(<)h Fn(+)p Fg(1)k Ff(and)734 1833 y Fe(R)756 1886 y Fh(Q)784 1876 y Fc(?)811 1870 y Fn(T)l(r)858 1882 y Fh(L)882 1870 y Ft(2)882 1893 y Fc(\030)899 1882 y Fv(\()p Fh(Q)p Fv(\))956 1833 y Fe(\002)975 1870 y Fg(\000)p Fn(\001)1048 1877 y Fh(\030)1067 1870 y Fi(K)1106 1877 y Fh(\030)1125 1833 y Fe(\003)1151 1870 y Fi(d\030)e(<)e Fn(+)p Fg(1)p Fi(:)37 1971 y Ff(T)m(o)j(every)f(such)g(family)g(of)g(op)n(er)n(ators)h(is)e (asso)n(ciate)n(d,)h(in)g(a)g(unique)g(way,)h(a)f(self-adjoint)g(op)n (er)n(ator)h Fi(K)i Ff(in)37 2028 y Fi(L)68 2011 y Fv(2)88 2028 y Fn(\()p Fj(R)145 2011 y Fv(3)165 2028 y Fn(\))p Ff(,)e(denote)n(d)g(by)g Fi(K)f Fn(=)540 1991 y Fe(R)562 2044 y Fh(Q)590 2034 y Fc(?)617 2028 y Fi(K)656 2035 y Fh(\030)717 2007 y(d\030)p 692 2017 84 2 v 692 2045 a Fv(\(2)p Fh(\031)q Fv(\))759 2035 y Ft(3)781 2028 y Ff(,)h(such)h(that)37 2132 y Fj(\(H1\))24 b Fi(K)19 b Ff(c)n(ommutes)d(with)h(the)g(tr)n(anslations)e(of)h Fj(Z)944 2116 y Fv(3)964 2132 y Ff(;)37 2226 y Fj(\(H2\))24 b Fn(0)12 b Fg(\024)h Fi(K)j Fg(\024)d Fj(1)p Ff(.)37 2320 y(We)i(denote)f(by)f Fg(K)i Ff(the)f(set)f(of)h(op)n(er)n(ators)g Fi(K)i Fn(=)857 2283 y Fe(R)878 2336 y Fh(Q)906 2326 y Fc(?)934 2320 y Fi(K)973 2327 y Fh(\030)1033 2299 y(d\030)p 1009 2310 V 1009 2337 a Fv(\(2)p Fh(\031)q Fv(\))1076 2327 y Ft(3)1111 2320 y Ff(which)f(satisfy)e(the)h(c)n(onditions)j Fn(\(H1\){\(H4\))37 2383 y Ff(\(or)g(e)n(quivalently)i Fn(\(H2'\))p Ff(,)c Fn(\(H3\))g Ff(and)21 b Fn(\(H4\))p Ff(\),)15 b(and)i(we)f(shal)r(l)g(c)n(al)r(l)f Fi(K)20 b Ff(a)c(p)n(erio)n(dic)g(density)g(matrix.)108 2481 y Fn(In)h(all)h(that)e(follo)o(ws,)h(w)o(e)f(shall)i(denote)f(b)o(y)g Fi(\032)p Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fi(;)g Fg(\001)p Fn(\))14 b(the)i(Hilb)q(ert-Sc)o(hmidt)k(k)o(ernel)d(of)f Fi(K)1690 2488 y Fh(\030)1709 2481 y Fn(.)24 b(Owing)37 2538 y(to)14 b(the)g(fact)g(that)f Fi(K)394 2545 y Fh(\030)427 2538 y Fn(is)i(a)f(trace-class)g(op)q(erator,)f(w)o(e)h(ma)o(y)f(giv)o (e)i(a)f(sense)g(to)g Fi(\032)p Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))k(as)i(a)g Fi(Q)p Fn(-p)q(erio)q(dic)37 2594 y(function)j(in)f Fi(L)300 2578 y Fv(1)300 2608 y(lo)q(c)346 2594 y Fn(\()p Fj(R)403 2578 y Fv(3)422 2594 y Fn(\),)f(and)g(to)689 2720 y Fi(\032)p Fn(\()p Fi(x)p Fn(\))d(=)834 2658 y Fe(Z)860 2761 y Fh(Q)888 2752 y Fc(?)915 2720 y Fi(\032)p Fn(\()p Fi(\030)r(;)c(x;)g(x)p Fn(\))1131 2689 y Fi(d\030)p 1101 2710 106 2 v 1101 2751 a Fn(\(2)p Fi(\031)r Fn(\))1188 2738 y Fv(3)1211 2720 y Fi(:)535 b Fn(\(2.16\))p eop %%Page: 10 10 10 9 bop 53 50 a Fm(2.)23 b Fl(General)17 b(setting)e(of)h(the)g (models)g(and)h(main)f(resul)m(ts)677 b Fn(10)37 199 y(Moreo)o(v)o(er,)22 b Fi(\032)p Fn(\()p Fi(x)p Fn(\))f(is)h(also)g(a)f Fi(Q)p Fn(-p)q(erio)q(dic)j(function)e(in)h Fi(L)1065 183 y Fv(1)1065 213 y(lo)q(c)1111 199 y Fn(\()p Fj(R)1168 183 y Fv(3)1187 199 y Fn(\),)g(whic)o(h)f(will)h(pla)o(y)f(the)g(role)g (of)f(the)37 256 y(electronic)16 b(densit)o(y)f(in)g(crystals.)k(Let)c (us)f(emphasize)h(once)g(more)f(the)g(fact)g(that)f(the)h (de\014nitions)i(of)e(the)37 312 y(v)m(arious)k(quan)o(tities)g(app)q (earing)f(in)h(the)f(ab)q(o)o(v)o(e)g(de\014nitions)h(are)f(made)g (precise)h(in)g(Section)g(3.2)e(b)q(elo)o(w,)37 369 y(and,)g(more)e(sp) q(eci\014cally)l(,)k(in)e(Prop)q(osition)g(3.2)e(therein.)108 450 y(With)f(the)g(help)i(of)d(the)i(ab)q(o)o(v)o(e)e(de\014nitions,)j (w)o(e)e(are)g(no)o(w)f(able)i(to)e(state)h(the)g(p)q(erio)q(dic)i (minimization)37 506 y(problems)20 b(asso)q(ciated)f(to)f(the)g(ab)q(o) o(v)o(e)g(RHF)h(and)g(HF)f(mo)q(dels.)31 b(First,)19 b(for)f(the)h(RHF)g(mo)q(del)g(\(2.10\){)37 563 y(\(2.11\),)13 b(w)o(e)i(set)g(:)609 677 y Fi(I)633 658 y Fh(RH)s(F)629 688 y(per)733 677 y Fn(=)e(inf)836 613 y Fe(\032)870 677 y Fi(E)907 658 y Fh(RH)s(F)904 688 y(per)994 677 y Fn(\()p Fi(K)s Fn(\))f(;)20 b Fi(K)15 b Fg(2)e(K)1250 613 y Fe(\033)1283 677 y Fi(;)463 b Fn(\(2.17\))339 852 y Fi(E)376 833 y Fh(RH)s(F)373 863 y(per)463 852 y Fn(\()p Fi(K)s Fn(\))11 b(=)600 790 y Fe(Z)626 893 y Fh(Q)654 883 y Fc(?)681 852 y Fn(T)l(r)728 863 y Fh(L)752 852 y Ft(2)752 875 y Fc(\030)769 863 y Fv(\()p Fh(Q)p Fv(\))827 815 y Fe(\002)845 852 y Fg(\000)p Fn(\001)p Fi(K)957 859 y Fh(\030)976 815 y Fe(\003)1038 821 y Fi(d\030)p 1008 841 106 2 v 1008 883 a Fn(\(2)p Fi(\031)r Fn(\))1095 870 y Fv(3)1128 852 y Fg(\000)1174 790 y Fe(Z)1199 893 y Fh(Q)1236 852 y Fi(G)d(\032)i Fn(+)1364 821 y(1)p 1364 841 23 2 v 1364 883 a(2)1391 852 y Fi(D)1429 859 y Fh(G)1458 852 y Fn(\()p Fi(\032;)e(\032)p Fn(\))p Fi(:)185 b Fn(\(2.18\))37 970 y(with)15 b Fi(\032)g Fn(b)q(eing)h(de\014ned)f(b)o(y)g(\(2.16\).)i (The)e(analogous)f(mo)q(del)h(in)h(the)e(smeared)g(n)o(uclei)j(setting) d(is)h(written)592 1059 y Fi(I)616 1040 y Fh(m;RH)s(F)612 1070 y(per)757 1059 y Fn(=)e(inf)t Fg(f)p Fi(E)920 1040 y Fh(m;RH)s(F)917 1070 y(per)1048 1059 y Fn(\()p Fi(K)s Fn(\);)8 b Fi(K)13 b Fg(2)g(K)q(g)p Fi(;)445 b Fn(\(2.19\))112 1208 y Fi(E)149 1190 y Fh(m;RH)s(F)146 1220 y(per)277 1208 y Fn(\()p Fi(K)s Fn(\))12 b(=)415 1147 y Fe(Z)440 1250 y Fh(Q)468 1240 y Fc(?)495 1208 y Fn(T)l(r)542 1220 y Fh(L)566 1209 y Ft(2)566 1232 y Fc(\030)583 1220 y Fv(\()p Fh(Q)p Fv(\))641 1172 y Fe(\002)660 1208 y Fg(\000)p Fn(\001)p Fi(K)772 1215 y Fh(\030)790 1172 y Fe(\003)852 1178 y Fi(d\030)p 822 1198 106 2 v 822 1240 a Fn(\(2)p Fi(\031)r Fn(\))909 1226 y Fv(3)942 1208 y Fn(+)993 1178 y(1)p 993 1198 23 2 v 993 1240 a(2)1020 1208 y Fi(D)1058 1215 y Fh(G)1088 1208 y Fn(\()p Fi(\032)e Fg(\000)g Fi(m;)e(\032)i Fg(\000)g Fi(m)p Fn(\))g Fg(\000)1442 1178 y Fn(1)p 1442 1198 V 1442 1240 a(2)1469 1208 y Fi(D)1507 1215 y Fh(G)1536 1208 y Fn(\()p Fi(m;)e(m)p Fn(\))p Fi(:)73 b Fn(\(2.20\))37 1366 y(W)l(e)16 b(shall)g(pro)o(v)o(e)f(in)h(Section)g(4)f(the)g(follo) o(wing)h(results.)37 1449 y Fj(Theorem)i(2.1)24 b(\(W)l(ell-p)q (osedness)18 b(of)g(the)g(RHF)f(p)q(erio)q(dic)j(problem\))108 1505 y Ff(The)15 b(minimization)e(pr)n(oblem)h(de\014ne)n(d)e(by)i (\(2.17\))g(and)g(\(2.18\))g(\(r)n(esp)n(e)n(ctively)d(by)j(\(2.19\))g (and)g(\(2.20\)\))37 1562 y(admits)i(a)f(minimum.)21 b(In)16 b(addition,)g(the)f(c)n(orr)n(esp)n(onding)f(minimizing)g (density)g Fi(\032)i Ff(is)f(unique)g(and,)h(thus,)37 1618 y(shar)n(es)g(the)h(symmetries)f(of)g(the)g(unit)h(cub)n(e.)37 1738 y Fj(Theorem)h(2.2)24 b(\(Thermo)q(dynamic)c(limit)g(for)f(the)g (RHF)f(energy\))g Ff(We)f(assume)g(that)h(the)f(V)m(an)37 1794 y(Hove)g(se)n(quenc)n(e)d Fn(\003)i Ff(satis\014es)648 1910 y Fn(lim)621 1940 y Fv(\003)p Fu(\000)-5 b(!1)750 1879 y Fg(j)p Fn(\003)795 1863 y Fh(h)817 1879 y Fg(j)p 750 1900 80 2 v 761 1941 a(j)p Fn(\003)p Fg(j)842 1910 y Fn(Log)7 b Fg(j)p Fn(\003)968 1891 y Fh(h)990 1910 y Fg(j)20 b Fn(=)13 b(0)p Fi(;)29 b Fg(8)p Fi(h)14 b(>)f Fn(0)p Fi(;)474 b Fn(\(2.21\))37 2029 y Ff(wher)n(e)14 b Fn(\003)195 2012 y Fh(h)230 2029 y Ff(is)f(de\014ne)n(d)g(in)g (De\014nition)f(1.)20 b(In)14 b(addition,)g(we)g(assume)f(that)h(the)g (unit)f(c)n(el)r(l)g Fi(Q)g Ff(of)g(the)h(p)n(erio)n(dic)37 2085 y(lattic)n(e)i(is)g(a)h(cub)n(e.)108 2142 y(In)f(the)h(p)n(oint)f (nuclei)g(c)n(ase,)f(we)h(have)682 2249 y Fn(lim)656 2279 y Fv(\003)p Fu(\000)-5 b(!1)802 2218 y Fn(1)p 785 2239 57 2 v 785 2280 a Fg(j)p Fn(\003)p Fg(j)847 2249 y Fi(I)871 2230 y Fh(RH)s(F)867 2260 y Fv(\003)970 2249 y Fn(=)13 b Fi(I)1042 2230 y Fh(RH)s(F)1038 2260 y(per)1140 2249 y Fn(+)1190 2218 y Fi(M)p 1190 2239 50 2 v 1203 2280 a Fn(2)1244 2249 y Fi(;)37 2362 y Ff(wher)n(e)i(the)f(c)n(onstant) f Fi(M)20 b Ff(is)13 b(de\014ne)n(d)h(by)g(\(2.15\).)20 b(R)n(esp)n(e)n(ctively,)13 b(in)g(the)i(sme)n(ar)n(e)n(d)e(nuclei)h(c) n(ase,)g(assuming)37 2419 y(in)i(addition)h(that)g Fi(m)f Ff(shar)n(es)g(the)g(symmetries)g(of)h(the)f(unit)g(cub)n(e)g Fi(Q)p Ff(,)g(we)h(have)641 2526 y Fn(lim)615 2557 y Fv(\003)p Fu(\000)-5 b(!1)761 2495 y Fn(1)p 744 2516 57 2 v 744 2557 a Fg(j)p Fn(\003)p Fg(j)806 2526 y Fi(I)830 2504 y Fh(m;RH)s(F)826 2541 y Fv(\003)970 2526 y Fn(=)13 b Fi(I)1042 2507 y Fh(m;RH)s(F)1038 2537 y(per)1181 2526 y Fn(+)1231 2495 y Fi(M)p 1231 2516 50 2 v 1244 2557 a Fn(2)1285 2526 y Fi(;)37 2637 y Ff(wher)n(e)k(the)f(c)n(onstant)g Fi(M)21 b Ff(is)16 b(this)g(time)g(de\014ne)n(d)g(by)448 2749 y Fi(M)i Fn(=)558 2687 y Fe(Z)5 b(Z)633 2790 y Fh(Q)p Fu(\002)p Fh(Q)726 2749 y Fi(m)p Fn(\()p Fi(x)p Fn(\))p Fi(m)p Fn(\()p Fi(y)r Fn(\))928 2684 y Fe(\024)950 2749 y Fi(G)p Fn(\()p Fi(x)10 b Fg(\000)g Fi(y)r Fn(\))g Fg(\000)1241 2718 y Fn(1)p 1187 2738 131 2 v 1187 2780 a Fg(j)p Fi(x)g Fg(\000)h Fi(y)r Fg(j)1323 2684 y Fe(\025)1355 2749 y Fi(dxdy)r(:)293 b Fn(\(2.22\))p eop %%Page: 11 11 11 10 bop 53 50 a Fm(2.)23 b Fl(General)17 b(setting)e(of)h(the)g (models)g(and)h(main)f(resul)m(ts)677 b Fn(11)37 199 y(Some)16 b(commen)o(ts)g(are)f(in)i(order.)k(The)16 b(reader)g(has)f(remark)o(ed)h(that)f(some)g(tec)o(hnical)i (assumptions)f(\()p Fi(Q)37 256 y Fn(is)k(a)e(cub)q(e,)j(\(2.21\),)c (and)i Fi(m)f Fn(has)h(cubic)h(symmetry\))e(ha)o(v)o(e)g(b)q(een)i (made)f(in)h(the)e(ab)q(o)o(v)o(e)h(theorem.)30 b(W)l(e)37 312 y(need)16 b(these)g(tec)o(hnical)g(assumptions)g(in)f(Section)h(4,) f(and)g(more)g(precisely)i(in)f(Subsection)g(4.3)e(to)h(pro)o(v)o(e)37 369 y(that)j(the)g(upp)q(er)h(limit)h(of)538 351 y Fv(1)p 525 358 45 2 v 525 385 a Fu(j)p Fv(\003)p Fu(j)574 369 y Fi(I)598 352 y Fh(RH)s(F)594 382 y Fv(\003)703 369 y Fn(ma)o(y)d(b)q(e)i(compared)f(from)g(ab)q(o)o(v)o(e)f(b)o(y)i Fi(I)1412 352 y Fh(RH)s(F)1408 380 y(per)1511 369 y Fn(+)1563 351 y Fh(M)p 1563 358 38 2 v 1573 384 a Fv(2)1606 369 y Fn(.)28 b(A)18 b(tec)o(hnical)37 431 y(assumption)23 b(suc)o(h)f(as)f(\(2.21\),)g(that)h(is)g(satis\014ed)g(b)o(y)g(all)h(V) l(an)f(Ho)o(v)o(e)f(sequences)i(except)g(some)e(v)o(ery)37 488 y(pathological)15 b(ones,)f(already)h(app)q(ears)f(in)h([11)o(].)k (Ho)o(w)o(ev)o(er,)13 b(in)i([11)o(],)f(w)o(e)g(manage)f(to)h(get)f (rid)i(of)f(all)h(these)37 544 y(tec)o(hnical)20 b(assumptions)f(using) g(another)e(strategy)g(of)h(pro)q(of)g(for)g(results)g(lik)o(e)i (Theorem)e(2.2;)g(namely)37 601 y(the)f(\\energy)f(via)g(densit)o(y")h (strategy)l(.)22 b(Here,)16 b(suc)o(h)g(a)g(strategy)l(,)f(based)i(up)q (on)f(the)h(con)o(v)o(ergence)f(of)g(the)37 657 y(minimizers,)23 b(is)d(out)g(of)f(reac)o(h.)34 b(Of)20 b(course)g(w)o(e)f(b)q(eliev)o (e)j(they)e(are)g(not)f(necessary)h(here)g(either.)35 b(W)l(e)37 714 y(b)q(eliev)o(e)17 b(there)e(is)g(ro)q(om)f(for)g(impro) o(v)o(emen)o(t)g(in)h(our)g(pro)q(ofs)f(and)h(some)f(other)g(strategy)f (could)j(allo)o(w)f(one)37 770 y(to)h(do)f(without)h(these)g (assumptions.)22 b(Unfortunately)l(,)16 b(w)o(e)f(ha)o(v)o(e)h(not)f(b) q(een)i(able)f(to)g(do)f(without)h(them)37 827 y(so)f(far.)108 883 y(On)e(the)f(con)o(trary)l(,)f(no)h(additional)h(assumption)g(at)e (all)i(is)f(necessary)g(for)g(the)g(other)f(results)i(stated)e(in)37 940 y(this)i(w)o(ork.)18 b(In)13 b(particular,)g(the)f(fact)g(that)g (the)g(unit)h(cell)h(is)f(a)e(cub)q(e)j(is)f(not)e(imp)q(ortan)o(t)h (for)g(Theorems)g(2.1)37 996 y(and)20 b(2.3.)32 b(W)l(e)19 b(shall)i(not)e(rep)q(eat)h(this)f(observ)m(ation)h(in)h(the)e (forthcoming)g(sections,)i(but)e(the)h(reader)37 1052 y(should)d(k)o(eep)e(it)h(in)g(mind.)21 b(F)l(or)14 b(further)h(commen) o(ts,)f(w)o(e)h(refer)g(the)h(reader)f(to)f(Section)i(6.)108 1150 y(In)c(view)h(of)e(the)h(ab)q(o)o(v)o(e)f(theorem,)h(and)f(in)i (view)f(of)f(calculations)j(that)c(will)k(b)q(e)e(detailed)h(in)g (Sections)f(3)37 1207 y(and)21 b(5)e(b)q(elo)o(w,)i(w)o(e)f(\014nd)g (it)g(natural)g(to)f(in)o(tro)q(duce)i(in)g(the)f(Hartree-F)l(o)q(c)o (k)f(framew)o(ork)f(the)i(follo)o(wing)37 1263 y(p)q(erio)q(dic)e (minimization)f(problem)f(:)636 1379 y Fi(I)660 1360 y Fh(H)s(F)656 1390 y(per)733 1379 y Fn(=)d(inf)836 1315 y Fe(\032)870 1379 y Fi(E)907 1360 y Fh(H)s(F)904 1390 y(per)967 1379 y Fn(\()p Fi(K)s Fn(\))f(;)20 b Fi(K)15 b Fg(2)e(K)1223 1315 y Fe(\033)1257 1379 y Fi(;)489 b Fn(\(2.23\))174 1562 y Fi(E)211 1543 y Fh(H)s(F)208 1573 y(per)271 1562 y Fn(\()p Fi(K)s Fn(\))12 b(=)409 1500 y Fe(Z)434 1603 y Fh(Q)462 1594 y Fc(?)490 1562 y Fn(T)l(r)536 1574 y Fh(L)560 1562 y Ft(2)560 1586 y Fc(\030)578 1574 y Fv(\()p Fh(Q)p Fv(\))635 1525 y Fe(\002)654 1562 y Fg(\000)p Fn(\001)p Fi(K)766 1569 y Fh(\030)785 1525 y Fe(\003)846 1531 y Fi(d\030)p 816 1551 106 2 v 816 1593 a Fn(\(2)p Fi(\031)r Fn(\))903 1580 y Fv(3)937 1562 y Fg(\000)982 1500 y Fe(Z)1007 1603 y Fh(Q)1045 1562 y Fi(G)c(\032)i Fn(+)1172 1531 y(1)p 1172 1551 23 2 v 1172 1593 a(2)1200 1562 y Fi(D)1238 1569 y Fh(G)1267 1562 y Fn(\()p Fi(\032;)e(\032)p Fn(\))i Fg(\000)1430 1531 y Fn(1)p 1430 1551 V 1430 1593 a(2)1458 1562 y Fi(E)1492 1569 y Fh(exc)1545 1562 y Fn(\()p Fi(K)s Fn(\))136 b(\(2.24\))37 1684 y(where)21 b Fi(\032)p Fn(\()p Fi(x)p Fn(\))g(is)g(still)h (de\014ned)g(b)o(y)f(\(2.16\).)34 b(With)21 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fi(;)g Fg(\001)p Fn(\))18 b(b)q(eing)k(the)e(Hilb)q(ert-Sc)o(hmidt)k(of)c Fi(K)1759 1691 y Fh(\030)1777 1684 y Fn(,)i(the)37 1746 y(Sc)o(h)o(w)o(arz)15 b(k)o(ernel)h(of)f Fi(K)k Fn(is)c(giv)o(en)h(b)o (y)g Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))j(=)875 1710 y Fe(R)896 1762 y Fh(Q)924 1753 y Fc(?)952 1746 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)d Fi(x;)g(y)r Fn(\))1165 1726 y Fh(d\030)p 1141 1736 84 2 v 1141 1763 a Fv(\(2)p Fh(\031)q Fv(\))1208 1754 y Ft(3)1229 1746 y Fn(,)15 b(and)h(b)q(elongs)g(to)f Fi(L)1596 1730 y Fv(2)1616 1746 y Fn(\()p Fi(Q)9 b Fg(\002)i Fj(R)1764 1730 y Fv(3)1784 1746 y Fn(\))k(\(at)37 1810 y(least)d(;)g(see)f(Prop)q(osition)h(3.2)e(in)i(Section)g(3.2)f(b)q (elo)o(w\).)18 b(F)l(or)11 b(some)g(reasons)g(whic)o(h)h(are)e(made)i (precise)g(later)37 1866 y(in)18 b(Section)f(5,)e(the)i(p)q(erio)q(dic) h(exc)o(hange)e(term)g Fg(\000)903 1848 y Fv(1)p 903 1855 18 2 v 903 1882 a(2)926 1866 y Fi(E)960 1873 y Fh(exc)1013 1866 y Fn(\()p Fi(K)s Fn(\))g(is)g(then)h(de\014ned)g(b)o(y)f(an)o(y)g (of)g(the)g(follo)o(wing)37 1922 y(t)o(w)o(o)e(equiv)m(alen)o(t)j(quan) o(tities)f(\(see)f(Lemma)h(5.1)e(in)i(Section)g(5)f(b)q(elo)o(w\))g(:) 212 2043 y Fi(E)246 2050 y Fh(exc)299 2043 y Fn(\()p Fi(K)s Fn(\))41 b(=)540 1981 y Fe(Z)539 2113 y Fh(Q)593 2043 y Fi(dx)652 1981 y Fe(Z)640 2116 y Fd(R)670 2106 y Ft(3)711 2013 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)867 1996 y Fv(2)p 711 2033 174 2 v 732 2074 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)897 2043 y Fi(dy)418 2206 y Fn(=)539 2144 y Fe(Z)5 b(Z)g(Z)h(Z)516 2285 y Fv(\()p Fh(Q)558 2273 y Fc(?)575 2285 y Fv(\))589 2273 y Ft(2)606 2285 y Fu(\002)p Fh(Q)661 2273 y Ft(2)719 2206 y Fi(\032)p Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))g Fi(W)943 2213 y Fu(1)978 2206 y Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)1095 2187 y Fu(0)1107 2206 y Fi(;)e(x)h Fg(\000)h Fi(y)r Fn(\))e Fi(\032)1281 2187 y Fu(\003)1301 2206 y Fn(\()p Fi(\030)1341 2187 y Fu(0)1352 2206 y Fi(;)g(x;)g(y)r Fn(\))g Fi(dxdy)1579 2175 y(d\030)r(d\030)1671 2159 y Fu(0)p 1577 2196 106 2 v 1577 2237 a Fn(\(2)p Fi(\031)r Fn(\))1664 2224 y Fv(6)1688 2206 y Fi(:)37 2380 y Fn(The)16 b(in)o(teraction)g(p)q(oten)o(tial)g Fi(W)595 2387 y Fu(1)647 2380 y Fn(is)g(de\014ned,)g(for)e(ev)o(ery)i Fi(\021)g Fn(and)f Fi(z)j Fn(in)e Fj(R)1310 2364 y Fv(3)1329 2380 y Fn(,)f(b)o(y)712 2500 y Fi(W)755 2507 y Fu(1)793 2500 y Fn(\()p Fi(\021)r(;)8 b(z)r Fn(\))i(=)966 2457 y Fe(X)956 2558 y Fh(k)q Fu(2)p Fd(Z)1024 2549 y Ft(3)1076 2470 y Fi(e)1097 2453 y Fh(ik)q Fu(\001)p Fh(\021)p 1054 2490 130 2 v 1054 2531 a Fg(j)p Fi(z)h Fn(+)g Fi(k)q Fg(j)1188 2500 y Fi(:)558 b Fn(\(2.25\))37 2637 y(The)16 b(analogous)f(problem)h(in)g(the)f(smeared)g(n)o(uclei)i(case)e(reads) 595 2753 y Fi(I)619 2734 y Fh(m;H)s(F)615 2764 y(per)733 2753 y Fn(=)e(inf)836 2689 y Fe(\032)870 2753 y Fi(E)907 2734 y Fh(m;H)s(F)904 2764 y(per)1008 2753 y Fn(\()p Fi(K)s Fn(\))f(;)20 b Fi(K)15 b Fg(2)e(K)1264 2689 y Fe(\033)1298 2753 y Fi(;)448 b Fn(\(2.26\))p eop %%Page: 12 12 12 11 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(12)335 216 y Fi(E)372 197 y Fh(m;H)s(F)369 227 y(per)473 216 y Fn(\()p Fi(K)s Fn(\))12 b(=)611 154 y Fe(Z)636 257 y Fh(Q)664 248 y Fc(?)692 216 y Fn(T)l(r)738 228 y Fh(L)762 216 y Ft(2)762 240 y Fc(\030)780 228 y Fv(\()p Fh(Q)p Fv(\))837 179 y Fe(\002)856 216 y Fg(\000)p Fn(\001)p Fi(K)968 223 y Fh(\030)987 179 y Fe(\003)1048 185 y Fi(d\030)p 1018 205 106 2 v 1018 247 a Fn(\(2)p Fi(\031)r Fn(\))1105 234 y Fv(3)1139 216 y Fn(+)1189 185 y(1)p 1189 205 23 2 v 1189 247 a(2)1217 216 y Fi(D)1255 223 y Fh(G)1284 216 y Fn(\()p Fi(\032)e Fg(\000)h Fi(m;)d(\032)h Fg(\000)i Fi(m)p Fn(\))606 340 y Fg(\000)656 309 y Fn(1)p 656 330 V 656 371 a(2)684 340 y Fi(E)718 347 y Fh(exc)771 340 y Fn(\()p Fi(K)s Fn(\))e Fg(\000)909 309 y Fn(1)p 909 330 V 909 371 a(2)937 340 y Fi(D)975 347 y Fh(G)1004 340 y Fn(\()p Fi(m;)f(m)p Fn(\))p Fi(:)1759 274 y Fn(\(2.27\))37 453 y(F)l(or)j(the)h(Hartree-F)l(o)q(c)o(k)f(mo)q(del,)i(w)o(e)e(only)h (ha)o(v)o(e)f(hin)o(ts)h(whic)o(h)g(indicate)h(that)e(the)g(limit)i(w)o (e)e(suggest)g(ab)q(o)o(v)o(e)37 509 y(is)18 b(the)e(correct)h(one.)24 b(In)18 b(order)e(to)g(prepare)h(and)g(stim)o(ulate)g(future)f(w)o (orks)g(on)h(the)g(sub)s(ject,)f(w)o(e)h(pro)o(v)o(e)37 566 y(in)g(Section)f(5)f(that)f(the)h(p)q(erio)q(dic)j(Hartree-F)l(o)q (c)o(k)c(problem)i(is)g(w)o(ell-p)q(osed,)g(in)g(the)g(follo)o(wing)g (sense.)37 660 y Fj(Theorem)i(2.3)24 b(\(W)l(ell-p)q(osedness)18 b(of)g(the)g(HF)f(p)q(erio)q(dic)j(problem\))108 716 y Ff(The)15 b(minimization)e(pr)n(oblem)h(de\014ne)n(d)e(by)i(\(2.23\)) g(and)g(\(2.24\))g(\(r)n(esp)n(e)n(ctively)d(by)j(\(2.26\))g(and)g (\(2.27\)\))37 772 y(admits)j(a)g(minimum.)37 866 y Fn(Moreo)o(v)o(er,) d(w)o(e)h(establish)h(in)g(Subsection)h(5.2)d(the)h(follo)o(wing.)37 960 y Fj(Prop)q(osition)20 b(2.1)k Ff(We)15 b(assume)f(that)i(the)f(V)m (an)f(Hove)g(se)n(quenc)n(e)g Fn(\003)g Ff(satis\014es)f(\(2.21\).)21 b(In)14 b(addition,)i(we)37 1017 y(assume)g(that)g(the)f(unit)h(c)n(el) r(l)e Fi(Q)h Ff(is)g(a)h(cub)n(e,)f(and)g(that)h(ther)n(e)g(exists)e(a) i(minimizer)f Fi(K)g Fg(2)e(K)j Ff(of)f Fi(I)1683 1000 y Fh(H)s(F)1679 1028 y(per)1759 1017 y Ff(whose)37 1073 y(density)h Fi(\032)h Ff(shar)n(es)f(the)g(symmetries)g(of)g(the)h (unit)f(cub)n(e.)21 b(Then,)696 1202 y Fn(lim)8 b(sup)718 1241 y Fv(\003)p Fu(!1)848 1171 y Fi(I)872 1154 y Fh(H)s(F)868 1184 y Fv(\003)p 848 1192 85 2 v 861 1233 a Fg(j)p Fn(\003)p Fg(j)950 1202 y(\024)13 b Fi(I)1022 1183 y Fh(H)s(F)1018 1213 y(per)1092 1202 y Fn(+)1142 1171 y Fi(M)p 1142 1192 50 2 v 1156 1233 a Fn(2)1196 1202 y Fi(;)550 b Fn(\(2.28\))37 1335 y Ff(wher)n(e)17 b Fi(I)190 1318 y Fh(H)s(F)186 1346 y(per)267 1335 y Ff(is)e(de\014ne)n(d)h(by)g(\(2.23\){\(2.24\).)37 1453 y Fn(Finally)l(,)h(w)o(e)d(de\014ne)i(the)f(follo)o(wing)g(useful) h(functional)g(transformation)d(whic)o(h)j(is)f(a)g(particular)g(con)o (v)o(ex)37 1510 y(com)o(bination,)22 b(and)e(that)f(w)o(e)g(ha)o(v)o(e) h(already)g(used)g(in)h([11)o(].)33 b(It)20 b(will)i(b)q(e)e(again)g(v) o(ery)g(e\016cien)o(t)g(in)h(the)37 1566 y(presen)o(t)14 b(w)o(ork)f(in)h(Subsection)h(4.1,)d(b)o(y)i(allo)o(wing)g(to)f(tak)o (e)g(adv)m(an)o(tage)g(of)g(the)h(con)o(v)o(exit)o(y)f(of)g(the)h (reduced)37 1623 y(Hartree-F)l(o)q(c)o(k)k(functional,)i(in)g(order)e (to)g(compare)g(from)g(b)q(elo)o(w)h(the)f(lo)o(w)o(er)g(limit)i(of)e (the)h(energy)f(p)q(er)37 1679 y(unit)e(v)o(olume)g(b)o(y)f(the)g (corresp)q(onding)i(reduced)f(Hartree-F)l(o)q(c)o(k)f(p)q(erio)q(dic)i (mo)q(del.)37 1777 y Fj(De\014nition)i(3)14 b Ff(F)m(or)h(a)h(given)f (se)n(quenc)n(e)f Fn(\003)h Ff(and)h(a)g(se)n(quenc)n(e)e Fi(\032)1127 1784 y Fv(\003)1169 1777 y Ff(of)i(densities,)e(we)i(c)n (al)r(l)f(the)h Fg(\030)p Ff(tr)n(ansform)37 1834 y(of)h Fi(\032)114 1841 y Fv(\003)157 1834 y Ff(and)g(denote)f(by)j Fe(e)-28 b Fi(\032)473 1841 y Fv(\003)516 1834 y Ff(the)17 b(fol)r(lowing)f(se)n(quenc)n(e)e(of)j(functions)734 1952 y Fe(e)-28 b Fi(\032)754 1959 y Fv(\003)793 1952 y Fn(=)863 1922 y(1)p 846 1942 57 2 v 846 1983 a Fg(j)p Fn(\003)p Fg(j)916 1909 y Fe(X)916 2008 y Fh(k)q Fu(2)p Fv(\003)991 1952 y Fi(\032)1014 1959 y Fv(\003)1041 1952 y Fn(\()p Fg(\001)9 b Fn(+)h Fi(k)q Fn(\))p Fi(:)108 2162 y Fn(W)l(e)16 b(shall)h(mak)o(e)e(use)h(in)h(the)f(sequel)h(of)e (the)h(follo)o(wing)h(notation.)k(If)16 b Fi(H)k Fn(is)c(a)f (functional)i(space,)f(w)o(e)37 2219 y(denote)g(b)o(y)f Fi(H)285 2226 y Fv(unif)349 2219 y Fn(\()p Fj(R)406 2202 y Fv(3)426 2219 y Fn(\))g(the)g(space)210 2321 y Fi(H)248 2328 y Fv(unif)313 2321 y Fn(\()p Fj(R)370 2302 y Fv(3)389 2321 y Fn(\))d(=)h Fg(f)p Fi( )h Fg(2)f(D)613 2302 y Fu(0)625 2321 y Fn(\()p Fj(R)682 2302 y Fv(3)702 2321 y Fn(\))f(;)20 b Fi( )14 b Fg(2)e Fi(H)t Fn(\()p Fi(x)e Fn(+)g Fi(Q)p Fn(\))17 b Fg(8)p Fi(x)c Fg(2)g Fj(R)1209 2302 y Fv(3)1229 2321 y Fi(;)19 b Fn(sup)1250 2363 y Fh(x)p Fu(2)p Fd(R)1324 2353 y Ft(3)1348 2321 y Fg(k)p Fi( )r Fg(k)1426 2330 y Fh(H)s Fv(\()p Fh(x)p Fv(+)p Fh(Q)p Fv(\))1573 2321 y Fi(<)13 b Fg(1g)p Fi(;)37 2451 y Fn(and)268 2553 y Fi(H)310 2534 y Fv(1)306 2564 y(p)q(er)358 2553 y Fn(\()p Fi(Q)p Fn(\))f(=)h Fg(f)p Fi(u)g Fg(2)g Fi(H)637 2534 y Fv(1)633 2565 y(lo)q(c)678 2553 y Fn(\()p Fj(R)735 2534 y Fv(3)755 2553 y Fn(\))f Fi(;)20 b(u)k Fn(p)q(erio)q(dic)17 b(in)25 b Fi(x)1130 2560 y Fh(i)1144 2553 y Fi(;)8 b(i)j Fn(=)i(1)p Fi(;)8 b Fn(2)p Fi(;)g Fn(3)p Fi(;)29 b Fn(of)14 b(p)q(erio)q(d)j(1)p Fg(g)p Fi(:)108 2680 y Fn(As)d(announced)i(in)f(the)f(in)o(tro)q(duction,)i (the)e(sequel)h(of)f(this)h(pap)q(er)g(is)g(dev)o(oted)f(to)g(the)g (pro)q(ofs)g(of)g(the)37 2736 y(ab)q(o)o(v)o(e)h(results.)p eop %%Page: 13 13 13 12 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(13)37 199 y Fk(3)67 b(Preliminari)q(es)37 344 y Fb(3.1)57 b(A)15 b(priori)f(estimates)j(for)e(the)g(Reduced)h(Hartree-F)-5 b(o)r(c)n(k)17 b(and)e(the)g(Hartree-F)-5 b(o)r(c)n(k)165 402 y(mo)r(dels)37 488 y Fn(W)l(e)18 b(b)q(egin)g(this)f(section)h(b)o (y)e(recalling)j(the)e(existence)h(results)g(of)e(minima)i(for)e(the)h (Hartree-F)l(o)q(c)o(k)g(and)37 544 y(the)i(reduced)h(Hartree-F)l(o)q (c)o(k)e(mo)q(dels)i(de\014ned)g(in)f(Section)h(2)e(through)g(the)h (form)o(ula)g(\(2.3\){\(2.5)o(\))d(and)37 601 y(\(2.10\){\(2.11\))f (resp)q(ectiv)o(ely)l(.)32 b(W)l(e)18 b(shall)i(only)f(state)e(the)i (results)g(and)f(mak)o(e)g(the)h(pro)q(ofs)f(in)h(the)g(case)37 657 y(of)d(p)q(oin)o(t)g(n)o(uclei)h(;)f(stating)f(the)h(analogues)g (in)g(the)g(case)g(of)f(the)h(smeared)f(n)o(uclei)j(brings)e(no)g (additional)37 714 y(di\016cult)o(y)h(and)e(the)h(pro)q(ofs)e(are)h(ev) o(en)h(easier)f(in)h(that)f(case)g(\(see)g([11)o(]\).)108 770 y(In)20 b(the)g(Hartree-F)l(o)q(c)o(k)f(setting,)i(the)e(existence) i(of)e(a)h(minimizer)h(for)e(neutral)i(molecules)g(for)e(the)37 827 y(standard)14 b(Hartree-F)l(o)q(c)o(k)f(mo)q(del)i(\(2.7\){\(2.8\)) c(has)j(b)q(een)h(pro)o(v)o(ed)e(b)o(y)h(E.)g(H.)f(Lieb)j(and)e(B.)g (Simon)g(in)h([33)o(])37 883 y(and)20 b(b)o(y)g(P)l(.-L.)g(Lions)g(in)h ([37)o(].)33 b(Moreo)o(v)o(er,)18 b(the)i(equiv)m(alence)j(b)q(et)o(w)o (een)d(the)f(standard)h(Hartree-F)l(o)q(c)o(k)37 939 y(mo)q(del)15 b(\(2.7\){\(2.8\))c(and)j(the)g(Hartree-F)l(o)q(c)o(k)g (mo)q(del)h(stated)e(in)i(terms)e(of)h(densit)o(y)g(matrices)g (\(2.3\){\(2.5\))37 996 y(\(without)g(restricting)g(the)f(minimization) j(to)d(pro)s(jectors\))f(is)i(due)g(to)f(E.H.)f(Lieb)j([30].)j(Lieb's)d (pro)q(of)e(has)37 1052 y(b)q(een)20 b(simpli\014ed)i(later)d(b)o(y)f (V.)h(Bac)o(h)g([5)o(].)30 b(A)19 b(similar)h(pro)q(of)f(b)o(y)f(P)l (.-L.)h(Lions)h(ma)o(y)e(also)h(b)q(e)g(found)g(in)37 1109 y([37].)f(In)13 b(the)g(reduced)g(Hartree-F)l(o)q(c)o(k)f(setting) g(\(2.10\){\(2.11\))o(,)e(the)i(existence)i(of)e(a)g(minimizer)i Fi(K)1732 1116 y Fv(\003)1771 1109 y Fg(2)f(K)1849 1116 y Fv(\003)37 1165 y Fn(for)k(neutral)h(molecules)g(is)g(due)g(to)e(J.)h (P)l(.)g(Solo)o(v)o(ej)g([52)o(].)25 b(It)18 b(is)f(imp)q(ortan)o(t)g (to)f(notice)i(that,)f(the)g(energy)37 1222 y(functional)f(\(2.11\))c (is)i(con)o(v)o(ex)g(with)h(resp)q(ect)f(to)f(the)i(densit)o(y)f (matrix.)19 b(Moreo)o(v)o(er,)13 b(thanks)h(to)f(the)h(strict)37 1278 y(con)o(v)o(exit)o(y)i(of)f Fi(\032)f Fg(7!)f Fi(D)q Fn(\()p Fi(\032;)8 b(\032)p Fn(\))15 b(\(this)h(is)g(standard\))f(and)g (of)g(the)h(con)o(v)o(exit)o(y)f(of)g(the)h(set)f Fg(K)q Fn(,)g(an)o(y)g(minimizer)37 1335 y Fi(K)76 1342 y Fv(\003)118 1335 y Fn(of)h(\(2.10\))d(leads)k(to)e(the)h(same)f(densit)o(y)h(whic)o (h)h(is)f(uniquely)i(de\014ned)f(\(it)f Ff(do)n(es)g(not)k Fn(dep)q(end)d(on)f(the)37 1391 y(minimizer)i Fi(K)289 1398 y Fv(\003)315 1391 y Fn(\))d(\(see)g([52)o(]\).)k(Let)d(us)f (henceforth)g(denote)h(b)o(y)f Fi(\032)1156 1398 y Fv(\003)1183 1391 y Fn(\()p Fi(x)p Fn(\))g(this)g(densit)o(y)l(.)108 1472 y(Let)20 b(us)f(no)o(w)g(b)q(egin)i(our)e(study)g(of)g(the)g (thermo)q(dynamic)h(limit)h(for)e(these)g(mo)q(dels)h(with)g(getting)37 1529 y(b)q(ounds)d(on)e(the)g(energy)g(p)q(er)h(unit)g(v)o(olume.)37 1635 y Fj(Lemma)i(3.1)23 b Ff(L)n(et)16 b Fn(\003)f Ff(b)n(e)g(a)g(V)m (an)g(Hove)g(se)n(quenc)n(e,)f(then)1055 1617 y Fv(1)p 1042 1624 45 2 v 1042 1651 a Fu(j)p Fv(\003)p Fu(j)1091 1635 y Fi(I)1115 1619 y Fh(RH)s(F)1111 1649 y Fv(\003)1218 1635 y Ff(and)1324 1617 y Fv(1)p 1310 1624 V 1310 1651 a Fu(j)p Fv(\003)p Fu(j)1359 1635 y Fi(I)1383 1619 y Fh(H)s(F)1379 1649 y Fv(\003)1459 1635 y Ff(ar)n(e)i(b)n(ounde)n(d)f (indep)n(en-)37 1692 y(dently)h(of)h Fn(\003)p Ff(.)37 1823 y Fj(Pro)q(of)i(of)e(Lemma)h(3.1)g(:)108 1879 y Fn(Since)i(the)e(so-called)h(exc)o(hange)g(term)e Fg(\000)847 1861 y Fv(1)p 847 1868 18 2 v 847 1895 a(2)878 1843 y Fe(R)r(R)931 1895 y Fd(R)961 1886 y Ft(3)979 1895 y Fu(\002)p Fd(R)1036 1886 y Ft(3)1069 1857 y Fu(j)p Fh(\032)p Fv(\()p Fh(x)p Fv(;)p Fh(y)q Fv(\))p Fu(j)1183 1845 y Ft(2)p 1069 1869 131 2 v 1091 1896 a Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j)1222 1879 y Fi(dxdy)i Fn(app)q(earing)g(in)g(the)f (de\014nition)37 1947 y(\(2.5\))c(of)h Fi(E)235 1931 y Fh(H)s(F)232 1961 y Fv(\003)310 1947 y Fn(is)h(non-p)q(ositiv)o(e,)g (it)f(is)h(ob)o(vious,)f(from)g(\(2.3\){\(2.5)o(\))d(and)k (\(2.10\){\(2.1)o(1\))c(that)777 2040 y(1)p 760 2060 57 2 v 760 2102 a Fg(j)p Fn(\003)p Fg(j)822 2071 y Fi(I)846 2052 y Fh(H)s(F)842 2082 y Fv(\003)919 2071 y Fg(\024)989 2040 y Fn(1)p 972 2060 V 972 2102 a Fg(j)p Fn(\003)p Fg(j)1034 2071 y Fi(I)1058 2052 y Fh(RH)s(F)1054 2082 y Fv(\003)1145 2071 y Fi(:)37 2202 y Fn(Th)o(us)17 b(in)f(order)g(to)g (pro)o(v)o(e)f(the)h(ab)q(o)o(v)o(e)g(lemma,)g(w)o(e)g(shall)h(\014rst) f(c)o(hec)o(k)g(that)1387 2184 y Fv(1)p 1373 2191 45 2 v 1373 2218 a Fu(j)p Fv(\003)p Fu(j)1422 2202 y Fi(I)1446 2186 y Fh(RH)s(F)1442 2216 y Fv(\003)1550 2202 y Fn(is)g(b)q(ounded)i (from)37 2270 y(ab)q(o)o(v)o(e,)f(and)h(then,)f(that)509 2252 y Fv(1)p 496 2259 V 496 2286 a Fu(j)p Fv(\003)p Fu(j)545 2270 y Fi(I)569 2254 y Fh(H)s(F)565 2284 y Fv(\003)646 2270 y Fn(is)h(b)q(ounded)h(from)d(b)q(elo)o(w,)i(with)f(b)q(ounds)i (that)d(are)h(indep)q(enden)o(t)i(of)37 2327 y(\003.)108 2408 y(Let)13 b(us)g(b)q(egin)h(with)g(the)e(b)q(ound)i(from)e(ab)q(o)o (v)o(e,)h(whic)o(h)h(is)f(simpler.)20 b(Let)14 b Fi(')e Fg(2)h(D)q Fn(\()p Fi(Q)p Fn(\))f(with)1656 2371 y Fe(R)1678 2424 y Fh(Q)1715 2408 y Fi(')1745 2392 y Fv(2)1777 2408 y Fi(dx)h Fn(=)37 2472 y(1.)23 b(F)l(or)15 b(eac)o(h)h Fi(k)h Fn(in)g(\003,)e(w)o(e)h(set)g Fi(')606 2479 y Fh(k)641 2472 y Fn(=)f Fi(')p Fn(\()p Fg(\001)9 b(\000)i Fi(k)q Fn(\).)22 b(Then,)17 b(the)f(trace-class)g(op)q(erator)f Fi(K)1544 2456 y Fv(0)1541 2486 y(\003)1583 2472 y Fn(whose)h(Hilb)q (ert-)37 2529 y(Sc)o(hmidt)h(k)o(ernel)g(is)g(de\014ned)g(b)o(y)619 2495 y Fe(P)667 2542 y Fh(k)q Fu(2)p Fv(\003)744 2529 y Fi(')774 2536 y Fh(k)795 2529 y Fn(\()p Fi(x)p Fn(\))d Fi(')901 2512 y Fu(\003)901 2543 y Fh(k)922 2529 y Fn(\()p Fi(y)r Fn(\))h(is)i(clearly)g(a)f(test)f(function)i(for)e Fi(I)1591 2512 y Fh(RH)s(F)1587 2542 y Fv(\003)1679 2529 y Fn(,)g(with)i(the)37 2585 y(electronic)h(densit)o(y)e(b)q(eing)i Fi(\032)547 2569 y Fv(0)547 2599 y(\003)588 2585 y Fn(=)637 2551 y Fe(P)685 2599 y Fh(k)q Fu(2)p Fv(\003)762 2585 y Fg(j)p Fi(')805 2592 y Fh(k)825 2585 y Fg(j)838 2569 y Fv(2)858 2585 y Fn(.)k(Arguing)16 b(as)g(in)g([11)o(],)g(Chapter)f (3,)h(Section)h(3.2)e(for)g(the)37 2642 y(electrostatic)h(terms,)e(w)o (e)h(c)o(hec)o(k)h(successiv)o(ely)h(that)715 2732 y(1)p 698 2752 57 2 v 698 2794 a Fg(j)p Fn(\003)p Fg(j)760 2762 y Fn(T)l(r[)p Fg(\000)p Fn(\001)p Fi(K)935 2744 y Fv(0)932 2774 y(\003)958 2762 y Fn(])12 b(=)1031 2700 y Fe(Z)1057 2804 y Fh(Q)1094 2762 y Fg(jr)p Fi(')p Fg(j)1188 2744 y Fv(2)1206 2762 y Fi(;)p eop %%Page: 14 14 14 13 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(14)37 199 y(and)16 b(that)247 305 y(lim)231 335 y Fv(\003)p Fu(!1)355 274 y Fn(1)p 338 294 57 2 v 338 336 a Fg(j)p Fn(\003)p Fg(j)408 241 y Fe(\022)446 274 y Fn(1)p 446 294 23 2 v 446 336 a(2)474 305 y Fi(U)505 312 y Fv(\003)541 305 y Fg(\000)587 243 y Fe(Z)612 346 y Fd(R)642 337 y Ft(3)670 305 y Fi(V)696 312 y Fv(\003)735 305 y Fi(\032)758 312 y Fv(\003)795 305 y Fn(+)846 274 y(1)p 846 294 V 846 336 a(2)873 305 y Fi(D)q Fn(\()p Fi(\032)953 312 y Fv(\003)980 305 y Fi(;)8 b(\032)1024 312 y Fv(\003)1050 305 y Fn(\))1068 241 y Fe(\023)1114 305 y Fn(=)1167 274 y Fi(M)p 1167 294 50 2 v 1180 336 a Fn(2)1231 305 y Fg(\000)1277 243 y Fe(Z)1302 346 y Fh(Q)1339 305 y Fi(G)13 b(\032)d Fn(+)1472 274 y(1)p 1472 294 23 2 v 1472 336 a(2)1499 305 y Fi(D)1537 312 y Fh(G)1567 305 y Fn(\()p Fi(\032;)e(\032)p Fn(\))p Fi(:)108 449 y Fn(Let)15 b(us)h(c)o(hec)o(k)f(no)o(w)g(the)g (lo)o(w)o(er)g(b)q(ound)h(for)e(the)i(Hartree-F)l(o)q(c)o(k)e(energy)h (p)q(er)h(unit)g(v)o(olume.)k(W)l(e)15 b(\014rst)37 505 y(recall)i(that,)d(b)o(y)h(virtue)h(of)f(the)g(so-called)i (Lieb{Thirring)h(inequalit)o(y)f([35)o(])e(and)g(the)g(generalization)i (b)o(y)37 561 y(P)l(.-L.)g(Lions)h(and)e(T.)g(P)o(aul)h([39)o(])g(to)f (the)g(case)h(of)f(general)h(densit)o(y)h(matrices\),)e(there)h(exists) g(a)f(p)q(ositiv)o(e)37 618 y(constan)o(t)f Fi(C)253 625 y Ft(L)n(T)293 618 y Fn(,)g(that)g(is)g Ff(indep)n(endent)k Fn(of)14 b(\003,)h(suc)o(h)g(that,)f(for)h(an)o(y)g Fi(K)j Fn(in)e Fg(K)1346 625 y Fv(\003)1387 618 y Fn(:)705 728 y Fi(C)737 735 y Ft(L)n(T)786 666 y Fe(Z)811 769 y Fd(R)841 760 y Ft(3)868 728 y Fi(\032)891 709 y Fv(5)p Fh(=)p Fv(3)959 728 y Fg(\024)d Fn(T)l(r[)p Fg(\000)p Fn(\001)p Fi(K)s Fn(])p Fi(:)574 b Fn(\(3.1\))37 843 y(On)21 b(the)f(other)f (hand,)i(the)f(Lieb{Oxford)h(inequalit)o(y)g([31)o(])e(giv)o(es)h(a)g (lo)o(w)o(er)f(b)q(ound)i(for)e(the)h(exc)o(hange)37 900 y(term)f(in)h(the)f(follo)o(wing)g(w)o(a)o(y)l(.)31 b(There)19 b(exists)g(a)g(p)q(ositiv)o(e)h(constan)o(t)e Fi(C)1299 907 y Ft(LO)1362 900 y Fn(that)h(is)g(indep)q(enden)o(t)i(of) e(\003,)37 956 y(suc)o(h)d(that,)e(for)h(an)o(y)g Fi(K)j Fn(in)e Fg(K)552 963 y Fv(\003)593 956 y Fn(with)f(densit)o(y)h Fi(\032)507 1073 y Fg(\000)p Fi(C)574 1080 y Ft(LO)626 1011 y Fe(Z)652 1114 y Fd(R)682 1105 y Ft(3)709 1073 y Fi(\032)732 1054 y Fv(4)p Fh(=)p Fv(3)800 1073 y Fg(\024)d(\000)891 1011 y Fe(Z)5 b(Z)967 1114 y Fd(R)997 1105 y Ft(3)1015 1114 y Fu(\002)p Fd(R)1072 1105 y Ft(3)1104 1042 y Fg(j)p Fi(\032)p Fn(\()p Fi(x)p Fn(;)j Fi(y)r Fn(\))p Fg(j)1260 1026 y Fv(2)p 1104 1063 174 2 v 1126 1104 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1296 1073 y Fi(dxdy)r(:)375 b Fn(\(3.2\))37 1192 y(Whence,)18 b(with)e(the)h(help)g(of)f(\(3.1\))f(and)h(\(3.2\),)f (the)h(HF)g(mo)q(del)i(ma)o(y)d(b)q(e)i(compared)g(from)e(b)q(elo)o(w)i (b)o(y)f(a)37 1248 y(Thomas-F)l(ermi-Dirac)g(t)o(yp)q(e)f(mo)q(del)h (\(see)f(E.H.)g(Lieb)h([29)o(]\),)f(as)f(follo)o(ws)i(:)329 1361 y Fi(E)366 1342 y Fh(H)s(F)363 1372 y Fv(\003)426 1361 y Fn(\()p Fi(K)s Fn(\))c Fg(\025)h Fi(C)596 1368 y Ft(L)n(T)644 1299 y Fe(Z)669 1402 y Fd(R)699 1393 y Ft(3)727 1361 y Fi(\032)750 1342 y Fv(5)p Fh(=)p Fv(3)815 1361 y Fg(\000)861 1299 y Fe(Z)886 1402 y Fd(R)916 1393 y Ft(3)943 1361 y Fi(V)969 1368 y Fv(\003)1009 1361 y Fi(\032)e Fn(+)1093 1330 y(1)p 1093 1350 23 2 v 1093 1392 a(2)1121 1361 y Fi(D)q Fn(\()p Fi(\032;)d(\032)p Fn(\))h Fg(\000)i Fi(C)1350 1368 y Ft(LO)1402 1299 y Fe(Z)1427 1402 y Fd(R)1457 1393 y Ft(3)1485 1361 y Fi(\032)1508 1342 y Fv(4)p Fh(=)p Fv(3)1563 1361 y Fi(;)206 b Fn(\(3.3\))37 1481 y(for)11 b(ev)o(ery)f Fi(K)16 b Fg(2)d(K)351 1488 y Fv(\003)377 1481 y Fn(.)18 b(The)11 b(pro)q(of)g(of)f(the)h(lo)o(w)o (er)g(b)q(ound)h(for)1067 1463 y Fv(1)p 1054 1470 45 2 v 1054 1497 a Fu(j)p Fv(\003)p Fu(j)1103 1481 y Fi(I)1127 1464 y Fh(H)s(F)1123 1494 y Fv(\003)1198 1481 y Fn(is)f(then)h(a)e (consequence)i(of)f(the)g(results)37 1544 y(obtained)16 b(b)o(y)e(E.H.)g(Lieb)i(and)e(B.)g(Simon)i([32)o(])e(and)g(b)o(y)h(the) f(authors)g([11)o(])g(for)g(the)h(Thomas-F)l(ermi)f(t)o(yp)q(e)37 1600 y(mo)q(dels.)20 b(W)l(e)12 b(\014rst)g(notice)g(that,)g(when)g Fi(K)j Fn(lies)e(in)g Fg(K)941 1607 y Fv(\003)967 1600 y Fn(,)f(b)q(ecause)h(of)f(the)g(Lieb-Thirring)i(inequalit)o(y)g (\(3.1\),)37 1657 y(the)19 b(corresp)q(onding)g(electronic)h(densit)o (y)f Fi(\032)g Fn(b)q(elongs)g(to)f(the)g(set)g Fg(f)p Fi(\032)h Fg(\025)f Fn(0)p Fi(;)c(\032)19 b Fg(2)f Fi(L)1493 1640 y Fv(5)p Fh(=)p Fv(3)1548 1657 y Fn(\()p Fj(R)1605 1640 y Fv(3)1625 1657 y Fn(\))p Fi(;)24 b(D)q Fn(\()p Fi(\032;)8 b(\032)p Fn(\))18 b Fi(<)37 1713 y Fn(+)p Fg(1)p Fi(;)151 1677 y Fe(R)173 1729 y Fd(R)203 1720 y Ft(3)230 1713 y Fi(\032)13 b Fn(=)g Fg(j)p Fn(\003)p Fg(jg)p Fi(:)f Fn(In)j(particular,)f(with)g(the)g(help)h(of)f(the)g (H\177)-23 b(older)14 b(inequalit)o(y)l(,)i(and)e(since)h(1)e Fg(\024)1752 1695 y Fv(4)p 1752 1702 18 2 v 1752 1729 a(3)1787 1713 y Fg(\024)1840 1695 y Fv(5)p 1840 1702 V 1840 1729 a(3)1862 1713 y Fn(,)37 1770 y(w)o(e)i(get)381 1818 y Fe(Z)406 1921 y Fd(R)436 1912 y Ft(3)464 1880 y Fi(\032)487 1861 y Fv(4)p Fh(=)p Fv(3)555 1880 y Fg(\024)603 1843 y Fe(\000)624 1818 y(Z)649 1921 y Fd(R)679 1912 y Ft(3)706 1880 y Fi(\032)729 1861 y Fv(5)p Fh(=)p Fv(3)785 1843 y Fe(\001)806 1854 y Fv(1)p Fh(=)p Fv(2)873 1843 y Fe(\000)894 1818 y(Z)919 1921 y Fd(R)949 1912 y Ft(3)977 1880 y Fi(\032)1000 1843 y Fe(\001)1021 1854 y Fv(1)p Fh(=)p Fv(2)1089 1880 y Fg(\024)1137 1843 y Fe(\000)1157 1818 y(Z)1183 1921 y Fd(R)1213 1912 y Ft(3)1240 1880 y Fi(\032)1263 1861 y Fv(5)p Fh(=)p Fv(3)1318 1843 y Fe(\001)1339 1854 y Fv(1)p Fh(=)p Fv(2)1407 1880 y Fg(j)p Fn(\003)p Fg(j)1465 1861 y Fv(1)p Fh(=)p Fv(2)1519 1880 y Fi(:)250 b Fn(\(3.4\))37 2000 y(In)16 b(addition,)g(with)g Fi(f)412 2007 y Fv(\003)451 2000 y Fn(=)d Fi(V)525 2007 y Fv(\003)562 2000 y Fg(\000)e Fi(\037)636 2009 y Fv(\000\(\003\))722 2000 y Fi(?)771 1982 y Fv(1)p 760 1989 40 2 v 760 2016 a Fu(j)p Fh(x)p Fu(j)805 2000 y Fn(,)j(w)o(e)h(recall)i(from)d([11)o(]) h(that,)f(for)h(ev)o(ery)g(1)d Fg(\024)h Fi(p)g(<)g Fn(3,)733 2108 y Fg(k)p Fi(f)778 2115 y Fv(\003)804 2108 y Fg(k)827 2118 y Fh(L)851 2108 y Fc(p)869 2118 y Fv(\()p Fd(R)913 2108 y Ft(3)930 2118 y Fv(\))959 2108 y Fg(\024)g Fi(C)j Fg(j)p Fn(\003)p Fg(j)1113 2089 y Fv(1)p Fh(=p)1167 2108 y Fi(:)602 b Fn(\(3.5\))37 2200 y(Whence,)16 b(going)f(bac)o(k)g(to)g (\(3.3\),)190 2313 y Fi(E)227 2294 y Fh(H)s(F)224 2324 y Fv(\003)287 2313 y Fn(\()p Fi(K)s Fn(\))9 b(+)425 2282 y Fi(U)456 2289 y Fv(\003)p 425 2302 58 2 v 442 2344 a Fn(2)529 2313 y Fg(\025)42 b Fi(C)638 2320 y Ft(L)n(T)686 2251 y Fe(Z)711 2354 y Fd(R)741 2344 y Ft(3)768 2313 y Fi(\032)791 2294 y Fv(5)p Fh(=)p Fv(3)857 2313 y Fg(\000)903 2251 y Fe(Z)928 2354 y Fd(R)958 2344 y Ft(3)985 2313 y Fi(f)1007 2320 y Fv(\003)1047 2313 y Fi(\032)10 b Fn(+)1131 2282 y(1)p 1131 2302 23 2 v 1131 2344 a(2)1158 2313 y Fi(D)q Fn(\()p Fi(\037)1243 2322 y Fv(\000\(\003\))1329 2313 y Fg(\000)h Fi(\032;)d(\037)1447 2322 y Fv(\000\(\003\))1533 2313 y Fg(\000)j Fi(\032)p Fn(\))842 2432 y Fg(\000)p Fi(C)909 2439 y Ft(LO)961 2370 y Fe(Z)987 2474 y Fd(R)1017 2464 y Ft(3)1044 2432 y Fi(\032)1067 2414 y Fv(4)p Fh(=)p Fv(3)1133 2432 y Fn(+)1183 2402 y Fi(U)1214 2409 y Fv(\003)p 1183 2422 58 2 v 1201 2464 a Fn(2)1256 2432 y Fg(\000)1306 2402 y Fn(1)p 1306 2422 23 2 v 1306 2464 a(2)1334 2432 y Fi(D)q Fn(\()p Fi(\037)1419 2442 y Fv(\000\(\003\))1495 2432 y Fi(;)d(\037)1544 2442 y Fv(\000\(\003\))1620 2432 y Fn(\))144 b(\(3.6\))529 2552 y Fg(\025)42 b Fi(C)638 2559 y Ft(L)n(T)686 2490 y Fe(Z)711 2593 y Fd(R)741 2584 y Ft(3)768 2552 y Fi(\032)791 2533 y Fv(5)p Fh(=)p Fv(3)857 2552 y Fg(\000)11 b(k)p Fi(f)948 2559 y Fv(\003)974 2552 y Fg(k)997 2565 y Fh(L)1021 2555 y Ft(5)p Fc(=)p Ft(2)1084 2552 y Fg(k)p Fi(\032)p Fg(k)1153 2565 y Fh(L)1177 2555 y Ft(5)p Fc(=)p Ft(3)842 2688 y Fg(\000)p Fi(C)909 2695 y Ft(LO)954 2623 y Fe(\022)987 2626 y(Z)1013 2729 y Fd(R)1043 2719 y Ft(3)1070 2688 y Fi(\032)1093 2669 y Fv(5)p Fh(=)p Fv(3)1148 2623 y Fe(\023)1182 2635 y Fv(1)p Fh(=)p Fv(2)1250 2688 y Fg(j)p Fn(\003)p Fg(j)1308 2669 y Fv(1)p Fh(=)p Fv(2)1371 2688 y Fg(\000)g Fi(C)1449 2695 y Fv(0)1469 2688 y Fg(j)p Fn(\003)p Fg(j)529 2796 y(\025)42 b Fi(C)638 2803 y Ft(L)n(T)678 2796 y Fg(k)p Fi(\032)p Fg(k)747 2772 y Fv(5)p Fh(=)p Fv(3)747 2816 y Fh(L)771 2806 y Ft(5)p Fc(=)p Ft(3)832 2796 y Fg(\000)10 b(j)p Fn(\003)p Fg(j)935 2777 y Fv(2)p Fh(=)p Fv(5)1002 2796 y Fg(k)p Fi(\032)p Fg(k)1071 2809 y Fh(L)1095 2799 y Ft(5)p Fc(=)p Ft(3)1155 2796 y Fg(\000)h Fi(C)1233 2803 y Ft(LO)1277 2796 y Fg(k)p Fi(\032)p Fg(k)1346 2772 y Fv(5)p Fh(=)p Fv(6)1346 2816 y Fh(L)1370 2806 y Ft(5)p Fc(=)p Ft(3)1433 2796 y Fg(j)p Fn(\003)p Fg(j)1491 2777 y Fv(1)p Fh(=)p Fv(2)1555 2796 y Fg(\000)g Fi(C)1633 2803 y Fv(0)1653 2796 y Fg(j)p Fn(\003)p Fg(j)p Fi(;)58 b Fn(\(3.7\))p eop %%Page: 15 15 15 14 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(15)37 199 y(for)19 b(ev)o(ery)f Fi(K)k Fg(2)c(K)378 206 y Fv(\003)404 199 y Fn(,)i(where,)f(in)h(addition)f(to)g(\(3.4\))e (and)i(\(3.5\))e(|)i(with)g Fi(p)f Fn(=)1471 181 y Fv(5)p 1471 188 18 2 v 1471 215 a(2)1494 199 y Fn(,)h(w)o(e)f(ha)o(v)o(e)h (used)g(the)37 256 y(follo)o(wing)d(t)o(w)o(o)e(facts)h(:)89 385 y Fg(\000)135 323 y Fe(Z)160 426 y Fd(R)190 416 y Ft(3)217 385 y Fi(V)243 392 y Fv(\003)283 385 y Fi(\032)10 b Fn(+)367 354 y(1)p 367 374 23 2 v 367 416 a(2)395 385 y Fi(D)q Fn(\()p Fi(\032;)e(\032)p Fn(\))h(+)597 354 y Fi(U)628 361 y Fv(\003)p 597 374 58 2 v 614 416 a Fn(2)392 504 y(=)k Fg(\000)483 443 y Fe(Z)508 546 y Fd(R)538 536 y Ft(3)565 504 y Fi(f)587 511 y Fv(\003)627 504 y Fi(\032)d Fn(+)711 474 y(1)p 711 494 23 2 v 711 536 a(2)739 504 y Fi(D)q Fn(\()p Fi(\037)824 514 y Fv(\000\(\003\))910 504 y Fg(\000)g Fi(\032;)e(\037)1027 514 y Fv(\000\(\003\))1113 504 y Fg(\000)j Fi(\032)p Fn(\))f(+)1261 474 y Fi(U)1292 481 y Fv(\003)p 1261 494 58 2 v 1278 536 a Fn(2)1333 504 y Fg(\000)1384 474 y Fn(1)p 1384 494 23 2 v 1384 536 a(2)1411 504 y Fi(D)q Fn(\()p Fi(\037)1496 514 y Fv(\000\(\003\))1572 504 y Fi(;)e(\037)1621 514 y Fv(\000\(\003\))1697 504 y Fn(\))g Fi(;)37 623 y Fn(whic)o(h)17 b(follo)o(ws)e(from)f(the)i (de\014nition)h(of)d Fi(f)778 630 y Fv(\003)805 623 y Fn(,)h(and)634 721 y Fg(j)p Fi(U)678 728 y Fv(\003)714 721 y Fg(\000)c Fi(D)q Fn(\()p Fi(\037)845 730 y Fv(\000\(\003\))921 721 y Fi(;)d(\037)970 730 y Fv(\000\(\003\))1046 721 y Fn(\))p Fg(j)k(\024)h Fi(C)1169 728 y Fv(0)1201 721 y Fg(j)p Fn(\003)p Fg(j)p Fi(;)510 b Fn(\(3.8\))37 819 y(for)22 b(some)g(p)q(ositiv)o(e)h(constan)o(t)e Fi(C)632 826 y Fv(0)674 819 y Fn(that)h(is)h(indep)q(enden)o(t)h(of)e(\003)f ([11)o(].)41 b(F)l(rom)21 b(\(3.7\))g(and)h(b)o(y)g(setting)37 900 y Fi(X)16 b Fn(=)144 869 y Fg(k)p Fi(\032)p Fg(k)213 882 y Fh(L)237 872 y Ft(5)p Fc(=)p Ft(3)p 144 889 144 2 v 173 933 a Fn(\003)205 920 y Fv(3)p Fh(=)p Fv(5)293 900 y Fn(,)f(w)o(e)f(\014nally)j(obtain)561 1008 y(1)p 544 1028 57 2 v 544 1070 a Fg(j)p Fn(\003)p Fg(j)606 1039 y Fi(I)630 1020 y Fh(H)s(F)626 1050 y Fv(\003)703 1039 y Fg(\025)c Fi(C)783 1046 y Ft(L)n(T)824 1039 y Fi(X)866 1020 y Fv(5)p Fh(=)p Fv(3)930 1039 y Fg(\000)e Fi(X)i Fg(\000)d Fi(C)1104 1046 y Ft(LO)1149 1039 y Fi(X)1191 1020 y Fv(5)p Fh(=)p Fv(6)1255 1039 y Fg(\000)h Fi(C)1333 1046 y Fv(0)1353 1039 y Fi(;)416 b Fn(\(3.9\))37 1159 y(for)12 b(an)o(y)f Fi(X)16 b Fg(\025)d Fn(0.)18 b(The)12 b(function)h(of)e Fi(X)k Fn(whic)o(h)d(app)q(ears)g(in)g(the)g(righ)o (t-hand)g(side)h(of)e(the)h(ab)q(o)o(v)o(e)f(inequalit)o(y)37 1215 y(is)22 b(b)q(ounded)h(from)d(b)q(elo)o(w)i(b)o(y)f(some)g (constan)o(t)f(\(indep)q(enden)o(t)j(of)e(\003\))f(on)h(the)g(set)g Fg(f)p Fi(X)26 b Fg(\025)c Fn(0)p Fg(g)p Fn(.)37 b(This)37 1272 y(concludes)17 b(the)f(pro)q(of)e(of)h(the)g(lemma.)1840 1328 y Fg(})108 1426 y Fn(F)l(rom)i(no)o(w)h(on,)g(w)o(e)f(shall)i (denote)f(b)o(y)g Fi(K)839 1433 y Fv(\003)883 1426 y Fn(a)g(minimizer)h(of)f Fi(I)1217 1410 y Fh(RH)s(F)1213 1439 y Fv(\003)1322 1426 y Fn(or)f Fi(I)1404 1410 y Fh(H)s(F)1400 1439 y Fv(\003)1482 1426 y Fn(indiscriminatel)q(y)l(,)k(b)o(y)37 1482 y Fi(\032)60 1489 y Fv(\003)88 1482 y Fn(\()p Fg(\001)p Fn(;)8 b Fg(\001)p Fn(\))14 b(its)i(k)o(ernel,)i(and)e(b)o(y)h Fi(\032)575 1489 y Fv(\003)617 1482 y Fn(=)e Fi(\032)690 1489 y Fv(\003)717 1482 y Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))15 b(the)h(corresp)q(onding)i(electronic)g(densit)o(y)l(.)24 b(As)17 b(a)f(corollary)37 1539 y(of)f(Lemma)h(3.1)e(and)h(its)h(pro)q (of,)e(w)o(e)h(ha)o(v)o(e)g(the)g(follo)o(wing)h(:)37 1641 y Fj(Prop)q(osition)k(3.1)k Ff(Ther)n(e)16 b(exist)g(p)n(ositive)f (c)n(onstants)g Fi(C)20 b Ff(that)c(ar)n(e)g(indep)n(endent)f(of)h Fn(\003)d Fg(\032)f Fj(Z)1659 1625 y Fv(3)1695 1641 y Ff(such)k(that)37 1698 y(the)h(fol)r(lowing)f(estimates)g(hold)g(:)598 1782 y Fn(1)p 581 1802 V 581 1844 a Fg(j)p Fn(\003)p Fg(j)643 1812 y Fn(T)l(r[)p Fg(\000)p Fn(\001)p Fi(K)815 1819 y Fv(\003)841 1812 y Fn(])c Fg(\024)h Fi(C)s Fn(;)774 b(\(3.10a\))598 1903 y(1)p 581 1923 V 581 1965 a Fg(j)p Fn(\003)p Fg(j)651 1872 y Fe(Z)676 1975 y Fd(R)706 1965 y Ft(3)733 1933 y Fg(jr)784 1903 y(p)p 822 1903 51 2 v 30 x Fi(\032)845 1940 y Fv(\003)872 1933 y Fg(j)885 1915 y Fv(2)917 1933 y Fg(\024)13 b Fi(C)s Fn(;)721 b(\(3.10b\))598 2024 y(1)p 581 2044 57 2 v 581 2086 a Fg(j)p Fn(\003)p Fg(j)651 1993 y Fe(Z)676 2096 y Fd(R)706 2086 y Ft(3)733 2054 y Fi(\032)756 2031 y Fv(5)p Fh(=)p Fv(3)756 2069 y(\003)824 2054 y Fg(\024)13 b Fi(C)s Fn(;)819 b(\(3.10c\))598 2145 y(1)p 581 2165 V 581 2207 a Fg(j)p Fn(\003)p Fg(j)651 2114 y Fe(Z)676 2217 y Fd(R)706 2207 y Ft(3)733 2175 y Fi(\032)762 2153 y Fh(p)756 2190 y Fv(\003)796 2175 y Fg(\024)13 b Fi(C)q(;)117 b Ff(for)16 b(every)g Fn(1)d Fg(\024)g Fi(p)f Fg(\024)1371 2145 y Fn(5)p 1371 2165 23 2 v 1371 2207 a(3)1399 2175 y(;)322 b(\(3.10d\))598 2264 y(1)p 581 2284 57 2 v 581 2326 a Fg(j)p Fn(\003)p Fg(j)643 2295 y Fi(D)q Fn(\()p Fi(\037)728 2304 y Fv(\000\(\003\))814 2295 y Fg(\000)11 b Fi(\032)883 2302 y Fv(\003)910 2295 y Fi(;)d(\037)959 2304 y Fv(\000\(\003\))1045 2295 y Fg(\000)i Fi(\032)1113 2302 y Fv(\003)1140 2295 y Fn(\))i Fg(\024)h Fi(C)s Fn(;)8 b Fi(and)390 b Fn(\(3.10e\))576 2422 y(0)13 b Fg(\024)682 2392 y Fn(1)p 665 2412 V 665 2454 a Fg(j)p Fn(\003)p Fg(j)734 2361 y Fe(Z)5 b(Z)810 2464 y Fd(R)840 2454 y Ft(3)858 2464 y Fu(\002)p Fd(R)915 2454 y Ft(3)947 2392 y Fg(j)p Fi(\032)983 2399 y Fv(\003)1010 2392 y Fn(\()p Fi(x)p Fn(;)j Fi(y)r Fn(\))p Fg(j)1130 2375 y Fv(2)p 947 2412 201 2 v 982 2454 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1165 2422 y Fi(dxdy)k Fg(\024)f Fi(C)q(:)372 b Fn(\(3.10f)t(\))37 2627 y Fj(Remark)18 b(3.1)24 b Ff(The)16 b(b)n(ound)h(\(3.10f\))e(on)h(the)h(exchange)e (term)i(was)f(p)n(ostulate)n(d)g(in)g(the)g(chemistry)g(liter-)37 2683 y(atur)n(e)i(\(se)n(e)d([42]\))i(but,)g(to)g(the)g(b)n(est)e(of)i (our)h(know)r(le)n(dge,)e(it)h(was)f(not)h(che)n(cke)n(d)f(rigor)n (ously)h(so)f(far)h(exc)n(ept)37 2740 y(in)g(the)g(simpli\014e)n(d)f (fr)n(amework)i(of)f(the)g(fr)n(e)n(e)g(ele)n(ctr)n(on)f(gas)g(by)h(G.) h(F)m(riese)n(cke)d([21].)23 b(This)18 b(b)n(ound)f(implies)37 2796 y(that)h(the)e(exchange)g(term)h(has)g(to)f(b)n(e)g(asymptotic)n (al)r(ly)h(of)f(the)h(same)f(or)n(der)h(as)f(the)h(volume)g(o)n(c)n (cupie)n(d)f(by)p eop %%Page: 16 16 16 15 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(16)37 199 y Ff(the)18 b(mole)n(cule,)f(her)n(e)g Fg(j)p Fn(\003)p Fg(j)p Ff(.)22 b(In)17 b(p)n(articular,)h(the)g(exchange)f (term)g(exhibits)g(an)g(asymptotic)h(b)n(ehaviour)g(in)37 256 y(the)g(thermo)n(dynamic)f(limit)g(which)g(is)f(c)n(ompletely)h (di\013er)n(ent)f(fr)n(om)h(the)g(one)f(of)h(the)g(other)h(ele)n(ctr)n (ostatic)37 312 y(terms.)23 b(\(Note)16 b(that)i(e)n(ach)e(of)h(them)g (b)n(ehave)g(sep)n(ar)n(ately)f(like)g Fg(j)p Fn(\003)p Fg(j)1173 296 y Fv(5)p Fh(=)p Fv(3)1244 312 y Ff(while)h(their)g(sum)g (glob)n(al)r(ly)f(b)n(ehaves)37 369 y(like)g Fg(j)p Fn(\003)p Fg(j)f Ff(\(se)n(e)g([11)q(]\)\).)37 493 y Fj(Pro)q(of)22 b(of)e(Prop)q(osition)j(3.1)e(:)k Fn(W)l(e)18 b(argue)g(only)g(in)g (the)g(framew)o(ork)e(of)i(the)f(Hartree-F)l(o)q(c)o(k)g(mo)q(del,)37 549 y(the)i(case)g(of)f(the)h(reduced)h(Hartree-F)l(o)q(c)o(k)e(mo)q (del)i(b)q(eing)g(ev)o(en)f(easier)g(to)f(deal)i(with.)30 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y(\014)839 1268 y(Z)865 1371 y Fd(R)895 1362 y Ft(3)922 1330 y Fi(f)944 1337 y Fv(\003)983 1330 y Fi(\032)1006 1337 y Fv(\003)1033 1264 y Fe(\014)1033 1291 y(\014)1033 1319 y(\014)1033 1346 y(\014)1061 1330 y Fg(\024)e Fi(C)q(:)604 b Fn(\(3.11\))37 1444 y(W)l(e)18 b(then)f(deduce)h(\(3.10e\))d(b)o(y)i(comparing)g(\(3.10c\),)e (\(3.10f\))g(and)j(\(3.11\))d(with)i(\(3.6\))e(and)i(\(3.8\).)24 b(Col-)37 1500 y(lecting)18 b(the)f(previous)h(b)q(ounds)f(and)g (comparing)g(with)g(the)f(de\014nition)j(\(2.5\))c(of)h Fi(E)1509 1484 y Fh(H)s(F)1506 1514 y Fv(\003)1569 1500 y Fn(\()p Fi(K)1626 1507 y Fv(\003)1652 1500 y Fn(\),)g(w)o(e)h(c)o (hec)o(k)37 1557 y(that)e(the)g(last)f(remaining)i(term)f(in)g(the)g (de\014nition)i(of)d(the)h(functional,)h(namely)f(T)l(r[)p Fg(\000)p Fn(\001)p Fi(K)1636 1564 y Fv(\003)1662 1557 y Fn(],)f(is)i(also)e(of)37 1613 y(the)i(order)f(of)f Fg(j)p Fn(\003)p Fg(j)p Fn(.)19 b(This)d(giv)o(es)f(\(3.10a\).)108 1694 y(W)l(e)g(next)h(observ)o(e)f(that)f(for)h(ev)o(ery)g 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(w)o(arz)f(inequalit)o(y)j(to)d(obtain)h(\(3.14\))e(and)i(with)h(the)f (con)o(v)o(en)o(tion)g(that)37 256 y(the)k(quan)o(tities)g(inside)i (the)e(in)o(tegrals)f(in)i(the)f(righ)o(t-hand)g(sides)g(of)f(\(3.13\)) f(to)h(\(3.14\))f(are)h(zero)g(almost)37 312 y(ev)o(erywhere)f(in)h (the)e(region)h(where)g Fi(\032)g Fn(itself)h(v)m(anishes.)21 b(Finally)l(,)16 b(thanks)e(to)g(\(3.12\),)f(\(3.10b\))g(is)i(a)f (direct)37 369 y(consequence)j(of)e(\(3.10a\).)j(This)d(concludes)i (the)e(pro)q(of)g(of)g(the)g(prop)q(osition.)451 b Fg(})108 467 y Fn(F)l(rom)14 b(these)g(b)q(ounds)i(on)e(the)g(energy)h(p)q(er)g (unit)g(v)o(olume,)f(w)o(e)h(deduce)g(as)f(in)h([11,)f(13)o(],)g(the)g (follo)o(wing)37 523 y(t)o(w)o(o)g(corollaries.)37 626 y Fj(Corollary)20 b(3.1)k(\(Compactness\))16 b Ff(F)m(or)g(any)g(V)m (an)f(Hove)i(se)n(quenc)n(e)d Fn(\003)p Ff(,)i(we)g(have)775 684 y Fe(Z)800 787 y Fv(\000\(\003\))874 777 y Fc(c)900 746 y Fi(\032)923 753 y Fv(\003)963 746 y Fn(=)d Fi(o)p Fn(\()p Fg(j)p Fn(\003)p 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(b)n(e)f(p)n(articularly)g(r)n(elevant)g(for)g(numeric)n(al)g(c)n (omputations.)37 1306 y Fj(Pro)q(of)24 b(of)f(Corollary)i(3.1)e(:)30 b Fn(This)20 b(is)h(a)e(direct)i(consequence)g(of)e(\(3.10e\))f(\(see)i ([11)o(],)g(Section)h(3)e(in)37 1363 y(Chapter)c(3\).)1572 b Fg(})108 1461 y Fn(The)22 b(second)h(corollary)f(mak)o(es)f(use)h(of) g(the)g(notion)g(of)f Fg(\030)p Fn(-transform,)h(in)o(tro)q(duced)i(in) e([11)o(])g(and)37 1517 y(recalled)17 b(in)f(De\014nition)h(3)e(in)h (Section)g(2.)37 1608 y Fj(Corollary)k(3.2)k Ff(F)m(or)18 b(any)g(V)m(an)f(Hove)i(se)n(quenc)n(e)d Fn(\003)p Ff(,)j(the)g(se)n (quenc)n(e)1266 1569 y Fe(p)p 1312 1569 51 2 v 1315 1608 a(e)-28 b Fi(\032)1335 1615 y Fv(\003)1380 1608 y Ff(is)18 b(b)n(ounde)n(d)h(in)f Fi(H)1707 1592 y Fv(1)1703 1622 y(unif)1767 1608 y Fn(\()p Fj(R)1824 1592 y Fv(3)1843 1608 y Fn(\))p Ff(,)37 1665 y(indep)n(endently)d(of)i Fn(\003)p Ff(.)j(Mor)n(e)n(over,)817 1782 y Fn(lim)801 1812 y Fv(\003)p Fu(!1)903 1720 y Fe(Z)928 1823 y Fh(Q)969 1782 y Fe(e)-28 b Fi(\032)989 1789 y Fv(\003)1029 1782 y Fn(=)13 b(1)p Fi(:)646 b Fn(\(3.16\))37 1909 y(The)11 b(ab)q(o)o(v)o(e)f(b)q(ounds)h(on)j Fe(e)-28 b Fi(\032)485 1916 y Fv(\003)512 1909 y Fn(,)11 b(whic)o(h)g(are)f(easily)i(deduced)g (b)o(y)e(a)g(con)o(v)o(exit)o(y)g(argumen)o(t)g(from)g(the)g (de\014nition)37 1965 y(of)20 b(the)f Fg(\030)p Fn(-transform)g(and)g (from)g(the)h(b)q(ounds)g(\(3.10b\))e(and)h(\(3.10d\),)g(will)i(b)q(e)f (useful)h(while)g(passing)37 2022 y(to)g(the)h(lo)o(w)o(er)f(limit)i (on)e(the)h(energy)f(p)q(er)h(unit)g(v)o(olume)g(for)f(the)g(reduced)i (Hartree-F)l(o)q(c)o(k)e(mo)q(del)h(in)37 2078 y(Subsection)17 b(4.1.)108 2176 y(Let)i(us)f(end)h(this)g(subsection)g(b)o(y)f (recalling)i(the)e(follo)o(wing)h(result)g(whic)o(h)g(asserts)f(that)f (the)i(limit)37 2233 y(of)f(a)g(sequence)h(of)f Fg(\030)p Fn(-transforms)f(is)i(necessarily)g(p)q(erio)q(dic.)31 b(W)l(e)18 b(skip)h(its)g(easy)e(pro)q(of,)i(for)e(whic)o(h)i(all)37 2289 y(argumen)o(ts)c(ma)o(y)f(b)q(e)i(found)g(in)g([11)o(].)37 2380 y Fj(Lemma)i(3.2)23 b Ff(L)n(et)14 b Fn(\003)g Ff(b)n(e)f(a)h(V)m (an)f(Hove)h(se)n(quenc)n(e)e(in)i(the)g(sense)e(of)i(De\014nition)f (1.)20 b(L)n(et)14 b Fi(f)1575 2387 y Fv(\003)1616 2380 y Ff(b)n(e)f(a)i(se)n(quenc)n(e)37 2437 y(of)f(function)f(such)h(that,) h(either)f Fg(k)p Fi(f)642 2444 y Fv(\003)668 2437 y Fg(k)691 2448 y Fh(L)715 2432 y Fc(p)715 2460 y Ft(unif)771 2448 y Fv(\()p Fd(R)815 2438 y Ft(3)833 2448 y Fv(\))861 2437 y Fg(\024)f Fi(C)k Ff(or)d Fg(k)p Fi(f)1059 2444 y Fv(\003)1086 2437 y Fg(k)1109 2446 y Fh(L)1133 2437 y Fc(p)1151 2446 y Fv(\()p Fd(R)1195 2437 y Ft(3)1212 2446 y Fv(\))1240 2437 y Fg(\024)f Fi(C)s Fg(j)p Fn(\003)p Fg(j)1381 2420 y Fv(1)p Fh(=p)1436 2437 y Ff(,)h(for)g(some)g Fi(p)e Fg(2)h Fn([1)p Fi(;)8 b Fn(+)p Fg(1)p Fn(])37 2506 y Ff(and)18 b(some)f(c)n(onstant)f Fi(C)21 b Ff(that)d(is)e(indep) n(endent)g(of)h Fn(\003)p Ff(.)24 b(L)n(et)17 b(us)g(assume)g(that)1402 2494 y Fn(~)1392 2506 y Fi(f)1414 2513 y Fv(\003)1458 2506 y Ff(c)n(onver)n(ges)e(to)j(some)1844 2494 y Fn(~)1834 2506 y Fi(f)5 b Ff(,)37 2563 y(almost)15 b(everywher)n(e)f(on)h Fj(R)514 2546 y Fv(3)534 2563 y Ff(,)f(or)h(we)n(akly)f(in)g Fi(L)845 2541 y Fh(p)845 2577 y Fv(lo)q(c)906 2563 y Ff(|)h(when)f Fn(1)e Fg(\024)h Fi(p)g(<)g Fn(+)p Fg(1)p Ff(,)i(or)g(in)f Fi(L)1499 2546 y Fu(1)1543 2563 y Fg(\000)6 b Fi(?)p Ff(we)n(ak)14 b(|)h(when)37 2624 y Fi(p)e Fn(=)g(+)p Fg(1)p Ff(.)22 b(Then,)379 2612 y Fn(~)369 2624 y Fi(f)f Ff(is)16 b(p)n(erio)n(dic.)37 2740 y Fn(In)k(order)e(to)h(state)f (rigorously)h(the)g(p)q(erio)q(dic)i(mo)q(dels)e(w)o(e)g(shall)h (consider)f(b)q(elo)o(w,)h(w)o(e)f(extend)g(in)h(the)37 2796 y(forthcoming)d(section)g(the)f(classical)i(notion)f(of)f (one-particle)i(densit)o(y)f(matrix)f(used)h(for)f(molecules)i(to)p eop %%Page: 18 18 18 17 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(18)37 199 y(its)17 b(analogue)f(for)f(crystals.)22 b(This)17 b(construction)f(will)i(allo)o(w)e(us)g(to)f(set)h(the)g(RHF) g(and)g(HF)g(mo)q(dels)h(for)37 256 y(crystals)g(in)h(terms)e(of)g(suc) o(h)h(\\p)q(erio)q(dic)i(densit)o(y)e(matrices".)24 b(These)17 b(new)g(ob)s(jects)f(are)h(closely)h(related)37 312 y(to)13 b(the)g(so-called)i(Blo)q(c)o(h)f(w)o(a)o(v)o(es)e(decomp)q(osition)j (classically)g(used)f(in)g(Solid)h(State)d(Ph)o(ysics,)i(as)f(w)o(e)g (shall)37 369 y(see)j(b)q(elo)o(w.)37 490 y Fb(3.2)57 b(Blo)r(c)n(h)18 b(w)n(a)n(v)n(es)k(decomp)r(osition)37 576 y Fn(Let)16 b Fi(Q)d Fn(=)h([)p Fg(\000)270 558 y Fv(1)p 270 565 18 2 v 270 592 a(2)292 576 y Fn(;)8 b(+)353 558 y Fv(1)p 353 565 V 353 592 a(2)376 576 y Fn([)395 560 y Fv(3)429 576 y Fn(b)q(e)16 b(the)g(unit)g(cub)q(e)h(of)e Fj(R)865 560 y Fv(3)900 576 y Fn(cen)o(tered)h(at)f(0.)21 b(W)l(e)15 b(denote)h(b)o(y)f Fi(Q)1518 560 y Fh(?)1551 576 y Fn(=)f([)p Fg(\000)p Fi(\031)r Fn(;)8 b(+)p Fi(\031)r Fn([)1779 560 y Fv(3)1812 576 y Fn(the)37 633 y(unit)17 b(cell)g(of)e(the)g(dual)i(\(or)d(recipro)q(cal\))j(lattice)f(asso)q (ciated)g(to)e Fj(Z)1190 616 y Fv(3)1210 633 y Fn(.)21 b(In)16 b(full)g(generalit)o(y)l(,)g(while)h(w)o(orking)37 689 y(with)d(a)f(general)h(p)q(erio)q(dic)h(lattice)f(\(with)g(unit)g (cell)h(still)f(denoted)g(b)o(y)g Fi(Q)p Fn(\),)e Fi(Q)1365 673 y Fh(?)1398 689 y Fn(is)i(the)g(so-called)g(Brillouin)37 746 y(zone)i(asso)q(ciated)f(to)g(the)g(dual)h(lattice)g(\(see)f(for)g (example)h([45)o(,)f(Section)h(XI)q(I)q(I-16]\).)108 843 y(In)k(the)g(sequel,)i(w)o(e)d(shall)i(denote)f(b)o(y)f Fi(K)k Fn(a)c(self-adjoin)o(t)h(op)q(erator)f(in)h Fi(L)1424 827 y Fv(2)1444 843 y Fn(\()p Fj(R)1501 827 y Fv(3)1521 843 y Fn(\),)g(whic)o(h)g(is)g(aimed)37 900 y(at)i(b)q(eing)h(the)e (\\p)q(erio)q(dic)j(densit)o(y)e(matrix")f(w)o(e)h(are)f(lo)q(oking)i (for,)f(and)g(that)f(enjo)o(ys)g(the)h(follo)o(wing)37 956 y(prop)q(erties)16 b(:)108 1038 y(\(H1\))k Fi(K)k Fn(comm)o(utes)d(with)g(the)g(translations)g(whic)o(h)h(lea)o(v)o(e)f (the)g(p)q(erio)q(dic)i(lattice)f Fj(Z)1646 1021 y Fv(3)1687 1038 y Fn(in)o(v)m(arian)o(t;)37 1094 y(namely)661 1196 y Fg(8)45 b Fi(k)14 b Fg(2)f Fj(Z)844 1178 y Fv(3)864 1196 y Fi(;)98 b(\034)995 1203 y Fh(k)1029 1196 y Fi(K)15 b Fn(=)e Fi(K)i(\034)1205 1203 y Fh(k)1239 1196 y Fi(;)37 1298 y Fn(with)h Fi(\034)161 1305 y Fh(k)197 1298 y Fn(b)q(eing)h (de\014ned)f(b)o(y)811 1401 y Fi(\034)831 1408 y Fh(k)852 1401 y Fi(')d Fn(=)g Fi(')p Fn(\()p Fg(\001)c Fn(+)h Fi(k)q Fn(\))37 1503 y(for)15 b(an)o(y)g(function)h Fi(')f Fn(on)g Fj(R)518 1486 y Fv(3)538 1503 y Fn(.)108 1584 y(\(H2\))d(0)g Fg(\024)h Fi(K)i Fg(\024)e Fj(1)p Fn(,)g(in)g(the)g (sense)f(of)g(self-adjoin)o(t)h(op)q(erators)f(in)h Fi(L)1249 1568 y Fv(2)1268 1584 y Fn(\()p Fj(R)1325 1568 y Fv(3)1345 1584 y Fn(\),)f(with)h Fj(1)f Fn(b)q(eing)i(the)e(iden)o(tit)o(y)37 1640 y(op)q(erator)j(on)g Fi(L)315 1624 y Fv(2)335 1640 y Fn(\()p Fj(R)392 1624 y Fv(3)411 1640 y Fn(\).)108 1722 y(Because)23 b(of)f(\(H1\),)g Fi(K)j Fn(is)e(not)f(a)g(compact)g (op)q(erator.)40 b(Ho)o(w)o(ev)o(er,)22 b(generally)i(sp)q(eaking,)g (taking)37 1778 y(adv)m(an)o(tage)12 b(of)f(this)h(in)o(v)m(ariance)i (prop)q(ert)o(y)d(of)g Fi(K)s Fn(,)h(w)o(e)g(shall)h(b)q(e)f(able)g(to) g(decomp)q(ose)g Fi(K)i Fn(in)o(to)e(a)g(con)o(tin)o(uous)37 1835 y(family)j(of)f(compact,)f(and)i(ev)o(en)f(trace-class,)g(op)q (erators,)f(whose)h(sp)q(ectral)g(decomp)q(osition)i(is)e(therefore)37 1891 y(v)o(ery)k(simple.)30 b(It)18 b(is)g(classical)i(to)d(study)h (the)g(sp)q(ectral)h(resolution)f(of)g Fi(K)j Fn(as)c(an)h(op)q(erator) f(on)h Fi(L)1761 1875 y Fv(2)1781 1891 y Fn(\()p Fj(R)1838 1875 y Fv(3)1857 1891 y Fn(\))37 1948 y(with)h(the)g(help)h(of)e(the)g (so-called)i(Blo)q(c)o(h)g(w)o(a)o(v)o(es)d(decomp)q(osition)j(of)e Fi(K)j Fn(whic)o(h)f(has)e(b)q(een)i(in)o(tro)q(duced)37 2004 y(b)o(y)i(G.)f(Flo)q(quet)i([20)o(])e(in)i(the)f(one)g (dimensional)i(case)e(and)g(b)o(y)g(F.)f(Blo)q(c)o(h)h([7])f(in)i(the)f (general)g(case.)37 2061 y(W)l(e)e(shall)h(explain)g(no)o(w)d(the)i (main)g(ingredien)o(ts)g(of)f(this)h(metho)q(d)g(follo)o(wing)g(mainly) g(the)g(formalism)37 2117 y(of)i(M.)f(Reed)i(and)f(B.)g(Simon)g([45,)h (Section)g(XI)q(I)q(I-16])f(together)g(with)g(the)g(b)q(o)q(ok)g(b)o(y) g(C.)f(Conca,)i(J.)37 2173 y(Planc)o(hard)f(and)f(M.)f(V)l(anninathan)i ([16)o(].)36 b(Among)20 b(the)h(wide)h(literature)f(whic)o(h)h(is)f (dev)o(oted)g(to)f(the)37 2230 y(Blo)q(c)o(h)d(w)o(a)o(v)o(es)d(decomp) q(osition)j(\(and)f(some)f(applications\),)h(w)o(e)g(refer)f(more)g(sp) q(eci\014cally)j(the)e(reader)g(to)37 2286 y([2,)f(15)o(,)g(17)o(,)g (18,)f(19,)h(22)o(,)g(24)o(,)g(56)o(].)108 2384 y(The)21 b(spirit)h(of)e(this)h(decomp)q(osition)h(is)g(the)f(follo)o(wing)g(:) 31 b(w)o(e)21 b(ma)o(y)f(construct)g(a)h(decomp)q(osition)37 2441 y(of)h Fi(L)127 2424 y Fv(2)147 2441 y Fn(\()p Fj(R)204 2424 y Fv(3)223 2441 y Fn(\))g(according)g(to)f(this)h(in)o(v)m (ariance)i(b)o(y)e(translation.)39 b(F)l(or)22 b(this)g(purp)q(ose,)i (w)o(e)d(de\014ne)i Fg(H)i Fn(=)37 2497 y Fi(L)68 2481 y Fv(2)88 2497 y Fn(\()p Fi(Q)142 2481 y Fh(?)161 2497 y Fn(;)8 b Fi(L)213 2481 y Fv(2)232 2497 y Fn(\()p Fi(Q)p Fn(\)\).)23 b(Then,)17 b(there)g(is)g(an)f(isometry)g Fi(U)22 b Fn(b)q(et)o(w)o(een)17 b Fi(L)1168 2481 y Fv(2)1187 2497 y Fn(\()p Fj(R)1244 2481 y Fv(3)1264 2497 y Fn(\))f(and)h Fg(H)p Fn(,)g(the)g(so-called)h(Flo)q(quet)37 2554 y(op)q(erator,)c (de\014ned)j(b)o(y)e Fi(U)j Fn(:)12 b Fi(L)558 2537 y Fv(2)578 2554 y Fn(\()p Fj(R)635 2537 y Fv(3)654 2554 y Fn(\))h Fg(!)g(H)j Fn(and)379 2663 y(\()p Fi(U)5 b(')p Fn(\))481 2670 y Fh(\030)499 2663 y Fn(\()p Fi(x)p Fn(\))12 b(=)631 2619 y Fe(X)621 2721 y Fh(k)q Fu(2)p Fd(Z)689 2711 y Ft(3)713 2663 y Fi(e)734 2644 y Fu(\000)7 b Fh(ik)g Fu(\001)f Fh(\030)840 2663 y Fi(')p Fn(\()p Fi(x)j Fn(+)i Fi(k)q Fn(\))p Fi(;)34 b Fn(for)15 b(a.e.)32 b Fi(\030)14 b Fg(2)f Fi(Q)1343 2644 y Fh(?)1363 2663 y Fi(;)20 b(x)12 b Fg(2)h Fi(Q;)233 b Fn(\(3.17\))p eop %%Page: 19 19 19 18 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(19)37 199 y(for)20 b(an)o(y)f Fi(')h Fn(in)h(the)f(Sc)o(h)o(w)o (artz)f(class)h Fg(S)s Fn(\()p Fj(R)789 183 y Fv(3)810 199 y Fn(\).)33 b(One)21 b(ma)o(y)e(c)o(hec)o(k)i(\(see)f([45)o(]\))f (that)g(U)h(is)g(unitary)h(from)37 256 y Fi(L)68 239 y Fv(2)88 256 y Fn(\()p Fj(R)145 239 y Fv(3)165 256 y Fn(\))15 b(on)o(to)f Fg(H)i Fn(and)f(that)g(the)g(in)o(v)o(erse)h(of)e Fi(U)21 b Fn(is)15 b Fi(U)954 239 y Fu(\003)989 256 y Fn(de\014ned,)h(for)f(all)h Fi(\030)f Fg(7!)e Fi(g)1406 263 y Fh(\030)1439 256 y Fn(in)j Fg(H)p Fn(,)g(b)o(y)f(:)278 371 y(\()p Fi(U)332 352 y Fu(\003)351 371 y Fi(g)r Fn(\)\()p Fi(x)9 b Fn(+)h Fi(k)q Fn(\))j(=)595 309 y Fe(Z)620 412 y Fh(Q)648 403 y Fc(?)676 371 y Fi(e)697 352 y Fh(ik)7 b Fu(\001)f Fh(\030)768 371 y Fi(g)790 378 y Fh(\030)809 371 y Fn(\()p Fi(x)p Fn(\))913 340 y Fi(d\030)p 883 361 106 2 v 883 402 a Fn(\(2)p Fi(\031)r Fn(\))970 389 y Fv(3)993 371 y Fi(;)35 b Fn(for)15 b(all)28 b Fi(k)14 b Fg(2)f Fj(Z)1299 352 y Fv(3)1319 371 y Fi(;)22 b Fn(for)15 b(a.e.)d Fi(x)g Fg(2)h Fi(Q:)124 b Fn(\(3.18\))37 492 y(W)l(e)21 b(write)f(do)o(wn)g(explicitly)j(the)d(fact)g(that)f Fi(U)25 b Fn(is)c(an)f(isometry)l(,)h(using)g(\(3.17\))d(and)j (\(3.18\),)e(and)h(w)o(e)37 548 y(obtain)c(the)f(follo)o(wing)h(iden)o (tit)o(y)500 659 y(\()p Fi(')p Fn(;)8 b Fi( )r Fn(\))619 669 y Fh(L)643 659 y Ft(2)658 669 y Fv(\()p Fd(R)702 659 y Ft(3)719 669 y Fv(\))748 659 y Fn(=)796 597 y Fe(Z)821 700 y Fh(Q)849 690 y Fc(?)869 659 y Fn(\(\()p Fi(U)d(')p Fn(\))989 666 y Fh(\030)1006 659 y Fn(;)j(\()p Fi(U)d( )r Fn(\))1131 666 y Fh(\030)1148 659 y Fn(\))1166 669 y Fh(L)1190 659 y Ft(2)1207 669 y Fv(\()p Fh(Q)p Fv(\))1312 628 y Fi(d\030)p 1282 648 V 1282 690 a Fn(\(2)p Fi(\031)r Fn(\))1369 677 y Fv(3)1392 659 y Fi(;)354 b Fn(\(3.19\))37 783 y(for)15 b(an)o(y)g(functions)h Fi(')f Fn(and)g Fi( )i Fn(in)f Fi(L)653 767 y Fv(2)672 783 y Fn(\()p Fj(R)729 767 y Fv(3)749 783 y Fn(\),)e(from)h(whic)o(h)h(w)o(e)f(also)g(infer)h (that,)e(in)i(particular)g(:)587 896 y Fg(k)p Fi(')p Fg(k)663 877 y Fv(2)663 910 y Fh(L)687 901 y Ft(2)703 910 y Fv(\()p Fd(R)747 901 y Ft(3)764 910 y Fv(\))793 896 y Fn(=)841 834 y Fe(Z)866 937 y Fh(Q)894 928 y Fc(?)921 896 y Fg(k)p Fn(\()p Fi(U)5 b(')p Fn(\))1046 903 y Fh(\030)1064 896 y Fg(k)1087 877 y Fv(2)1087 910 y Fh(L)1111 901 y Ft(2)1128 910 y Fv(\()p Fh(Q)p Fv(\))1233 865 y Fi(d\030)p 1203 886 V 1203 927 a Fn(\(2)p Fi(\031)r Fn(\))1290 914 y Fv(3)1313 896 y Fi(:)433 b Fn(\(3.20\))108 1017 y(Let)12 b(us)g(mak)o(e)f(a)h(few)f(commen)o(ts)g(on)h(the)g(de\014nition)h (\(3.17\).)k(First)11 b(of)h(all,)h(the)f(expression)g(app)q(earing)37 1074 y(in)i(the)e(righ)o(t-hand)h(side)h(of)d(\(3.17\))g(ma)o(y)h(b)q (e)h(seen)g(as)f(a)g(F)l(ourier)h(series)g(expansion)g(with)g(resp)q (ect)g(to)f(the)g Fi(\030)37 1130 y Fn(v)m(ariable,)j(and)f(whose)f(co) q(e\016cien)o(ts)h(lie)h(in)f Fi(L)807 1114 y Fv(2)827 1130 y Fn(\()p Fi(Q)p Fn(\).)k(Next,)c(it)f(is)h(clear)g(from)f (\(3.17\))e(that)i Fi(e)1600 1114 y Fu(\000)p Fh(i)6 b(x)p Fu(\001)p Fh(\030)1694 1130 y Fn(\()p Fi(U)f(')p Fn(\))1796 1137 y Fh(\030)1814 1130 y Fn(\()p Fi(x)p Fn(\))37 1187 y(is)15 b Fi(Q)p Fn(-p)q(erio)q(dic.)22 b(Suc)o(h)15 b(functions)g(are)f(often)g(called)h(quasi-p)q(erio)q(dic) i(functions)e(with)g(quasi-momen)o(tum)37 1243 y Fi(\030)r Fn(.)20 b(They)13 b(are)g(kno)o(wn)h(as)f Ff(Blo)n(ch)h(waves)f Fn(in)h(the)g(Solid)g(State)f(Ph)o(ysics)h(literature.)20 b(It)14 b(is)f(more)g(con)o(v)o(enien)o(t)37 1299 y(\(and)18 b(w)o(e)g(shall)h(alw)o(a)o(ys)e(do)g(it)h(in)h(the)f(follo)o(wing\))g (to)f(lo)q(ok)h(at)g(\()p Fi(U)5 b(')p Fn(\))1255 1306 y Fh(\030)1290 1299 y Fn(as)18 b(a)f(function)i(lying)g(in)f Fi(L)1771 1283 y Fv(2)1771 1313 y Fh(\030)1791 1299 y Fn(\()p Fi(Q)p Fn(\),)37 1356 y(with)228 1447 y Fi(L)259 1429 y Fv(2)259 1459 y Fh(\030)278 1447 y Fn(\()p Fi(Q)p Fn(\))12 b(=)h Fg(f)p Fi(')f Fg(2)h Fi(L)549 1429 y Fv(2)549 1459 y(lo)q(c)596 1447 y Fn(\()p Fj(R)653 1429 y Fv(3)672 1447 y Fn(\))f Fi(=)h(')p Fn(\()p Fi(x)c Fn(+)i Fi(k)q Fn(\))h(=)h Fi(e)991 1429 y Fh(i)6 b(k)q Fu(\001)p Fh(\030)1057 1447 y Fi(')p Fn(\()p Fi(x)p Fn(\))p Fi(;)i Fg(8)j Fi(k)j Fg(2)f Fj(Z)1319 1429 y Fv(3)1339 1447 y Fi(;)22 b Fn(for)14 b(a.e.)20 b Fi(x)12 b Fg(2)h Fi(Q)p Fg(g)p Fi(;)37 1539 y Fn(or,)i(equiv)m(alen)o(tly)l(,)437 1630 y Fi(L)468 1611 y Fv(2)468 1642 y Fh(\030)488 1630 y Fn(\()p Fi(Q)p Fn(\))d(=)h Fg(f)p Fi(')f Fg(2)h Fi(L)759 1611 y Fv(2)759 1642 y(lo)q(c)805 1630 y Fn(\()p Fj(R)862 1611 y Fv(3)882 1630 y Fn(\))f Fi(=)g(e)968 1611 y Fu(\000)p Fh(i)7 b(\030)q Fu(\001)p Fh(x)1062 1630 y Fi(')p Fn(\()p Fi(x)p Fn(\))15 b(is)g Fi(Q)p Fn(-p)q(erio)q(dic)j Fg(g)p Fi(:)37 1722 y Fn(It)g(is)h(clear)f(from)g(the)g(second)g(form)o(ulation,)g(that)f Fi(L)983 1705 y Fv(2)983 1736 y Fh(\030)1003 1722 y Fn(\()p Fi(Q)p Fn(\),)h(endo)o(w)o(ed)g(with)g(the)g(usual)h(Hilb)q(ert)g (scalar)37 1784 y(pro)q(duct)d(on)f Fi(L)302 1768 y Fv(2)322 1784 y Fn(\()p Fi(Q)p Fn(\),)f(is)i(a)f(Hilb)q(ert)h(space)g(whic)o(h)g (is)f(isomorphic)i(to)d Fi(L)1275 1768 y Fv(2)1295 1784 y Fn(\()p Fi(Q)p Fn(\).)108 1882 y(With)e(the)g(help)h(of)e(the)h (isomorphism)g Fi(U)17 b Fn(b)q(et)o(w)o(een)12 b Fi(L)1028 1865 y Fv(2)1048 1882 y Fn(\()p Fj(R)1105 1865 y Fv(3)1124 1882 y Fn(\))f(and)h Fg(H)p Fn(,)h(w)o(e)f(no)o(w)f(return)g(to)h(the)f (sp)q(ectral)37 1938 y(analysis)16 b(of)f(the)h(op)q(erators)e Fi(K)k Fn(satisfying)d(\(H1\))g(b)o(y)g(follo)o(wing)h([45)o(].)108 2020 y(T)l(o)i(the)g(ab)q(o)o(v)o(e)g(decomp)q(osition)i(of)e (functions)h(in)g Fi(L)1035 2003 y Fv(2)1055 2020 y Fn(\()p Fj(R)1112 2003 y Fv(3)1131 2020 y Fn(\))f(in)o(to)g(Blo)q(c)o(h)i(w)o (a)o(v)o(es)d(corresp)q(onds)h(a)g(so-)37 2076 y(called)k(direct)f(in)o (tegral)g(decomp)q(osition)g(of)f Fi(K)j Fn(in)e(the)f(sense)h(that)e (there)h(exists)h(a)e(unique)j(function)37 2133 y Fi(\030)g Fg(7!)f Fi(K)184 2140 y Fh(\030)222 2133 y Fn(in)f Fi(L)310 2116 y Fu(1)348 2133 y Fn(\()p Fi(Q)402 2116 y Fh(?)421 2133 y Fn(;)8 b Fg(L)p Fn(\()p Fi(L)522 2116 y Fv(2)522 2147 y Fh(\030)541 2133 y Fn(\()p Fi(Q)p Fn(\)\)\))18 b(\(in)i(that)f(follo)o(ws,)i Fg(L)p Fn(\()p Fi(X)t Fn(\))d(denotes)i (the)g(space)g(of)f(b)q(ounded)i(linear)37 2195 y(op)q(erators)e(from)g (X)h(in)o(to)g(itself)t(\))g(suc)o(h)g(that,)g(for)f(an)o(y)g(function) i Fi(')e Fn(in)i Fi(L)1351 2179 y Fv(2)1370 2195 y Fn(\()p Fj(R)1427 2179 y Fv(3)1447 2195 y Fn(\))e(and)h(almost)f(ev)o(ery)h Fi(\030)37 2252 y Fn(in)d Fi(Q)127 2235 y Fh(?)161 2252 y Fn(:)740 2343 y(\()p Fi(U)5 b(K)s(')p Fn(\))884 2350 y Fh(\030)914 2343 y Fn(=)13 b Fi(K)1001 2350 y Fh(\030)1027 2343 y Fn(\()p Fi(U)5 b(')p Fn(\))1129 2350 y Fh(\030)1160 2343 y Fi(:)586 b Fn(\(3.21\))37 2434 y(Moreo)o(v)o(er,)14 b(w)o(e)h(also)g(ha)o(v)o(e)g(:)626 2526 y(sup)617 2565 y Fh(\030)q Fu(2)p Fh(Q)686 2556 y Fc(?)711 2526 y Fg(k)p Fi(K)773 2533 y Fh(\030)791 2526 y Fg(k)814 2538 y Fu(L)p Fv(\()p Fh(L)876 2526 y Ft(2)876 2550 y Fc(\030)893 2538 y Fv(\()p Fh(Q)p Fv(\)\))977 2526 y Fn(=)e Fg(k)p Fi(K)s Fg(k)1113 2536 y Fu(L)p Fv(\()p Fh(L)1175 2526 y Ft(2)1191 2536 y Fv(\()p Fd(R)1235 2526 y Ft(3)1253 2536 y Fv(\)\))1282 2526 y Fi(:)464 b Fn(\(3.22\))37 2647 y(And)16 b(w)o(e)f(shall)h(write) 772 2749 y Fi(K)f Fn(=)874 2687 y Fe(Z)900 2790 y Fh(Q)928 2780 y Fc(?)955 2749 y Fi(K)994 2756 y Fh(\030)1060 2718 y Fi(d\030)p 1030 2738 V 1030 2780 a Fn(\(2)p Fi(\031)r Fn(\))1117 2767 y Fv(3)1759 2749 y Fn(\(3.23\))p eop %%Page: 20 20 20 19 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(20)37 199 y(in)17 b(order)d(to)h(refer)g(to)g(the)g(decomp)q (osition)h(\(3.21\))e(of)g Fi(K)s Fn(.)108 281 y(The)e(sp)q(ectral)h (analysis)g(of)e Fi(K)k Fn(no)o(w)c(reduces)i(to)e(the)h(sp)q(ectral)h (analysis)g(of)e(the)h(family)h(of)e(self-adjoin)o(t)37 337 y(op)q(erators)j Fi(K)277 344 y Fh(\030)308 337 y Fg(2)f(L)p Fn(\()p Fi(L)431 321 y Fv(2)431 351 y Fh(\030)450 337 y Fn(\()p Fi(Q)p Fn(\)\),)g(the)h(parameter)f Fi(\030)j Fn(v)m(arying)f(in)g Fi(Q)1144 321 y Fh(?)1163 337 y Fn(.)20 b(W)l(e)14 b(no)o(w)f(en)o(ter)h(the)g(details)h(of)f(suc)o(h)g (a)37 394 y(decomp)q(osition)f(for)d(a)g(sp)q(ecial)j(class)e(of)f(op)q (erators)g(satisfying)h(\(H1\))f(and)h(\(H2\))e(whic)o(h)j(will)g(app)q (ear)f(b)q(elo)o(w)37 450 y(in)17 b(the)e(setting)g(of)g(the)g(p)q (erio)q(dic)i(reduced)g(Hartree-F)l(o)q(c)o(k)d(and)i(the)f(p)q(erio)q (dic)i(Hartree-F)l(o)q(c)o(k)e(mo)q(dels.)108 531 y(F)l(rom)i(no)o(w)f (on,)i(let)f(us)h(denote)f(b)o(y)h Fi(K)i Fn(an)d(arbitrary)f (self-adjoin)o(t)i(op)q(erator)f(satisfying)g(\(H1\))g(and)37 588 y(\(H2\).)j(W)l(e)15 b(assume)g(that)f(there)i(exists)f(a)g(k)o (ernel)h(represen)o(tation)f(of)g Fi(K)j Fn(of)d(the)g(form)673 699 y Fi(K)s(')p Fn(\()p Fi(x)p Fn(\))c(=)866 637 y Fe(Z)892 740 y Fd(R)922 731 y Ft(3)949 699 y Fi(\032)p Fn(\()p Fi(x)p Fn(;)d Fi(y)r Fn(\))p Fi(')p Fn(\()p Fi(y)r Fn(\))j Fi(dy)r(;)37 816 y Fn(sa)o(y)k(for)f(an)o(y)g(function)i Fi(')e Fn(in)h Fg(S)s Fn(\()p Fj(R)631 800 y Fv(3)652 816 y Fn(\),)f(with)h Fi(\032)p Fn(\()p Fg(\001)p Fn(;)8 b Fg(\001)p Fn(\))j Fg(2)i Fi(L)991 800 y Fv(2)991 830 y(lo)q(c)1037 816 y Fn(\()p Fj(R)1094 800 y Fv(3)1123 816 y Fg(\002)c Fj(R)1206 800 y Fv(3)1226 816 y Fn(\))p Fi(:)14 b Fn(Note)h(that)e(\(H1\))h(is)i(then)f(equiv)m(a-)37 873 y(len)o(tly)i(written)332 963 y Fi(\032)p Fn(\()p Fi(x)11 b Fn(+)f Fi(k)q Fn(;)e Fi(y)j Fn(+)g Fi(k)q Fn(\))h(=)h Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fi(;)52 b Fn(for)14 b(ev)o(ery)h Fi(k)f Fg(2)f Fj(Z)1180 945 y Fv(3)1200 963 y Fi(;)22 b Fn(a.e.)d(on)d Fj(R)1426 945 y Fv(3)1456 963 y Fg(\002)10 b Fj(R)1540 945 y Fv(3)1560 963 y Fi(;)186 b Fn(\(3.24\))37 1054 y(while)17 b(the)e(self-adjoin)o (tness)i(of)d Fi(K)k Fn(simply)f(reads)781 1144 y Fi(\032)804 1126 y Fu(\003)824 1144 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))j(=)i Fi(\032)p Fn(\()p Fi(y)r Fn(;)8 b Fi(x)p Fn(\))p Fi(;)37 1235 y Fn(where)16 b Fi(z)192 1218 y Fu(\003)226 1235 y Fn(denotes)f(the)g(complex)h(conjugate)e(of)h Fi(z)i Fn(\()p Fg(2)12 b Fj(C)p Fn(\).)19 b(W)l(e)c(shall)h(no)o(w)e (imp)q(ose)i(further)f(conditions)37 1291 y(on)h(the)f(k)o(ernel)h Fi(\032)p Fn(.)108 1373 y(As)h(a)g(consequence)i(of)e(the)g (de\014nition)i(and)e(of)g(the)g(uniqueness)i(of)e(the)h(decomp)q (osition)g(\(3.23\))d(of)37 1429 y Fi(K)s Fn(,)g(w)o(e)g(deduce)i(that) d(eac)o(h)h(op)q(erator)g Fi(K)749 1436 y Fh(\030)782 1429 y Fn(is)h(self-adjoin)o(t)g([45)o(].)j(Moreo)o(v)o(er,)14 b(w)o(e)h(ha)o(v)o(e)f(the)i(follo)o(wing)37 1518 y Fj(Lemma)i(3.3)23 b Ff(L)n(et)17 b Fi(K)e Fn(=)493 1482 y Fe(R)515 1534 y Fh(Q)543 1525 y Fc(?)570 1518 y Fi(K)609 1525 y Fh(\030)670 1498 y(d\030)p 645 1508 84 2 v 645 1535 a Fv(\(2)p Fh(\031)q Fv(\))712 1526 y Ft(3)734 1518 y Ff(.)21 b(Then,)16 b(\(H2\))g(is)g(e)n (quivalent)f(to)i(:)37 1613 y Fj(\(H2\)')117 b Fn(0)18 b Fg(\024)g Fi(K)408 1620 y Fh(\030)444 1613 y Fg(\024)g Fj(1)i Ff(in)e(the)i(sense)d(of)j(self-adjoint)e(op)n(er)n(ators)h(in)g Fg(L)p Fn(\()p Fi(L)1437 1597 y Fv(2)1437 1627 y Fh(\030)1457 1613 y Fn(\()p Fi(Q)p Fn(\)\))p Ff(,)f(and)i(for)f(almost)151 1670 y(every)d Fi(\030)j Ff(in)c Fi(Q)401 1653 y Fh(?)421 1670 y Ff(.)37 1754 y Fn(W)l(e)i(are)e(searc)o(hing)i(for)e(op)q (erators)h Fi(K)i Fn(for)e(whic)o(h)h(the)f Fi(K)1043 1761 y Fh(\030)1061 1754 y Fn('s)g(are)g(Hilb)q(ert-Sc)o(hmidt,)i(and)e (more)g(sp)q(eci\014-)37 1810 y(cally)l(,)h(w)o(e)e(shall)h(rely)g(up)q (on)g(the)f(follo)o(wing.)37 1894 y Fj(Lemma)j(3.4)23 b Ff(L)n(et)17 b Fi(K)e Fn(=)493 1858 y Fe(R)515 1910 y Fh(Q)543 1901 y Fc(?)570 1894 y Fi(K)609 1901 y Fh(\030)670 1874 y(d\030)p 645 1884 V 645 1911 a Fv(\(2)p Fh(\031)q Fv(\))712 1902 y Ft(3)734 1894 y Ff(.)21 b(Then)16 b(the)h(fol)r (lowing)f(two)h(pr)n(op)n(erties)f(ar)n(e)g(e)n(quivalent)f(:)37 1985 y Fj(\(i\))706 2076 y Fi(\032)e Fg(2)g Fi(L)816 2057 y Fv(2)835 2076 y Fn(\()p Fi(Q)d Fg(\002)g Fj(R)983 2057 y Fv(3)1003 2076 y Fn(\))g Fg(\\)h Fi(L)1103 2057 y Fv(2)1122 2076 y Fn(\()p Fj(R)1179 2057 y Fv(3)1209 2076 y Fg(\002)f Fi(Q)p Fn(\))p Fi(:)438 b Fn(\(3.25\))37 2183 y Fj(\(ii\))25 b Ff(F)m(or)20 b(almost)g(every)g Fi(\030)i Ff(in)e Fi(Q)630 2167 y Fh(?)649 2183 y Ff(,)i Fi(K)724 2190 y Fh(\030)762 2183 y Ff(is)e(a)g(Hilb)n(ert-Schmidt)g(op) n(er)n(ator)h(with)f(kernel)g Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\))18 b Ff(and)151 2239 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\))j Fg(2)i Fi(L)409 2223 y Fv(2)428 2239 y Fn(\()p Fi(Q)482 2223 y Fh(?)502 2239 y Fn(;)8 b Fi(L)554 2223 y Fv(2)573 2239 y Fn(\()p Fi(Q)h Fg(\002)i Fi(Q)p Fn(\)\))p Ff(.)37 2349 y(Mor)n(e)n(over,)16 b(if)g Fi(K)k Ff(satis\014es)e Fj(\(i\))g Ff(or)j Fj(\(ii\))p Ff(,)d(we)e(have)335 2398 y Fe(Z)360 2502 y Fh(Q)388 2492 y Fc(?)450 2430 y Fi(d\030)p 420 2450 106 2 v 420 2492 a Fn(\(2)p Fi(\031)r Fn(\))507 2478 y Fv(3)538 2398 y Fe(Z)6 b(Z)614 2502 y Fh(Q)p Fu(\002)p Fh(Q)707 2460 y Fg(j)p Fi(\032)p Fn(\()p Fi(\030)r Fn(;)i Fi(x)p Fn(;)g Fi(y)r Fn(\))p Fg(j)906 2442 y Fv(2)935 2460 y Fi(dxdy)14 b Fn(=)1093 2398 y Fe(Z)5 b(Z)1169 2502 y Fh(Q)p Fu(\002)p Fd(R)1254 2492 y Ft(3)1281 2460 y Fg(j)p Fi(\032)p Fn(\()p Fi(x)p Fn(;)j Fi(y)r Fn(\))p Fg(j)1437 2442 y Fv(2)1468 2460 y Fi(dxdy)r(:)180 b Fn(\(3.26\))37 2580 y Ff(In)18 b(addition,)h Fi(\032)f Ff(and)g Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(;)g Fg(\001)p Fn(\))15 b Ff(ar)n(e)j(r)n(elate)n(d)f(as)h(fol)r (lows)f(:)24 b(for)18 b(almost)g(every)g Fi(x)f Ff(and)h Fi(y)i Ff(in)d Fi(Q)p Ff(,)h(and)g Fi(\030)i Ff(in)37 2637 y Fi(Q)73 2620 y Fh(?)93 2637 y Ff(,)389 2731 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\))j(=)631 2688 y Fe(X)621 2790 y Fh(k)q Fu(2)p Fd(Z)689 2780 y Ft(3)714 2731 y Fi(e)735 2713 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)822 2731 y Fi(\032)p Fn(\()p Fi(x)f Fn(+)h Fi(k)q Fn(;)d Fi(y)r Fn(\))j(=)1102 2688 y Fe(X)1092 2790 y Fh(k)q Fu(2)p Fd(Z)1160 2780 y Ft(3)1185 2731 y Fi(e)1206 2713 y Fv(+)p Fh(ik)q Fu(\001)p Fh(\030)1293 2731 y Fi(\032)p Fn(\()p Fi(x)p Fn(;)d Fi(y)k Fn(+)e Fi(k)q Fn(\))p Fi(;)243 b Fn(\(3.27\))p eop %%Page: 21 21 21 20 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(21)37 199 y Ff(henc)n(e)668 303 y Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))j(=)857 241 y Fe(Z)883 344 y Fh(Q)911 335 y Fc(?)938 303 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)d Fi(x)p Fn(;)g Fi(y)r Fn(\))1152 272 y Fi(d\030)p 1122 292 106 2 v 1122 334 a Fn(\(2)p Fi(\031)r Fn(\))1209 321 y Fv(3)1232 303 y Fi(:)514 b Fn(\(3.28\))37 451 y Fj(Pro)q(of)15 b(of)f(Lemma)g(3.3)g(:)41 b Fn(The)12 b(pro)q(of)g(mimics)h(that)e(of)g(\(3.22\),)g(whic)o(h)i(ma)o(y)e(b)q (e)i(found)f(in)h([45)o(,)f(Section)37 507 y(XI)q(I)q(I.16)j(])f(\(pro) q(of)f(of)g(Theorem)h(XI)q(I)q(I-83\).)20 b(W)l(e)14 b(shall)h(partially)g(repro)q(duce)g(the)e(argumen)o(t)g(here)i(for)e (the)37 564 y(sak)o(e)i(of)g(consistency)l(.)108 620 y(In)h(virtue)g(of)e(\(3.22\),)f(and)j(since)g(\(H2\))e(implies)k(in)e (particular)g(that)780 714 y Fg(k)p Fi(K)s Fg(k)868 724 y Fu(L)p Fv(\()p Fh(L)930 714 y Ft(2)946 724 y Fv(\()p Fd(R)990 714 y Ft(3)1008 724 y Fv(\)\))1050 714 y Fg(\024)d Fn(1)p Fi(;)37 807 y Fn(it)k(just)f(remains)g(to)g(c)o(hec)o(k)g(that)g Fi(K)664 814 y Fh(\030)697 807 y Fg(\025)e Fn(0)i(for)g(almost)f(ev)o (ery)h Fi(\030)h Fg(2)d Fi(Q)1241 791 y Fh(?)1277 807 y Fn(as)i(so)q(on)g(as)g Fi(K)h Fg(\025)d Fn(0)i(\(the)g(rev)o(erse)37 864 y(implication)i(b)q(eing)e(ev)o(en)g(easier)g(to)e(pro)o(v)o(e\).) 108 920 y(F)l(ollo)o(wing)g([45)o(],)f(w)o(e)g(c)o(ho)q(ose)g(a)g (dense)h(subset)g Fg(f)p Fi(\014)959 927 y Fh(k)980 920 y Fg(g)1003 927 y Fh(k)q Fu(\025)p Fv(1)1082 920 y Fn(of)f(the)g(unit)h (sphere)g(of)f Fi(L)1523 904 y Fv(2)1543 920 y Fn(\()p Fi(Q)p Fn(\),)f(and)i(w)o(e)f(tak)o(e)37 977 y(an)j(arbitrary)e (function)i Fi(f)i Fg(\025)13 b Fn(0)i(in)h Fi(L)682 960 y Fv(1)702 977 y Fn(\()p Fi(Q)756 960 y Fh(?)775 977 y Fn(\).)j(W)l(e)d(c)o(hec)o(k)f(no)o(w)g(that,)f(for)h(ev)o(ery)g Fi(k)e Fg(\025)g Fn(1,)605 1030 y Fe(Z)630 1133 y Fh(Q)658 1123 y Fc(?)686 1092 y Fi(f)5 b Fn(\()p Fi(\030)r Fn(\)\()p Fi(K)828 1099 y Fh(\030)853 1092 y Fi(\014)879 1099 y Fh(k)900 1092 y Fn(;)j Fi(\014)947 1099 y Fh(k)967 1092 y Fn(\))985 1101 y Fh(L)1009 1092 y Ft(2)1026 1101 y Fv(\()p Fh(Q)p Fv(\))1131 1061 y Fi(d\030)p 1101 1081 V 1101 1123 a Fn(\(2)p Fi(\031)r Fn(\))1188 1110 y Fv(3)1224 1092 y Fg(\025)13 b Fn(0)p Fi(:)37 1215 y Fn(Our)k(claim)g(will)g (follo)o(w)f(then)h(immediately)l(,)g(since)g(w)o(e)f(already)g(kno)o (w)g(that)f(the)h(\()p Fi(K)1541 1222 y Fh(\030)1567 1215 y Fi(\014)1593 1222 y Fh(k)1614 1215 y Fn(;)8 b Fi(\014)1661 1222 y Fh(k)1681 1215 y Fn(\)'s)15 b(b)q(elong)37 1271 y(to)i Fi(L)126 1255 y Fu(1)163 1271 y Fn(\()p Fi(Q)217 1255 y Fh(?)236 1271 y Fn(\).)25 b(Let)17 b(us)g(\014rst)f(note)h(that) 733 1237 y Fg(p)p 771 1237 28 2 v 34 x Fi(f)22 b Fn(b)q(elongs)c(to)e Fi(L)1068 1255 y Fv(2)1088 1271 y Fn(\()p Fi(Q)1142 1255 y Fh(?)1161 1271 y Fn(\).)25 b(Then,)17 b(if)g(w)o(e)g(set)g Fi(g)1557 1278 y Fh(k)1593 1271 y Fn(=)f Fi(U)1680 1255 y Fu(\003)1700 1271 y Fn(\()1718 1237 y Fg(p)p 1755 1237 V 1755 1271 a Fi(f)21 b(\014)1824 1278 y Fh(k)1845 1271 y Fn(\),)37 1327 y Fi(g)59 1334 y Fh(k)96 1327 y Fg(2)15 b Fi(L)172 1311 y Fv(2)192 1327 y Fn(\()p Fj(R)249 1311 y Fv(3)269 1327 y Fn(\),)h(and)h(b)o(y)g(using)h(the)e(de\014nition)j (\(3.21\))c(of)h Fi(K)1100 1334 y Fh(\030)1136 1327 y Fn(together)g(with)h(the)g(de\014nition)h(\(3.19\))d(of)37 1384 y(the)h(scalar)f(pro)q(duct)h(on)f Fg(H)p Fn(,)g(w)o(e)g(ha)o(v)o (e)207 1434 y Fe(Z)232 1538 y Fh(Q)260 1528 y Fc(?)287 1496 y Fi(f)5 b Fn(\()p Fi(\030)r Fn(\)\()p Fi(K)429 1503 y Fh(\030)455 1496 y Fi(\014)481 1503 y Fh(k)502 1496 y Fn(;)j Fi(\014)549 1503 y Fh(k)569 1496 y Fn(\))587 1506 y Fh(L)611 1497 y Ft(2)628 1506 y Fv(\()p Fh(Q)p Fv(\))733 1466 y Fi(d\030)p 703 1486 106 2 v 703 1528 a Fn(\(2)p Fi(\031)r Fn(\))790 1514 y Fv(3)855 1496 y Fn(=)932 1434 y Fe(Z)957 1538 y Fh(Q)985 1528 y Fc(?)1005 1496 y Fn(\()p Fi(K)1062 1503 y Fh(\030)1087 1455 y Fe(p)p 1133 1455 85 2 v 1133 1496 a Fi(f)d Fn(\()p Fi(\030)r Fn(\))o Fi(\014)1243 1503 y Fh(k)1264 1496 y Fn(;)1285 1455 y Fe(p)p 1330 1455 V 41 x Fi(f)g Fn(\()p Fi(\030)r Fn(\))o Fi(\014)1440 1503 y Fh(k)1461 1496 y Fn(\))1479 1506 y Fh(L)1503 1497 y Ft(2)1520 1506 y Fv(\()p Fh(Q)p Fv(\))1625 1466 y Fi(d\030)p 1595 1486 106 2 v 1595 1528 a Fn(\(2)p Fi(\031)r Fn(\))1682 1514 y Fv(3)855 1623 y Fn(=)932 1561 y Fe(Z)957 1664 y Fh(Q)985 1655 y Fc(?)1005 1623 y Fn(\()p Fi(K)1062 1630 y Fh(\030)1087 1623 y Fn(\()p Fi(U)g(g)1163 1630 y Fh(k)1184 1623 y Fn(\))1202 1630 y Fh(\030)1221 1623 y Fn(;)j(\()p Fi(U)d(g)1318 1630 y Fh(k)1337 1623 y Fn(\))1355 1630 y Fh(\030)1374 1623 y Fn(\))1392 1633 y Fh(L)1416 1623 y Ft(2)1433 1633 y Fv(\()p Fh(Q)p Fv(\))1538 1592 y Fi(d\030)p 1508 1612 V 1508 1654 a Fn(\(2)p Fi(\031)r Fn(\))1595 1641 y Fv(3)855 1750 y Fn(=)932 1688 y Fe(Z)957 1791 y Fh(Q)985 1781 y Fc(?)1005 1750 y Fn(\(\()p Fi(U)g(K)s(g)1141 1757 y Fh(k)1161 1750 y Fn(\))1179 1757 y Fh(\030)1197 1750 y Fn(;)j(\()p Fi(U)d(g)1294 1757 y Fh(k)1314 1750 y Fn(\))1332 1757 y Fh(\030)1351 1750 y Fn(\))1369 1759 y Fh(L)1393 1750 y Ft(2)1410 1759 y Fv(\()p Fh(Q)p Fv(\))1515 1719 y Fi(d\030)p 1485 1739 V 1485 1781 a Fn(\(2)p Fi(\031)r Fn(\))1572 1768 y Fv(3)855 1848 y Fn(=)42 b(\()p Fi(K)10 b(g)1021 1855 y Fh(k)1042 1848 y Fn(;)e Fi(g)1085 1855 y Fh(k)1105 1848 y Fn(\))1123 1858 y Fh(L)1147 1848 y Ft(2)1164 1858 y Fv(\()p Fd(R)1208 1848 y Ft(3)1225 1858 y Fv(\))1299 1848 y Fg(\025)59 b Fn(0)p Fi(;)37 1941 y Fn(b)q(ecause)17 b(of)d(\(H2\).)1477 b Fg(})108 2023 y Fn(The)14 b(pro)q(of)g(of)f(Lemma)h(3.4)f(is)h(based)g(up)q(on)g(the) g(follo)o(wing)h(result)f(that)f(w)o(e)h(shall)h(use)f(sev)o(eral)g (times)37 2079 y(in)j(the)e(sequel)h(:)37 2177 y Fj(Lemma)i(3.5)23 b Ff(L)n(et)16 b Fn(\()p Fi(u)434 2184 y Fh(n)457 2177 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\)\))567 2184 y Fh(n)p Fu(\025)p Fv(1)648 2177 y Ff(b)n(e)15 b(a)g(Hilb)n(ert)f(b)n (asis)g(of)h Fi(L)1080 2160 y Fv(2)1080 2191 y Fh(\030)1100 2177 y Fn(\()p Fi(Q)p Fn(\))f Ff(for)h(almost)g(every)g Fi(\030)i Ff(in)d Fi(Q)1647 2160 y Fh(?)1667 2177 y Ff(,)h(such)g(that) 37 2233 y Fi(\030)j Fg(7!)e Fi(u)162 2240 y Fh(n)186 2233 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\))16 b Ff(\(that)i(we)g(shal)r(l)f(simply)h(denote)g(by)g Fi(u)963 2240 y Fh(n)1004 2233 y Ff(in)f(the)h(fol)r(lowing\))g(b)n (elongs)e(to)i Fg(H)p Ff(.)27 b(Then,)18 b(if)g(we)37 2290 y(set)e Fi(')138 2297 y Fh(n)174 2290 y Fn(=)d Fi(U)258 2273 y Fu(\003)278 2290 y Fi(u)304 2297 y Fh(n)344 2290 y Ff(and,)j(for)g(every)h Fi(p)f Ff(in)g Fj(Z)766 2273 y Fv(3)785 2290 y Ff(,)572 2402 y Fi(')602 2409 y Fh(n;p)665 2402 y Fn(=)d Fi(\034)733 2409 y Fh(p)765 2402 y Fi(')795 2409 y Fh(n)831 2402 y Fn(=)879 2340 y Fe(Z)905 2444 y Fh(Q)933 2434 y Fc(?)960 2402 y Fi(e)981 2384 y Fh(ip)p Fu(\001)p Fh(\030)1040 2402 y Fi(u)1066 2409 y Fh(n)1089 2402 y Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(\))1240 2372 y Fi(d\030)p 1210 2392 V 1210 2433 a Fn(\(2)p Fi(\031)r Fn(\))1297 2420 y Fv(3)1321 2402 y Fi(;)425 b Fn(\(3.29\))37 2529 y Ff(the)17 b(family)f Fn(\()p Fi(')299 2536 y Fh(n;p)350 2529 y Fn(\))368 2539 y Fh(n)p Fu(\025)p Fv(1)p Fh(;p)p Fu(2)p Fd(Z)511 2530 y Ft(3)545 2529 y Ff(is)g(a)h(Hilb)n(ert)e(b)n (asis)g(of)i Fi(L)977 2513 y Fv(2)996 2529 y Fn(\()p Fj(R)1053 2513 y Fv(3)1073 2529 y Fn(\))p Ff(.)37 2627 y Fj(Remark)h(3.3)24 b Ff(Befor)n(e)17 b(giving)f(the)i(pr)n(o)n(of)f (of)g(L)n(emma)h(3.5,)g(and)g(then)f(the)g(one)g(of)g(L)n(emma)h(3.4,)g (let)f(us)37 2683 y(\014rst)c(note)g(that)h(such)f(a)h(b)n(asis)e (exists.)19 b(Inde)n(e)n(d,)13 b(if)g Fn(\()p Fi(u)941 2690 y Fh(n)964 2683 y Fn(\))g Ff(is)g(a)g(given)g(Hilb)n(ert)f(b)n (asis)g(of)i Fi(L)1527 2667 y Fv(2)1546 2683 y Fn(\()p Fi(Q)p Fn(\))f Ff(c)n(onsisting)e(of)37 2740 y Fi(Q)p Ff(-p)n(erio)n(dic)17 b(functions)f(\(think,)g(for)i(example,)e(of)h Fi(u)948 2747 y Fh(n)985 2740 y Fn(=)d Fi(e)1055 2723 y Fv(2)p Fh(i\031)q(n)p Fu(\001)p Fh(x)1160 2740 y Fi(;)8 b(n)13 b Fg(2)g Fj(Z)1296 2723 y Fv(3)1316 2740 y Ff(\),)k(then)f Fi(u)1493 2747 y Fh(n)1517 2740 y Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(\))k(=)h Fi(e)1703 2723 y Fh(i\030)q Fu(\001)p Fh(x)1764 2740 y Fi(u)1790 2747 y Fh(n)1814 2740 y Fn(\()p Fi(x)p Fn(\))37 2796 y Ff(pr)n(ovides)k(the)f(desir)n(e)n(d)g(example.) p eop %%Page: 22 22 22 21 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(22)37 199 y Fj(Pro)q(of)19 b(of)f(Lemma)f(3.5)i(:)45 b Fn(Let)15 b(\()p Fi(u)696 206 y Fh(n)719 199 y Fn(\))737 206 y Fh(n)774 199 y Fg(2)e(H)j Fn(and)g(\()p Fi(')1008 206 y Fh(n;p)1058 199 y Fn(\))1076 206 y Fh(n;p)1142 199 y Fn(b)q(e)g(de\014ned)h(as)e(in)h(the)g(statemen)o(t)e(of)h(the)37 256 y(ab)q(o)o(v)o(e)h(lemma.)22 b(W)l(e)16 b(\014rst)f(sho)o(w)h(that) f(the)h(\()p Fi(')846 263 y Fh(n;p)896 256 y Fn(\))914 266 y Fh(n)p Fu(\025)p Fv(1)p Fh(;p)p Fu(2)p Fd(Z)1057 256 y Ft(3)j Fn(form)c(an)h(orthonormal)f(family)i(in)f Fi(L)1748 239 y Fv(2)1768 256 y Fn(\()p Fj(R)1825 239 y Fv(3)1845 256 y Fn(\).)37 312 y(Indeed,)j(let)e Fi(n;)8 b(m)15 b Fg(\025)h Fn(1)h(and)g(let)g Fi(p;)8 b(q)16 b Fg(2)g Fj(Z)776 296 y Fv(3)796 312 y Fn(.)25 b(Then,)17 b(using)h(\014rst)e(the)h(de\014nition)i(\(3.19\))c(of)h(the)h(Hilb)q (ert)37 369 y(scalar)h(pro)q(duct)f(on)g Fg(H)p Fn(,)h(the)f (de\014nition)i(\(3.29\))c(of)i(\()p Fi(')p Fn(\))1012 376 y Fh(n;p)1062 369 y Fn(,)g(next)g(the)g(orthonormalit)o(y)f(of)h Fi(u)1671 376 y Fh(n)1695 369 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\))15 b(and)37 425 y Fi(u)63 432 y Fh(m)97 425 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\))13 b(,)i(and)g (\014nally)i(the)e(fact)g(that)742 487 y Fe(Z)767 590 y Fh(Q)795 580 y Fc(?)822 549 y Fi(e)843 530 y Fh(ip)p Fu(\001)p Fh(\030)950 518 y Fi(d\030)p 920 538 106 2 v 920 580 a Fn(\(2)p Fi(\031)r Fn(\))1007 567 y Fv(3)1043 549 y Fn(=)e Fi(\016)1111 556 y Fh(p;)p Fv(0)1158 549 y Fi(;)37 684 y Fn(for)i(ev)o(ery)g Fi(p)e Fg(2)g Fj(Z)338 667 y Fv(3)372 684 y Fn(\()p Fi(\016)410 691 y Fu(\001)p Fh(;)p Fu(\001)457 684 y Fn(b)q(eing)j(the)g(Kronec)o(k)o(er)f(sym)o(b) q(ol\),)g(w)o(e)g(ha)o(v)o(e)224 747 y Fe(Z)250 851 y Fd(R)280 841 y Ft(3)307 809 y Fi(')337 816 y Fh(n)360 809 y Fn(\()p Fi(x)10 b Fn(+)g Fi(p)p Fn(\))p Fi(')530 791 y Fu(\003)530 821 y Fh(m)563 809 y Fn(\()p Fi(x)g Fn(+)g Fi(q)r Fn(\))e Fi(dx)40 b Fn(=)877 747 y Fe(Z)903 851 y Fh(Q)931 841 y Fc(?)950 809 y Fn(\(\()p Fi(U)5 b(')1052 816 y Fh(n;p)1102 809 y Fn(\))1120 816 y Fh(\030)1139 809 y Fn(;)j(\()p Fi(U)d(')1244 816 y Fh(m;q)1303 809 y Fn(\))1321 816 y Fh(\030)1339 809 y Fn(\))1357 821 y Fh(L)1381 810 y Ft(2)1381 833 y Fc(\030)1398 821 y Fv(\()p Fh(Q)p Fv(\))1498 779 y Fi(d\030)p 1468 799 V 1468 841 a Fn(\(2)p Fi(\031)r Fn(\))1555 827 y Fv(3)800 936 y Fn(=)877 874 y Fe(Z)903 977 y Fh(Q)931 968 y Fc(?)958 936 y Fi(e)979 917 y Fh(i)p Fv(\()p Fh(p)p Fu(\000)p Fh(q)q Fv(\))p Fu(\001)p Fh(\030)1152 905 y Fi(d\030)p 1122 925 V 1122 967 a Fn(\(2)p Fi(\031)r Fn(\))1209 954 y Fv(3)1240 874 y Fe(Z)1265 977 y Fh(Q)1303 936 y Fi(u)1329 943 y Fh(n)1352 936 y Fn(\()p Fi(\030)r Fn(;)j Fi(x)p Fn(\))j Fi(u)1494 917 y Fu(\003)1494 947 y Fh(m)1527 936 y Fn(\()p Fi(\030)r Fn(;)d Fi(x)p Fn(\))g Fi(dx)800 1063 y Fn(=)42 b Fi(\016)897 1070 y Fh(n;m)970 1001 y Fe(Z)995 1104 y Fh(Q)1023 1094 y Fc(?)1050 1063 y Fi(e)1071 1044 y Fh(i)p Fv(\()p Fh(p)p Fu(\000)p Fh(q)q Fv(\))p Fu(\001)p Fh(\030)1244 1032 y Fi(d\030)p 1214 1052 V 1214 1094 a Fn(\(2)p Fi(\031)r Fn(\))1301 1081 y Fv(3)1383 1063 y Fn(=)58 b Fi(\016)1496 1070 y Fh(n;m)1573 1063 y Fi(\016)1593 1070 y Fh(p;q)1640 1063 y Fi(:)37 1192 y Fn(W)l(e)21 b(c)o(hec)o(k)g(no)o(w)g(that)f(the)h(P)o(arsev)m(al)f (iden)o(tit)o(y)i(holds,)g(th)o(us)f(pro)o(ving)g(our)f(claim.)38 b(Indeed,)23 b(let)f Fi( )h Fg(2)37 1248 y Fi(L)68 1232 y Fv(2)88 1248 y Fn(\()p Fj(R)145 1232 y Fv(3)165 1248 y Fn(\),)14 b(then)100 1340 y Fe(X)100 1438 y Fh(n)p Fu(\025)p Fv(1)195 1340 y Fe(X)186 1441 y Fh(p)p Fu(2)p Fd(Z)253 1432 y Ft(3)315 1317 y Fe(\014)315 1345 y(\014)315 1372 y(\014)315 1399 y(\014)330 1322 y(Z)356 1425 y Fd(R)386 1415 y Ft(3)413 1383 y Fi( )r Fn(\()p Fi(x)p Fn(\))d Fi(')548 1365 y Fu(\003)548 1395 y Fh(n)571 1383 y Fn(\()p Fi(x)f Fn(+)h Fi(p)p Fn(\))d Fi(dx)770 1317 y Fe(\014)770 1345 y(\014)770 1372 y(\014)770 1399 y(\014)784 1331 y Fv(2)141 1543 y Fn(=)218 1500 y Fe(X)218 1598 y Fh(n)p Fu(\025)p Fv(1)314 1500 y Fe(X)305 1601 y Fh(p)p Fu(2)p Fd(Z)372 1592 y Ft(3)434 1477 y Fe(\014)434 1505 y(\014)434 1532 y(\014)434 1559 y(\014)449 1481 y(Z)474 1584 y Fh(Q)502 1575 y Fc(?)522 1543 y Fn(\(\()p Fi(U)d( )r Fn(\))644 1550 y Fh(\030)661 1543 y Fn(;)j(\()p Fi(U)d(')766 1550 y Fh(n;p)816 1543 y Fn(\))834 1550 y Fh(\030)852 1543 y Fn(\))870 1555 y Fh(L)894 1544 y Ft(2)894 1567 y Fc(\030)911 1555 y Fv(\()p Fh(Q)p Fv(\))1016 1513 y Fi(d\030)p 986 1533 V 986 1574 a Fn(\(2)p Fi(\031)r Fn(\))1073 1561 y Fv(3)1096 1477 y Fe(\014)1096 1505 y(\014)1096 1532 y(\014)1096 1559 y(\014)1112 1490 y Fv(2)141 1695 y Fn(=)218 1651 y Fe(X)218 1749 y Fh(n)p Fu(\025)p Fv(1)314 1651 y Fe(X)305 1753 y Fh(p)p Fu(2)p Fd(Z)372 1743 y Ft(3)441 1633 y Fe(Z)h(Z)517 1736 y Fh(Q)545 1726 y Fc(?)563 1736 y Fu(\002)p Fh(Q)618 1726 y Fc(?)646 1695 y Fi(e)667 1676 y Fh(ip)p Fu(\001)p Fv(\()p Fh(\030)q Fu(\000)p Fh(\030)782 1664 y Fp(0)792 1676 y Fv(\))808 1695 y Fn(\(\()p Fi(U)f( )r Fn(\))930 1702 y Fh(\030)947 1695 y Fn(;)j Fi(u)994 1702 y Fh(n)1017 1695 y Fn(\()p Fi(\030)r Fn(;)g Fg(\001)p Fn(\)\))1127 1706 y Fh(L)1151 1695 y Ft(2)1151 1718 y Fc(\030)1166 1706 y Fv(\()p Fh(Q)p Fv(\))1236 1695 y Fn(\(\()p Fi(U)d( )r Fn(\))1358 1702 y Fh(\030)1375 1693 y Fp(0)1386 1695 y Fn(;)j Fi(u)1433 1702 y Fh(n)1456 1695 y Fn(\()p Fi(\030)1496 1676 y Fu(0)1507 1695 y Fn(;)g Fg(\001)p Fn(\)\))1577 1676 y Fu(\003)1577 1711 y Fh(L)1601 1699 y Ft(2)1601 1726 y Fc(\030)1616 1719 y Fp(0)1628 1711 y Fv(\()p Fh(Q)p Fv(\))1704 1664 y Fi(d\030)r(d\030)1796 1647 y Fu(0)p 1703 1684 V 1703 1726 a Fn(\(2)p Fi(\031)r Fn(\))1790 1713 y Fv(6)141 1854 y Fn(=)218 1811 y Fe(X)218 1909 y Fh(n)p Fu(\025)p Fv(1)338 1793 y Fe(Z)363 1896 y Fh(Q)391 1886 y Fc(?)466 1824 y Fi(d\030)p 436 1844 V 436 1886 a Fn(\(2)p Fi(\031)r Fn(\))523 1873 y Fv(3)546 1789 y Fe(\014)546 1816 y(\014)546 1843 y(\014)546 1870 y(\014)561 1793 y(Z)587 1896 y Fh(Q)616 1854 y Fn(\()p Fi(U)d( )r Fn(\))720 1861 y Fh(\030)738 1854 y Fn(\()p Fi(x)p Fn(\))12 b Fi(u)838 1836 y Fu(\003)838 1866 y Fh(n)861 1854 y Fn(\()p Fi(\030)r Fn(;)c Fi(x)p Fn(\))j Fi(dx)1027 1789 y Fe(\014)1027 1816 y(\014)1027 1843 y(\014)1027 1870 y(\014)1042 1802 y Fv(2)1759 1854 y Fn(\(3.30\))141 1998 y(=)218 1936 y Fe(Z)243 2040 y Fh(Q)271 2030 y Fc(?)299 1998 y Fg(k)p Fn(\()p Fi(U)5 b( )r Fn(\))426 2005 y Fh(\030)443 1998 y Fg(k)466 1980 y Fv(2)466 2014 y Fh(L)490 2003 y Ft(2)490 2026 y Fc(\030)507 2014 y Fv(\()p Fh(Q)p Fv(\))612 1968 y Fi(d\030)p 582 1988 V 582 2030 a Fn(\(2)p Fi(\031)r Fn(\))669 2016 y Fv(3)1759 1998 y Fn(\(3.31\))141 2103 y(=)42 b Fg(k)p Fi( )r Fg(k)296 2084 y Fv(2)296 2117 y Fh(L)320 2107 y Ft(2)336 2117 y Fv(\()p Fd(R)380 2107 y Ft(3)397 2117 y Fv(\))459 2103 y Fi(;)1287 b Fn(\(3.32\))37 2205 y(where)17 b(\(3.30\))e(follo)o(ws)h (from)g(the)h(P)o(oisson)f(form)o(ula,)g(\(3.31\))f(from)h(the)g(P)o (arsev)m(al)h(iden)o(tit)o(y)l(,)g(and)g(\(3.32\))37 2261 y(from)e(\(3.20\).)1564 b Fg(})37 2343 y Fn(W)l(e)16 b(ma)o(y)e(turn)h(no)o(w)g(to)g(the)37 2424 y Fj(Pro)q(of)j(of)g(Lemma) f(3.4)g(:)k Fn(Let)15 b(\()p Fi(u)669 2431 y Fh(n)692 2424 y Fn(\))710 2431 y Fh(n)748 2424 y Fn(and)g(\()p Fi(')884 2431 y Fh(n;p)935 2424 y Fn(\))953 2431 y Fh(n;p)1018 2424 y Fn(b)q(e)h(de\014ned)g(as)e(in)i(Lemma)f(3.5.)k(With)c(the)g (help)37 2481 y(of)f(Lemma)g(3.5,)f(and)h(using)g(\014rst)g(the)f(P)o (arsev)m(al)h(Iden)o(tit)o(y)h(in)f Fi(L)1139 2464 y Fv(2)1159 2481 y Fn(\()p Fj(R)1216 2464 y Fv(3)1236 2481 y Fn(\),)f(and)h(then)g(the)g(de\014nition)i(of)d Fi(\032)h Fn(as)p eop %%Page: 23 23 23 22 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(23)37 199 y(the)16 b(k)o(ernel)g(of)f Fi(K)s Fn(,)f(w)o(e)h(ha)o(v)o (e)265 248 y Fe(Z)5 b(Z)340 351 y Fh(Q)p Fu(\002)p Fd(R)425 342 y Ft(3)453 310 y Fg(j)p Fi(\032)p Fn(\()p Fi(x)p Fn(;)j Fi(y)r Fn(\))p Fg(j)609 291 y Fv(2)635 310 y Fi(dxdy)42 b Fn(=)850 248 y Fe(Z)875 351 y Fh(Q)913 310 y Fi(dx)970 267 y Fe(X)971 365 y Fh(n)p Fu(\025)p Fv(1)1053 267 y Fe(X)1044 368 y Fh(p)p Fu(2)p Fd(Z)1111 358 y Ft(3)1128 244 y Fe(\014)1128 271 y(\014)1128 298 y(\014)1128 326 y(\014)1143 248 y(Z)1168 351 y Fd(R)1198 342 y Ft(3)1225 310 y Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))k Fi(')1397 291 y Fu(\003)1397 321 y Fh(n)1420 310 y Fn(\()p Fi(y)g Fn(+)e Fi(p)p Fn(\))e Fi(dy)1614 244 y Fe(\014)1614 271 y(\014)1614 298 y(\014)1614 326 y(\014)1628 257 y Fv(2)773 443 y Fn(=)851 400 y Fe(X)850 498 y Fh(n)p Fu(\025)p Fv(1)933 400 y Fe(X)924 501 y Fh(p)p Fu(2)p Fd(Z)991 492 y Ft(3)1015 443 y Fg(k)p Fi(K)s(')1110 450 y Fh(n;p)1161 443 y Fg(k)1184 424 y Fv(2)1184 457 y Fh(L)1208 448 y Ft(2)1225 457 y Fv(\()p Fh(Q)p Fv(\))773 577 y Fn(=)851 533 y Fe(X)850 631 y Fh(n)p Fu(\025)p Fv(1)924 577 y Fg(k)p Fi(K)s(')1019 584 y Fh(n)1042 577 y Fg(k)1065 558 y Fv(2)1065 591 y Fh(L)1089 581 y Ft(2)1106 591 y Fv(\()p Fd(R)1150 581 y Ft(3)1167 591 y Fv(\))1759 577 y Fn(\(3.33\))773 721 y(=)851 677 y Fe(X)850 775 y Fh(n)p Fu(\025)p Fv(1)924 659 y Fe(Z)949 762 y Fh(Q)977 752 y Fc(?)1005 721 y Fg(k)p Fi(K)1067 728 y Fh(\030)1093 721 y Fi(u)1119 728 y Fh(n)1142 721 y Fn(\()p Fi(\030)r Fn(;)g Fg(\001)p Fn(\))p Fg(k)1257 702 y Fv(2)1257 735 y Fh(L)1281 725 y Ft(2)1296 735 y Fv(\()p Fh(Q)p Fv(\))1401 690 y Fi(d\030)p 1371 710 106 2 v 1371 752 a Fn(\(2)p Fi(\031)r Fn(\))1458 739 y Fv(3)1482 721 y Fi(;)264 b Fn(\(3.34\))37 847 y(where)13 b(\(3.33\))e(comes)i(from)f(\(H1\),)g (and)h(\(3.34\))d(from)i(the)h(de\014nitions)i(of)d Fi(K)1351 854 y Fh(\030)1382 847 y Fn(and)h(of)f(the)h(scalar)g(pro)q(duct)37 904 y(on)k Fg(H)p Fn(.)23 b(F)l(rom)15 b(\(3.34\),)f(w)o(e)i(obtain)g (in)h(particular,)g(that)1028 869 y Fe(P)1076 917 y Fh(n)p Fu(\025)p Fv(1)1152 904 y Fg(k)p Fi(K)1214 911 y Fh(\030)1232 904 y Fi(u)1258 911 y Fh(n)1282 904 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\))p Fg(k)1397 887 y Fv(2)1397 920 y Fh(L)1421 910 y Ft(2)1436 920 y Fv(\()p Fh(Q)p Fv(\))1509 904 y Fn(is)17 b(\014nite)g(for)e(almost)37 966 y(ev)o(ery)g Fi(\030)h Fn(in)f Fi(Q)281 950 y Fh(?)314 966 y Fn(as)f(so)q(on)g(as)g(\(3.25\))e(holds)j(true;)f(this)g(is)h (precisely)h(the)e(de\014nition)i(of)e Fi(K)1569 973 y Fh(\030)1601 966 y Fn(as)g(an)g(Hilb)q(ert-)37 1023 y(Sc)o(hmidt)f(op)q(erator)d(on)h Fi(L)481 1006 y Fv(2)481 1037 y Fh(\030)501 1023 y Fn(\()p Fi(Q)p Fn(\))f(\(see,)i(for)e (example,)j([44)o(]\),)d(whose)i(k)o(ernel)f Fi(\032)p Fn(\()p Fi(\030)r Fn(;)d Fg(\001)p Fn(;)g Fg(\001)p Fn(\))h(b)q(elongs) j(to)f Fi(L)1709 1006 y Fv(2)1728 1023 y Fn(\()p Fi(Q)r Fg(\002)r Fi(Q)p Fn(\))37 1079 y(for)k(almost)g(ev)o(ery)g Fi(\030)i Fn(in)f Fi(Q)500 1063 y Fh(?)520 1079 y Fn(.)k(Therefore,)14 b(\(ii\))i(holds.)21 b(If)15 b(w)o(e)g(go)g(bac)o(k)g(to)f(\(3.34\),)f (w)o(e)i(no)o(w)g(ha)o(v)o(e)261 1119 y Fe(Z)287 1222 y Fh(Q)315 1212 y Fc(?)342 1137 y Fe(X)342 1235 y Fh(n)p Fu(\025)p Fv(1)416 1181 y Fg(k)p Fi(K)478 1188 y Fh(\030)504 1181 y Fi(u)530 1188 y Fh(n)553 1181 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\))p Fg(k)668 1162 y Fv(2)668 1195 y Fh(L)692 1185 y Ft(2)708 1195 y Fv(\()p Fh(Q)p Fv(\))812 1150 y Fi(d\030)p 782 1170 V 782 1212 a Fn(\(2)p Fi(\031)r Fn(\))869 1199 y Fv(3)905 1181 y Fn(=)953 1119 y Fe(Z)979 1222 y Fh(Q)1007 1212 y Fc(?)1069 1150 y Fi(d\030)p 1039 1170 V 1039 1212 a Fn(\(2)p Fi(\031)r Fn(\))1126 1199 y Fv(3)1149 1119 y Fe(Z)e(Z)1225 1222 y Fh(Q)p Fu(\002)p Fh(Q)1318 1181 y Fg(j)p Fi(\032)p Fn(\()p Fi(\030)r Fn(;)i Fi(x)p Fn(;)g Fi(y)r Fn(\))p Fg(j)1517 1162 y Fv(2)1541 1181 y Fi(dxdy)r(;)37 1307 y Fn(whence)19 b(\(3.26\).)27 b(It)18 b(is)h(easily)g(seen)f(that)g(the)g(same)f(pro)q(of)h(giv)o(es) g(in)h(fact)f(the)g(pro)q(of)f(of)h(the)g(con)o(v)o(erse)37 1364 y(implication)g(\(ii\))e Fg(\))f Fn(\(i\))g(since)i(at)d(eac)o(h)i (step)f(of)g(the)g(pro)q(of)g(w)o(e)g(ha)o(v)o(e)f(argued)h(b)o(y)h (equiv)m(alence.)108 1445 y(Let)g(us)h(no)o(w)e(pro)o(v)o(e)h(\(3.27\)) e(and)i(\(3.28\).)21 b(F)l(or)16 b(almost)f(ev)o(ery)h Fi(y)i Fn(\014xed)f(in)g Fi(Q)p Fn(,)f(w)o(e)g(kno)o(w)f(from)h (\(3.25\))37 1501 y(that)j Fi(\032)p Fn(\()p Fg(\001)p Fn(;)8 b Fi(y)r Fn(\))18 b(lies)j(in)f Fi(L)446 1485 y Fv(2)466 1501 y Fn(\()p Fj(R)523 1485 y Fv(3)542 1501 y Fn(\).)33 b(Next,)20 b(\(3.27\))d(and)j(\(3.28\))d(are)i(t)o(w)o(o)g (equiv)m(alen)o(t)i(form)o(ulations)e(of)g(the)37 1558 y(claim)14 b(that)e Fi(\032)p Fn(\()p Fi(\030)r Fn(;)c Fi(x)p Fn(;)g Fi(y)r Fn(\))j(is)j(obtained)f(b)o(y)g(applying)h(the)f (transformation)f Fi(U)18 b Fn(in)13 b Fi(x)g Fn(to)f Fi(\032)p Fn(\()p Fi(x)p Fn(;)c Fi(y)r Fn(\).)18 b(By)13 b(the)g(w)o(a)o(y)l(,)37 1614 y(let)k(us)f(note)g(that,)f(b)q(ecause)i (of)e(\(3.20\),)f(this)j(claim)g(pro)o(vides)f(another)g(pro)q(of)f(of) h(\(3.26\).)k(Let)d(us)f(c)o(hec)o(k)37 1671 y(that)f(\(3.27\))e (holds.)21 b(Let)15 b Fi(')g Fn(b)q(e)h(\014xed,)g(sa)o(y)e(in)i Fg(S)s Fn(\()p Fj(R)933 1654 y Fv(3)954 1671 y Fn(\),)e(w)o(e)h(c)o (hec)o(k)h(successiv)o(ely)h(that)178 1757 y(\()p Fi(U)5 b(K)s(')p Fn(\))322 1764 y Fh(\030)340 1757 y Fn(\()p Fi(x)p Fn(\))41 b(=)529 1714 y Fe(X)520 1816 y Fh(k)q Fu(2)p Fd(Z)588 1806 y Ft(3)612 1757 y Fi(e)633 1739 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)721 1757 y Fn(\()p Fi(K)s(')p Fn(\)\()p Fi(x)8 b Fn(+)j Fi(k)q Fn(\))443 1901 y(=)529 1857 y Fe(X)520 1959 y Fh(k)q Fu(2)p Fd(Z)588 1949 y Ft(3)612 1901 y Fi(e)633 1882 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)728 1839 y Fe(Z)753 1942 y Fd(R)783 1932 y Ft(3)811 1901 y Fi(\032)p Fn(\()p Fi(x)f Fn(+)h Fi(k)q Fn(;)d Fi(y)r Fn(\))p Fi(')p Fn(\()p Fi(y)r Fn(\))g Fi(dy)443 2044 y Fn(=)529 2001 y Fe(X)520 2102 y Fh(k)q Fu(2)p Fd(Z)588 2093 y Ft(3)612 2044 y Fi(e)633 2025 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)728 1982 y Fe(Z)753 2085 y Fd(R)783 2076 y Ft(3)811 2044 y Fi(\032)834 2025 y Fu(\003)854 2044 y Fn(\()p Fi(y)r Fn(;)g Fi(x)h Fn(+)h Fi(k)q Fn(\))p Fi(')p Fn(\()p Fi(y)r Fn(\))e Fi(dy)443 2191 y Fn(=)529 2148 y Fe(X)520 2249 y Fh(k)q Fu(2)p Fd(Z)588 2240 y Ft(3)612 2191 y Fi(e)633 2172 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)728 2129 y Fe(Z)753 2232 y Fh(Q)781 2223 y Fc(?)838 2160 y Fi(d\030)884 2144 y Fu(0)p 814 2180 V 814 2222 a Fn(\(2)p Fi(\031)r Fn(\))901 2209 y Fv(3)932 2129 y Fe(Z)957 2232 y Fh(Q)1013 2148 y Fe(X)1007 2249 y Fh(l)p Fu(2)p Fd(Z)1067 2240 y Ft(3)1091 2191 y Fi(e)1112 2172 y Fh(il)p Fu(\001)p Fh(\030)1162 2160 y Fp(0)1175 2191 y Fi(\032)1198 2172 y Fu(\003)1218 2191 y Fn(\()p Fi(y)k Fn(+)f Fi(l)q Fn(;)d Fi(x)g Fn(+)j Fi(k)q Fn(\)\()p Fi(U)5 b(')p Fn(\))1577 2199 y Fh(\030)1594 2189 y Fp(0)1606 2191 y Fn(\()p Fi(y)r Fn(\))j Fi(dy)39 b Fn(\(3.35\))443 2338 y(=)529 2295 y Fe(X)520 2396 y Fh(k)q Fu(2)p Fd(Z)588 2387 y Ft(3)612 2338 y Fi(e)633 2319 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)728 2276 y Fe(Z)753 2379 y Fh(Q)781 2370 y Fc(?)838 2307 y Fi(d\030)884 2291 y Fu(0)p 814 2327 V 814 2369 a Fn(\(2)p Fi(\031)r Fn(\))901 2356 y Fv(3)932 2276 y Fe(Z)957 2379 y Fh(Q)1013 2295 y Fe(X)1007 2396 y Fh(l)p Fu(2)p Fd(Z)1067 2387 y Ft(3)1091 2338 y Fi(e)1112 2319 y Fh(il)p Fu(\001)p Fh(\030)1162 2307 y Fp(0)1175 2338 y Fi(\032)p Fn(\()p Fi(x)10 b Fn(+)h Fi(k)g Fg(\000)g Fi(l)q Fn(;)d Fi(y)r Fn(\)\()p Fi(U)d(')p Fn(\))1559 2346 y Fh(\030)1575 2336 y Fp(0)1586 2338 y Fn(\()p Fi(y)r Fn(\))j Fi(dy)443 2485 y Fn(=)529 2442 y Fe(X)520 2543 y Fh(k)q Fu(2)p Fd(Z)588 2534 y Ft(3)612 2485 y Fi(e)633 2466 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)728 2423 y Fe(Z)753 2526 y Fh(Q)781 2517 y Fc(?)838 2454 y Fi(d\030)884 2438 y Fu(0)p 814 2474 V 814 2516 a Fn(\(2)p Fi(\031)r Fn(\))901 2503 y Fv(3)932 2423 y Fe(Z)957 2526 y Fh(Q)994 2485 y Fi(e)1015 2466 y Fh(ik)q Fu(\001)p Fh(\030)1073 2454 y Fp(0)1100 2442 y Fe(X)1094 2543 y Fh(l)p Fu(2)p Fd(Z)1154 2534 y Ft(3)1178 2485 y Fi(e)1199 2466 y Fu(\000)p Fh(il)p Fu(\001)p Fh(\030)1276 2454 y Fp(0)1290 2485 y Fi(\032)p Fn(\()p Fi(x)i Fn(+)h Fi(l)q Fn(;)d Fi(y)r Fn(\)\()p Fi(U)d(')p Fn(\))1593 2493 y Fh(\030)1609 2483 y Fp(0)1620 2485 y Fn(\()p Fi(y)r Fn(\))j Fi(dy)443 2628 y Fn(=)520 2566 y Fe(Z)545 2669 y Fh(Q)592 2585 y Fe(X)582 2686 y Fh(k)q Fu(2)p Fd(Z)650 2677 y Ft(3)675 2628 y Fi(e)696 2609 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\030)783 2628 y Fi(\032)p Fn(\()p Fi(x)j Fn(+)f Fi(k)q Fn(;)e Fi(y)r Fn(\)\()p Fi(U)d(')p Fn(\))1096 2635 y Fh(\030)1113 2628 y Fn(\()p Fi(y)r Fn(\))j Fi(dy)532 b Fn(\(3.36\))443 2771 y(=)42 b Fi(K)559 2778 y Fh(\030)577 2771 y Fn(\()p Fi(U)5 b(')p Fn(\))679 2778 y Fh(\030)697 2771 y Fn(\()p Fi(x)p Fn(\))12 b(=)819 2709 y Fe(Z)844 2813 y Fh(Q)882 2771 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)c Fi(x)p Fn(;)g Fi(y)r Fn(\)\()p Fi(U)d(')p Fn(\))1157 2778 y Fh(\030)1173 2771 y Fn(\()p Fi(y)r Fn(\))j Fi(dy)r(;)p eop %%Page: 24 24 24 23 bop 53 50 a Fm(3.)23 b Fl(Preliminaries)1428 b Fn(24)37 199 y(with)14 b(\(3.19\))c(to)i(deduce)i(\(3.35\),)d(and)i (the)g(P)o(oisson)f(form)o(ula)h(to)f(obtain)g(\(3.36\).)18 b(This)13 b(pro)o(v)o(es)f(our)g(claim.)1840 256 y Fg(})108 354 y Fn(W)l(e)i(are)g(no)o(w)g(ready)g(to)f(state)g(the)i (de\014nition)h(of)d(the)i(admissible)h(\\p)q(erio)q(dic)g(densit)o(y)e (matrices")g(w)o(e)37 410 y(shall)j(w)o(ork)d(with.)108 491 y(Let)j Fi(K)g Fn(=)297 455 y Fe(R)318 507 y Fh(Q)346 498 y Fc(?)374 491 y Fi(K)413 498 y Fh(\030)475 471 y(d\030)p 451 481 84 2 v 451 508 a Fv(\(2)p Fh(\031)q Fv(\))518 499 y Ft(3)556 491 y Fn(b)q(e)g(a)f(self-adjoin)o(t)h(op)q(erator)e(in) i Fi(L)1162 475 y Fv(2)1182 491 y Fn(\()p Fj(R)1239 475 y Fv(3)1259 491 y Fn(\))f(satisfying)g(\(H1\).)23 b(W)l(e)16 b(shall)i(sa)o(y)37 555 y(that)12 b Fi(K)j Fn(is)d(an)25 b Ff(admissible)13 b(p)n(erio)n(dic)g(density)f(matrix)19 b Fn(if)12 b Fi(K)j Fn(satis\014es)d(in)h(addition)g(to)e(\(H1\))g(the) h(follo)o(wing)37 611 y(prop)q(erties)k(\(H2\){\(H4\))e(:)37 693 y(\(H2\))h(0)d Fg(\024)h Fi(K)j Fg(\024)d Fj(1)p Fn(,)37 774 y(\(H3\))i(for)f(almost)h(ev)o(ery)g Fi(\030)i Fn(in)f Fi(Q)607 758 y Fh(?)627 774 y Fn(,)f Fi(K)694 781 y Fh(\030)727 774 y Fn(is)h(a)f(trace-class)g(op)q(erator)g(on)g Fi(L)1309 758 y Fv(2)1309 788 y Fh(\030)1328 774 y Fn(\()p Fi(Q)p Fn(\),)g(and)699 829 y Fe(Z)724 932 y Fh(Q)752 922 y Fc(?)779 891 y Fn(T)l(r)826 902 y Fh(L)850 891 y Ft(2)850 914 y Fc(\030)868 902 y Fv(\()p Fh(Q)p Fv(\))925 891 y Fi(K)964 898 y Fh(\030)1030 860 y Fi(d\030)p 1000 880 106 2 v 1000 922 a Fn(\(2)p Fi(\031)r Fn(\))1087 909 y Fv(3)1123 891 y Fn(=)e(1)p Fi(;)552 b Fn(\(3.37\))37 1035 y(\(H4\))15 b(for)f(almost)h(ev)o(ery)g Fi(\030)i Fn(in)f Fi(Q)607 1019 y Fh(?)627 1035 y Fn(,)f Fg(\000)p Fn(\001)728 1042 y Fh(\030)760 1035 y Fi(K)799 1042 y Fh(\030)832 1035 y Fn(is)h(a)f(trace-class)g(op)q(erator)f(on)i Fi(L)1414 1019 y Fv(2)1414 1049 y Fh(\030)1433 1035 y Fn(\()p Fi(Q)p Fn(\),)e(and)605 1090 y Fe(Z)630 1193 y Fh(Q)658 1183 y Fc(?)685 1152 y Fn(T)l(r)732 1163 y Fh(L)756 1152 y Ft(2)756 1175 y Fc(\030)773 1163 y Fv(\()p Fh(Q)p Fv(\))831 1152 y Fn([)p Fg(\000)p Fn(\001)917 1159 y Fh(\030)948 1152 y Fi(K)987 1159 y Fh(\030)1006 1152 y Fn(])1066 1121 y Fi(d\030)p 1036 1141 V 1036 1183 a Fn(\(2)p Fi(\031)r Fn(\))1123 1170 y Fv(3)1159 1152 y Fi(<)f Fn(+)p Fg(1)p Fi(;)459 b Fn(\(3.38\))37 1273 y(where)12 b Fg(\000)p Fn(\001)238 1280 y Fh(\030)269 1273 y Fn(is)g(a)g(notation)f(for)g(the)h(op)q(erator)f Fi(e)866 1256 y Fh(i\030)q Fu(\001)p Fh(x)934 1273 y Fn(\()p Fg(\000)p Fn(\001)1025 1280 y Fh(per)1078 1273 y Fn(\))d Fi(e)1125 1256 y Fu(\000)p Fh(i\030)q Fu(\001)p Fh(x)1224 1273 y Fn(acting)k(on)g Fi(L)1448 1256 y Fv(2)1448 1287 y Fh(\030)1467 1273 y Fn(\()p Fi(Q)p Fn(\),)g(and)g(with)g Fg(\000)p Fn(\001)1822 1280 y Fh(per)37 1329 y Fn(denoting)20 b(the)g(Laplace)g(op)q(erator)f(asso)q(ciated)g(to)g(p)q(erio)q(dic)i (b)q(oundary)f(conditions)h(in)f Fi(Q)p Fn(.)32 b(Actually)l(,)37 1386 y Fg(\000)p Fn(\001)110 1393 y Fh(\030)146 1386 y Fn(is)18 b(equiv)m(alen)o(tly)h(de\014ned)f(b)o(y)e Fg(\000)p Fn(\001)g(=)810 1349 y Fe(R)831 1402 y Fh(Q)859 1392 y Fc(?)886 1386 y Fg(\000)p Fn(\001)959 1393 y Fh(\030)1023 1365 y(d\030)p 999 1375 84 2 v 999 1403 a Fv(\(2)p Fh(\031)q Fv(\))1066 1393 y Ft(3)1104 1386 y Fn(according)h(to)g(the)f (de\014nitions)j(\(3.21\))c(and)37 1449 y(\(3.23\))e(of)h(the)h(Blo)q (c)o(h)g(w)o(a)o(v)o(es)f(decomp)q(osition)i(\(see)e([45)o(]\).)19 b(The)c(set)f(of)h(all)g(admissible)i(p)q(erio)q(dic)f(densit)o(y)37 1506 y(matrices)i(is)f(denoted)h(b)o(y)f Fg(K)q Fn(.)25 b(W)l(e)17 b(collect)i(in)f(the)f(forthcoming)g(Prop)q(osition)g(3.2)g (v)m(arious)g(prop)q(erties)37 1562 y(of)i(the)g(p)q(erio)q(dic)i (densit)o(y)e(matrices)g(in)h Fg(K)q Fn(,)e(that)h(ha)o(v)o(e)f(b)q (een)i(pro)o(v)o(ed)f(in)g(the)g(course)g(of)f(this)i(section.)37 1618 y(But,)e(b)q(efore)f(that,)g(let)g(us)g(in)o(tro)q(duce)h(some)f (functional)h(spaces)g(:)23 b(for)17 b(ev)o(ery)g Fi(\030)i Fn(in)f Fi(Q)1559 1602 y Fh(?)1578 1618 y Fn(,)f(and)h(for)e(ev)o(ery) 37 1675 y(1)d Fg(\024)g Fi(p)f Fg(\024)h Fn(+)p Fg(1)p Fn(,)448 1761 y Fi(L)479 1739 y Fh(p)479 1776 y(\030)499 1761 y Fn(\()p Fi(Q)p Fn(\))e(=)i Fg(f)p Fi(')g Fg(2)f Fi(L)769 1739 y Fh(p)769 1776 y Fv(lo)q(c)816 1761 y Fn(\()p Fj(R)873 1742 y Fv(3)892 1761 y Fn(\))g Fi(=)h(\034)978 1768 y Fh(k)1009 1761 y Fg(\001)d Fi(')i Fn(=)h Fi(e)1143 1742 y Fh(i)6 b(k)q Fu(\001)p Fh(\030)1209 1761 y Fi(';)i Fg(8)k Fi(k)i Fg(2)f Fj(Z)1410 1742 y Fv(3)1429 1761 y Fg(g)p Fi(;)37 1847 y Fn(and)439 1934 y Fi(H)481 1915 y Fv(1)477 1945 y Fh(\030)500 1934 y Fn(\()p Fi(Q)p Fn(\))f(=)h Fg(f)p Fi(')f Fg(2)h Fi(H)782 1915 y Fv(1)778 1945 y(lo)q(c)824 1934 y Fn(\()p Fj(R)881 1915 y Fv(3)901 1934 y Fn(\))f Fi(=)g(\034)986 1941 y Fh(k)1018 1934 y Fg(\001)e Fi(')i Fn(=)h Fi(e)1152 1915 y Fh(i)6 b(k)q Fu(\001)p Fh(\030)1218 1934 y Fi(';)i Fg(8)k Fi(k)h Fg(2)g Fj(Z)1418 1915 y Fv(3)1438 1934 y Fg(g)p Fi(:)37 2054 y Fj(Prop)q(osition)20 b(3.2)k Ff(L)n(et)16 b Fi(K)j Ff(b)n(elong)c(to)i Fg(K)q Ff(.)j(Then,)37 2159 y Fj(\(i\))k Fi(K)c Ff(satis\014es)14 b(the)j(e)n(quivalent)e(pr)n(op)n(erties)h(given)g(in)g(L)n(emma)g (3.4.)151 2257 y(L)n(et)h Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(;)g Fg(\001)p Fn(\))i Fg(2)j Fi(L)463 2240 y Fv(2)482 2257 y Fn(\()p Fi(Q)536 2240 y Fh(?)556 2257 y Fn(;)8 b Fi(Q)h Fg(\002)h Fi(Q)p Fn(\))16 b Ff(denote)g(the)h (Hilb)n(ert-)f(Schmidt)g(kernel)f(of)i Fi(K)1527 2264 y Fh(\030)1545 2257 y Ff(.)37 2370 y Fj(\(ii\))25 b Ff(F)m(or)18 b(almost)g(every)g Fi(\030)j Ff(in)d Fi(Q)621 2354 y Fh(?)640 2370 y Ff(,)h(ther)n(e)f(exists)g(a)h(c)n(omplete)f(set)g(of)g (eigenfunctions)f Fn(\()p Fi(u)1620 2377 y Fh(n)1643 2370 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\)\))1753 2377 y Fh(n)p Fu(\025)p Fv(1)1838 2370 y Ff(of)151 2427 y Fi(K)190 2434 y Fh(\030)223 2427 y Ff(in)14 b Fi(L)308 2410 y Fv(2)308 2440 y Fh(\030)328 2427 y Fn(\()p Fi(Q)p Fn(\))g Ff(c)n(orr)n(esp)n(onding)f(to)i(the)g(non-incr)n(e)n(asing)e (se)n(quenc)n(e)g(of)h(eigenvalues)g Fn(0)e Fg(\024)h Fi(\025)1710 2434 y Fh(n)1734 2427 y Fn(\()p Fi(\030)r Fn(\))f Fg(\024)h Fn(1)151 2489 y Ff(\(c)n(ounte)n(d)h(with)i(their)f (multiplicity\))g(such)g(that)g Fi(u)1014 2496 y Fh(n)1038 2489 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\))j Fg(2)h Fi(H)1225 2472 y Fv(1)1221 2503 y Fh(\030)1245 2489 y Fn(\()p Fi(Q)p Fn(\))p Ff(,)i Fi(\030)h Fg(7!)e Fi(u)1464 2496 y Fh(n)1487 2489 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\))j Fg(2)i(H)p Ff(,)j(and)f(such)151 2545 y(that)632 2632 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\))j(=)864 2588 y Fe(X)864 2686 y Fh(n)p Fu(\025)p Fv(1)938 2632 y Fi(\025)964 2639 y Fh(n)988 2632 y Fn(\()p Fi(\030)r Fn(\))h Fi(u)1084 2639 y Fh(n)1107 2632 y Fn(\()p Fi(\030)r Fn(;)c Fi(x)p Fn(\))j Fi(u)1249 2613 y Fu(\003)1249 2643 y Fh(n)1273 2632 y Fn(\()p Fi(\030)r Fn(;)d Fi(y)r Fn(\))p Fi(;)151 2762 y Ff(for)17 b(almost)f(every)g Fi(\030)i Ff(in)e Fi(Q)619 2746 y Fh(?)639 2762 y Ff(.)p eop %%Page: 25 25 25 24 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(25)37 199 y Fj(\(iii\))25 b Ff(F)m(or)17 b(almost)g(every)g Fi(\030)i Ff(in)d Fi(Q)628 183 y Fh(?)648 199 y Ff(,)h Fi(x)c Fg(7!)i Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))15 b Ff(is)h(p)n(erio)n(dic,)h (non-ne)n(gative,)f(b)n(elongs)f(to)i Fi(L)1725 183 y Fv(1)1725 213 y(unif)1790 199 y Fn(\()p Fi(Q)p Fn(\))p Ff(,)151 256 y(and)f(may)h(b)n(e)f(written)566 355 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))i(=)800 312 y Fe(X)799 410 y Fh(n)p Fu(\025)p Fv(1)873 355 y Fi(\025)899 362 y Fh(n)923 355 y Fn(\()p Fi(\030)r Fn(\))i Fg(j)p Fi(u)1032 362 y Fh(n)1055 355 y Fn(\()p Fi(\030)r Fn(;)c Fi(x)p Fn(\))p Fg(j)1173 336 y Fv(2)1254 355 y Ff(a.e.)21 b(on)16 b Fi(Q:)151 524 y Ff(Henc)n(e,)f Fn(T)l(r)348 536 y Fh(L)372 525 y Ft(2)372 548 y Fc(\030)389 536 y Fv(\()p Fh(Q)p Fv(\))446 524 y Fi(K)485 531 y Fh(\030)517 524 y Fn(=)565 462 y Fe(Z)590 566 y Fh(Q)627 524 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))g Fi(dx)n(:)16 b Ff(In)g(addition,)h(we)f(may)h(de\014ne)e Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))15 b Ff(by)723 677 y Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))j(=)914 615 y Fe(Z)939 718 y Fh(Q)967 709 y Fc(?)995 677 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)d Fi(x)p Fn(;)g Fi(x)p Fn(\))1203 646 y Fi(d\030)p 1175 667 106 2 v 1175 708 a Fn(\(2)p Fi(\031)r Fn(\))1262 695 y Fv(3)1283 677 y Fi(;)463 b Fn(\(3.39\))151 805 y Ff(and)16 b Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))16 b Ff(is)f(a)i Fi(Q)p Ff(-p)n(erio)n(dic,)f (non-ne)n(gative)f(function)h(in)g Fi(L)1243 789 y Fv(1)1243 819 y(unif)1307 805 y Fn(\()p Fi(Q)p Fn(\))p Ff(.)k(A)o(nd,)c(we)g (also)g(have)297 862 y Fe(Z)322 965 y Fh(Q)350 956 y Fc(?)378 924 y Fn(T)l(r)424 936 y Fh(L)448 924 y Ft(2)448 948 y Fc(\030)466 936 y Fv(\()p Fh(Q)p Fv(\))523 924 y Fi(K)562 931 y Fh(\030)628 893 y Fi(d\030)p 598 914 V 598 955 a Fn(\(2)p Fi(\031)r Fn(\))685 942 y Fv(3)721 924 y Fn(=)769 862 y Fe(Z)794 965 y Fh(Q)822 956 y Fc(?)885 893 y Fi(d\030)p 855 914 V 855 955 a Fn(\(2)p Fi(\031)r Fn(\))942 942 y Fv(3)973 862 y Fe(Z)998 965 y Fh(Q)1035 924 y Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))g Fi(dx)768 1051 y Fn(=)816 989 y Fe(Z)841 1092 y Fh(Q)869 1082 y Fc(?)897 1007 y Fe(X)896 1105 y Fh(n)p Fu(\025)p Fv(1)970 1051 y Fi(\025)996 1058 y Fh(n)1020 1051 y Fn(\()p Fi(\030)r Fn(\))1125 1020 y Fi(d\030)p 1095 1040 V 1095 1082 a Fn(\(2)p Fi(\031)r Fn(\))1182 1069 y Fv(3)1218 1051 y Fn(=)1266 989 y Fe(Z)1291 1092 y Fh(Q)1329 1051 y Fi(\032)p Fn(\()p Fi(x)p Fn(;)g Fi(x)p Fn(\))g Fi(dx)j Fn(=)i(1)p Fi(:)1759 1000 y Fn(\(3.40\))37 1252 y Fj(\(iv\))24 b Fn(\(H4\))15 b Ff(writes)207 1307 y Fe(Z)232 1410 y Fh(Q)260 1400 y Fc(?)287 1368 y Fn(T)l(r)334 1380 y Fh(L)358 1369 y Ft(2)358 1392 y Fc(\030)375 1380 y Fv(\()p Fh(Q)p Fv(\))433 1368 y Fn([)p Fg(\000)p Fn(\001)519 1375 y Fh(\030)537 1368 y Fi(K)576 1375 y Fh(\030)595 1368 y Fn(])655 1338 y Fi(d\030)p 625 1358 V 625 1400 a Fn(\(2)p Fi(\031)r Fn(\))712 1387 y Fv(3)748 1368 y Fn(=)796 1307 y Fe(Z)821 1410 y Fh(Q)849 1400 y Fc(?)877 1325 y Fe(X)877 1423 y Fh(n)p Fu(\025)p Fv(1)951 1368 y Fi(\025)977 1375 y Fh(n)1001 1368 y Fn(\()p Fi(\030)r Fn(\))1106 1338 y Fi(d\030)p 1076 1358 V 1076 1400 a Fn(\(2)p Fi(\031)r Fn(\))1163 1387 y Fv(3)1194 1307 y Fe(Z)1219 1410 y Fh(Q)1256 1368 y Fg(jr)p Fi(u)1333 1375 y Fh(n)1356 1368 y Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(\))p Fg(j)1474 1350 y Fv(2)1500 1368 y Fi(dx)k(<)h Fn(+)p Fg(1)p Fi(:)56 b Fn(\(3.41\))37 1567 y(Let)16 b(us)f(no)o(w)g(turn)g(to)g(the)g(thermo)q(dynamic)h(limit)g(problem)g (for)f(the)g(RHF)h(mo)q(del.)37 1709 y Fk(4)67 b(The)23 b(Reduced)g(Hartree-F)-6 b(o)r(c)n(k)23 b(mo)r(del)37 1810 y Fn(This)18 b(section)g(is)g(dev)o(oted)f(to)g(the)h(pro)q(of)f (of)f(Theorem)i(2.2)e(whic)o(h)i(has)f(b)q(een)i(stated)e(in)h(Section) g(2.)26 b(It)37 1867 y(is)16 b(organized)g(as)e(follo)o(ws.)20 b(W)l(e)15 b(b)q(egin)i(with)e(the)h(hardest)e(part)h(of)g(the)g(w)o (ork)f(in)i(Subsection)g(4.1,)e(whic)o(h)37 1923 y(consists)h(in)f(v)o (erifying)h(that)e(the)h(lo)o(w)o(er)g(limit)h(of)f(the)g(energy)g(p)q (er)g(unit)h(v)o(olume)f(ma)o(y)g(b)q(e)g(b)q(ounded)i(from)37 1980 y(b)q(elo)o(w)c(b)o(y)g(the)f(p)q(erio)q(dic)j(RHF)d(mo)q(del.)20 b(Subsection)13 b(4.2)d(is)i(then)g(dev)o(oted)f(to)g(the)g(pro)q(of)g (of)g(Theorem)g(2.1;)37 2036 y(that)18 b(is)g(of)g(the)g(w)o(ell-p)q (osedness)h(of)f(the)g(p)q(erio)q(dic)i(RHF)e(mo)q(del.)29 b(A)o(t)17 b(last,)h(with)h(a)e(minimizer)j(of)e(this)37 2092 y(p)q(erio)q(dic)i(mo)q(del)f(at)e(hand,)i(w)o(e)e(are)h(able)h (to)e(c)o(hec)o(k)h(in)g(Subsection)i(4.3)c(that)i(the)f(upp)q(er)i (limit)g(of)f(the)37 2149 y(energy)e(p)q(er)g(unit)f(v)o(olume)h(ma)o (y)f(b)q(e)g(compared)h(from)e(ab)q(o)o(v)o(e)h(b)o(y)g(the)g(p)q(erio) q(dic)j(RHF)d(mo)q(del.)37 2269 y Fb(4.1)57 b(Lo)n(w)n(er)20 b(limit)e(of)h(the)g(energy)i(p)r(er)d(unit)h(v)n(olume)37 2367 y Fn(In)f(this)f(section,)g(w)o(e)g(b)q(ound)h(from)e(b)q(elo)o(w) h(the)g(lo)o(w)o(er)f(limit)i(of)1170 2341 y Fh(I)1188 2330 y Fc(RH)r(F)1185 2353 y Ft(\003)p 1170 2356 94 2 v 1195 2383 a Fu(j)p Fv(\003)p Fu(j)1285 2367 y Fn(as)f(de\014ned)h(in) f(\(2.10\){\(2.11\))c(b)o(y)37 2430 y(the)i(energy)f(of)g(the)g(p)q (erio)q(dic)i(RHF)f(mo)q(del)g(\(2.17\){\(2.18)o(\).)h(F)l(or)e(the)g (sak)o(e)g(of)g(clarit)o(y)l(,)g(let)h(us)f(recall)i(here)37 2486 y(the)g(de\014nitions)h(of)e(these)g(problems)h(:)527 2603 y Fi(I)551 2585 y Fh(RH)s(F)547 2615 y Fv(\003)651 2603 y Fn(=)d(inf)754 2539 y Fe(\032)788 2603 y Fi(E)825 2585 y Fh(RH)s(F)822 2615 y Fv(\003)912 2603 y Fn(\()p Fi(K)s Fn(\))c(+)1050 2573 y(1)p 1050 2593 23 2 v 1050 2635 a(2)1077 2603 y Fi(U)1108 2610 y Fv(\003)1148 2603 y Fn(;)f Fi(K)27 b Fg(2)13 b(K)1313 2610 y Fv(\003)1339 2539 y Fe(\033)1373 2603 y Fi(;)532 2788 y(E)569 2769 y Fh(RH)s(F)566 2799 y Fv(\003)656 2788 y Fn(\()p Fi(K)s Fn(\))f(=)h(T)l(r)841 2751 y Fe(\002)860 2788 y Fn(\()p Fg(\000)p Fn(\001)d Fg(\000)g Fi(V)1032 2795 y Fv(\003)1059 2788 y Fn(\))p Fi(K)1119 2751 y Fe(\003)1148 2788 y Fn(+)1198 2757 y(1)p 1198 2777 V 1198 2819 a(2)1226 2788 y Fi(D)q Fn(\()p Fi(\032;)e(\032)p Fn(\);)p eop %%Page: 26 26 26 25 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(26)613 304 y Fi(I)637 286 y Fh(RH)s(F)633 316 y(per)737 304 y Fn(=)13 b(inf)840 240 y Fe(\032)874 304 y Fi(E)911 286 y Fh(RH)s(F)908 316 y(per)998 304 y Fn(\()p Fi(K)s Fn(\))f(;)19 b Fi(K)d Fg(2)d(K)1254 240 y Fe(\033)1287 304 y Fi(;)343 484 y(E)380 465 y Fh(RH)s(F)377 495 y(per)466 484 y Fn(\()p Fi(K)s Fn(\))f(=)604 422 y Fe(Z)630 525 y Fh(Q)658 516 y Fc(?)685 484 y Fn(T)l(r)732 496 y Fh(L)756 484 y Ft(2)756 508 y Fc(\030)773 496 y Fv(\()p Fh(Q)p Fv(\))830 447 y Fe(\002)849 484 y Fg(\000)p Fn(\001)p Fi(K)961 491 y Fh(\030)980 447 y Fe(\003)1041 453 y Fi(d\030)p 1011 474 106 2 v 1011 515 a Fn(\(2)p Fi(\031)r Fn(\))1098 502 y Fv(3)1132 484 y Fg(\000)1177 422 y Fe(Z)1203 525 y Fh(Q)1240 484 y Fi(G\032)e Fn(+)1360 453 y(1)p 1360 474 23 2 v 1360 515 a(2)1387 484 y Fi(D)1425 491 y Fh(G)1455 484 y Fn(\()p Fi(\032;)e(\032)p Fn(\))p Fi(;)37 605 y Fn(where)16 b(the)f(meaning)h(of)f Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))14 b(is)i(made)f(precise)h(in)g (Prop)q(osition)g(3.2.)108 661 y(Of)j(course,)h(our)f(argumen)o(t)f (will)j(still)g(apply)e Ff(mutatis)i(mutandis)e Fn(\(this)g(is)h(ev)o (en)f(simpler\))h(to)f(the)37 718 y(smeared)14 b(n)o(uclei)i(case)d (\(see)h(\(2.12\),)e(\(2.19\),)g(\(2.20\)\).)17 b(W)l(e)d(concen)o (trate)f(ourselv)o(es)h(in)h(the)e(sequel)i(on)f(the)37 774 y(p)q(oin)o(t)i(n)o(uclei)h(case.)108 831 y(W)l(e)f(shall)g(use)g (in)g(a)f(crucial)i(w)o(a)o(y)e(the)g(fact)g(that)g(the)h(RHF)f (functional)i Fi(E)1412 814 y Fh(RH)s(F)1409 844 y Fv(\003)1499 831 y Fn(\()p Fi(K)s Fn(\))d(is)i(con)o(v)o(ex)g(with)37 887 y(resp)q(ect)k(to)f(the)g(densit)o(y)h(matrix)f Fi(K)s Fn(,)h(when)f Fi(K)j Fn(b)q(elongs)f(to)d(the)h(con)o(v)o(ex)h(set)f Fg(K)1483 894 y Fv(\003)1509 887 y Fn(,)h(for)e(an)o(y)h(\003)g Fg(\032)h Fj(Z)1843 871 y Fv(3)1862 887 y Fn(.)37 944 y(Indeed,)26 b(this)d(con)o(v)o(exit)o(y)g(prop)q(ert)o(y)f(will)i (allo)o(w)f(us)g(to)f(use)h(the)g Fg(\030)p Fn(-transform)e(tric)o(k)i (that)f(w)o(e)g(ha)o(v)o(e)37 1000 y(previously)d(applied)g(to)d(the)i (TFW)e(mo)q(del)i(in)g([11)o(].)25 b(Let)17 b(us)h(emphasize)g(the)f (fact)g(that,)f(since)i(w)o(e)f(fail)37 1057 y(in)g(obtaining)f(lo)q (cal)g(b)q(ounds)g(on)g(the)f(electronic)i(densit)o(y)f(in)g(the)f(RHF) h(mo)q(del)g(\(sa)o(y)l(,)e(for)h(example,)h Fi(L)1838 1040 y Fu(1)37 1113 y Fn(b)q(ounds)i(on)f Fi(\032)285 1120 y Fv(\003)329 1113 y Fn(indep)q(enden)o(t)j(of)c(\003\),)g(this)i (is)f(the)g(only)h(metho)q(d)f(among)f(all)i(the)f(metho)q(ds)g(presen) o(ted)37 1170 y(in)g([11)o(])d(whic)o(h)i(seems)g(to)e(go)h(through)g (to)f(the)i(RHF)f(mo)q(del.)108 1251 y(The)h(sequel)g(of)f(this)g (subsection)h(is)g(dev)o(oted)f(to)g(the)g(pro)q(of)g(of)g(the)g(follo) o(wing)37 1336 y Fj(Prop)q(osition)20 b(4.1)k Ff(L)n(et)16 b Fn(\003)g Ff(b)n(e)g(a)g(V)m(an)g(Hove)g(se)n(quenc)n(e.)j(Then,)679 1454 y Fn(lim)8 b(inf)695 1485 y Fv(\003)p Fu(!1)818 1423 y Fi(I)842 1407 y Fh(RH)s(F)838 1437 y Fv(\003)p 818 1444 112 2 v 845 1485 a Fg(j)p Fn(\003)p Fg(j)947 1454 y(\025)13 b Fi(I)1019 1436 y Fh(RH)s(F)1015 1466 y(per)1116 1454 y Fn(+)1166 1424 y Fi(M)p 1166 1444 50 2 v 1180 1485 a Fn(2)1220 1454 y Fi(;)37 1575 y Ff(wher)n(e)k Fi(I)190 1559 y Fh(RH)s(F)186 1586 y(per)293 1575 y Ff(is)f(de\014ne)n (d)f(by)i(\(2.17\){\(2.18\).)37 1690 y Fj(Pro)q(of)h(of)f(Prop)q (osition)i(4.1)f(:)i Fn(F)l(rom)14 b(no)o(w)g(on,)g(w)o(e)h(shall)g (denote)g(b)o(y)g Fi(K)1348 1697 y Fv(\003)1388 1690 y Fn(a)g(minimizer)h(of)f Fi(I)1713 1673 y Fh(RH)s(F)1709 1703 y Fv(\003)1800 1690 y Fn(,)f(b)o(y)37 1746 y Fi(\032)60 1753 y Fv(\003)88 1746 y Fn(\()p Fg(\001)p Fn(;)8 b Fg(\001)p Fn(\))j(its)i(Hilb)q(ert-Sc)o(hmidt)j(k)o(ernel)e(and)g(b)o(y)f Fi(\032)879 1753 y Fv(\003)919 1746 y Fn(=)g Fi(\032)990 1753 y Fv(\003)1017 1746 y Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))j(the)j(\(unique\))g(corresp)q(onding)g(electronic)37 1803 y(densit)o(y)l(.)29 b(In)19 b(particular,)g(using)f(the)g(fact)f (that)h Fi(K)937 1810 y Fv(\003)981 1803 y Fn(admits)g(a)f(complete)i (set)e(of)h(orthonormal)f(eigen-)37 1859 y(functions)h(\()p Fi( )283 1866 y Fh(n)306 1859 y Fn(\))324 1866 y Fh(n)p Fu(\025)p Fv(1)410 1859 y Fn(b)q(elonging)g(to)f Fi(H)717 1843 y Fv(1)736 1859 y Fn(\()p Fj(R)793 1843 y Fv(3)813 1859 y Fn(\),)g(asso)q(ciated)g(to)g(the)g(eigen)o(v)m(alues)i(0)d Fg(\024)g Fi(\026)1571 1866 y Fh(n)1611 1859 y Fg(\024)g Fn(1)h(\(coun)o(ted)37 1915 y(with)f(their)g(m)o(ultiplicit)o(y\),)h(w) o(e)e(ma)o(y)f(write)669 2014 y Fi(\032)692 2021 y Fv(\003)719 2014 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))j(=)885 1971 y Fe(X)885 2069 y Fh(n)p Fu(\025)p Fv(1)959 2014 y Fi(\026)986 2021 y Fh(n)1010 2014 y Fi( )1040 2021 y Fh(n)1063 2014 y Fn(\()p Fi(x)p Fn(\))d Fi( )1165 1995 y Fu(\003)1163 2025 y Fh(n)1185 2014 y Fn(\()p Fi(y)r Fn(\))37 2149 y(and)699 2206 y Fi(\032)722 2213 y Fv(\003)749 2206 y Fn(\()p Fi(x)p Fn(;)g Fi(x)p Fn(\))j(=)917 2163 y Fe(X)917 2261 y Fh(n)p Fu(\025)p Fv(1)991 2206 y Fi(\026)1018 2213 y Fh(n)1042 2206 y Fg(j)p Fi( )1085 2213 y Fh(n)1108 2206 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1183 2187 y Fv(2)1201 2206 y Fi(;)37 2328 y Fn(where,)17 b(here)g(and)f(in)h(all)h(that)d (follo)o(ws,)h(w)o(e)g(ha)o(v)o(e)g(on)h(purp)q(ose)f(omitted)h(to)e (men)o(tion)i(the)f(dep)q(endence)37 2385 y(of)f(the)h Fi(\026)195 2392 y Fh(n)218 2385 y Fn('s)f(and)h(of)e(the)i Fi( )513 2392 y Fh(n)536 2385 y Fn('s)e(on)h(\003,)g(in)h(order)f(to)f (simplify)j(the)f(notation.)j(Let)d(us)f(recall)h(that)526 2495 y(T)l(r)573 2504 y Fh(L)597 2495 y Ft(2)614 2504 y Fv(\()p Fd(R)658 2495 y Ft(3)676 2504 y Fv(\))691 2495 y Fi(K)730 2502 y Fv(\003)769 2495 y Fn(=)818 2451 y Fe(X)817 2549 y Fh(n)p Fu(\025)p Fv(1)891 2495 y Fi(\026)918 2502 y Fh(n)950 2433 y Fe(Z)975 2536 y Fd(R)1005 2526 y Ft(3)1032 2495 y Fg(j)p Fi( )1075 2502 y Fh(n)1098 2495 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1173 2476 y Fv(2)1199 2495 y Fi(dx)c Fn(=)h Fg(j)p Fn(\003)p Fg(j)p Fi(;)402 b Fn(\(4.1\))37 2630 y(and)16 b(that)330 2731 y(0)58 b Fg(\024)h Fn(T)l(r)552 2741 y Fh(L)576 2732 y Ft(2)593 2741 y Fv(\()p Fd(R)637 2732 y Ft(3)654 2741 y Fv(\))670 2731 y Fn([)p Fg(\000)p Fn(\001)8 b Fi(K)803 2738 y Fv(\003)829 2731 y Fn(])k(=)902 2688 y Fe(X)902 2786 y Fh(n)p Fu(\025)p Fv(1)976 2731 y Fi(\026)1003 2738 y Fh(n)1034 2669 y Fe(Z)1060 2772 y Fd(R)1090 2763 y Ft(3)1117 2731 y Fg(jr)p Fi( )1198 2738 y Fh(n)1221 2731 y Fg(j)1234 2712 y Fv(2)1261 2731 y Fi(dx)57 b Fg(\024)i Fi(C)11 b Fg(j)p Fn(\003)p Fg(j)p Fi(;)206 b Fn(\(4.2\))p eop %%Page: 27 27 27 26 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(27)37 199 y(thanks)19 b(to)g(\(3.10a\).)29 b(Generally)20 b(sp)q(eaking,)h(the)e(idea)h(of)e(the)h(strategy)f (detailed)j(b)q(elo)o(w,)f(and)f(whic)o(h)37 256 y(dra)o(ws)10 b(its)g(inspiration)i(from)e([11)o(],)g(is)h(the)f(follo)o(wing.)19 b(W)l(e)11 b(shall)g(build)h(a)e(particular)h(con)o(v)o(ex)f(com)o (bination)37 312 y(from)20 b(the)g(op)q(erators)g Fi(K)479 319 y Fv(\003)505 312 y Fn(,)h(whic)o(h)g(is)g(more)f(or)f(less)i(a)f (minimizing)j(sequence)f(of)d Fi(I)1550 296 y Fh(RH)s(F)1546 326 y Fv(\003)1638 312 y Fn(,)i(but)f(whic)o(h)37 369 y(con)o(v)o(erges)15 b(to)g(a)g(p)q(erio)q(dic)j(densit)o(y)e(matrix,)e (as)h(\003)g(go)q(es)h(to)e(in\014nit)o(y)l(.)22 b(Moreo)o(v)o(er,)14 b(this)i(p)q(erio)q(dic)h(densit)o(y)37 425 y(matrix)12 b(will)h(turn)e(to)g(b)q(e)h(a)f(minimizer)i(of)e Fi(I)785 409 y Fh(RH)s(F)781 436 y(per)873 425 y Fn(.)18 b(By)12 b(analogy)f(with)h(the)f(de\014nition)i(of)e(the)h Fg(\030)p Fn(-transform)37 482 y(for)j(functions)h(\(see)f(De\014nition)i(3)e(in) h(Section)g(2\),)e(w)o(e)h(set)703 591 y Fe(e)692 603 y Fi(K)731 610 y Fv(\003)770 603 y Fn(=)840 572 y(1)p 823 592 57 2 v 823 634 a Fg(j)p Fn(\003)p Fg(j)893 560 y Fe(X)892 658 y Fh(k)q Fu(2)p Fv(\003)967 603 y Fi(\034)987 610 y Fh(k)1018 603 y Fg(\001)10 b Fi(K)1080 610 y Fv(\003)1116 603 y Fg(\001)g Fi(\034)1159 610 y Fu(\000)p Fh(k)1208 603 y Fi(:)561 b Fn(\(4.3\))37 759 y(Then,)16 b(it)f(is)h(easy)f(to)g (c)o(hec)o(k)g(that)647 748 y Fe(e)636 759 y Fi(K)675 766 y Fv(\003)717 759 y Fn(b)q(elongs)h(to)e Fg(K)970 766 y Fv(\003)997 759 y Fn(,)g(that)h(the)g(Hilb)q(ert-Sc)o(hmidt)j(k)o (ernel)e(of)1732 748 y Fe(e)1722 759 y Fi(K)1761 766 y Fv(\003)1802 759 y Fn(is)612 878 y Fe(e)-28 b Fi(\032)632 885 y Fv(\003)659 878 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))j(=)847 847 y(1)p 830 867 V 830 909 a Fg(j)p Fn(\003)p Fg(j)900 835 y Fe(X)899 933 y Fh(k)q Fu(2)p Fv(\003)974 878 y Fi(\032)997 885 y Fv(\003)1024 878 y Fn(\()p Fi(x)f Fn(+)g Fi(k)q Fn(;)e Fi(y)k Fn(+)e Fi(k)q Fn(\))p Fi(;)37 1024 y Fn(while)18 b Fe(e)-28 b Fi(\032)177 1031 y Fv(\003)204 1024 y Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))k(coincides)j(with)e(the)h (usual)g Fg(\030)p Fn(-transform)e(of)h Fi(\032)1140 1031 y Fv(\003)1180 1024 y Fn(as)g(in)o(tro)q(duced)h(in)g([11)o(].)19 b(In)14 b(particular,)48 1069 y Fe(e)37 1080 y Fi(K)76 1087 y Fv(\003)118 1080 y Fn(also)i(admits)g(a)f(complete)h(set)g(of)f (orthonormal)g(eigenfunctions)i(\()p Fi(')1320 1087 y Fh(n)1343 1080 y Fn(\))1361 1087 y Fh(n)p Fu(\025)p Fv(1)1445 1080 y Fn(b)q(elonging)g(to)e Fi(H)1749 1064 y Fv(1)1768 1080 y Fn(\()p Fj(R)1825 1064 y Fv(3)1845 1080 y Fn(\),)37 1137 y(asso)q(ciated)g(to)e(the)i(eigen)o(v)m(alues)h(0)c Fg(\024)h Fi(\025)728 1144 y Fh(n)765 1137 y Fg(\024)g Fn(1)h(\(coun)o(ted)g(with)g(their)h(m)o(ultiplicit)o(y\),)h(and,)e (therefore,)g(w)o(e)37 1193 y(ma)o(y)h(write)g(as)g(for)g Fi(K)417 1200 y Fv(\003)672 1254 y Fe(e)-28 b Fi(\032)692 1261 y Fv(\003)719 1254 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))j(=)885 1211 y Fe(X)885 1309 y Fh(n)p Fu(\025)p Fv(1)959 1254 y Fi(\025)985 1261 y Fh(n)1009 1254 y Fi(')1039 1261 y Fh(n)1062 1254 y Fn(\()p Fi(x)p Fn(\))d Fi(')1162 1235 y Fu(\003)1162 1265 y Fh(n)1184 1254 y Fn(\()p Fi(y)r Fn(\))37 1382 y(and)702 1438 y Fe(e)-28 b Fi(\032)722 1445 y Fv(\003)749 1438 y Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))j(=)918 1395 y Fe(X)917 1493 y Fh(n)p Fu(\025)p Fv(1)991 1438 y Fi(\025)1017 1445 y Fh(n)1041 1438 y Fg(j)p Fi(')1084 1445 y Fh(n)1107 1438 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1182 1419 y Fv(2)1201 1438 y Fi(:)37 1579 y Fn(Of)16 b(course,)f(the)g(analogues)g(of)g(\(4.1\))f(and)h(\(4.2\))f (remain)h(true)g(for)1225 1567 y Fe(e)1214 1579 y Fi(K)1253 1586 y Fv(\003)1279 1579 y Fn(.)108 1660 y(The)i(pro)q(of)g(is)g (organized)g(as)g(follo)o(ws.)25 b(W)l(e)17 b(\014rst)f(c)o(hec)o(k)h (that)g(the)f(sequence)1487 1648 y Fe(e)1477 1660 y Fi(K)1516 1667 y Fv(\003)1559 1660 y Fn(de\014ned)i(b)o(y)f(\(4.3\))37 1716 y(con)o(v)o(erges)e(in)h(a)f(sense)h(to)e(b)q(e)i(made)g(precise)g (later)f(to)g(some)g(op)q(erator)1307 1705 y Fe(e)1296 1716 y Fi(K)j Fn(b)q(elonging)f(to)e Fg(K)q Fn(;)f(that)g(is,)i(to)37 1773 y(a)i(p)q(erio)q(dic)h(densit)o(y)g(matrix)e(\()p Ff(Step)h(1)6 b Fn(\).)27 b(Moreo)o(v)o(er,)16 b(using)i(the)g(t)o(w)o (o)e(facts)h(that)1488 1733 y Fe(p)p 1534 1733 51 2 v 1537 1773 a(e)-28 b Fi(\032)1557 1780 y Fv(\003)1601 1773 y Fn(is)18 b(b)q(ounded)h(in)37 1829 y Fi(H)79 1813 y Fv(1)75 1843 y(unif)140 1829 y Fn(\()p Fj(R)197 1813 y Fv(3)216 1829 y Fn(\))d(\(Corollary)g(3.2\))f(and)i(that)e(its)i (limit)h(is)e(necessarily)i(p)q(erio)q(dic)h(\(Lemma)d(3.2\),)f(w)o(e)h (already)37 1890 y(infer)21 b(that)250 1851 y Fe(p)p 296 1851 V 299 1890 a(e)-28 b Fi(\032)319 1897 y Fv(\003)366 1890 y Fn(con)o(v)o(erges)19 b(w)o(eakly)h(in)h Fi(H)829 1874 y Fv(1)825 1904 y(lo)q(c)871 1890 y Fn(\()p Fj(R)928 1874 y Fv(3)947 1890 y Fn(\),)g(strongly)e(in)i Fi(L)1268 1868 y Fh(p)1268 1905 y Fv(lo)q(c)1314 1890 y Fn(\()p Fj(R)1371 1874 y Fv(3)1391 1890 y Fn(\),)f(for)g(ev)o(ery)f(1)i Fg(\024)f Fi(p)h(<)g Fn(6,)37 1952 y(and)c(almost)f(ev)o(erywhere)h(on) f Fj(R)616 1936 y Fv(3)653 1952 y Fn(to)710 1913 y Fe(p)p 755 1913 24 2 v 758 1952 a(e)-28 b Fi(\032)q Fn(,)16 b(with)k Fe(e)-28 b Fi(\032)17 b Fn(b)q(eing)g Fi(Q)p Fn(-p)q(erio)q(dic,)i(non-negativ)o(e,)e(and)g(suc)o(h)g(that)37 1969 y Fe(p)p 83 1969 V 86 2009 a(e)-28 b Fi(\032)13 b Fg(2)g Fi(H)204 1992 y Fv(1)200 2020 y(p)q(er)252 2009 y Fn(\()p Fi(Q)p Fn(\).)19 b(The)c(second)h(step,)f(whic)o(h)h(is)g(m)o (uc)o(h)f(more)f(in)o(v)o(olv)o(ed,)i(consists)f(in)h(v)o(erifying)g (that)f(the)37 2075 y(limit)21 b Fe(e)-28 b Fi(\032)16 b Fn(of)g(the)g(densit)o(y)j Fe(e)-28 b Fi(\032)498 2082 y Fv(\003)541 2075 y Fn(asso)q(ciated)16 b(to)825 2064 y Fe(e)815 2075 y Fi(K)854 2082 y Fv(\003)896 2075 y Fn(is)g(also)g(the)g Fi(Q)p Fn(-p)q(erio)q(dic)i(densit)o(y)f(whic)o(h) f(is)h(asso)q(ciated)37 2131 y(to)d(the)g(p)q(erio)q(dic)j(densit)o(y)e (matrix)657 2120 y Fe(e)646 2131 y Fi(K)i Fn(according)e(to)e(\(3.39\)) g(in)i(Subsection)g(3.2,)e(Prop)q(osition)i(3.2|\(iii\))37 2188 y(\()p Ff(Step)20 b(2)6 b Fn(\).)31 b(Finally)l(,)22 b(w)o(e)d(b)q(ound)h(from)e(b)q(elo)o(w)i(the)f(lo)o(w)o(er)g(limit)h (of)f(the)g(energy)g(p)q(er)h(unit)g(v)o(olume)f(b)o(y)37 2244 y Fi(E)74 2228 y Fh(RH)s(F)71 2256 y(per)161 2244 y Fn(\()190 2233 y Fe(e)179 2244 y Fi(K)s Fn(\))d(\()p Ff(Step)h(3)6 b Fn(\),)17 b(thereb)o(y)g(concluding)h(the)f(pro)q(of)f (of)g(Prop)q(osition)i(4.1.)23 b(Let)17 b(us)f(already)h(sa)o(y)l(,)g (at)37 2301 y(this)g(stage,)e(that)h(thanks)f(to)h(the)g(pro)q(of)g(of) f(the)i(upp)q(er)g(limit)g(\(and)f(th)o(us)g(of)g(the)g(limit\),)h(of)f (the)g(energy)37 2357 y(p)q(er)j(unit)g(v)o(olume)g(w)o(e)f(shall)h (\014nally)h(deduce)f(that)966 2346 y Fe(e)955 2357 y Fi(K)i Fn(is)e(a)f(minimizer)i(of)e Fi(I)1399 2341 y Fh(RH)s(F)1395 2369 y(per)1486 2357 y Fn(,)g(and)h(therefore)i Fe(e)-28 b Fi(\032)18 b Fn(is)37 2414 y(simply)28 b Fi(\032)217 2421 y Fh(per)270 2414 y Fn(,)19 b(with)27 b Fi(\032)440 2421 y Fh(per)512 2414 y Fn(denoting)19 b(the)f(unique)i(p)q(erio)q (dic)h(electronic)f(densit)o(y)f(that)f(corresp)q(onds)h(to)37 2470 y(an)o(y)f(minimizer)i(of)e Fi(I)420 2454 y Fh(RH)s(F)416 2481 y(per)507 2470 y Fn(.)28 b(In)19 b(particular,)g(the)f(whole)g (sequence)1237 2431 y Fe(p)p 1283 2431 51 2 v 1286 2470 a(e)-28 b Fi(\032)1306 2477 y Fv(\003)1351 2470 y Fn(con)o(v)o(erges)17 b(\(and)h(not)g(only)g(a)37 2527 y(subsequence\))f(and)e(its)h(limit)g (is)g(indep)q(enden)o(t)i(of)c(the)i(c)o(hoice)g(of)f(the)g(V)l(an)g (Ho)o(v)o(e)g(sequence)h(\003.)108 2608 y Fj(Step)i(1)p 108 2624 162 2 v 15 w Fn(:)27 b Ff(We)20 b(\014rst)g(che)n(ck)f(that)h (the)g(se)n(quenc)n(e)992 2597 y Fe(e)981 2608 y Fi(K)1020 2615 y Fv(\003)1066 2608 y Ff(c)n(onver)n(ges)e(to)i(some)g(op)n(er)n (ator)1643 2597 y Fe(e)1632 2608 y Fi(K)j Ff(b)n(elonging)37 2664 y(to)d(the)f(set)g(of)g(p)n(erio)n(dic)g(density)f(matric)n(es)h Fg(K)q Ff(,)g(which)g(is)g(de\014ne)n(d)f(thr)n(ough)i(pr)n(op)n (erties)f(\(H1\){\(H4\))f(in)37 2721 y(Subse)n(ction)e(3.2)g(\(e)n (quivalent)f(to)i(De\014nition)e(2)i(in)e(Se)n(ction)h(2\).)p eop %%Page: 28 28 28 27 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(28)37 199 y(Since)19 b(the)d(sequence)i(of)e(op)q (erators)694 188 y Fe(e)683 199 y Fi(K)722 206 y Fv(\003)765 199 y Fn(is)h(b)q(ounded)h(in)f(op)q(erators)f(norm,)g(w)o(e)g(ma)o(y)g (extract)g(a)g(subse-)37 256 y(quence)i(if)f(necessary)g(in)g(suc)o(h)g (w)o(a)o(y)f(that)797 244 y Fe(e)787 256 y Fi(K)826 263 y Fv(\003)868 256 y Fn(con)o(v)o(erges)g(to)g(some)h(b)q(ounded)h(op)q (erator)1628 244 y Fe(e)1617 256 y Fi(K)i Fn(in)d Fi(L)1761 239 y Fv(2)1781 256 y Fn(\()p Fj(R)1838 239 y Fv(3)1857 256 y Fn(\))37 312 y(for)e(the)g(w)o(eak)g(con)o(v)o(ergence)g(of)g(op) q(erators;)f(that)h(is)576 414 y(\()604 403 y Fe(e)594 414 y Fi(K)633 421 y Fv(\003)659 414 y Fi(')p Fn(;)8 b Fi( )r Fn(\))i Fg(\000)-7 b(!)13 b Fn(\()885 403 y Fe(e)874 414 y Fi(K)s(')p Fn(;)8 b Fi( )r Fn(\))103 b(as)15 b(\003)d Fg(!)i(1)p Fi(;)37 516 y Fn(for)d(all)h Fi(')e Fn(and)h Fi( )i Fn(in)e Fi(L)408 500 y Fv(2)428 516 y Fn(\()p Fj(R)485 500 y Fv(3)505 516 y Fn(\).)18 b(In)11 b(particular,)836 505 y Fe(e)825 516 y Fi(K)j Fn(is)d(a)g(self-adjoin)o (t)h(op)q(erator)e(in)i Fi(L)1442 500 y Fv(2)1461 516 y Fn(\()p Fj(R)1518 500 y Fv(3)1538 516 y Fn(\))e(and)h(0)i Fg(\024)1745 505 y Fe(e)1734 516 y Fi(K)i Fg(\024)e Fj(1)p Fn(.)37 573 y(Therefore,)267 561 y Fe(e)256 573 y Fi(K)18 b Fn(enjo)o(ys)d(\(H2\).)108 654 y(Let)h(us)f(c)o(hec)o(k)g(no)o(w)g (that)f(it)i(also)f(satis\014es)h(\(H1\);)d(that)i(is,)g(let)h(us)f (pro)o(v)o(e)g(that)108 736 y(a\))174 724 y Fe(e)164 736 y Fi(K)j Fn(comm)o(utes)c(with)i(the)f(translations)g(whic)o(h)h (lea)o(v)o(e)g Fj(Z)1139 719 y Fv(3)1173 736 y Fn(in)o(v)m(arian)o(t.)p 164 743 1199 2 v 37 817 a(Let)g Fi(\034)k Fn(b)q(e)c(suc)o(h)g(a)e (translation.)20 b(W)l(e)c(\014x)f Fi(')g Fn(and)g Fi( )i Fn(in)f Fi(L)1016 800 y Fv(2)1035 817 y Fn(\()p Fj(R)1092 800 y Fv(3)1112 817 y Fn(\),)e(and)i(w)o(e)f(in)o(tend)h(to)e(pro)o(v)o (e)h(that)735 919 y(\([)776 908 y Fe(e)766 919 y Fi(K)r(;)8 b(\034)d Fn(])p Fi(';)j( )r Fn(\))966 929 y Fh(L)990 919 y Ft(2)1005 929 y Fv(\()p Fd(R)1049 919 y Ft(3)1066 929 y Fv(\))1095 919 y Fn(=)13 b(0)p Fi(:)37 1021 y Fn(F)l(or)k(this)g (purp)q(ose,)h(w)o(e)e(mak)o(e)h(use)g(of)f(a)h(standard)f(argumen)o(t) h(of)f([11)o(])h(whic)o(h)g(is)h(based)f(up)q(on)h(the)f(fact)37 1078 y(that)e(the)g(sequence)i(\003)d(is)i(a)f(V)l(an)g(Ho)o(v)o(e)g (sequence.)21 b(W)l(e)15 b(just)g(outline)i(this)e(argumen)o(t)g(here.) 20 b(W)l(e)15 b(ha)o(v)o(e)392 1202 y([)415 1191 y Fe(e)405 1202 y Fi(K)444 1209 y Fv(\003)470 1202 y Fi(;)8 b(\034)d Fn(])40 b(=)668 1171 y(1)p 651 1192 57 2 v 651 1233 a Fg(j)p Fn(\003)p Fg(j)713 1138 y Fe(\022)747 1159 y(X)746 1258 y Fh(k)q Fu(2)p Fv(\003)821 1202 y Fi(\034)841 1209 y Fu(\000)p Fh(k)890 1202 y Fi(K)929 1209 y Fv(\003)955 1202 y Fi(\034)975 1209 y Fh(k)996 1202 y Fi(\034)15 b Fg(\000)1077 1159 y Fe(X)1076 1258 y Fh(k)q Fu(2)p Fv(\003)1151 1202 y Fi(\034)5 b(\034)1196 1209 y Fu(\000)p Fh(k)1245 1202 y Fi(K)1284 1209 y Fv(\003)1310 1202 y Fi(\034)1330 1209 y Fh(k)1351 1138 y Fe(\023)569 1347 y Fn(=)668 1316 y(1)p 651 1337 V 651 1378 a Fg(j)p Fn(\003)p Fg(j)713 1283 y Fe(\022)778 1304 y(X)746 1404 y Fh(k)q Fu(2)p Fh(\034)t Fv(\003)p Fu(n)p Fv(\003)883 1347 y Fi(\034)g(\034)928 1354 y Fu(\000)p Fh(k)976 1347 y Fi(K)1015 1354 y Fv(\003)1041 1347 y Fi(\034)1061 1354 y Fh(k)1093 1347 y Fg(\000)1170 1304 y Fe(X)1138 1404 y Fh(k)q Fu(2)p Fv(\003)p Fu(n)p Fh(\034)t Fv(\003)1275 1347 y Fi(\034)g(\034)1320 1354 y Fu(\000)p Fh(k)1368 1347 y Fi(K)1407 1354 y Fv(\003)1433 1347 y Fi(\034)1453 1354 y Fh(k)1475 1283 y Fe(\023)1508 1347 y Fi(:)37 1498 y Fn(Hence)338 1548 y Fe(\014)338 1575 y(\014)338 1603 y(\014)338 1630 y(\014)354 1614 y Fn(\([)395 1602 y Fe(e)385 1614 y Fi(K)424 1621 y Fv(\003)449 1614 y Fi(;)j(\034)d Fn(])p Fi(';)j( )r Fn(\))609 1624 y Fh(L)633 1614 y Ft(2)647 1624 y Fv(\()p Fd(R)691 1614 y Ft(3)708 1624 y Fv(\))724 1548 y Fe(\014)724 1575 y(\014)724 1603 y(\014)724 1630 y(\014)781 1614 y Fg(\024)863 1583 y(j)p Fn(\003)h Fg(n)h Fi(\034)5 b Fn(\003)p Fg(j)p 863 1603 157 2 v 913 1645 a(j)p Fn(\003)p Fg(j)1037 1614 y(k)p Fi(K)1099 1621 y Fv(\003)1125 1614 y Fg(k)25 b(k)p Fi(')p Fg(k)1249 1624 y Fh(L)1273 1614 y Ft(2)1289 1624 y Fv(\()p Fd(R)1333 1614 y Ft(3)1350 1624 y Fv(\))1379 1614 y Fg(k)p Fi( )r Fg(k)1457 1624 y Fh(L)1481 1614 y Ft(2)1497 1624 y Fv(\()p Fd(R)1541 1614 y Ft(3)1558 1624 y Fv(\))781 1708 y Fn(=)42 b Fi(o)p Fn(\(1\))37 1810 y(as)19 b(\003)g(go)q(es)g(to)f(in\014nit)o(y)l(.)33 b(As)19 b(the)g(left-hand)h(side)g(con)o(v)o(erges)f(to)f(\([)1246 1798 y Fe(e)1236 1810 y Fi(K)r(;)8 b(\034)d Fn(])p Fi(';)j( )r Fn(\))1437 1820 y Fh(L)1461 1810 y Ft(2)1475 1820 y Fv(\()p Fd(R)1519 1810 y Ft(3)1537 1820 y Fv(\))1552 1810 y Fn(,)20 b(this)f(sho)o(ws)g(the)37 1866 y(exp)q(ected)e(in)o(v)m(ariance.)108 1948 y(According)f(to)f(the)g(results)h(and)g(the)f(notation)g(of)g (Section)h(3.2,)e(w)o(e)h(ma)o(y)g(write)1542 1936 y Fe(e)1531 1948 y Fi(K)h Fn(=)1634 1911 y Fe(R)1655 1964 y Fh(Q)1683 1954 y Fc(?)1722 1936 y Fe(e)1711 1948 y Fi(K)1750 1955 y Fh(\030)1811 1927 y(d\030)p 1786 1937 84 2 v 1786 1965 a Fv(\(2)p Fh(\031)q Fv(\))1853 1955 y Ft(3)37 2021 y Fn(with)153 2010 y Fe(e)142 2021 y Fi(K)181 2028 y Fh(\030)216 2021 y Fn(b)q(eing)i(a)e(self-adjoin)o(t)h(op)q (erator)e(in)j Fi(L)884 2005 y Fv(2)884 2035 y Fh(\030)903 2021 y Fn(\()p Fi(Q)p Fn(\))e(suc)o(h)h(that)e(0)g Fg(\024)1292 2010 y Fe(e)1281 2021 y Fi(K)1320 2028 y Fh(\030)1354 2021 y Fg(\024)f Fj(1)q Fn(,)i(for)g(almost)g(ev)o(ery)g Fi(\030)i Fn(in)37 2078 y Fi(Q)73 2061 y Fh(?)93 2078 y Fn(.)108 2159 y(Ha)o(ving)h(c)o(hec)o(k)o(ed)g(that)f(\(H1\))g(and)h (\(H2\))f(are)g(satis\014ed)h(b)o(y)1188 2148 y Fe(e)1177 2159 y Fi(K)s Fn(,)g(w)o(e)g(next)f(w)o(an)o(t)g(to)g(v)o(erify)h(that) 1844 2148 y Fe(e)1833 2159 y Fi(K)37 2216 y Fn(satis\014es)d(\(H3\).)j (Namely)l(,)c(w)o(e)g(no)o(w)g(w)o(an)o(t)f(to)h(c)o(hec)o(k)g(that,) 108 2297 y(b\))g(for)g(almost)f(ev)o(ery)i Fi(\030)h Fn(in)f Fi(Q)629 2280 y Fh(?)648 2297 y Fn(,)f(the)g(op)q(erator)949 2286 y Fe(e)938 2297 y Fi(K)p 166 2313 813 2 v 978 2304 a Fh(\030)1012 2297 y Fn(has)g(a)g(\014nite)i(trace)d(on)h Fi(L)1455 2280 y Fv(2)p 997 2316 478 2 v 1475 2304 a Fh(\030)1494 2297 y Fn(\()p Fi(Q)p Fn(\),)f(and)p 1494 2316 188 2 v 703 2368 a Fe(Z)728 2471 y Fh(Q)756 2461 y Fc(?)783 2430 y Fn(T)l(r)830 2441 y Fh(L)854 2430 y Ft(2)854 2453 y Fc(\030)871 2441 y Fv(\()p Fh(Q)p Fv(\))939 2418 y Fe(e)929 2430 y Fi(K)968 2437 y Fh(\030)1034 2399 y Fi(d\030)p 1004 2419 106 2 v 1004 2461 a Fn(\(2)p Fi(\031)r Fn(\))1091 2448 y Fv(3)1127 2430 y Fn(=)f(1)p Fi(:)37 2586 y Fn(According)19 b(to)e(Lemma)g(3.5)g(in)h(Subsection)h(3.2,)e(w) o(e)g(denote)h(b)o(y)f(\()p Fi(u)1255 2593 y Fh(n)1278 2586 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\)\))1388 2593 y Fh(n)p Fu(\025)p Fv(1)1472 2586 y Fn(an)18 b(arbitrary)e(Hilb)q (ert)37 2643 y(basis)e(of)e Fi(L)227 2626 y Fv(2)227 2656 y Fh(\030)246 2643 y Fn(\()p Fi(Q)p Fn(\))g(for)g(almost)h(ev)o (ery)f Fi(\030)j Fn(in)e Fi(Q)780 2626 y Fh(?)800 2643 y Fn(,)g(and)f(b)o(y)h Fi(')1002 2650 y Fh(n;p)1065 2643 y Fn(=)g Fi(\034)1133 2650 y Fu(\000)p Fh(p)1193 2643 y Fi(')1223 2650 y Fh(n)1259 2643 y Fn(the)g(corresp)q(onding)g(Hilb)q (ert)i(basis)37 2711 y(of)h Fi(L)121 2695 y Fv(2)141 2711 y Fn(\()p Fj(R)198 2695 y Fv(3)217 2711 y Fn(\).)23 b(\(Note)15 b(that)g(in)i(fact,)f(since)h Fi(K)i Fn(and)d(the)1005 2700 y Fe(e)994 2711 y Fi(K)1033 2718 y Fh(\030)1052 2711 y Fn('s)f(are)h(non-negativ)o(e,)h(it)f(w)o(ould)g(b)q(e)h (su\016cien)o(t)p eop %%Page: 29 29 29 28 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(29)37 199 y(to)15 b(mak)o(e)f(the)h(follo)o(wing)g (argumen)o(t)f(for)g Ff(one)k Fn(giv)o(en)d(basis)g(|)h(see)f([44)o (].\))k(Then,)c(w)o(e)f(c)o(hec)o(k)h(successiv)o(ely)37 256 y(that)g(:)300 365 y(1)41 b(=)463 334 y(1)p 446 355 57 2 v 446 396 a Fg(j)p Fn(\003)p Fg(j)508 365 y Fn(T)l(r)555 375 y Fh(L)579 365 y Ft(2)596 375 y Fv(\()p Fd(R)640 365 y Ft(3)658 375 y Fv(\))673 365 y Fi(K)712 372 y Fv(\003)751 365 y Fn(=)821 334 y(1)p 804 355 V 804 396 a Fg(j)p Fn(\003)p Fg(j)874 322 y Fe(X)874 420 y Fh(n)p Fu(\025)p Fv(1)957 322 y Fe(X)948 423 y Fh(p)p Fu(2)p Fd(Z)1015 414 y Ft(3)1031 365 y Fn(\()p Fi(K)1088 372 y Fv(\003)1114 365 y Fi(')1144 372 y Fh(n;p)1195 365 y Fn(;)8 b Fi(')1246 372 y Fh(n;p)1296 365 y Fn(\))364 511 y Fg(\025)463 480 y Fn(1)p 446 500 V 446 542 a Fg(j)p Fn(\003)p Fg(j)516 468 y Fe(X)516 566 y Fh(n)p Fu(\025)p Fv(1)590 468 y Fe(X)590 566 y Fh(p)p Fu(2)p Fv(\003)656 511 y Fn(\()p Fi(K)713 518 y Fv(\003)738 511 y Fi(')768 518 y Fh(n;p)819 511 y Fn(;)g Fi(')870 518 y Fh(n;p)920 511 y Fn(\))k(=)999 468 y Fe(X)998 566 y Fh(n)p Fu(\025)p Fv(1)1095 480 y Fn(1)p 1077 500 V 1077 542 a Fg(j)p Fn(\003)p Fg(j)1147 468 y Fe(X)1147 566 y Fh(p)p Fu(2)p Fv(\003)1213 511 y Fn(\()p Fi(\034)1251 518 y Fh(p)1280 511 y Fg(\001)e Fi(K)1342 518 y Fv(\003)1368 511 y Fi(\034)1388 518 y Fu(\000)p Fh(p)1445 511 y Fg(\001)g Fi(')1498 518 y Fh(n)1521 511 y Fn(;)e Fi(')1572 518 y Fh(n)1595 511 y Fn(\))364 641 y(=)442 598 y Fe(X)441 696 y Fh(n)p Fu(\025)p Fv(1)508 641 y Fn(\()536 630 y Fe(e)526 641 y Fi(K)565 648 y Fv(\003)591 641 y Fi(')621 648 y Fh(n)644 641 y Fn(;)g Fi(')695 648 y Fh(n)717 641 y Fn(\))p Fi(;)1034 b Fn(\(4.4\))37 787 y(the)20 b(inequalit)o(y)i(in)e (the)g(ab)q(o)o(v)o(e)f(string)h(of)f(equalities)j(coming)e(from)f(the) g(p)q(ositiv)o(eness)i(of)f Fi(K)s Fn(.)33 b(Then,)37 844 y(b)q(ecause)17 b(of)f(the)g(w)o(eak)f(con)o(v)o(ergence)h(of)767 832 y Fe(e)756 844 y Fi(K)795 851 y Fv(\003)837 844 y Fn(to)904 832 y Fe(e)894 844 y Fi(K)r Fn(,)g(\()992 832 y Fe(e)982 844 y Fi(K)1021 851 y Fv(\003)1047 844 y Fi(')1077 851 y Fh(n)1100 844 y Fn(;)8 b Fi(')1151 851 y Fh(n)1173 844 y Fn(\))16 b(con)o(v)o(erges)f(to)g(\()1496 832 y Fe(e)1485 844 y Fi(K)s(')1557 851 y Fh(n)1580 844 y Fn(;)8 b Fi(')1631 851 y Fh(n)1654 844 y Fn(\),)15 b(for)g(ev)o(ery)37 900 y Fi(n)23 b Fg(\025)f Fn(1,)f(as)g(\003)f(go)q(es)g(to)h(in\014nit) o(y)l(.)38 b(No)o(w,)21 b(since)h(0)f Fg(\024)1021 889 y Fe(e)1011 900 y Fi(K)1050 907 y Fv(\003)1076 900 y Fn(,)g(w)o(e)g(kno)o(w)f(that)g(the)h(terms)f(of)g(the)h(series)37 956 y(app)q(earing)f(in)f(the)f(righ)o(t-hand)h(side)g(of)f(\(4.4\))e (are)i(all)h(non-negativ)o(e.)30 b(W)l(e)18 b(ma)o(y)f(then)i(app)q (eal)g(to)f(the)37 1013 y(discrete)f(v)o(ersion)e(of)g(the)g(F)l(atou)f (lemma)i(to)e(infer)i(that)1024 979 y Fe(P)1072 1026 y Fh(n)p Fu(\025)p Fv(1)1140 1013 y Fn(\()1169 1001 y Fe(e)1158 1013 y Fi(K)s(')1230 1020 y Fh(n)1253 1013 y Fn(;)8 b Fi(')1304 1020 y Fh(n)1327 1013 y Fn(\))k Fi(<)h Fn(+)p Fg(1)p Fn(,)i(and)h(that)571 1129 y(lim)8 b(inf)586 1159 y Fv(\003)p Fu(!1)705 1086 y Fe(X)704 1184 y Fh(n)p Fu(\025)p Fv(1)771 1129 y Fn(\()799 1117 y Fe(e)789 1129 y Fi(K)828 1136 y Fv(\003)854 1129 y Fi(')884 1136 y Fh(n)907 1129 y Fn(;)g Fi(')958 1136 y Fh(n)980 1129 y Fn(\))13 b Fg(\025)1059 1086 y Fe(X)1059 1184 y Fh(n)p Fu(\025)p Fv(1)1125 1129 y Fn(\()1154 1117 y Fe(e)1143 1129 y Fi(K)s(')1215 1136 y Fh(n)1238 1129 y Fn(;)8 b Fi(')1289 1136 y Fh(n)1311 1129 y Fn(\))p Fi(:)37 1287 y Fn(Owing)21 b(to)d(the)i(de\014nitions)h(\(3.21\))c(of)i (the)h(notation)1022 1276 y Fe(e)1011 1287 y Fi(K)j Fn(=)1128 1251 y Fe(R)1149 1303 y Fh(Q)1177 1294 y Fc(?)1215 1276 y Fe(e)1205 1287 y Fi(K)1244 1294 y Fh(\030)1311 1267 y(d\030)p 1287 1277 84 2 v 1287 1304 a Fv(\(2)p Fh(\031)q Fv(\))1354 1295 y Ft(3)1395 1287 y Fn(and)c(\(3.19\))f(of)h(the)g (scalar)37 1348 y(pro)q(duct)d(on)f Fg(H)p Fn(,)h(w)o(e)f(ha)o(v)o(e) 379 1426 y Fe(X)379 1524 y Fh(n)p Fu(\025)p Fv(1)445 1469 y Fn(\()474 1457 y Fe(e)463 1469 y Fi(K)s(')535 1476 y Fh(n)558 1469 y Fn(;)8 b Fi(')609 1476 y Fh(n)632 1469 y Fn(\))41 b(=)768 1426 y Fe(X)768 1524 y Fh(n)p Fu(\025)p Fv(1)842 1407 y Fe(Z)867 1510 y Fh(Q)895 1501 y Fc(?)915 1469 y Fn(\(\()p Fi(U)997 1457 y Fe(e)987 1469 y Fi(K)r(')1058 1476 y Fh(n)1081 1469 y Fn(\))1099 1476 y Fh(\030)1118 1469 y Fn(;)8 b(\()p Fi(U)d(')1223 1476 y Fh(n)1245 1469 y Fn(\))1263 1476 y Fh(\030)1281 1469 y Fn(\))1299 1479 y Fh(L)1323 1469 y Ft(2)1340 1479 y Fv(\()p Fh(Q)p Fv(\))1445 1438 y Fi(d\030)p 1415 1458 106 2 v 1415 1500 a Fn(\(2)p Fi(\031)r Fn(\))1502 1487 y Fv(3)691 1613 y Fn(=)768 1569 y Fe(X)768 1667 y Fh(n)p Fu(\025)p Fv(1)842 1551 y Fe(Z)867 1654 y Fh(Q)895 1644 y Fc(?)915 1613 y Fn(\()943 1601 y Fe(e)933 1613 y Fi(K)972 1620 y Fh(\030)990 1613 y Fi(u)1016 1620 y Fh(n)1039 1613 y Fn(\()p Fi(\030)r Fn(;)j Fg(\001)p Fn(\);)g Fi(u)1178 1620 y Fh(n)1199 1613 y Fn(\()p Fi(\030)r Fn(;)g Fg(\001)p Fn(\)\))1309 1622 y Fh(L)1333 1613 y Ft(2)1348 1622 y Fv(\()p Fh(Q)p Fv(\))1453 1582 y Fi(d\030)p 1423 1602 V 1423 1644 a Fn(\(2)p Fi(\031)r Fn(\))1510 1631 y Fv(3)691 1756 y Fn(=)768 1695 y Fe(Z)793 1798 y Fh(Q)821 1788 y Fc(?)849 1713 y Fe(X)848 1811 y Fh(n)p Fu(\025)p Fv(1)915 1756 y Fn(\()943 1745 y Fe(e)933 1756 y Fi(K)972 1763 y Fh(\030)990 1756 y Fi(u)1016 1763 y Fh(n)1039 1756 y Fn(\()p Fi(\030)r Fn(;)g Fg(\001)p Fn(\);)g Fi(u)1178 1763 y Fh(n)1199 1756 y Fn(\()p Fi(\030)r Fn(;)g Fg(\001)p Fn(\)\))1309 1766 y Fh(L)1333 1757 y Ft(2)1348 1766 y Fv(\()p Fh(Q)p Fv(\))1453 1726 y Fi(d\030)p 1423 1746 V 1423 1788 a Fn(\(2)p Fi(\031)r Fn(\))1510 1774 y Fv(3)691 1900 y Fn(=)768 1838 y Fe(Z)793 1941 y Fh(Q)821 1932 y Fc(?)848 1900 y Fn(T)l(r)895 1912 y Fh(L)919 1901 y Ft(2)919 1924 y Fc(\030)936 1912 y Fv(\()p Fh(Q)p Fv(\))1004 1889 y Fe(e)994 1900 y Fi(K)1033 1907 y Fh(\030)1099 1870 y Fi(d\030)p 1069 1890 V 1069 1931 a Fn(\(2)p Fi(\031)r Fn(\))1156 1918 y Fv(3)1179 1900 y Fi(:)37 2029 y Fn(In)13 b(particular,)g(collecting)g(with)g(the)e(ab)q(o)o(v)o(e)h(string)g(of) f(inequalities,)k(w)o(e)c(already)i(kno)o(w)e(that,)g(for)g(almost)37 2085 y(ev)o(ery)16 b Fi(\030)h Fn(in)f Fi(Q)284 2069 y Fh(?)303 2085 y Fn(,)342 2074 y Fe(e)331 2085 y Fi(K)370 2092 y Fh(\030)404 2085 y Fn(is)g(a)e(trace-class)i(op)q(erator)e(on)h Fi(L)985 2069 y Fv(2)985 2099 y Fh(\030)1005 2085 y Fn(\()p Fi(Q)p Fn(\))f(and)i(that)703 2155 y Fe(Z)728 2258 y Fh(Q)756 2249 y Fc(?)783 2217 y Fn(T)l(r)830 2229 y Fh(L)854 2218 y Ft(2)854 2241 y Fc(\030)871 2229 y Fv(\()p Fh(Q)p Fv(\))939 2206 y Fe(e)929 2217 y Fi(K)968 2224 y Fh(\030)1034 2186 y Fi(d\030)p 1004 2207 V 1004 2248 a Fn(\(2)p Fi(\031)r Fn(\))1091 2235 y Fv(3)1127 2217 y Fg(\024)d Fn(1)p Fi(:)571 b Fn(\(4.5\))37 2358 y(Moreo)o(v)o(er,)17 b(denoting)i(b)o(y)h Fe(e)-28 b Fi(\032)q Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\),)16 b(the)h(Hilb)q(ert-Sc)o(hmidt)k(k)o(ernel)d(of) 1330 2347 y Fe(e)1320 2358 y Fi(K)1359 2365 y Fh(\030)1377 2358 y Fn(,)g(w)o(e)g(ma)o(y)f(giv)o(e)h(a)f(sense)i(to)40 2421 y Fe(e)-28 b Fi(\032)q Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))h(as)j(a)f(non-negativ)o(e)h(p)q(erio)q (dic)i(function)f(in)g Fi(L)1000 2404 y Fv(1)1019 2421 y Fn(\()p Fi(Q)p Fn(\),)e(suc)o(h)i(that)e(T)l(r)1356 2433 y Fh(L)1380 2421 y Ft(2)1380 2444 y Fc(\030)1398 2433 y Fv(\()p Fh(Q)p Fv(\))1466 2409 y Fe(e)1455 2421 y Fi(K)1494 2428 y Fh(\030)1525 2421 y Fn(=)1573 2384 y Fe(R)1595 2437 y Fh(Q)1635 2421 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))g Fi(dx)p Fn(.)37 2498 y(In)15 b(addition,)g(thanks)f(to)f(\(4.5\),)f(w)o (e)i(ma)o(y)f(asso)q(ciate)h(to)1023 2486 y Fe(e)1013 2498 y Fi(K)s Fn(,)f(the)h(non-negativ)o(e)h Fi(Q)p Fn(-p)q(erio)q(dic) h(densit)o(y)f Fi(\032)1836 2506 y Fa(e)1828 2514 y Fh(K)1862 2498 y Fn(,)37 2562 y(whic)o(h)j(is)g(de\014ned)g(b)o(y)f Fi(\032)471 2570 y Fa(e)463 2579 y Fh(K)497 2562 y Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))14 b(=)671 2526 y Fe(R)692 2578 y Fh(Q)720 2569 y Fc(?)751 2562 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))958 2542 y Fh(d\030)p 933 2552 84 2 v 933 2579 a Fv(\(2)p Fh(\031)q Fv(\))1000 2570 y Ft(3)1022 2562 y Fn(,)17 b(and)g(whic)o(h)h(b)q(elongs)g(to)e Fi(L)1527 2546 y Fv(1)1546 2562 y Fn(\()p Fi(Q)p Fn(\).)25 b(In)17 b(order)g(to)37 2626 y(conclude)g(the)f(pro)q(of)e(of)h(\(H3\),)f(it)i(remains)f(to)g (sho)o(w)f(that)703 2687 y Fe(Z)728 2790 y Fh(Q)756 2780 y Fc(?)783 2749 y Fn(T)l(r)830 2760 y Fh(L)854 2749 y Ft(2)854 2772 y Fc(\030)871 2760 y Fv(\()p Fh(Q)p Fv(\))939 2737 y Fe(e)929 2749 y Fi(K)968 2756 y Fh(\030)1034 2718 y Fi(d\030)p 1004 2738 106 2 v 1004 2780 a Fn(\(2)p Fi(\031)r Fn(\))1091 2767 y Fv(3)1127 2749 y Fn(=)f(1)p Fi(:)571 b Fn(\(4.6\))p eop %%Page: 30 30 30 29 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(30)37 199 y(T)l(o)15 b(pro)o(v)o(e)g(this)h(claim)g(w)o (e)f(shall)h(actually)g(pro)o(v)o(e)f(in)h(Step)f(2)g(b)q(elo)o(w)h (that)585 299 y Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)p Fn(\))12 b(=)h Fi(\032)759 307 y Fa(e)751 316 y Fh(K)785 299 y Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))k(=)954 237 y Fe(Z)979 340 y Fh(Q)1007 331 y Fc(?)1037 299 y Fe(e)-28 b Fi(\032)q Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))1250 268 y Fi(d\030)p 1220 288 106 2 v 1220 330 a Fn(\(2)p Fi(\031)r Fn(\))1307 317 y Fv(3)1782 299 y Fn(\(4.7\))37 409 y(for)22 b(almost)f(ev)o(ery)h Fi(x)g Fn(in)h Fi(Q)p Fn(,)g(where)i Fe(e)-28 b Fi(\032)23 b Fn(is)f(the)g(limit)i(of)g Fe(e)-28 b Fi(\032)1092 416 y Fv(\003)1119 409 y Fn(.)40 b(Whence)23 b(\(4.6\),)f(thanks)g(to)f(\(3.16\))f(in)37 466 y(Corollary)c(3.2.)108 547 y(Admitting)g(\(4.7\),)d(for)i(a)g (while,)h(w)o(e)f(no)o(w)g(claim)h(that)1081 536 y Fe(e)1070 547 y Fi(K)i Fn(satis\014es)e(\(H4\))e Ff(i.e.)20 b Fn(that)108 628 y(c\))15 b(for)g(almost)f(ev)o(ery)h Fi(\030)j Fn(in)e Fi(Q)624 612 y Fh(?)643 628 y Fn(,)f Fg(\000)p Fn(\001)p 161 644 586 2 v 746 635 a Fh(\030)789 617 y Fe(e)778 628 y Fi(K)p 765 642 54 2 v 819 635 a Fh(\030)853 628 y Fn(is)g(a)g(trace-class)g(op)q(erator)g(on)g Fi(L)1434 612 y Fv(2)p 837 647 617 2 v 1454 635 a Fh(\030)1472 628 y Fn(\()p Fi(Q)p Fn(\),)g(suc)o(h)g(that)p 1472 647 301 2 v 605 678 a Fe(Z)630 781 y Fh(Q)658 771 y Fc(?)685 740 y Fn(T)l(r)732 751 y Fh(L)756 740 y Ft(2)756 763 y Fc(\030)773 751 y Fv(\()p Fh(Q)p Fv(\))831 740 y Fn([)p Fg(\000)p Fn(\001)917 747 y Fh(\030)959 728 y Fe(e)948 740 y Fi(K)987 747 y Fh(\030)1006 740 y Fn(])1066 709 y Fi(d\030)p 1036 729 106 2 v 1036 771 a Fn(\(2)p Fi(\031)r Fn(\))1123 758 y Fv(3)1159 740 y Fi(<)e Fn(+)p Fg(1)p Fi(;)482 b Fn(\(4.8\))37 847 y(b)o(y)16 b(pro)o(ving)f(that)384 947 y(lim)8 b(inf)400 977 y Fv(\003)p Fu(!1)540 916 y Fn(1)p 523 937 57 2 v 523 978 a Fg(j)p Fn(\003)p Fg(j)585 947 y Fn(T)l(r)631 957 y Fh(L)655 947 y Ft(2)673 957 y Fv(\()p Fd(R)717 947 y Ft(3)734 957 y Fv(\))750 947 y Fn([)p Fg(\000)p Fn(\001)p Fi(K)875 954 y Fv(\003)901 947 y Fn(])k Fg(\025)974 885 y Fe(Z)999 988 y Fh(Q)1027 979 y Fc(?)1055 947 y Fn(T)l(r)1102 959 y Fh(L)1126 947 y Ft(2)1126 971 y Fc(\030)1143 959 y Fv(\()p Fh(Q)p Fv(\))1200 947 y Fn([)p Fg(\000)p Fn(\001)1286 954 y Fh(\030)1328 936 y Fe(e)1318 947 y Fi(K)1357 954 y Fh(\030)1375 947 y Fn(])1435 916 y Fi(d\030)p 1405 937 106 2 v 1405 978 a Fn(\(2)p Fi(\031)r Fn(\))1492 965 y Fv(3)1516 947 y Fi(:)253 b Fn(\(4.9\))37 1082 y(The)18 b(pro)q(of)e(of)h(\(4.9\))e (follo)o(ws)i(the)g(same)f(lines)j(as)d(the)h(pro)q(of)g(of)f(\(4.5\).) 24 b(Indeed,)18 b(let)f(\()p Fi(u)1585 1089 y Fh(n)1609 1082 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\)\))1719 1089 y Fh(n)p Fu(\025)p Fv(1)1802 1082 y Fn(and)37 1139 y Fi(')67 1146 y Fh(n;p)139 1139 y Fn(=)22 b Fi(\034)216 1146 y Fu(\000)p Fh(p)284 1139 y Fi(')314 1146 y Fh(n)357 1139 y Fn(b)q(e)f(de\014ned)h(as)d(b)q(efore,)j(with)e(the)h (additional)g(assumptions)g(that)e(\()p Fi(u)1624 1146 y Fh(n)1647 1139 y Fn(\()p Fi(\030)r Fn(;)8 b Fg(\001)p Fn(\)\))1757 1146 y Fh(n)p Fu(\025)p Fv(1)1845 1139 y Fg(2)37 1195 y Fi(H)79 1179 y Fv(1)75 1209 y Fh(\030)99 1195 y Fn(\()p Fi(Q)p Fn(\))15 b(for)h(almost)g(ev)o(ery)g Fi(\030)i Fn(in)f Fi(Q)654 1179 y Fh(?)674 1195 y Fn(,)f(and)g(that)g (the)g Fi(')1001 1202 y Fh(n;p)1052 1195 y Fn('s)g(b)q(elong)h(to)f Fi(H)1344 1179 y Fv(1)1363 1195 y Fn(\()p Fj(R)1420 1179 y Fv(3)1440 1195 y Fn(\).)23 b(Then,)16 b(for)g(the)g(same)37 1252 y(reasons)f(as)g(b)q(efore,)g(w)o(e)g(\014nd)289 1318 y(1)p 272 1339 57 2 v 272 1380 a Fg(j)p Fn(\003)p Fg(j)334 1349 y Fn(T)l(r)381 1359 y Fh(L)405 1349 y Ft(2)422 1359 y Fv(\()p Fd(R)466 1349 y Ft(3)483 1359 y Fv(\))499 1349 y Fn([)p Fg(\000)p Fn(\001)8 b Fi(K)632 1356 y Fv(\003)658 1349 y Fn(])41 b(=)811 1318 y(1)p 794 1339 V 794 1380 a Fg(j)p Fn(\003)p Fg(j)864 1306 y Fe(X)863 1404 y Fh(n)p Fu(\025)p Fv(1)946 1306 y Fe(X)937 1407 y Fh(p)p Fu(2)p Fd(Z)1004 1398 y Ft(3)1021 1349 y Fn(\()p Fi(K)1078 1356 y Fv(\003)1111 1349 y Fn(\()p Fg(\000)p Fn(\001\))1220 1330 y Fv(1)p Fh(=)p Fv(2)1275 1349 y Fi(')1305 1356 y Fh(n;p)1356 1349 y Fn(;)8 b(\()p Fg(\000)p Fn(\001\))1486 1330 y Fv(1)p Fh(=)p Fv(2)1547 1349 y Fi(')1577 1356 y Fh(n;p)1628 1349 y Fn(\))712 1482 y Fg(\025)789 1439 y Fe(X)789 1537 y Fh(n)p Fu(\025)p Fv(1)855 1482 y Fn(\()884 1471 y Fe(e)873 1482 y Fi(K)912 1489 y Fv(\003)946 1482 y Fn(\()p Fg(\000)p Fn(\001\))1055 1464 y Fv(1)p Fh(=)p Fv(2)1109 1482 y Fi(')1139 1489 y Fh(n)1163 1482 y Fn(;)g(\()p Fg(\000)p Fn(\001\))1293 1464 y Fv(1)p Fh(=)p Fv(2)1354 1482 y Fi(')1384 1489 y Fh(n)1407 1482 y Fn(\))p Fi(:)37 1620 y Fn(Then,)16 b(thanks)f(to)f(the)i(w)o(eak)e(con)o(v)o(ergence)i (of)878 1609 y Fe(e)867 1620 y Fi(K)906 1627 y Fv(\003)948 1620 y Fn(to)1014 1609 y Fe(e)1003 1620 y Fi(K)s Fn(,)f(w)o(e)g(ha)o(v) o(e,)f(for)h(ev)o(ery)g Fi(n)e Fg(\025)g Fn(1,)212 1701 y(lim)196 1731 y Fv(\003)p Fu(!1)291 1701 y Fn(\()320 1690 y Fe(e)309 1701 y Fi(K)348 1708 y Fv(\003)382 1701 y Fn(\()p Fg(\000)p Fn(\001\))491 1682 y Fv(1)p Fh(=)p Fv(2)545 1701 y Fi(')575 1708 y Fh(n)598 1701 y Fn(;)8 b(\()p Fg(\000)p Fn(\001\))728 1682 y Fv(1)p Fh(=)p Fv(2)790 1701 y Fi(')820 1708 y Fh(n)843 1701 y Fn(\))474 1797 y(=)42 b(\()579 1785 y Fe(e)569 1797 y Fi(K)10 b Fn(\()p Fg(\000)p Fn(\001\))727 1778 y Fv(1)p Fh(=)p Fv(2)781 1797 y Fi(')811 1804 y Fh(n)835 1797 y Fn(;)e(\()p Fg(\000)p Fn(\001\))965 1778 y Fv(1)p Fh(=)p Fv(2)1026 1797 y Fi(')1056 1804 y Fh(n)1079 1797 y Fn(\))13 b(=)g(\(\()p Fg(\000)p Fn(\001\))1285 1778 y Fv(1)p Fh(=)p Fv(2)1357 1785 y Fe(e)1346 1797 y Fi(K)e Fn(\()p Fg(\000)p Fn(\001\))1505 1778 y Fv(1)p Fh(=)p Fv(2)1559 1797 y Fi(')1589 1804 y Fh(n)1613 1797 y Fn(;)d Fi(')1664 1804 y Fh(n)1686 1797 y Fn(\))p Fi(:)37 1878 y Fn(Moreo)o(v)o(er,)21 b(F)l(atou's)f (lemma)h(still)h(applies)g(since,)h(on)e(the)g(one)g(hand,)h(for)f(ev)o (ery)f(\003)h(and)g Fi(n)p Fn(,)h(w)o(e)e(get)37 1934 y(\(\()p Fg(\000)p Fn(\001\))164 1918 y Fv(1)p Fh(=)p Fv(2)237 1923 y Fe(e)226 1934 y Fi(K)265 1941 y Fv(\003)299 1934 y Fn(\()p Fg(\000)p Fn(\001\))408 1918 y Fv(1)p Fh(=)p Fv(2)463 1934 y Fi(')493 1941 y Fh(n)516 1934 y Fn(;)8 b Fi(')567 1941 y Fh(n)589 1934 y Fn(\))13 b Fg(\025)g Fn(0)g(;)h(while,)i(on)e(the)g(other)g(hand,)g(w)o(e)g(kno)o (w)g(from)f(\(3.10a\))f(in)j(Prop)q(o-)37 1990 y(sition)g(3.1,)e(that) 360 1973 y Fv(1)p 347 1980 45 2 v 347 2007 a Fu(j)p Fv(\003)p Fu(j)396 1990 y Fn(T)l(r)443 2000 y Fh(L)467 1991 y Ft(2)484 2000 y Fv(\()p Fd(R)528 1991 y Ft(3)546 2000 y Fv(\))561 1990 y Fn([)p Fg(\000)p Fn(\001)8 b Fi(K)694 1997 y Fv(\003)720 1990 y Fn(])13 b(is)i(b)q(ounded)g(indep)q(enden)o(tly)i(of)c(\003.)19 b(Therefore,)13 b(passing)i(to)e(the)37 2047 y(lo)o(w)o(er)i(limit)i (as)e(\003)f(go)q(es)h(to)g(in\014nit)o(y)i(in)f(the)f(ab)q(o)o(v)o(e)g (inequalities,)i(w)o(e)e(get)g(\014rst)g(that)704 2089 y Fe(X)704 2187 y Fh(n)p Fu(\025)p Fv(1)770 2132 y Fn(\()p Fg(\000)p Fn(\001)879 2121 y Fe(e)869 2132 y Fi(K)r(')940 2139 y Fh(n)964 2132 y Fn(;)8 b Fi(')1015 2139 y Fh(n)1037 2132 y Fn(\))k Fi(<)h Fn(+)p Fg(1)p Fi(;)37 2257 y Fn(and)j(next,)f (that)56 2354 y(lim)8 b(inf)71 2385 y Fv(\003)p Fu(!1)211 2324 y Fn(1)p 194 2344 57 2 v 194 2386 a Fg(j)p Fn(\003)p Fg(j)256 2354 y Fn(T)l(r)303 2364 y Fh(L)327 2355 y Ft(2)344 2364 y Fv(\()p Fd(R)388 2355 y Ft(3)405 2364 y Fv(\))421 2354 y Fn([)p Fg(\000)p Fn(\001)p Fi(K)546 2361 y Fv(\003)572 2354 y Fn(])41 b Fg(\025)h Fn(lim)8 b(inf)719 2385 y Fv(\003)p Fu(!1)837 2311 y Fe(X)837 2409 y Fh(n)p Fu(\025)p Fv(1)903 2354 y Fn(\(\()p Fg(\000)p Fn(\001\))1030 2336 y Fv(1)p Fh(=)p Fv(2)1103 2343 y Fe(e)1092 2354 y Fi(K)1131 2361 y Fv(\003)1165 2354 y Fn(\()p Fg(\000)p Fn(\001\))1274 2336 y Fv(1)p Fh(=)p Fv(2)1336 2354 y Fi(')1366 2361 y Fh(n)1389 2354 y Fn(;)g Fi(')1440 2361 y Fh(n)1463 2354 y Fn(\))626 2484 y Fg(\025)704 2440 y Fe(X)703 2538 y Fh(n)p Fu(\025)p Fv(1)770 2484 y Fn(\()798 2472 y Fe(e)788 2484 y Fi(K)i Fn(\()p Fg(\000)p Fn(\001\))946 2465 y Fv(1)p Fh(=)p Fv(2)1008 2484 y Fi(')1038 2491 y Fh(n)1061 2484 y Fn(;)e(\()p Fg(\000)p Fn(\001\))1191 2465 y Fv(1)p Fh(=)p Fv(2)1253 2484 y Fi(')1283 2491 y Fh(n)1306 2484 y Fn(\))626 2627 y(=)703 2566 y Fe(Z)729 2669 y Fh(Q)757 2659 y Fc(?)784 2584 y Fe(X)784 2682 y Fh(n)p Fu(\025)p Fv(1)850 2627 y Fn(\()879 2616 y Fe(e)868 2627 y Fi(K)907 2634 y Fh(\030)933 2627 y Fn(\()p Fg(\000)p Fn(\001)1024 2634 y Fh(\030)1043 2627 y Fn(\))1061 2609 y Fv(1)p Fh(=)p Fv(2)1123 2627 y Fi(u)1149 2634 y Fh(n)1173 2627 y Fn(\()p Fi(\030)r Fn(;)g Fg(\001)p Fn(\);)g(\()p Fg(\000)p Fn(\001)1377 2634 y Fh(\030)1393 2627 y Fn(\))1411 2609 y Fv(1)p Fh(=)p Fv(2)1473 2627 y Fi(u)1499 2634 y Fh(n)1523 2627 y Fn(\()p Fi(\030)r Fn(;)g Fg(\001)p Fn(\)\))1633 2637 y Fh(L)1657 2628 y Ft(2)1672 2637 y Fv(\()p Fh(Q)p Fv(\))1776 2597 y Fi(d\030)p 1747 2617 106 2 v 1747 2659 a Fn(\(2)p Fi(\031)r Fn(\))1834 2646 y Fv(3)626 2771 y Fn(=)703 2709 y Fe(Z)729 2813 y Fh(Q)757 2803 y Fc(?)784 2771 y Fn(T)l(r)831 2783 y Fh(L)855 2772 y Ft(2)855 2795 y Fc(\030)872 2783 y Fv(\()p Fh(Q)p Fv(\))929 2771 y Fn([)p Fg(\000)p Fn(\001)1015 2778 y Fh(\030)1045 2760 y Fe(e)1034 2771 y Fi(K)1073 2778 y Fh(\030)1092 2771 y Fn(])1152 2741 y Fi(d\030)p 1122 2761 V 1122 2803 a Fn(\(2)p Fi(\031)r Fn(\))1209 2789 y Fv(3)1232 2771 y Fi(;)p eop %%Page: 31 31 31 30 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(31)37 199 y(since)17 b(\()p Fg(\000)p Fn(\001\))258 183 y Fv(1)p Fh(=)p Fv(2)325 199 y Fn(=)373 163 y Fe(R)395 215 y Fh(Q)423 206 y Fc(?)442 199 y Fn(\()p Fg(\000)p Fn(\001)533 206 y Fh(\030)552 199 y Fn(\))570 183 y Fv(1)p Fh(=)p Fv(2)667 179 y Fh(d\030)p 642 189 84 2 v 642 216 a Fv(\(2)p Fh(\031)q Fv(\))709 207 y Ft(3)731 199 y Fn(.)j(Hence)c(\(4.8\))e(and)h(\(4.9\).)37 304 y(W)l(e)h(turn)f(no)o(w)g(to)f(the)h(pro)q(of)g(of)g(\(4.7\)\(thereb)o (y)e(pro)o(ving)j(\(4.6\)\);)d(that)h(is,)h(the)h(fact)e(that)108 402 y Fj(Step)k(2)p 108 418 146 2 v 16 w Fn(:)45 b Ff(The)18 b(limit)g(of)i Fe(e)-28 b Fi(\032)609 409 y Fv(\003)636 402 y Ff(,)21 b Fe(e)-28 b Fi(\032)p Ff(,)18 b(and)f(the)h(p)n(erio)n (dic)f(density)g(asso)n(ciate)n(d)f(to)1497 391 y Fe(e)1486 402 y Fi(K)s Ff(,)h(which)h(is)f(de\014ne)n(d)37 468 y(by)g Fi(\032)128 476 y Fa(e)120 485 y Fh(K)167 468 y Fn(=)215 431 y Fe(R)236 484 y Fh(Q)264 474 y Fc(?)295 468 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))507 447 y Fh(d\030)p 483 457 84 2 v 483 485 a Fv(\(2)p Fh(\031)q Fv(\))550 475 y Ft(3)571 468 y Ff(,)17 b(c)n(oincide.)108 554 y Fn(W)l(e)e(are)g(\014rst)g(lo)q (oking)h(for)108 635 y(a\))f(A)g(priori)h(estimates)f(on)j Fe(e)-28 b Fi(\032)623 642 y Fv(\003)650 635 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\).)p 164 654 605 2 v 37 722 a(Since)17 b Fi(K)198 706 y Fv(2)195 736 y(\003)234 722 y Fg(\024)c Fi(K)321 729 y Fv(\003)347 722 y Fn(,)i(w)o(e)g(ha)o(v)o(e) 443 848 y Fi(\032)466 855 y Fv(\003)493 848 y Fn(\()p Fi(x)p Fn(\))d Fg(\025)615 786 y Fe(Z)640 889 y Fd(R)670 879 y Ft(3)697 848 y Fi(\032)720 855 y Fv(\003)747 848 y Fn(\()p Fi(x)p Fn(;)c Fi(y)r Fn(\))g Fi(\032)885 855 y Fv(\003)910 848 y Fn(\()p Fi(y)r Fn(;)g Fi(x)p Fn(\))g Fi(dy)13 b Fn(=)1132 786 y Fe(Z)1157 889 y Fd(R)1187 879 y Ft(3)1214 848 y Fg(j)p Fi(\032)1250 855 y Fv(\003)1277 848 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fg(j)1397 829 y Fv(2)1422 848 y Fi(dy)291 b Fn(\(4.10\))37 976 y(almost)15 b(ev)o(erywhere)h(on)f Fj(R)523 960 y Fv(3)543 976 y Fn(.)k(Thanks)d(to)e(\(4.10\),)f(w)o(e)i(\014rst)g(c)o(hec)o(k)g (that)694 1100 y Fe(e)-28 b Fi(\032)714 1107 y Fv(\003)741 1100 y Fn(\()p Fi(x)p Fn(\))12 b Fg(\025)863 1038 y Fe(Z)888 1141 y Fd(R)918 1131 y Ft(3)946 1100 y Fg(j)s Fe(e)-28 b Fi(\032)982 1107 y Fv(\003)1008 1100 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fg(j)1128 1081 y Fv(2)1154 1100 y Fi(dy)r(;)544 b Fn(\(4.11\))37 1222 y(whic)o(h)17 b(in)f(particular)f (implies)j(that)462 1324 y Fe(e)-27 b Fi(\032)483 1331 y Fv(\003)510 1324 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))28 b(is)16 b(b)q(ounded)h(in)f Fi(L)961 1305 y Fv(2)980 1324 y Fn(\()p Fi(Q)10 b Fg(\002)g Fj(R)1128 1305 y Fv(3)1148 1324 y Fn(\))g Fg(\\)h Fi(L)1248 1305 y Fv(2)1267 1324 y Fn(\()p Fj(R)1324 1305 y Fv(3)1354 1324 y Fg(\002)f Fi(Q)p Fn(\))306 b(\(4.12\))37 1426 y(indep)q(endently)24 b(of)c(\003,)i(for)h Fe(e)-28 b Fi(\032)560 1433 y Fv(\003)587 1426 y Fn(\()p Fi(x)p Fn(\))21 b(is)g(b)q(ounded)i(in)f Fi(L)1003 1410 y Fv(1)1022 1426 y Fn(\()p Fi(Q)p Fn(\).)37 b(Indeed,)24 b(b)q(ecause)e(of)f(\(4.10\),)f(w)o(e)h(c)o(hec)o(k)37 1483 y(successiv)o(ely)c(that)206 1602 y Fe(e)-28 b Fi(\032)226 1609 y Fv(\003)253 1602 y Fn(\()p Fi(x)p Fn(\))41 b(=)454 1571 y(1)p 437 1591 57 2 v 437 1633 a Fg(j)p Fn(\003)p Fg(j)508 1558 y Fe(X)507 1657 y Fh(k)q Fu(2)p Fv(\003)582 1602 y Fi(\032)605 1609 y Fv(\003)632 1602 y Fn(\()p Fi(x)9 b Fn(+)i Fi(k)q Fn(;)d Fi(x)h Fn(+)i Fi(k)q Fn(\))356 1742 y Fg(\025)454 1712 y Fn(1)p 437 1732 V 437 1773 a Fg(j)p Fn(\003)p Fg(j)508 1699 y Fe(X)507 1798 y Fh(k)q Fu(2)p Fv(\003)582 1680 y Fe(Z)607 1783 y Fd(R)637 1774 y Ft(3)664 1742 y Fg(j)p Fi(\032)700 1749 y Fv(\003)727 1742 y Fn(\()p Fi(x)f Fn(+)g Fi(k)q Fn(;)e Fi(y)r Fn(\))p Fg(j)927 1723 y Fv(2)953 1742 y Fi(dy)14 b Fn(=)1083 1712 y(1)p 1066 1732 V 1066 1773 a Fg(j)p Fn(\003)p Fg(j)1136 1699 y Fe(X)1135 1798 y Fh(k)q Fu(2)p Fv(\003)1210 1680 y Fe(Z)1236 1783 y Fd(R)1266 1774 y Ft(3)1293 1742 y Fg(j)p Fi(\032)1329 1749 y Fv(\003)1356 1742 y Fn(\()p Fi(x)9 b Fn(+)i Fi(k)q Fn(;)d Fi(y)j Fn(+)g Fi(k)q Fn(\))p Fg(j)1636 1723 y Fv(2)1662 1742 y Fi(dy)356 1883 y Fg(\025)432 1821 y Fe(Z)458 1924 y Fd(R)488 1915 y Ft(3)515 1883 y Fg(j)s Fe(e)-28 b Fi(\032)551 1890 y Fv(\003)578 1883 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fg(j)698 1864 y Fv(2)723 1883 y Fi(dy)r(;)37 2006 y Fn(b)o(y)16 b(con)o(v)o(exit)o(y) l(.)k(W)l(e)15 b(pro)o(v)o(e)f(no)o(w)h(that)400 2108 y Fe(e)-28 b Fi(\032)420 2115 y Fv(\003)447 2108 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))29 b(is)15 b(b)q(ounded)i(in)f Fi(L)898 2089 y Fv(2)898 2119 y Fh(x)920 2108 y Fn(\()p Fi(Q)p Fn(;)8 b Fi(H)1037 2089 y Fv(1)1033 2119 y Fh(y)1055 2108 y Fn(\()p Fj(R)1112 2089 y Fv(3)1131 2108 y Fn(\)\))i Fg(\\)g Fi(L)1248 2089 y Fv(2)1248 2119 y Fh(y)1269 2108 y Fn(\()p Fi(Q)p Fn(;)e Fi(H)1386 2089 y Fv(1)1382 2119 y Fh(x)1404 2108 y Fn(\()p Fj(R)1461 2089 y Fv(3)1480 2108 y Fn(\)\))243 b(\(4.13\))37 2210 y(indep)q(endently)18 b(of)c(\003,)h(or,)f(more)h(precisely)l(,)i(that)567 2268 y Fe(Z)592 2372 y Fh(Q)630 2330 y Fi(dy)685 2268 y Fe(Z)710 2372 y Fd(R)740 2362 y Ft(3)767 2330 y Fg(j)p Fn(\(1)9 b Fg(\000)i Fn(\001\))932 2312 y Fv(1)p Fh(=)p Fv(2)932 2342 y Fh(x)989 2330 y Fe(e)-28 b Fi(\032)1009 2337 y Fv(\003)1037 2330 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fg(j)1157 2312 y Fv(2)1182 2330 y Fi(dx)k Fg(\024)h Fi(C)q(;)421 b Fn(\(4.14\))37 2459 y(where)15 b Fi(C)j Fn(denotes)d(here)f(and)h(b)q(elo)o(w)g(a)f(p)q(ositiv)o(e)h(constan)o (t)e(that)h(is)h(indep)q(enden)o(t)h(of)e(\003.)19 b(W)l(e)c(emphasize) 37 2521 y(the)j(fact)e(that,)h(in)h(the)g(follo)o(wing,)g(w)o(e)f (shall)h(use)f(the)h(notation)f(\(1)11 b Fg(\000)g Fn(\001\))1342 2497 y Fv(1)p Fh(=)p Fv(2)1342 2527 y Fh(x)1400 2521 y Fe(e)-28 b Fi(\032)1420 2528 y Fv(\003)1447 2521 y Fn(\()p Fg(\001)p Fi(;)8 b Fg(\001)p Fn(\))15 b(for)i(the)g(Sc)o(h)o(w) o(artz)37 2578 y(k)o(ernel)h(of)e(the)h(op)q(erator)f(\(1)11 b Fg(\000)g Fn(\001\))645 2561 y Fv(1)p Fh(=)p Fv(2)711 2566 y Fe(e)700 2578 y Fi(K)739 2585 y Fv(\003)765 2578 y Fn(,)17 b(whic)o(h)h(is)f(also)1067 2544 y Fe(P)1115 2591 y Fh(n)p Fu(\025)p Fv(1)1191 2578 y Fi(\025)1217 2585 y Fh(n)1248 2578 y Fn([\(1)10 b Fg(\000)i Fn(\001\))1415 2561 y Fv(1)p Fh(=)p Fv(2)1470 2578 y Fi(')1500 2585 y Fh(n)1523 2578 y Fn(]\()p Fg(\001)p Fn(\))c Fi(')1623 2561 y Fu(\003)1623 2589 y Fh(n)1644 2578 y Fn(\()p Fg(\001)p Fn(\).)24 b(Indeed,)p eop %%Page: 32 32 32 31 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(32)37 199 y(w)o(e)15 b(ha)o(v)o(e)100 256 y Fe(Z)125 359 y Fh(Q)162 318 y Fi(dy)218 256 y Fe(Z)243 359 y Fd(R)273 350 y Ft(3)300 318 y Fg(j)p Fn(\(1)9 b Fg(\000)i Fn(\001\))465 299 y Fv(1)p Fh(=)p Fv(2)465 329 y Fh(x)522 318 y Fe(e)-28 b Fi(\032)542 325 y Fv(\003)569 318 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fg(j)689 299 y Fv(2)715 318 y Fi(dx)k Fn(=)825 256 y Fe(Z)850 359 y Fh(Q)888 318 y Fi(dy)943 256 y Fe(Z)968 359 y Fd(R)998 350 y Ft(3)1018 252 y Fe(\014)1018 280 y(\014)1018 307 y(\014)1018 334 y(\014)1055 287 y Fn(1)p 1038 308 57 2 v 1038 349 a Fg(j)p Fn(\003)p Fg(j)1108 275 y Fe(X)1107 374 y Fh(k)q Fu(2)p Fv(\003)1175 318 y Fn(\(1)d Fg(\000)i Fn(\001\))1327 299 y Fv(1)p Fh(=)p Fv(2)1327 329 y Fh(x)1381 318 y Fi(\032)1404 325 y Fv(\003)1431 318 y Fn(\()p Fi(x)f Fn(+)g Fi(k)q Fn(;)e Fi(y)k Fn(+)e Fi(k)q Fn(\))1698 252 y Fe(\014)1698 280 y(\014)1698 307 y(\014)1698 334 y(\014)1713 265 y Fv(2)1740 318 y Fi(dx)378 459 y Fg(\024)476 428 y Fn(1)p 459 449 V 459 490 a Fg(j)p Fn(\003)p Fg(j)530 416 y Fe(X)529 515 y Fh(k)q Fu(2)p Fv(\003)604 397 y Fe(Z)629 500 y Fh(Q)666 459 y Fi(dy)721 397 y Fe(Z)747 500 y Fd(R)777 491 y Ft(3)804 459 y Fg(j)p Fn(\(1)f Fg(\000)i Fn(\001\))969 440 y Fv(1)p Fh(=)p Fv(2)969 470 y Fh(x)1023 459 y Fi(\032)1046 466 y Fv(\003)1073 459 y Fn(\()p Fi(x)f Fn(+)g Fi(k)q Fn(;)e Fi(y)k Fn(+)e Fi(k)q Fn(\))p Fg(j)1353 440 y Fv(2)1380 459 y Fi(dx)378 600 y Fn(=)476 569 y(1)p 459 589 V 459 631 a Fg(j)p Fn(\003)p Fg(j)529 538 y Fe(Z)554 641 y Fv(\000\(\003\))638 600 y Fi(dy)693 538 y Fe(Z)718 641 y Fd(R)748 632 y Ft(3)775 600 y Fg(j)p Fn(\(1)f Fg(\000)i Fn(\001\))940 581 y Fv(1)p Fh(=)p Fv(2)940 611 y Fh(x)994 600 y Fi(\032)1017 607 y Fv(\003)1045 600 y Fn(\()p Fi(x)p Fn(;)d Fi(y)r Fn(\))p Fg(j)1165 581 y Fv(2)1190 600 y Fi(dx)378 728 y Fg(\024)476 697 y Fn(1)p 459 717 V 459 759 a Fg(j)p Fn(\003)p Fg(j)529 666 y Fe(Z)554 769 y Fd(R)584 760 y Ft(3)611 728 y Fi(dy)666 666 y Fe(Z)692 769 y Fd(R)722 760 y Ft(3)749 728 y Fg(j)p Fn(\(1)h Fg(\000)i Fn(\001\))914 709 y Fv(1)p Fh(=)p Fv(2)914 739 y Fh(x)968 728 y Fi(\032)991 735 y Fv(\003)1018 728 y Fn(\()p Fi(x)p Fn(;)d Fi(y)r Fn(\))p Fg(j)1138 709 y Fv(2)1164 728 y Fi(dx)378 861 y Fn(=)476 830 y(1)p 459 851 V 459 892 a Fg(j)p Fn(\003)p Fg(j)529 799 y Fe(Z)d(Z)604 902 y Fd(R)634 893 y Ft(3)652 902 y Fu(\002)p Fd(R)709 893 y Ft(3)729 795 y Fe(\014)729 822 y(\014)729 850 y(\014)729 877 y(\014)745 818 y(X)745 916 y Fh(n)p Fu(\025)p Fv(1)819 861 y Fi(\026)846 868 y Fh(n)869 861 y Fn([\(1)10 b Fg(\000)g Fn(\001\))1034 842 y Fv(1)p Fh(=)p Fv(2)1089 861 y Fi( )1119 868 y Fh(n)1142 861 y Fn(]\()p Fi(x)p Fn(\))e Fi( )1257 842 y Fu(\003)1255 872 y Fh(n)1276 861 y Fn(\()p Fi(y)r Fn(\))1336 795 y Fe(\014)1336 822 y(\014)1336 850 y(\014)1336 877 y(\014)1351 808 y Fv(2)1378 861 y Fi(dxdy)378 1004 y Fn(=)476 974 y(1)p 459 994 V 459 1036 a Fg(j)p Fn(\003)p Fg(j)529 943 y Fe(Z)554 1046 y Fd(R)584 1036 y Ft(3)612 961 y Fe(X)611 1059 y Fh(n)p Fu(\025)p Fv(1)685 1004 y Fi(\026)712 986 y Fv(2)712 1016 y Fh(n)744 1004 y Fg(j)p Fn(\(1)h Fg(\000)h Fn(\001\))908 986 y Fv(1)p Fh(=)p Fv(2)963 1004 y Fi( )993 1011 y Fh(n)1016 1004 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1091 986 y Fv(2)1117 1004 y Fi(dx)58 b Fg(\024)1341 974 y Fn(1)p 1323 994 V 1323 1036 a Fg(j)p Fn(\003)p Fg(j)1385 1004 y Fn(T)l(r)1432 1014 y Fh(L)1456 1005 y Ft(2)1473 1014 y Fv(\()p Fd(R)1517 1005 y Ft(3)1535 1014 y Fv(\))1551 1004 y Fn([\(1)9 b Fg(\000)h Fn(\001\))e Fi(K)1762 1011 y Fv(\003)1787 1004 y Fn(])p Fi(:)37 1148 y Fn(The)17 b(\014rst)f(inequalit)o(y)h(is)g(deduced)h(b)o(y)e(a)g(con) o(v)o(exit)o(y)g(argumen)o(t,)f(and)h(the)g(last)g(one)h(comes)f(from)f (\(4.1\),)37 1205 y(\(4.2\))g(and)i(the)g(fact)f(that)f(0)g Fg(\024)g Fi(\026)622 1212 y Fh(n)661 1205 y Fg(\024)g Fn(1.)24 b(Finally)l(,)18 b(w)o(e)e(conclude)i(with)f(the)g(help)g(of)f (\(3.10a\))f(in)i(Prop)q(o-)37 1261 y(sition)22 b(3.1.)34 b(An)21 b(easy)f(consequence)i(of)e(\(4.13\))f(is)i(that)h Fe(e)-28 b Fi(\032)1091 1268 y Fv(\003)1118 1261 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))19 b(is)i(b)q(ounded)h(in)f Fi(H)1586 1245 y Fv(1)1582 1275 y(unif)1647 1261 y Fn(\()p Fj(R)1704 1245 y Fv(3)1737 1261 y Fg(\002)14 b Fj(R)1825 1245 y Fv(3)1845 1261 y Fn(\),)37 1318 y(th)o(us,)19 b(up)g(to)f(a)g(subsequence,)i(it)f(con)o(v)o(erges)f(to)f(some)h (function)24 b(\026)-27 b Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))17 b(w)o(eakly)h(in)h Fi(H)1607 1301 y Fv(1)1603 1332 y(lo)q(c)1649 1318 y Fn(\()p Fj(R)1706 1301 y Fv(3)1738 1318 y Fg(\002)13 b Fj(R)1825 1301 y Fv(3)1845 1318 y Fn(\),)37 1374 y(strongly)i(in)h Fi(L)297 1352 y Fh(p)297 1389 y Fv(lo)q(c)343 1374 y Fn(\()p Fj(R)400 1358 y Fv(3)429 1374 y Fg(\002)9 b Fj(R)512 1358 y Fv(3)532 1374 y Fn(\),)15 b(for)f(ev)o(ery)g(1)f Fg(\024)g Fi(p)f(<)h Fn(3)i(\(b)o(y)f(the)h(Rellic)o(h)j(theorem)c(for)g(b)q(ounded)j (domains)37 1437 y(of)g Fj(R)130 1421 y Fv(6)150 1437 y Fn(\),)f(and)h(almost)f(ev)o(erywhere)h(on)g Fj(R)777 1421 y Fv(6)796 1437 y Fn(.)25 b(Actually)l(,)18 b(b)q(ecause)g(of)e (the)h(w)o(eak)f(con)o(v)o(ergence)h(of)1764 1426 y Fe(e)1753 1437 y Fi(K)1792 1444 y Fv(\003)1835 1437 y Fn(to)48 1482 y Fe(e)37 1494 y Fi(K)s Fn(,)j(\026)-27 b Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))15 b(is)h(nothing)g(but)j Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\),)14 b(the)i(Sc)o(h)o(w)o(artz)f(k)o(ernel)i(of)1179 1482 y Fe(e)1168 1494 y Fi(K)s Fn(.)22 b(Note)15 b(that,)g(in)i(particular,) f(w)o(e)g(ob-)37 1550 y(tain)e(from)e(\(4.12\))g(\(resp)q(ectiv)o(ely)i (\(4.14\)\))d(that,)i(up)h(to)e(a)h(further)g(subsequence,)18 b Fe(e)-28 b Fi(\032)1479 1557 y Fv(\003)1506 1550 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))k(\(resp)q(ectiv)o(ely)37 1615 y(\(1)s Fg(\000)s Fn(\001\))175 1591 y Fv(1)p Fh(=)p Fv(2)175 1620 y Fh(x)233 1615 y Fe(e)-28 b Fi(\032)253 1622 y Fv(\003)280 1615 y Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\)\))i(con)o(v)o(erges)h(to)j Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))i(\(resp)q(ectiv)o(ely)j(\(1)s Fg(\000)s Fn(\001\))1208 1591 y Fv(1)p Fh(=)p Fv(2)1208 1620 y Fh(x)1265 1615 y Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\)\))i(w)o(eakly)h(in)i Fi(L)1648 1598 y Fv(2)1668 1615 y Fn(\()p Fi(Q)1722 1622 y Fh(x)1746 1615 y Fg(\002)s Fj(R)1823 1598 y Fv(3)1823 1626 y Fh(y)1845 1615 y Fn(\).)37 1671 y(Therefore,)i(w)o(e)g(ha)o(v)o(e)663 1725 y Fe(Z)689 1828 y Fh(Q)726 1787 y Fi(dx)784 1725 y Fe(Z)809 1828 y Fd(R)839 1818 y Ft(3)866 1787 y Fg(j)s Fe(e)-28 b Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fg(j)1022 1768 y Fv(2)1047 1787 y Fi(dy)15 b(<)e Fn(+)p Fg(1)p Fi(;)37 1911 y Fn(and)560 1955 y Fe(Z)585 2059 y Fh(Q)623 2017 y Fi(dy)678 1955 y Fe(Z)703 2059 y Fd(R)733 2049 y Ft(3)761 2017 y Fg(j)p Fn(\(1)c Fg(\000)h Fn(\001\))925 1999 y Fv(1)p Fh(=)p Fv(2)925 2029 y Fh(x)983 2017 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))p Fg(j)1123 1999 y Fv(2)1149 2017 y Fi(dx)k(<)h Fn(+)p Fg(1)p Fi(:)37 2154 y Fn(The)k(\014rst)f(b)q(ound)i(pro)o(vides)f (another)f(pro)q(of)g(of)g(the)g(fact)g(that)1172 2143 y Fe(e)1161 2154 y Fi(K)1200 2161 y Fh(\030)1235 2154 y Fn(is)h(Hilb)q(ert-Sc)o(hmidt)i(on)d Fi(L)1713 2138 y Fv(2)1713 2168 y Fh(\030)1733 2154 y Fn(\()p Fi(Q)p Fn(\))g(for)37 2222 y(almost)i(ev)o(ery)g Fi(\030)i Fg(2)e Fi(Q)434 2206 y Fh(?)454 2222 y Fn(,)g(with)h(k)o(ernel)j Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\),)17 b(and)h(that)i Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))16 b(=)1324 2186 y Fe(R)1346 2238 y Fh(Q)1374 2229 y Fc(?)1404 2222 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\))1609 2202 y Fh(d\030)p 1585 2212 84 2 v 1585 2239 a Fv(\(2)p Fh(\031)q Fv(\))1652 2230 y Ft(3)1673 2222 y Fn(,)19 b(b)o(y)f(using)37 2279 y(Lemma)e(3.4.)108 2360 y(Our)g(next)f(step)g(consists)h(no)o(w)e(in)i(sho)o(wing)f(that) 427 2457 y(\(1)9 b Fg(\000)i Fn(\001\))579 2438 y Fv(1)p Fh(=)p Fv(2)579 2468 y Fh(x)636 2457 y Fe(e)-28 b Fi(\032)656 2464 y Fv(\003)683 2457 y Fn(\()p Fi(x)10 b Fn(+)g Fi(y)r(;)e(y)r Fn(\))14 b(is)i(b)q(ounded)h(in)f Fi(L)1199 2438 y Fu(1)1199 2468 y Fh(x)1236 2457 y Fn(\()p Fj(R)1293 2438 y Fv(3)1312 2457 y Fn(;)8 b Fi(L)1364 2438 y Fv(1)1364 2468 y Fh(y)1384 2457 y Fn(\()p Fi(Q)p Fn(\)\))p Fi(:)272 b Fn(\(4.15\))37 2554 y(This)17 b(claim)g(will)h(b)q(e)f(a)f(consequence)h(of)f(the)g (follo)o(wing)h(t)o(w)o(o)e(b)q(ounds.)24 b(First,)15 b(since)21 b Fe(e)-28 b Fi(\032)1561 2561 y Fv(\003)1604 2554 y Fn(is)17 b(b)q(ounded)g(in)37 2611 y Fi(L)68 2594 y Fv(1)68 2625 y(unif)133 2611 y Fn(\()p Fj(R)190 2594 y Fv(3)210 2611 y Fn(\),)d(w)o(e)h(clearly)h(ha)o(v)o(e)639 2731 y(sup)632 2773 y Fh(t)p Fu(2)p Fd(R)699 2764 y Ft(3)724 2688 y Fe(X)723 2786 y Fh(n)p Fu(\025)p Fv(1)797 2731 y Fi(\025)823 2738 y Fh(n)855 2669 y Fe(Z)880 2772 y Fh(t)p Fv(+)p Fh(Q)958 2731 y Fg(j)p Fi(')1001 2738 y Fh(n)1024 2731 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1099 2712 y Fv(2)1125 2731 y Fi(dx)c Fg(\024)h Fi(C)q(:)478 b Fn(\(4.16\))p eop %%Page: 33 33 33 32 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(33)37 199 y(Next,)15 b(w)o(e)g(no)o(w)g(pro)o(v)o(e)f (that)561 314 y(sup)553 356 y Fh(t)p Fu(2)p Fd(R)620 347 y Ft(3)645 271 y Fe(X)645 369 y Fh(n)p Fu(\025)p Fv(1)719 314 y Fi(\025)745 321 y Fh(n)776 252 y Fe(Z)802 356 y Fh(t)p Fv(+)p Fh(Q)879 314 y Fg(j)p Fn(\(1)9 b Fg(\000)i Fn(\001\))1044 296 y Fv(1)p Fh(=)p Fv(2)1098 314 y Fi(')1128 321 y Fh(n)1152 314 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1227 296 y Fv(2)1265 314 y Fg(\024)i Fi(C)q(:)400 b Fn(\(4.17\))37 465 y(Indeed,)27 b(using)d(the)g(fact)e(that)h(the)g (self-adjoin)o(t)h(op)q(erators)f(\(1)15 b Fg(\000)h Fn(\001\))1327 449 y Fv(1)p Fh(=)p Fv(2)1381 465 y Fi(K)1420 472 y Fv(\003)1446 465 y Fn(\(1)f Fg(\000)h Fn(\001\))1609 449 y Fv(1)p Fh(=)p Fv(2)1687 465 y Fn(and)24 b(\(1)15 b Fg(\000)37 522 y Fn(\001\))93 505 y Fv(1)p Fh(=)p Fv(2)159 510 y Fe(e)148 522 y Fi(K)187 529 y Fv(\003)213 522 y Fn(\(1)q Fg(\000)q Fn(\001\))347 505 y Fv(1)p Fh(=)p Fv(2)413 522 y Fn(are)10 b(p)q(ositiv)o(e)i(and)f(trace-class)g(with)g (Hilb)q(ert-Sc)o(hmidt)i(k)o(ernels)e(b)q(eing)h(resp)q(ectiv)o(ely)37 578 y(de\014ned)17 b(b)o(y)257 544 y Fe(P)305 591 y Fh(n)p Fu(\025)p Fv(1)381 578 y Fi(\026)408 585 y Fh(n)432 578 y Fn([\(1)8 b Fg(\000)i Fn(\001\))595 561 y Fv(1)p Fh(=)p Fv(2)650 578 y Fi( )680 585 y Fh(n)703 578 y Fn(]\()p Fi(x)p Fn(\))e([\(1)f Fg(\000)j Fn(\001\))948 561 y Fv(1)p Fh(=)p Fv(2)1002 578 y Fi( )1032 585 y Fh(n)1056 578 y Fn(])1069 561 y Fu(\003)1088 578 y Fn(\()p Fi(y)r Fn(\))k(and)1250 544 y Fe(P)1298 591 y Fh(n)p Fu(\025)p Fv(1)1374 578 y Fi(\025)1400 585 y Fh(n)1424 578 y Fn([\(1)9 b Fg(\000)h Fn(\001\))1588 561 y Fv(1)p Fh(=)p Fv(2)1642 578 y Fi(')1672 585 y Fh(n)1696 578 y Fn(]\()p Fi(x)p Fn(\))e([\(1)f Fg(\000)37 643 y Fn(\001\))93 626 y Fv(1)p Fh(=)p Fv(2)148 643 y Fi(')178 650 y Fh(n)201 643 y Fn(])214 626 y Fu(\003)234 643 y Fn(\()p Fi(y)r Fn(\),)14 b(w)o(e)h(ma)o(y)f(observ)o(e)h(that)352 715 y Fe(X)352 813 y Fh(n)p Fu(\025)p Fv(1)426 758 y Fi(\025)452 765 y Fh(n)476 758 y Fg(j)p Fn(\(1)9 b Fg(\000)h Fn(\001\))640 740 y Fv(1)p Fh(=)p Fv(2)695 758 y Fi(')725 765 y Fh(n)748 758 y Fn(\()p Fi(x)p Fn(\))p Fg(j)823 740 y Fv(2)854 758 y Fn(=)924 728 y(1)p 907 748 57 2 v 907 790 a Fg(j)p Fn(\003)p Fg(j)977 715 y Fe(X)977 813 y Fh(n)p Fu(\025)p Fv(1)1051 758 y Fi(\026)1078 765 y Fh(n)1102 758 y Fg(j)p Fn(\(1)f Fg(\000)h Fn(\001\))1266 740 y Fv(1)p Fh(=)p Fv(2)1321 758 y Fi( )1351 765 y Fh(n)1374 758 y Fn(\()p Fi(x)g Fn(+)g Fi(k)q Fn(\))p Fg(j)1529 740 y Fv(2)1548 758 y Fi(;)37 912 y Fn(almost)15 b(ev)o(erywhere)h(on)f Fj(R)523 896 y Fv(3)543 912 y Fn(,)f(thanks)h(to)g(the)g(de\014nition)i (of)1117 901 y Fe(e)1106 912 y Fi(K)1145 919 y Fv(\003)1171 912 y Fn(.)j(Therefore,)15 b(for)f(ev)o(ery)h Fi(t)h Fn(in)g Fj(R)1736 896 y Fv(3)1756 912 y Fn(,)154 984 y Fe(X)153 1082 y Fh(n)p Fu(\025)p Fv(1)227 1027 y Fi(\025)253 1034 y Fh(n)285 965 y Fe(Z)310 1068 y Fh(t)p Fv(+)p Fh(Q)380 1027 y Fg(j)p Fn(\(1)9 b Fg(\000)i Fn(\001\))545 1008 y Fv(1)p Fh(=)p Fv(2)599 1027 y Fi(')629 1034 y Fh(n)653 1027 y Fn(\()p Fi(x)p Fn(\))p Fg(j)728 1008 y Fv(2)754 1027 y Fi(dx)h Fn(=)886 996 y(1)p 869 1017 V 869 1058 a Fg(j)p Fn(\003)p Fg(j)939 984 y Fe(X)938 1082 y Fh(n)p Fu(\025)p Fv(1)1012 1027 y Fi(\026)1039 1034 y Fh(n)1071 965 y Fe(Z)1096 1068 y Fh(t)p Fv(+\000\(\003\))1220 1027 y Fg(j)p Fn(\(1)d Fg(\000)i Fn(\001\))1385 1008 y Fv(1)p Fh(=)p Fv(2)1439 1027 y Fi( )1469 1034 y Fh(n)1492 1027 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1567 1008 y Fv(2)1593 1027 y Fi(dx)438 1171 y Fg(\024)508 1140 y Fn(1)p 491 1160 V 491 1202 a Fg(j)p Fn(\003)p Fg(j)561 1127 y Fe(X)561 1225 y Fh(n)p Fu(\025)p Fv(1)635 1171 y Fi(\026)662 1178 y Fh(n)693 1109 y Fe(Z)718 1212 y Fd(R)748 1202 y Ft(3)776 1171 y Fg(j)p Fn(\(1)e Fg(\000)h Fn(\001\))940 1152 y Fv(1)p Fh(=)p Fv(2)995 1171 y Fi( )1025 1178 y Fh(n)1048 1171 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1123 1152 y Fv(2)1149 1171 y Fi(dx)438 1312 y Fg(\024)j Fi(C)552 1281 y Fn(1)p 535 1302 V 535 1343 a Fg(j)p Fn(\003)p Fg(j)604 1312 y Fn(T)l(r[\(1)c Fg(\000)h Fn(\001\))e Fi(K)862 1319 y Fv(\003)888 1312 y Fn(])57 b Fg(\024)i Fi(C)q(;)37 1437 y Fn(b)q(ecause)19 b(of)e(\(3.10a\).)25 b(Let)18 b(us)g(no)o(w)f(pro)o(v)o(e)g(\(4.15\).)25 b(F)l(or)17 b(almost)g(ev)o(ery)g Fi(x)h Fn(in)g Fj(R)1457 1420 y Fv(3)1477 1437 y Fn(,)g(and)g(b)o(y)f(a)g(rep)q(eated)37 1493 y(use)f(of)f(the)g(Cauc)o(h)o(y-Sc)o(h)o(w)o(arz)g(inequalit)o(y)l (,)h(w)o(e)f(obtain)242 1547 y Fe(Z)267 1650 y Fh(Q)305 1608 y Fg(j)p Fn(\(1)9 b Fg(\000)h Fn(\001\))469 1590 y Fv(1)p Fh(=)p Fv(2)469 1620 y Fh(x)527 1608 y Fe(e)-28 b Fi(\032)547 1615 y Fv(\003)574 1608 y Fn(\()p Fi(x)9 b Fn(+)i Fi(y)r(;)d(y)r Fn(\))p Fg(j)g Fi(dy)283 1739 y Fn(=)360 1677 y Fe(Z)385 1780 y Fh(Q)415 1673 y Fe(\014)415 1700 y(\014)415 1727 y(\014)415 1755 y(\014)431 1695 y(X)430 1793 y Fh(n)p Fu(\025)p Fv(1)505 1739 y Fi(\025)531 1746 y Fh(n)562 1739 y Fn([\(1)h Fg(\000)i Fn(\001\))727 1720 y Fv(1)p Fh(=)p Fv(2)781 1739 y Fi(')811 1746 y Fh(n)834 1739 y Fn(]\()p Fi(x)f Fn(+)g Fi(y)r Fn(\))e Fi(')1026 1720 y Fu(\003)1026 1750 y Fh(n)1049 1739 y Fn(\()p Fi(y)r Fn(\))1109 1673 y Fe(\014)1109 1700 y(\014)1109 1727 y(\014)1109 1755 y(\014)1131 1739 y Fi(dy)283 1903 y Fg(\024)360 1841 y Fe(Z)385 1944 y Fh(Q)415 1838 y Fe(\022)449 1859 y(X)449 1957 y Fh(n)p Fu(\025)p Fv(1)523 1903 y Fi(\025)549 1910 y Fh(n)580 1903 y Fg(j)p Fn(\(1)i Fg(\000)g Fn(\001\))745 1884 y Fv(1)p Fh(=)p Fv(2)800 1903 y Fi(')830 1910 y Fh(n)853 1903 y Fn(\()p Fi(x)g Fn(+)g Fi(y)r Fn(\))p Fg(j)1007 1884 y Fv(2)1026 1838 y Fe(\023)1064 1836 y Ft(1)p 1064 1842 16 2 v 1064 1863 a(2)1094 1838 y Fe(\022)1128 1859 y(X)1128 1957 y Fh(n)p Fu(\025)p Fv(1)1202 1903 y Fi(\025)1228 1910 y Fh(n)1259 1903 y Fg(j)p Fi(')1302 1910 y Fh(n)1325 1903 y Fn(\()p Fi(y)r Fn(\))p Fg(j)1398 1884 y Fv(2)1417 1838 y Fe(\023)1455 1836 y Ft(1)p 1455 1842 V 1455 1863 a(2)1477 1903 y Fi(dy)283 2067 y Fg(\024)360 2002 y Fe(\022)394 2005 y(Z)419 2108 y Fh(Q)457 2023 y Fe(X)456 2121 y Fh(n)p Fu(\025)p Fv(1)530 2067 y Fi(\025)556 2074 y Fh(n)588 2067 y Fg(j)p Fn(\(1)f Fg(\000)i Fn(\001\))753 2048 y Fv(1)p Fh(=)p Fv(2)807 2067 y Fi(')837 2074 y Fh(n)860 2067 y Fn(\()p Fi(x)f Fn(+)h Fi(y)r Fn(\))p Fg(j)1015 2048 y Fv(2)1041 2067 y Fi(dy)1089 2002 y Fe(\023)1127 2000 y Ft(1)p 1127 2006 V 1127 2027 a(2)1157 2002 y Fe(\022)1190 2005 y(Z)1216 2108 y Fh(Q)1253 2023 y Fe(X)1253 2121 y Fh(n)p Fu(\025)p Fv(1)1327 2067 y Fi(\025)1353 2074 y Fh(n)1385 2067 y Fg(j)p Fi(')1428 2074 y Fh(n)1451 2067 y Fn(\()p Fi(y)r Fn(\))p Fg(j)1524 2048 y Fv(2)1550 2067 y Fi(dy)1598 2002 y Fe(\023)1636 2000 y Ft(1)p 1636 2006 V 1636 2027 a(2)1658 2067 y Fi(;)37 2210 y Fn(and)16 b(\(4.15\))d(follo)o(ws,)i (thanks)g(to)g(\(4.16\))e(and)i(\(4.17\).)108 2266 y(A)o(t)g(this)g (stage,)f(w)o(e)h(observ)o(e)g(that)g(\(4.14\))e(in)j(particular)g (yields)434 2363 y(\(1)10 b Fg(\000)g Fn(\001\))586 2344 y Fv(1)p Fh(=)p Fv(2)586 2374 y Fh(x)644 2363 y Fe(e)-28 b Fi(\032)664 2370 y Fv(\003)691 2363 y Fn(\()p Fi(x)10 b Fn(+)g Fi(y)r(;)e(y)r Fn(\))14 b(is)i(b)q(ounded)g(in)g Fi(L)1206 2344 y Fv(2)1206 2374 y Fh(x)1228 2363 y Fn(\()p Fj(R)1285 2344 y Fv(3)1305 2363 y Fn(;)8 b Fi(L)1357 2344 y Fv(2)1357 2374 y Fh(y)1377 2363 y Fn(\()p Fi(Q)p Fn(\)\))p Fi(:)279 b Fn(\(4.18\))37 2460 y(Therefore,)15 b(b)o(y)g(a)g(standard)g(in)o(terp)q(olation)h(argumen)o(t,)425 2556 y(\(1)10 b Fg(\000)g Fn(\001\))577 2538 y Fv(1)p Fh(=)p Fv(2)577 2568 y Fh(x)635 2556 y Fe(e)-28 b Fi(\032)655 2563 y Fv(\003)682 2556 y Fn(\()p Fi(x)10 b Fn(+)g Fi(y)r(;)e(y)r Fn(\))14 b(is)i(b)q(ounded)g(in)g Fi(L)1197 2538 y Fh(p)1197 2568 y(x)1219 2556 y Fn(\()p Fj(R)1276 2538 y Fv(3)1296 2556 y Fn(;)8 b Fi(L)1348 2538 y Fh(p)1366 2526 y Fp(0)1348 2568 y Fh(y)1378 2556 y Fn(\()p Fi(Q)p Fn(\)\))p Fi(;)278 b Fn(\(4.19\))37 2658 y(with)147 2640 y Fv(1)p 147 2647 18 2 v 147 2674 a Fh(p)180 2658 y Fn(+)236 2640 y Fv(1)p 231 2647 29 2 v 231 2674 a Fh(p)249 2664 y Fp(0)278 2658 y Fn(=)14 b(1,)h(and)h(for)f(ev)o(ery)h(2)d Fg(\024)h Fi(p)g Fg(\024)g Fn(+)p Fg(1)p Fn(.)22 b(In)16 b(particular,)h (extracting)e(a)h(further)f(subsequence)37 2733 y(if)j(necessary)l(,)h (w)o(e)e(ma)o(y)g(assume)g(that)g(\(1)11 b Fg(\000)h Fn(\001\))880 2709 y Fv(1)p Fh(=)p Fv(2)880 2739 y Fh(x)937 2733 y Fe(e)-28 b Fi(\032)957 2740 y Fv(\003)985 2733 y Fn(\()p Fi(x)11 b Fn(+)h Fi(y)r(;)c(y)r Fn(\))16 b(con)o(v)o(erges)h (to)g(\(1)11 b Fg(\000)h Fn(\001\))1609 2709 y Fv(1)p Fh(=)p Fv(2)1609 2739 y Fh(x)1666 2733 y Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)11 b Fn(+)h Fi(y)r(;)c(y)r Fn(\))37 2796 y(w)o(eakly)16 b(in)g Fi(L)272 2774 y Fh(p)272 2802 y(x)294 2796 y Fn(\()p Fj(R)351 2780 y Fv(3)370 2796 y Fn(;)8 b Fi(L)422 2774 y Fh(p)440 2762 y Fp(0)422 2802 y Fh(y)452 2796 y Fn(\()p Fi(Q)p Fn(\)\),)14 b(for)h(ev)o(ery)g(2)d Fg(\024)h Fi(p)g(<)g Fn(+)p Fg(1)p Fn(.)p eop %%Page: 34 34 34 33 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(34)108 199 y(b\))15 b(Pro)q(of)g(of)f(\(4.7\).)p 166 218 283 2 v 108 281 a(Let)f Fi(\022)i Fn(b)q(e)f(a)f(con)o(tin)o (uous)g(real-v)m(alued)i(function,)f(whic)o(h)g(is)g(compactly)f(supp)q (orted)h(in)g(the)f(unit)h(cub)q(e)37 337 y Fi(Q)p Fn(,)f(and)g(let)h (us)f(denote)g Fi(\022)470 344 y Fv(\003)510 337 y Fn(=)558 303 y Fe(P)606 351 y Fh(k)q Fu(2)p Fv(\003)682 337 y Fi(\022)q Fn(\()p Fg(\001)6 b(\000)g Fi(k)q Fn(\).)19 b(\(Note)12 b(that)g Fi(\022)1099 344 y Fv(\003)1139 337 y Fn(has)g(compact)h(supp)q(ort)g(in)h(\000\(\003\).\))k(Then,)37 394 y(on)e(the)f(one)g(hand,)h(w)o(e)e(ha)o(v)o(e)520 471 y(1)p 503 492 57 2 v 503 533 a Fg(j)p Fn(\003)p Fg(j)564 502 y Fn(T)l(r)611 512 y Fh(L)635 503 y Ft(2)653 512 y Fv(\()p Fd(R)697 503 y Ft(3)714 512 y Fv(\))730 502 y Fn([)p Fi(K)782 509 y Fv(\003)815 502 y Fi(\022)836 509 y Fv(\003)863 502 y Fn(])e(=)958 471 y(1)p 941 492 V 941 533 a Fg(j)p Fn(\003)p Fg(j)1011 440 y Fe(Z)1036 543 y Fd(R)1066 534 y Ft(3)1093 502 y Fi(\032)1116 509 y Fv(\003)1151 502 y Fi(\022)1172 509 y Fv(\003)1211 502 y Fn(=)1259 440 y Fe(Z)1285 543 y Fh(Q)1325 502 y Fe(e)-28 b Fi(\032)1345 509 y Fv(\003)1380 502 y Fi(\022)q(;)37 619 y Fn(and,)16 b(th)o(us,)547 728 y(lim)531 758 y Fv(\003)p Fu(!1)656 697 y Fn(1)p 639 718 V 639 759 a Fg(j)p Fn(\003)p Fg(j)701 728 y Fn(T)l(r)747 738 y Fh(L)771 729 y Ft(2)789 738 y Fv(\()p Fd(R)833 729 y Ft(3)850 738 y Fv(\))866 728 y Fn([)p Fi(K)918 735 y Fv(\003)951 728 y Fi(\022)972 735 y Fv(\003)999 728 y Fn(])c(=)1072 666 y Fe(Z)1098 769 y Fh(Q)1138 728 y Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)p Fn(\))8 b Fi(\022)q Fn(\()p Fi(x)p Fn(\))g Fi(dx:)375 b Fn(\(4.20\))37 845 y(On)16 b(the)g(other)e(hand,)i(w)o(e)f(no)o(w)f (pro)o(v)o(e)h(that)397 954 y(lim)381 985 y Fv(\003)p Fu(!1)506 924 y Fn(1)p 489 944 V 489 986 a Fg(j)p Fn(\003)p Fg(j)551 954 y Fn(T)l(r)598 964 y Fh(L)622 955 y Ft(2)639 964 y Fv(\()p Fd(R)683 955 y Ft(3)700 964 y Fv(\))716 954 y Fn([)p Fi(K)768 961 y Fv(\003)802 954 y Fi(\022)823 961 y Fv(\003)849 954 y Fn(])e(=)923 893 y Fe(Z)948 996 y Fh(Q)993 954 y Fi(dx)1051 893 y Fe(Z)1075 996 y Fh(Q)1103 986 y Fc(?)1134 954 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))g Fi(\022)q Fn(\()p Fi(x)p Fn(\))1438 924 y Fi(d\030)p 1408 944 106 2 v 1408 986 a Fn(\(2)p Fi(\031)r Fn(\))1495 973 y Fv(3)1518 954 y Fi(:)228 b Fn(\(4.21\))37 1074 y(Since)17 b Fi(\022)f Fn(is)f(arbitrary)l(,)f(comparing)h(\(4.20\))e(and)i(\(4.21\))e (completes)j(the)f(pro)q(of)f(of)g(\(4.7\).)k(W)l(e)d(no)o(w)g(pro)o(v) o(e)37 1131 y(\(4.21\),)g(and)h(w)o(e)g(b)q(egin)h(with)f(recalling)i (that)e(the)g(self-adjoin)o(t)h(op)q(erator)e(\(1)10 b Fg(\000)h Fn(\001\))1501 1114 y Fv(1)p Fh(=)p Fv(2)1556 1131 y Fi(K)1595 1138 y Fv(\003)1621 1131 y Fn(\(1)f Fg(\000)h Fn(\001\))1774 1114 y Fv(1)p Fh(=)p Fv(2)1844 1131 y Fn(is)37 1195 y(Hilb)q(ert-Sc)o(hmidt)16 b(\(and)d(ev)o(en)g (trace-class\))g(on)g Fi(L)903 1179 y Fv(2)923 1195 y Fn(\()p Fj(R)980 1179 y Fv(3)999 1195 y Fn(\),)g(with)g(k)o(ernel)h (\(1)6 b Fg(\000)g Fn(\001\))1420 1171 y Fv(1)p Fh(=)p Fv(2)1420 1201 y Fh(x)1474 1195 y Fn(\(1)g Fg(\000)g Fn(\001\))1618 1171 y Fv(1)p Fh(=)p Fv(2)1618 1201 y Fh(y)1672 1195 y Fi(\032)1695 1202 y Fv(\003)1722 1195 y Fn(\()p Fi(x;)i(y)r Fn(\))j(=)37 1217 y Fe(P)85 1265 y Fh(n)p Fu(\025)p Fv(1)162 1252 y Fi(\026)189 1259 y Fh(n)212 1252 y Fn([\(1)f Fg(\000)h Fn(\001\))378 1235 y Fv(1)p Fh(=)p Fv(2)433 1252 y Fi( )463 1259 y Fh(n)486 1252 y Fn(]\()p Fi(x)p Fn(\))d([\(1)g Fg(\000)j Fn(\001\))733 1235 y Fv(1)p Fh(=)p Fv(2)788 1252 y Fi( )818 1259 y Fh(n)841 1252 y Fn(])854 1235 y Fu(\003)873 1252 y Fn(\()p Fi(y)r Fn(\),)16 b(for)f(T)l(r)1079 1261 y Fh(L)1103 1252 y Ft(2)1120 1261 y Fv(\()p Fd(R)1164 1252 y Ft(3)1182 1261 y Fv(\))1198 1252 y Fn([\(1)9 b Fg(\000)j Fn(\001\))c Fi(K)1411 1259 y Fv(\003)1436 1252 y Fn(])14 b Fi(<)g Fn(+)p Fg(1)p Fn(.)24 b(Besides,)17 b(due)37 1308 y(to)d(the)f(fact)g (that)g Fi(\022)374 1315 y Fv(\003)415 1308 y Fn(is)h(a)f(con)o(tin)o (uous)h(function)h(with)f(compact)f(supp)q(ort,)h(it)g(is)g(a)f(kno)o (wn)h(fact)f(that)g(the)37 1364 y(self-adjoin)o(t)18 b(op)q(erator)e(\(1)11 b Fg(\000)h Fn(\001\))613 1348 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)695 1364 y Fi(\022)716 1371 y Fv(\003)743 1364 y Fn(\(1)e Fg(\000)i Fn(\001\))897 1348 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)996 1364 y Fn(is)18 b(Hilb)q(ert-Sc)o(hmidt)h(on)e Fi(L)1476 1348 y Fv(2)1496 1364 y Fn(\()p Fj(R)1553 1348 y Fv(3)1572 1364 y Fn(\),)g(whose)g(k)o (ernel)37 1421 y(is)f(denoted)g(b)o(y)f(\002)353 1428 y Fv(\003)380 1421 y Fn(\()p Fi(x;)8 b(y)r Fn(\).)18 b(With)e(these)f(observ)m(ations,)g(w)o(e)g(write)169 1511 y(T)l(r)216 1521 y Fh(L)240 1512 y Ft(2)257 1521 y Fv(\()p Fd(R)301 1512 y Ft(3)318 1521 y Fv(\))334 1511 y Fn([)p Fi(K)386 1518 y Fv(\003)420 1511 y Fi(\022)441 1518 y Fv(\003)467 1511 y Fn(])305 1590 y(=)42 b(T)l(r)429 1599 y Fh(L)453 1590 y Ft(2)470 1599 y Fv(\()p Fd(R)514 1590 y Ft(3)531 1599 y Fv(\))547 1590 y Fn([\(1)9 b Fg(\000)i Fn(\001\))712 1571 y Fv(1)p Fh(=)p Fv(2)766 1590 y Fi(K)805 1597 y Fv(\003)831 1590 y Fn(\(1)f Fg(\000)g Fn(\001\))983 1571 y Fv(1)p Fh(=)p Fv(2)1045 1590 y Fn(\(1)g Fg(\000)g Fn(\001\))1197 1571 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1279 1590 y Fi(\022)1300 1597 y Fv(\003)1327 1590 y Fn(\(1)g Fg(\000)g Fn(\001\))1479 1571 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1561 1590 y Fn(])305 1686 y(=)382 1624 y Fe(Z)5 b(Z)457 1727 y Fd(R)487 1718 y Ft(3)505 1727 y Fu(\002)p Fd(R)562 1718 y Ft(3)582 1649 y Fe(\002)601 1686 y Fn(\(1)10 b Fg(\000)g Fn(\001\))753 1667 y Fv(1)p Fh(=)p Fv(2)753 1697 y Fh(x)808 1686 y Fn(\(1)f Fg(\000)i Fn(\001\))960 1667 y Fv(1)p Fh(=)p Fv(2)960 1697 y Fh(y)1014 1686 y Fi(\032)1037 1693 y Fv(\003)1064 1686 y Fn(\()p Fi(x;)d(y)r Fn(\))1171 1649 y Fe(\003)1196 1686 y Fn(\002)1231 1693 y Fv(\003)1258 1686 y Fn(\()p Fi(x;)g(y)r Fn(\))g Fi(dxdy)305 1813 y Fn(=)382 1770 y Fe(X)382 1868 y Fh(n)p Fu(\025)p Fv(1)456 1813 y Fi(\026)483 1820 y Fh(n)514 1752 y Fe(Z)539 1855 y Fd(R)569 1845 y Ft(3)604 1813 y Fi(dx)p Fn([\(1)h Fg(\000)i Fn(\001\))819 1795 y Fv(1)p Fh(=)p Fv(2)873 1813 y Fi( )903 1820 y Fh(n)926 1813 y Fn(]\()p Fi(x)p Fn(\))1001 1749 y Fe(\022)1034 1752 y(Z)1059 1855 y Fd(R)1089 1845 y Ft(3)1116 1813 y Fn(\002)1151 1820 y Fv(\003)1178 1813 y Fn(\()p Fi(x;)d(y)r Fn(\))g([\(1)f Fg(\000)k Fn(\001\))1456 1795 y Fv(1)p Fh(=)p Fv(2)1510 1813 y Fi( )1540 1820 y Fh(n)1563 1813 y Fn(])1576 1795 y Fu(\003)1596 1813 y Fn(\()p Fi(y)r Fn(\))d Fi(dy)1712 1749 y Fe(\023)1759 1813 y Fn(\(4.22\))37 1948 y(thanks)23 b(to)f(F)l(ubini's)i(theorem.)43 b(W)l(e)22 b(shall)i(no)o(w)e(use)i(the)e(explicit)j(form)d(of)h(the)f (Hilb)q(ert-Sc)o(hmidt)37 2004 y(k)o(ernel)g(of)e(\(1)14 b Fg(\000)g Fn(\001\))394 1988 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)476 2004 y Fi(\022)497 2011 y Fv(\003)524 2004 y Fn(\(1)f Fg(\000)h Fn(\001\))683 1988 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)765 2004 y Fn(.)37 b(Indeed,)23 b(recalling)f(that,)f(b)o (y)g(de\014nition)i(of)d(\(1)13 b Fg(\000)h Fn(\001\))1780 1988 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1862 2004 y Fn(,)37 2061 y Fg(F)5 b Fn([\(1)i Fg(\000)h Fn(\001\))235 2044 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)317 2061 y Fi(')p Fn(]\()p Fi(x)p Fn(\))j(=)i(\(1)7 b(+)i Fg(j)p Fi(x)p Fg(j)625 2044 y Fv(2)644 2061 y Fn(\))662 2044 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)744 2061 y Fg(F)c Fi(')p Fn(\()p Fi(x)p Fn(\))12 b(\(where)i Fg(F)k Fn(denotes)d(the)f(F)l(ourier)g(transform\),)e(it)j (is)f(not)37 2117 y(di\016cult)j(to)e(v)o(erify)g(that,)f(if)i Fi(')f Fn(is,)g(sa)o(y)l(,)f(in)j(the)e(Sc)o(h)o(w)o(artz)f(class,)h(w) o(e)g(ha)o(v)o(e)720 2208 y(\(1)9 b Fg(\000)i Fn(\001\))872 2189 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)954 2208 y Fi(')h Fn(=)h Fi(G)1080 2215 y Fv(1)1110 2208 y Fi(?)d(';)573 b Fn(\(4.23\))37 2304 y(where)13 b Fi(G)202 2311 y Fv(1)234 2304 y Fn(is)f(a)g(function)h(in)g Fi(L)567 2287 y Fv(1)587 2304 y Fn(\()p Fj(R)644 2287 y Fv(3)663 2304 y Fn(\))f(whose)g(F)l (ourier)h(transform)e(is)h(simply)i(the)e(function)h(\(1)t(+)t Fg(j)p Fi(x)p Fg(j)1756 2287 y Fv(2)1775 2304 y Fn(\))1793 2287 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)37 2360 y Fn(\()p Fg(2)k Fi(L)133 2344 y Fh(p)153 2360 y Fn(\()p Fj(R)210 2344 y Fv(3)230 2360 y Fn(\),)g(for)g(ev)o(ery)h(3)e Fi(<)i(p)e Fg(\024)h Fn(+)p Fg(1)p Fn(\).)28 b(The)18 b(function)h Fi(G)1108 2367 y Fv(1)1145 2360 y Fn(is)f(a)f(sp)q(ecial)j (kind)f(of)e(Bessel)i(functions,)37 2417 y(and,)d(from)e([53)o(],)h (for)g(example,)g(w)o(e)h(kno)o(w)e(that)h Fi(G)926 2424 y Fv(1)961 2417 y Fn(is)h(a)f(non-negativ)o(e)g(radially)i(symmetric)e (function,)37 2473 y(suc)o(h)h(that)496 2571 y Fi(G)532 2578 y Fv(1)551 2571 y Fn(\()p Fi(x)p Fn(\))c Fg(\024)h Fi(C)705 2578 y Fv(1)741 2571 y Fn(exp\()p Fg(\000)868 2540 y Fn(1)p 868 2561 23 2 v 868 2602 a(2)903 2571 y Fg(j)p Fi(x)p Fg(j)p Fn(\))p Fi(;)19 b Fn(for)27 b Fg(j)p Fi(x)p Fg(j)12 b Fn(large)j(enough)q Fi(;)37 2672 y Fn(and)544 2776 y Fi(G)580 2783 y Fv(1)599 2776 y Fn(\()p Fi(x)p Fn(\))d(=)h Fi(C)753 2783 y Fv(2)810 2745 y Fn(1)p 786 2765 71 2 v 786 2807 a Fg(j)p Fi(x)p Fg(j)838 2794 y Fv(2)872 2776 y Fn(+)d Fi(o)947 2712 y Fe(\022)1010 2745 y Fn(1)p 986 2765 V 986 2807 a Fg(j)p Fi(x)p Fg(j)1038 2794 y Fv(2)1061 2712 y Fe(\023)1103 2776 y Fi(;)19 b Fn(as)28 b Fg(j)p Fi(x)p Fg(j)12 b(!)h Fn(0)8 b Fi(;)p eop %%Page: 35 35 35 34 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(35)37 199 y(for)17 b(some)g(p)q(ositiv)o(e)h(constan)o (ts)e Fi(C)630 206 y Fv(1)667 199 y Fn(and)i Fi(C)790 206 y Fv(2)810 199 y Fn(.)26 b(In)17 b(particular,)h Fi(G)1168 206 y Fv(1)1205 199 y Fn(actually)g(b)q(elongs)g(to)e Fi(L)1633 183 y Fv(3)p Fh(=)p Fv(2)p Fh(;)5 b Fu(1)1739 199 y Fn(\()p Fj(R)1796 183 y Fv(3)1816 199 y Fn(\))11 b Fg(\\)37 256 y Fi(L)68 239 y Fh(p)88 256 y Fn(\()p Fj(R)145 239 y Fv(3)165 256 y Fn(\),)j(for)h(ev)o(ery)g(1)d Fg(\024)h Fi(p)g(<)572 238 y Fv(3)p 572 245 18 2 v 572 271 a(2)594 256 y Fn(.)108 312 y(Therefore,)f(for)e(an)o(y)h(con)o(tin) o(uous)g(function)h Fi(#)f Fn(with,)h(sa)o(y)l(,)f(a)g(compact)g(supp)q (ort,)h(the)f(Sc)o(h)o(w)o(artz)f(k)o(ernel)37 369 y(of)15 b(\(1)10 b Fg(\000)g Fn(\001\))241 352 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)323 369 y Fi(#)p Fn(\(1)g Fg(\000)g Fn(\001\))502 352 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)599 369 y Fn(ma)o(y)15 b(b)q(e)h(written)f(as)975 332 y Fe(R)996 385 y Fd(R)1026 375 y Ft(3)1053 369 y Fi(G)1089 376 y Fv(1)1109 369 y Fn(\()p Fi(x)10 b Fg(\000)g Fi(z)r Fn(\))e Fi(#)p Fn(\()p Fi(z)r Fn(\))g Fi(G)1387 376 y Fv(1)1405 369 y Fn(\()p Fi(z)k Fg(\000)e Fi(y)r Fn(\))e Fi(dz)r Fn(.)108 425 y(Then,)15 b(b)o(y)h(de\014nition)h(of)d(\002)591 432 y Fv(\003)618 425 y Fn(,)418 484 y Fe(Z)443 587 y Fd(R)473 577 y Ft(3)500 546 y Fn(\002)535 553 y Fv(\003)562 546 y Fn(\()p Fi(x;)8 b(y)r Fn(\))g([\(1)f Fg(\000)k Fn(\001\))840 527 y Fv(1)p Fh(=)p Fv(2)894 546 y Fi( )924 553 y Fh(n)947 546 y Fn(])960 527 y Fu(\003)980 546 y Fn(\()p Fi(y)r Fn(\))d Fi(dy)459 656 y Fn(=)42 b(\(1)10 b Fg(\000)g Fn(\001\))688 637 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)770 656 y Fi(\022)791 663 y Fv(\003)825 656 y Fn(\(1)g Fg(\000)g Fn(\001\))977 637 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1067 605 y Fe(\020)1094 656 y Fn([\(1)f Fg(\000)i Fn(\001\))1259 637 y Fv(1)p Fh(=)p Fv(2)1313 656 y Fi( )1343 663 y Fh(n)1366 656 y Fn(])1379 637 y Fu(\003)1399 605 y Fe(\021)1433 656 y Fn(\()p Fi(x)p Fn(\))459 746 y(=)42 b(\(1)10 b Fg(\000)g Fn(\001\))688 727 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)770 746 y Fn(\()p Fi(\022)809 753 y Fv(\003)843 746 y Fi( )875 727 y Fu(\003)873 757 y Fh(n)896 746 y Fn(\))58 b(=)g Fi(G)1101 753 y Fv(1)1131 746 y Fi(?)10 b Fn(\()p Fi(\022)1203 753 y Fv(\003)1237 746 y Fi( )1269 727 y Fu(\003)1267 757 y Fh(n)1290 746 y Fn(\))p Fi(:)37 848 y Fn(Th)o(us,)15 b(using)h(again)f(F)l(ubini's)i(theorem,)260 929 y Fe(X)260 1027 y Fh(n)p Fu(\025)p Fv(1)334 972 y Fi(\026)361 979 y Fh(n)392 910 y Fe(Z)418 1013 y Fd(R)448 1004 y Ft(3)483 972 y Fi(dx)p Fn([\(1)8 b Fg(\000)j Fn(\001\))697 953 y Fv(1)p Fh(=)p Fv(2)751 972 y Fi( )781 979 y Fh(n)804 972 y Fn(]\()p Fi(x)p Fn(\))887 908 y Fe(\022)919 910 y(Z)945 1013 y Fd(R)975 1004 y Ft(3)1002 972 y Fn(\002)1037 979 y Fv(\003)1064 972 y Fn(\()p Fi(x;)d(y)r Fn(\))g([\(1)f Fg(\000)k Fn(\001\))1342 953 y Fv(1)p Fh(=)p Fv(2)1396 972 y Fi( )1426 979 y Fh(n)1449 972 y Fn(])1462 953 y Fu(\003)1482 972 y Fn(\()p Fi(y)r Fn(\))d Fi(dy)1598 908 y Fe(\023)301 1120 y Fn(=)378 1058 y Fe(Z)e(Z)454 1161 y Fd(R)484 1151 y Ft(3)502 1161 y Fu(\002)p Fd(R)559 1151 y Ft(3)586 1120 y Fi(G)622 1127 y Fv(1)642 1120 y Fn(\()p Fi(x)k Fg(\000)g Fi(y)r Fn(\))p Fi(\022)804 1127 y Fv(\003)831 1120 y Fn(\()p Fi(y)r Fn(\))891 1056 y Fe(\022)924 1076 y(X)923 1174 y Fh(n)p Fu(\025)p Fv(1)997 1120 y Fi(\026)1024 1127 y Fh(n)1048 1120 y Fn([\(1)f Fg(\000)i Fn(\001\))1213 1101 y Fv(1)p Fh(=)p Fv(2)1267 1120 y Fi( )1297 1127 y Fh(n)1321 1120 y Fn(]\()p Fi(x)p Fn(\))d Fi( )1436 1101 y Fu(\003)1434 1131 y Fh(n)1455 1120 y Fn(\()p Fi(y)r Fn(\))1515 1056 y Fe(\023)1555 1120 y Fi(dxdy)301 1263 y Fn(=)378 1201 y Fe(Z)e(Z)454 1304 y Fd(R)484 1295 y Ft(3)502 1304 y Fu(\002)p Fd(R)559 1295 y Ft(3)586 1263 y Fi(G)622 1270 y Fv(1)642 1263 y Fn(\()p Fi(x)k Fg(\000)g Fi(y)r Fn(\))p Fi(\022)804 1270 y Fv(\003)831 1263 y Fn(\()p Fi(y)r Fn(\)\(1)e Fg(\000)j Fn(\001\))1042 1244 y Fv(1)p Fh(=)p Fv(2)1042 1274 y Fh(x)1096 1263 y Fi(\032)1119 1270 y Fv(\003)1146 1263 y Fn(\()p Fi(x;)d(y)r Fn(\))g Fi(dxdy)r(:)37 1392 y Fn(Therefore,)15 b(w)o(e)g(deduce,)h(comparing)g(with)f(\(4.22\),)122 1484 y(1)p 105 1504 57 2 v 105 1546 a Fg(j)p Fn(\003)p Fg(j)167 1515 y Fn(T)l(r)214 1525 y Fh(L)238 1515 y Ft(2)255 1525 y Fv(\()p Fd(R)299 1515 y Ft(3)316 1525 y Fv(\))332 1515 y Fn([)p Fi(K)384 1522 y Fv(\003)418 1515 y Fi(\022)439 1522 y Fv(\003)466 1515 y Fn(])41 b(=)619 1484 y(1)p 602 1504 V 602 1546 a Fg(j)p Fn(\003)p Fg(j)671 1453 y Fe(Z)5 b(Z)747 1556 y Fd(R)777 1546 y Ft(3)794 1556 y Fu(\002)p Fd(R)851 1546 y Ft(3)879 1515 y Fi(G)915 1522 y Fv(1)935 1515 y Fn(\()p Fi(x)k Fg(\000)i Fi(y)r Fn(\))p Fi(\022)1097 1522 y Fv(\003)1123 1515 y Fn(\()p Fi(y)r Fn(\)\(1)e Fg(\000)h Fn(\001\))1334 1496 y Fv(1)p Fh(=)p Fv(2)1334 1526 y Fh(x)1389 1515 y Fi(\032)1412 1522 y Fv(\003)1439 1515 y Fn(\()p Fi(x;)e(y)r Fn(\))g Fi(dxdy)520 1638 y Fn(=)619 1607 y(1)p 602 1627 V 602 1669 a Fg(j)p Fn(\003)p Fg(j)671 1576 y Fe(Z)696 1679 y Fv(\000\(\003\))787 1638 y Fi(dy)842 1576 y Fe(Z)868 1679 y Fd(R)898 1670 y Ft(3)925 1638 y Fi(G)961 1645 y Fv(1)981 1638 y Fn(\()p Fi(x)h Fg(\000)i Fi(y)r Fn(\))p Fi(\022)1143 1645 y Fv(\003)1169 1638 y Fn(\()p Fi(y)r Fn(\)\(1)e Fg(\000)h Fn(\001\))1380 1619 y Fv(1)p Fh(=)p Fv(2)1380 1649 y Fh(x)1435 1638 y Fi(\032)1458 1645 y Fv(\003)1485 1638 y Fn(\()p Fi(x;)e(y)r Fn(\))g Fi(dx)520 1766 y Fn(=)619 1735 y(1)p 602 1755 V 602 1797 a Fg(j)p Fn(\003)p Fg(j)672 1723 y Fe(X)671 1821 y Fh(k)q Fu(2)p Fv(\003)746 1704 y Fe(Z)771 1807 y Fh(Q)816 1766 y Fi(dy)871 1704 y Fe(Z)896 1807 y Fd(R)926 1798 y Ft(3)954 1766 y Fi(G)990 1773 y Fv(1)1009 1766 y Fn(\()p Fi(x)i Fg(\000)g Fi(k)i Fg(\000)e Fi(y)r Fn(\))p Fi(\022)q Fn(\()p Fi(y)r Fn(\)\(1)f Fg(\000)h Fn(\001\))1464 1747 y Fv(1)p Fh(=)p Fv(2)1464 1777 y Fh(x)1519 1766 y Fi(\032)1542 1773 y Fv(\003)1569 1766 y Fn(\()p Fi(x;)e(y)j Fn(+)f Fi(k)q Fn(\))e Fi(dx)520 1907 y Fn(=)619 1876 y(1)p 602 1896 V 602 1938 a Fg(j)p Fn(\003)p Fg(j)672 1863 y Fe(X)671 1962 y Fh(k)q Fu(2)p Fv(\003)746 1845 y Fe(Z)771 1948 y Fh(Q)816 1907 y Fi(dy)871 1845 y Fe(Z)896 1948 y Fd(R)926 1938 y Ft(3)954 1907 y Fi(G)990 1914 y Fv(1)1009 1907 y Fn(\()p Fi(x)i Fg(\000)g Fi(y)r Fn(\))p Fi(\022)q Fn(\()p Fi(y)r Fn(\)\(1)f Fg(\000)i Fn(\001\))1384 1888 y Fv(1)p Fh(=)p Fv(2)1384 1918 y Fh(x)1438 1907 y Fi(\032)1461 1914 y Fv(\003)1488 1907 y Fn(\()p Fi(x)f Fn(+)h Fi(k)q(;)d(y)j Fn(+)f Fi(k)q Fn(\))e Fi(dx)520 2047 y Fn(=)597 1985 y Fe(Z)622 2089 y Fh(Q)667 2047 y Fi(dy)722 1985 y Fe(Z)747 2089 y Fd(R)777 2079 y Ft(3)805 2047 y Fi(G)841 2054 y Fv(1)860 2047 y Fn(\()p Fi(x)i Fg(\000)g Fi(y)r Fn(\))p Fi(\022)q Fn(\()p Fi(y)r Fn(\)\(1)f Fg(\000)i Fn(\001\))1235 2029 y Fv(1)p Fh(=)p Fv(2)1235 2059 y Fh(x)1292 2047 y Fe(e)-28 b Fi(\032)1312 2054 y Fv(\003)1339 2047 y Fn(\()p Fi(x;)8 b(y)r Fn(\))g Fi(dx)520 2173 y Fn(=)597 2112 y Fe(Z)622 2215 y Fh(Q)667 2173 y Fi(dy)722 2112 y Fe(Z)747 2215 y Fd(R)777 2205 y Ft(3)805 2173 y Fi(G)841 2180 y Fv(1)860 2173 y Fn(\()p Fi(x)p Fn(\))p Fi(\022)q Fn(\()p Fi(y)r Fn(\)\(1)h Fg(\000)h Fn(\001\))1155 2155 y Fv(1)p Fh(=)p Fv(2)1155 2185 y Fh(x)1213 2173 y Fe(e)-28 b Fi(\032)1233 2180 y Fv(\003)1260 2173 y Fn(\()p Fi(x)9 b Fn(+)i Fi(y)r(;)d(y)r Fn(\))g Fi(dx:)37 2320 y Fn(W)l(e)k(no)o(w)f (mak)o(e)h(use)g(of)f(the)h(t)o(w)o(o)e(facts)h(that)g(\(1)s Fg(\000)s Fn(\001\))936 2296 y Fv(1)p Fh(=)p Fv(2)936 2326 y Fh(x)994 2320 y Fe(e)-28 b Fi(\032)1014 2327 y Fv(\003)1041 2320 y Fn(\()p Fi(x)s Fn(+)s Fi(y)r(;)8 b(y)r Fn(\))i(con)o(v)o(erges)h(to)g(\(1)s Fg(\000)s Fn(\001\))1613 2296 y Fv(1)p Fh(=)p Fv(2)1613 2326 y Fh(x)1671 2320 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(x)s Fn(+)s Fi(y)r(;)8 b(y)r Fn(\),)37 2384 y(w)o(eakly)15 b(in,)g(sa)o(y)e Fi(L)360 2368 y Fv(4)360 2395 y Fh(x)382 2384 y Fn(\()p Fj(R)439 2368 y Fv(3)459 2384 y Fn(;)8 b Fi(L)511 2360 y Fv(4)p Fh(=)p Fv(3)511 2390 y Fh(y)565 2384 y Fn(\()p Fi(Q)p Fn(\)\),)13 b(and)i(that)e Fi(G)p Fn(\()p Fi(x)p Fn(\))8 b Fi(\022)q Fn(\()p Fi(y)r Fn(\))13 b(b)q(elongs)i(to)f Fi(L)1315 2360 y Fv(4)p Fh(=)p Fv(3)1315 2390 y Fh(x)1370 2384 y Fn(\()p Fj(R)1427 2368 y Fv(3)1446 2384 y Fn(\))8 b Fg(\002)h Fi(L)1547 2368 y Fv(4)1547 2395 y Fh(y)1568 2384 y Fn(\()p Fi(Q)p Fn(\).)18 b(Therefore,)233 2514 y(lim)217 2544 y Fv(\003)p Fu(!1)342 2483 y Fn(1)p 325 2503 V 325 2545 a Fg(j)p Fn(\003)p Fg(j)387 2514 y Fn(T)l(r)434 2523 y Fh(L)458 2514 y Ft(2)475 2523 y Fv(\()p Fd(R)519 2514 y Ft(3)536 2523 y Fv(\))552 2514 y Fn([)p Fi(K)604 2521 y Fv(\003)637 2514 y Fi(\022)658 2521 y Fv(\003)685 2514 y Fn(])41 b(=)816 2452 y Fe(Z)842 2555 y Fh(Q)887 2514 y Fi(dy)942 2452 y Fe(Z)967 2555 y Fd(R)997 2545 y Ft(3)1024 2514 y Fi(G)1060 2521 y Fv(1)1080 2514 y Fn(\()p Fi(x)p Fn(\))p Fi(\022)q Fn(\()p Fi(y)r Fn(\)\(1)8 b Fg(\000)j Fn(\001\))1375 2495 y Fv(1)p Fh(=)p Fv(2)1375 2525 y Fh(x)1432 2514 y Fe(e)-28 b Fi(\032)q Fn(\()p Fi(x)10 b Fn(+)g Fi(y)r(;)e(y)r Fn(\))g Fi(dx)739 2640 y Fn(=)816 2578 y Fe(Z)842 2681 y Fh(Q)887 2640 y Fi(dy)942 2578 y Fe(Z)967 2681 y Fd(R)997 2671 y Ft(3)1024 2640 y Fi(G)1060 2647 y Fv(1)1080 2640 y Fn(\()p Fi(x)i Fg(\000)g Fi(y)r Fn(\))p Fi(\022)q Fn(\()p Fi(y)r Fn(\)\(1)f Fg(\000)h Fn(\001\))1454 2621 y Fv(1)p Fh(=)p Fv(2)1454 2651 y Fh(x)1512 2640 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))g Fi(dx)739 2766 y Fn(=)816 2704 y Fe(Z)842 2808 y Fh(Q)887 2766 y Fi(dx)945 2704 y Fe(Z)969 2808 y Fh(Q)997 2798 y Fc(?)1024 2766 y Fi(\022)q Fn(\()p Fi(x)p Fn(\))j Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))1332 2736 y Fi(d\030)p 1302 2756 106 2 v 1302 2797 a Fn(\(2)p Fi(\031)r Fn(\))1389 2784 y Fv(3)1412 2766 y Fi(:)p eop %%Page: 36 36 36 35 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(36)37 199 y(Indeed,)16 b(w)o(e)e(\014rst)g(observ)o(e)g (that)f(the)i(self-adjoin)o(t)g(op)q(erator)e(\(1)7 b Fg(\000)i Fn(\001\))1255 183 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1351 199 y Fn(is)15 b(b)q(ounded)h(on)e Fi(L)1674 183 y Fv(2)1693 199 y Fn(\()p Fj(R)1750 183 y Fv(3)1770 199 y Fn(\))g(and)37 256 y(comm)o(utes)k(with)h(the)f(group)g(of)g(the) g(translations)g(of)g Fj(Z)1048 239 y Fv(3)1068 256 y Fn(.)29 b(Therefore,)18 b(w)o(e)g(ma)o(y)g(apply)h(the)f(abstract)37 312 y(Blo)q(c)o(h)j(w)o(a)o(v)o(e)e(decomp)q(osition)i(\(that)e(is)i (explained)h(in)e(Section)h(3.2\))e(to)g(\(1)12 b Fg(\000)i Fn(\001\))1505 296 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1587 312 y Fn(,)21 b(and)f(w)o(e)f(ha)o(v)o(e)37 377 y(\(1)10 b Fg(\000)g Fn(\001\))189 360 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)284 377 y Fn(=)332 340 y Fe(R)354 393 y Fh(Q)382 383 y Fc(?)401 377 y Fn(\(1)g Fg(\000)g Fn(\001\))553 353 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)553 391 y Fh(\030)672 356 y(d\030)p 648 366 84 2 v 648 393 a Fv(\(2)p Fh(\031)q Fv(\))715 384 y Ft(3)737 377 y Fn(,)k(with)i(the)f(op)q(erator)g(\(1)9 b Fg(\000)i Fn(\001\))1282 353 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1282 391 y Fh(\030)1379 377 y Fn(b)q(eing)16 b(de\014ned)h(b)o(y) 240 439 y Fe(\020)267 489 y Fi(U)5 b Fn(\(1)10 b Fg(\000)g Fn(\001\))455 470 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)537 489 y Fi(')567 439 y Fe(\021)594 521 y Fh(\030)621 489 y Fn(\()p Fi(x)p Fn(\))i(=)h(\(1)c Fg(\000)i Fn(\001\))895 465 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)895 504 y Fh(\030)977 489 y Fn(\()p Fi(U)5 b(')p Fn(\))1079 496 y Fh(\030)1109 489 y Fn(=)1157 427 y Fe(Z)1182 530 y Fh(Q)1220 489 y Fi(G)1256 496 y Fh(\030)1274 489 y Fn(\()p Fi(x)10 b Fg(\000)h Fi(y)r Fn(\))j(\()p Fi(U)5 b(')p Fn(\))1531 503 y Fh(\030)1558 489 y Fn(\()p Fi(y)r Fn(\))j Fi(dy)37 600 y Fn(thanks)14 b(to)g(\(4.23\),)e(for)i(ev)o(ery)g Fi(')g Fn(in)h Fi(L)697 584 y Fv(2)716 600 y Fn(\()p Fj(R)773 584 y Fv(3)793 600 y Fn(\),)f(and)g(for)g(almost)f(ev)o(ery)h Fi(x)g Fn(in)h Fi(Q)g Fn(and)f Fi(\030)i Fn(in)f Fi(Q)1612 584 y Fh(?)1632 600 y Fn(,)f(and)g(where,)37 657 y(according)i(to)f (\(4.23\),)669 742 y Fi(G)705 749 y Fh(\030)724 742 y Fn(\()p Fi(t)p Fn(\))d(=)846 698 y Fe(X)836 800 y Fh(k)q Fu(2)p Fd(Z)904 790 y Ft(3)929 742 y Fi(e)950 723 y Fu(\000)p Fh(i)6 b(k)q Fu(\001)p Fh(\030)1043 742 y Fi(G)1079 749 y Fv(1)1098 742 y Fn(\()p Fi(t)11 b Fn(+)f Fi(k)q Fn(\))p Fi(:)37 881 y Fn(\(Note)15 b(that,)g(from)f(the)i(de\014nition)h(of)e (\()p Fg(\000)p Fn(\001\))826 888 y Fh(\030)844 881 y Fn(,)h(\(1)9 b Fg(\000)i Fn(\001\))1025 857 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1025 895 y Fh(\030)1120 881 y Fn(=)i(\(1)d Fg(\000)g Fn(\001)1302 888 y Fh(\030)1321 881 y Fn(\))1339 864 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1421 881 y Fn(.\))20 b(Finally)l(,)d(w)o(e)e(conclude)37 937 y(as)g(follo)o(ws)124 963 y Fe(Z)150 1066 y Fh(Q)195 1025 y Fi(dy)250 963 y Fe(Z)275 1066 y Fd(R)305 1057 y Ft(3)332 1025 y Fi(G)368 1032 y Fv(1)388 1025 y Fn(\()p Fi(x)10 b Fg(\000)g Fi(y)r Fn(\))p Fi(\022)q Fn(\()p Fi(y)r Fn(\)\(1)f Fg(\000)h Fn(\001\))762 1006 y Fv(1)p Fh(=)p Fv(2)762 1036 y Fh(x)820 1025 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))g Fi(dx)255 1151 y Fn(=)303 1089 y Fe(Z)328 1193 y Fh(Q)365 1151 y Fi(\022)q Fn(\()p Fi(y)r Fn(\))g Fi(dy)510 1089 y Fe(Z)535 1193 y Fh(Q)580 1151 y Fi(dx)638 1089 y Fe(Z)663 1193 y Fh(Q)691 1183 y Fc(?)718 1151 y Fi(G)754 1158 y Fh(\030)773 1151 y Fn(\()p Fi(x)h Fg(\000)i Fi(y)r Fn(\)[\(1)e Fg(\000)h Fn(\001)1060 1158 y Fh(\030)1079 1151 y Fn(\))1097 1133 y Fv(1)p Fh(=)p Fv(2)1097 1163 y Fh(x)1154 1151 y Fe(e)-27 b Fi(\032)p Fn(]\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\))1378 1121 y Fi(d\030)p 1348 1141 106 2 v 1348 1183 a Fn(\(2)p Fi(\031)r Fn(\))1435 1169 y Fv(3)255 1278 y Fn(=)303 1216 y Fe(Z)328 1319 y Fh(Q)365 1278 y Fi(\022)q Fn(\()p Fi(y)r Fn(\))g Fi(dy)510 1235 y Fe(X)510 1333 y Fh(n)p Fu(\025)p Fv(1)604 1216 y Fe(Z)e(Z)680 1319 y Fh(Q)p Fu(\002)p Fh(Q)763 1310 y Fc(?)791 1278 y Fi(G)827 1285 y Fh(\030)845 1278 y Fn(\()p Fi(x)k Fg(\000)g Fi(y)r Fn(\))p Fi(\025)1012 1285 y Fh(n)1036 1278 y Fn(\()p Fi(\030)r Fn(\)[\(1)f Fg(\000)h Fn(\001\))1258 1254 y Fv(1)p Fh(=)p Fv(2)1258 1293 y Fh(\030)1313 1278 y Fi(u)1339 1285 y Fh(n)1362 1278 y Fn(\()p Fi(\030)r(;)e(x)p Fn(\)])g Fi(u)1514 1259 y Fu(\003)1514 1289 y Fh(n)1535 1278 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1654 1247 y Fi(d\030)r(dx)p 1649 1268 V 1649 1309 a Fn(\(2)p Fi(\031)r Fn(\))1736 1296 y Fv(3)255 1425 y Fn(=)303 1382 y Fe(X)303 1480 y Fh(n)p Fu(\025)p Fv(1)397 1364 y Fe(Z)d(Z)473 1467 y Fh(Q)p Fu(\002)p Fh(Q)556 1457 y Fc(?)583 1425 y Fi(\022)q Fn(\()p Fi(y)r Fn(\))p Fi(\025)691 1432 y Fh(n)715 1425 y Fn(\()p Fi(\030)r Fn(\))j Fi(u)807 1407 y Fu(\003)807 1437 y Fh(n)830 1425 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))933 1361 y Fe(\022)965 1364 y(Z)990 1467 y Fh(Q)1027 1425 y Fi(G)1063 1432 y Fh(\030)1082 1425 y Fn(\()p Fi(x)i Fg(\000)g Fi(y)r Fn(\)\(1)f Fg(\000)i Fn(\001\))1375 1401 y Fv(1)p Fh(=)p Fv(2)1375 1440 y Fh(\030)1429 1425 y Fi(u)1455 1432 y Fh(n)1479 1425 y Fn(\()p Fi(\030)r(;)d(x)p Fn(\))g Fi(dx)1642 1361 y Fe(\023)1684 1395 y Fi(d\030)r(dy)p 1678 1415 V 1678 1457 a Fn(\(2)p Fi(\031)r Fn(\))1765 1443 y Fv(3)255 1569 y Fn(=)303 1507 y Fe(Z)328 1610 y Fh(Q)365 1569 y Fi(\022)q Fn(\()p Fi(y)r Fn(\))g Fi(dy)510 1526 y Fe(X)510 1624 y Fh(n)p Fu(\025)p Fv(1)584 1507 y Fe(Z)609 1610 y Fh(Q)637 1601 y Fc(?)665 1569 y Fi(\025)691 1576 y Fh(n)715 1569 y Fn(\()p Fi(\030)r Fn(\))g Fg(j)p Fi(u)820 1576 y Fh(n)842 1569 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))p Fg(j)958 1550 y Fv(2)1018 1538 y Fi(d\030)p 988 1559 V 988 1600 a Fn(\(2)p Fi(\031)r Fn(\))1075 1587 y Fv(3)255 1713 y Fn(=)303 1651 y Fe(Z)328 1754 y Fh(Q)365 1713 y Fi(\022)q Fn(\()p Fi(y)r Fn(\))g Fi(dy)510 1651 y Fe(Z)535 1754 y Fh(Q)563 1745 y Fc(?)594 1713 y Fe(e)-28 b Fi(\032)p Fn(\()p Fi(\030)r Fn(;)8 b Fi(y)r Fn(;)g Fi(y)r Fn(\))807 1682 y Fi(d\030)p 777 1703 V 777 1744 a Fn(\(2)p Fi(\031)r Fn(\))864 1731 y Fv(3)888 1713 y Fi(:)37 1820 y Fn(This)16 b(completes)g(the)f(pro)q(of)g(of)g(\(4.21\).)108 1902 y(The)h(kinetic)g(energy)g(term)e(b)q(eing)j(settled)e(with)h(\(4.9\),) d(w)o(e)i(no)o(w)g(turn)g(to)108 2000 y Ff(Step)h(3)p 108 2016 125 2 v 17 w(:)k(L)n(ower)d(limit)f(of)h(the)f(sum)h(of)f(the) h(ele)n(ctr)n(ostatic)e(terms.)21 b(Conclusion.)37 2081 y Fn(W)l(e)e(shall)g(\014rst)f(rewrite)g(the)h(sum)f(of)g (electrostatic)g(terms)g(in)h(a)f(more)g(con)o(v)o(enien)o(t)g(\(and)h (equiv)m(alen)o(t\))37 2138 y(manner)f(whic)o(h)g(has)g(b)q(een)g(in)o (tro)q(duced)h(in)f([11)o(,)g(Section)g(3.4].)26 b(W)l(e)17 b(shall)i(only)f(sk)o(etc)o(h)f(the)h(argumen)o(t)37 2194 y(and)h(refer)e(the)h(reader)g(to)f([11])g(for)g(more)h(details.) 29 b(The)18 b(electrostatic)g(terms)f(in)i(the)f(energy)g(are)g(the)37 2251 y(follo)o(wing)140 2347 y Fg(\000)p Fn(T)l(r)222 2357 y Fh(L)246 2347 y Ft(2)264 2357 y Fv(\()p Fd(R)308 2347 y Ft(3)325 2357 y Fv(\))341 2310 y Fe(\002)360 2347 y Fi(V)386 2354 y Fv(\003)413 2347 y Fi(K)452 2354 y Fv(\003)478 2310 y Fe(\003)507 2347 y Fn(+)558 2316 y(1)p 558 2336 23 2 v 558 2378 a(2)585 2347 y Fi(D)q Fn(\()p Fi(\032)665 2354 y Fv(\003)692 2347 y Fi(;)8 b(\032)736 2354 y Fv(\003)762 2347 y Fn(\))i(+)840 2316 y(1)p 840 2336 V 840 2378 a(2)868 2347 y Fi(U)899 2354 y Fv(\003)938 2347 y Fn(=)j Fg(\000)1029 2285 y Fe(Z)1054 2388 y Fd(R)1084 2379 y Ft(3)1112 2347 y Fi(V)1138 2354 y Fv(\003)1165 2347 y Fi(\032)1188 2354 y Fv(\003)1225 2347 y Fn(+)1275 2316 y(1)p 1275 2336 V 1275 2378 a(2)1303 2347 y Fi(D)q Fn(\()p Fi(\032)1383 2354 y Fv(\003)1410 2347 y Fi(;)8 b(\032)1454 2354 y Fv(\003)1480 2347 y Fn(\))i(+)1558 2316 y(1)p 1558 2336 V 1558 2378 a(2)1586 2347 y Fi(U)1617 2354 y Fv(\003)1643 2347 y Fi(:)103 b Fn(\(4.24\))37 2466 y(Denoting)14 b(b)o(y)f Fi(f)315 2473 y Fv(\003)342 2466 y Fn(\()p Fi(x)p Fn(\))f(=)464 2432 y Fe(P)512 2479 y Fh(k)q Fu(2)p Fv(\003)589 2416 y Fe(\020)655 2448 y Fv(1)p 621 2455 87 2 v 621 2482 a Fu(j)p Fh(x)p Fu(\000)p Fh(k)q Fu(j)722 2466 y Fg(\000)768 2429 y Fe(R)789 2482 y Fh(Q)879 2445 y(dy)p 832 2456 133 2 v 832 2482 a Fu(j)p Fh(x)p Fu(\000)p Fh(k)q Fu(\000)p Fh(y)q Fu(j)969 2416 y Fe(\021)1009 2466 y Fn(=)h Fi(V)1083 2473 y Fv(\003)1116 2466 y Fg(\000)6 b Fi(\037)1185 2475 y Fv(\000\(\003\))1269 2466 y Fi(?)1314 2448 y Fv(1)p 1303 2455 40 2 v 1303 2482 a Fu(j)p Fh(x)p Fu(j)1347 2466 y Fn(,)14 b(w)o(e)f(rewrite)g(the)h (sum)f(of)g(the)37 2531 y(electrostatic)j(terms)f(as)f(follo)o(ws)394 2592 y(1)p 394 2612 23 2 v 394 2654 a(2)430 2580 y Fe(X)430 2678 y Fh(z)q Fu(2)p Fv(\003)529 2623 y Fn(lim)524 2651 y Fh(x)p Fu(!)p Fh(z)528 2683 y(x)p Fu(6)p Fv(=)p Fh(z)625 2559 y Fe(\024)649 2623 y Fi(f)671 2630 y Fv(\003)698 2623 y Fn(\()p Fi(x)p Fn(\))9 b Fg(\000)873 2592 y Fn(1)p 820 2612 130 2 v 820 2654 a Fg(j)p Fi(x)g Fg(\000)i Fi(z)r Fg(j)955 2559 y Fe(\025)989 2623 y Fn(+)1039 2592 y(1)p 1039 2612 23 2 v 1039 2654 a(2)1074 2561 y Fe(Z)1100 2664 y Fv(\000\(\003\))1183 2623 y Fi(f)1205 2630 y Fv(\003)608 2769 y Fg(\000)685 2708 y Fe(Z)710 2811 y Fv(\000\(\003\))794 2769 y Fi(f)816 2776 y Fv(\003)850 2769 y Fi(\032)873 2776 y Fv(\003)918 2769 y Fn(+)968 2739 y(1)p 968 2759 V 968 2801 a(2)996 2769 y Fi(D)q Fn(\()p Fi(\037)1081 2779 y Fv(\000\(\003\))1167 2769 y Fg(\000)g Fi(\032)1236 2776 y Fv(\003)1263 2769 y Fi(;)d(\037)1312 2779 y Fv(\000\(\003\))1397 2769 y Fg(\000)j Fi(\032)1466 2776 y Fv(\003)1493 2769 y Fn(\))p Fi(:)235 b Fn(\(4.25\))p eop %%Page: 37 37 37 36 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(37)37 199 y(W)l(e)16 b(next)f(remark)g(\(see)g([11)o(,) g(Chapter)g(2]\))f(that)g(w)o(e)h(ha)o(v)o(e)156 323 y(lim)129 353 y Fv(\003)p Fu(\000)-5 b(!1)275 292 y Fn(1)p 258 312 57 2 v 258 354 a Fg(j)p Fn(\003)p Fg(j)328 280 y Fe(X)328 378 y Fh(z)q Fu(2)p Fv(\003)427 323 y Fn(lim)422 351 y Fh(x)p Fu(!)p Fh(z)426 383 y(x)p Fu(6)p Fv(=)p Fh(z)523 259 y Fe(\024)547 323 y Fi(f)569 330 y Fv(\003)596 323 y Fn(\()p Fi(x)p Fn(\))9 b Fg(\000)771 292 y Fn(1)p 718 312 130 2 v 718 354 a Fg(j)p Fi(x)g Fg(\000)i Fi(z)r Fg(j)853 259 y Fe(\025)889 323 y Fn(=)40 b(lim)937 353 y Fv(\003)p Fu(\000)-5 b(!1)1087 323 y Fn(lim)1082 353 y Fh(x)p Fu(!)p Fv(0)1086 384 y Fh(x)p Fu(6)p Fv(=0)1183 259 y Fe(\024)1216 311 y(e)1207 323 y Fi(f)1229 330 y Fv(\003)1256 323 y Fn(\()p Fi(x)p Fn(\))10 b Fg(\000)1392 292 y Fn(1)p 1378 312 52 2 v 1378 354 a Fg(j)p Fi(x)p Fg(j)1434 259 y Fe(\025)1471 323 y Fn(=)j Fi(M)i Fn(+)10 b Fi(d;)99 b Fn(\(4.26\))37 471 y(where)17 b Fi(d)e Fn(is)h(some)g (constan)o(t)f(whic)o(h)h(is)h(related)f(to)f(our)h(c)o(hoice)h(of)e (normalization)i(for)e(the)h(p)q(oten)o(tial)g Fi(G)37 527 y Fn(and)g(whic)o(h)g(is)g(de\014ned)g(in)g([11)o(].)k(In)c (addition,)g(w)o(e)e(ha)o(v)o(e)468 647 y(lim)441 677 y Fv(\003)p Fu(\000)-5 b(!1)587 616 y Fn(1)p 570 636 57 2 v 570 678 a Fg(j)p Fn(\003)p Fg(j)640 585 y Fe(Z)665 688 y Fv(\000\(\003\))748 647 y Fi(f)770 654 y Fv(\003)817 647 y Fn(=)40 b(lim)865 677 y Fv(\003)p Fu(\000)-5 b(!1)990 585 y Fe(Z)1015 688 y Fh(Q)1061 635 y Fe(e)1052 647 y Fi(f)1074 654 y Fv(\003)1114 647 y Fn(=)1162 585 y Fe(Z)1187 688 y Fh(Q)1217 647 y Fn(\()p Fi(G)9 b Fn(+)i Fi(d)p Fn(\))h(=)h Fi(d;)294 b Fn(\(4.27\))37 774 y(and)561 882 y(lim)545 913 y Fv(\003)p Fu(!1)670 851 y Fn(1)p 653 872 V 653 913 a Fg(j)p Fn(\003)p Fg(j)722 820 y Fe(Z)748 923 y Fd(R)778 914 y Ft(3)805 882 y Fi(f)827 889 y Fv(\003)861 882 y Fi(\032)884 889 y Fv(\003)924 882 y Fn(=)13 b Fi(d)d Fn(+)1051 820 y Fe(Z)1077 923 y Fh(Q)1114 882 y Fi(G)p Fn(\()p Fi(y)r Fn(\))f Fe(e)-26 b Fi(\032)o Fn(\()p Fi(y)r Fn(\))8 b Fi(dy)r(:)390 b Fn(\(4.28\))37 1008 y(Therefore,)15 b(if)h(w)o(e)f(pro)o(v)o(e)f(that)473 1123 y(lim)8 b(inf)489 1153 y Fv(\003)p Fu(!1)629 1092 y Fn(1)p 612 1113 V 612 1154 a Fg(j)p Fn(\003)p Fg(j)674 1123 y Fi(D)q Fn(\()p Fi(\037)759 1132 y Fv(\000\(\003\))845 1123 y Fg(\000)i Fi(\032)913 1130 y Fv(\003)940 1123 y Fi(;)e(\037)989 1132 y Fv(\000\(\003\))1075 1123 y Fg(\000)j Fi(\032)1144 1130 y Fv(\003)1171 1123 y Fn(\))h Fg(\025)h Fi(D)1287 1130 y Fh(G)1316 1123 y Fn(\()s Fe(e)-28 b Fi(\032;)11 b Fe(e)-28 b Fi(\032)p Fn(\))p Fi(;)327 b Fn(\(4.29\))37 1246 y(w)o(e)18 b(shall)h(easily)g(deduce)g(from)e(\(4.26\),)g (\(4.27\),)f(\(4.28\),)g(and)i(the)g(form)o(ulation)g(\(4.25\))e(of)i (the)f(sum)h(of)37 1302 y(the)e(electrostatic)f(terms)g(\(4.24\))e (that)472 1426 y(lim)8 b(inf)477 1456 y Fv(\003)p Fu(\000)-5 b(!1)628 1395 y Fn(1)p 611 1416 V 611 1457 a Fg(j)p Fn(\003)p Fg(j)673 1362 y Fe(\022)706 1426 y Fg(\000)p Fn(T)l(r)788 1389 y Fe(\002)807 1426 y Fi(V)833 1433 y Fv(\003)860 1426 y Fi(K)899 1433 y Fv(\003)926 1389 y Fe(\003)955 1426 y Fn(+)1005 1395 y(1)p 1005 1416 23 2 v 1005 1457 a(2)1033 1426 y Fi(D)q Fn(\()p Fi(\032)1113 1433 y Fv(\003)1139 1426 y Fi(;)8 b(\032)1183 1433 y Fv(\003)1210 1426 y Fn(\))h(+)1288 1395 y(1)p 1288 1416 V 1288 1457 a(2)1316 1426 y Fi(U)1347 1433 y Fv(\003)1373 1362 y Fe(\023)632 1548 y Fg(\025)42 b(\000)752 1486 y Fe(Z)777 1589 y Fh(Q)814 1548 y Fi(G)p Fn(\()p Fi(y)r Fn(\))10 b Fe(e)-27 b Fi(\032)p Fn(\()p Fi(y)r Fn(\))8 b Fi(dy)i Fn(+)1115 1517 y(1)p 1115 1537 V 1115 1579 a(2)1143 1548 y Fi(D)1181 1555 y Fh(G)1210 1548 y Fn(\()s Fe(e)-28 b Fi(\032)q(;)10 b Fe(e)-27 b Fi(\032)o Fn(\))10 b(+)1373 1517 y Fi(M)p 1373 1537 50 2 v 1387 1579 a Fn(2)1428 1548 y Fi(:)318 b Fn(\(4.30\))37 1681 y(W)l(e)16 b(no)o(w)e(pro)o(v)o(e)h(\(4.29\).)j (Let)e(us)f(de\014ne)h Fi(g)775 1688 y Fv(\003)814 1681 y Fn(=)d(\()p Fi(\037)908 1690 y Fv(\000\(\003\))994 1681 y Fg(\000)e Fi(\032)1063 1688 y Fv(\003)1090 1681 y Fn(\))f Fi(?)1166 1663 y Fv(1)p 1155 1670 40 2 v 1155 1697 a Fu(j)p Fh(x)p Fu(j)1200 1681 y Fn(.)20 b(It)15 b(is)h(a)e(standard)h(fact)g(that)380 1783 y(1)p 363 1803 57 2 v 363 1845 a Fg(j)p Fn(\003)p Fg(j)425 1814 y Fi(D)q Fn(\()p Fi(\037)510 1823 y Fv(\000\(\003\))596 1814 y Fg(\000)10 b Fi(\032)664 1821 y Fv(\003)691 1814 y Fi(;)e(\037)740 1823 y Fv(\000\(\003\))826 1814 y Fg(\000)i Fi(\032)894 1821 y Fv(\003)921 1814 y Fn(\))j(=)1022 1783 y(1)p 1005 1803 V 1005 1845 a Fg(j)p Fn(\003)p Fg(j)1074 1752 y Fe(Z)1099 1855 y Fd(R)1129 1846 y Ft(3)1157 1814 y Fg(jr)p Fi(g)1230 1821 y Fv(\003)1255 1814 y Fg(j)1268 1795 y Fv(2)1300 1814 y Fg(\025)1348 1752 y Fe(Z)1374 1855 y Fh(Q)1411 1814 y Fg(jr)p Fe(e)-25 b Fi(g)1484 1821 y Fv(\003)1510 1814 y Fg(j)1523 1795 y Fv(2)1542 1814 y Fi(:)204 b Fn(\(4.31\))37 1942 y(F)l(rom)15 b(the)g(b)q(ound)h (\(3.10e\))e(in)i(Prop)q(osition)g(3.1,)d(w)o(e)i(deduce)806 2000 y Fe(Z)831 2103 y Fh(Q)869 2061 y Fg(jr)p Fe(e)-25 b Fi(g)942 2068 y Fv(\003)968 2061 y Fg(j)981 2043 y Fv(2)1013 2061 y Fg(\024)12 b Fi(C)q(;)37 2195 y Fn(where)h Fe(e)-25 b Fi(g)188 2202 y Fv(\003)226 2195 y Fn(is)12 b(de\014ned,)i(as)e(usual,)h(b)o(y)f Fe(e)-25 b Fi(g)699 2202 y Fv(\003)737 2195 y Fn(=)804 2177 y Fv(1)p 790 2184 45 2 v 790 2211 a Fu(j)p Fv(\003)p Fu(j)847 2161 y Fe(P)895 2208 y Fh(k)q Fu(2)p Fv(\003)972 2195 y Fi(g)994 2202 y Fv(\003)1020 2195 y Fn(\()p Fg(\001)t Fn(+)t Fi(k)q Fn(\).)18 b(Th)o(us,)12 b Fg(r)p Fe(e)-25 b Fi(g)1353 2202 y Fv(\003)1391 2195 y Fn(is)12 b(b)q(ounded)i(in)f Fi(L)1697 2178 y Fv(2)1697 2209 y(unif)1761 2195 y Fn(\()p Fj(R)1818 2178 y Fv(3)1838 2195 y Fn(\))1856 2178 y Fv(3)37 2251 y Fn(indep)q(endently)20 b(of)e(\003.)28 b(Therefore,)18 b(extracting)g(a)f(subsequence)j(if)e(necessary)l(,)h(w)o(e)f(ma)o(y)f (assume)h(that)37 2308 y(there)c(exists)275 2296 y Fe(e)276 2308 y Fi(h)g Fn(in)h Fi(L)399 2291 y Fv(2)399 2321 y(unif)463 2308 y Fn(\()p Fj(R)520 2291 y Fv(3)540 2308 y Fn(\))558 2291 y Fv(3)577 2308 y Fn(,)f(suc)o(h)g(that)f Fg(r)p Fe(e)-25 b Fi(g)862 2315 y Fv(\003)901 2308 y Fn(con)o(v)o(erges)13 b(to)1157 2296 y Fe(e)1158 2308 y Fi(h)f Fg(2)h Fi(L)1270 2291 y Fv(2)1270 2321 y(unif)1335 2308 y Fn(\()p Fj(R)1392 2291 y Fv(3)1411 2308 y Fn(\))1429 2291 y Fv(3)1449 2308 y Fn(,)g(for)g(the)h(w)o(eak)f(con)o(v)o(er-)37 2370 y(gence)k(in)h Fi(L)248 2354 y Fv(2)248 2384 y(lo)q(c)294 2370 y Fn(\()p Fj(R)351 2354 y Fv(3)370 2370 y Fn(\))388 2354 y Fv(3)408 2370 y Fn(,)e(and)g(curl)617 2358 y Fe(e)617 2370 y Fi(h)f Fn(=)g(0,)h(in)h(the)f(sense)h(of)f(distributions.)25 b(Moreo)o(v)o(er,)15 b(from)g(P)o(oincar)o(\023)-21 b(e's)37 2427 y(theorem,)19 b(there)g(exists)g Fe(e)-26 b Fi(g)20 b Fn(in)g Fg(D)614 2410 y Fu(0)626 2427 y Fn(\()p Fj(R)683 2410 y Fv(3)702 2427 y Fn(\))e(\(whic)o(h)h(is)g(uniquely)i(de\014ned,) f(up)f(to)f(a)g(constan)o(t\),)g(suc)o(h)h(that)37 2471 y Fe(e)37 2483 y Fi(h)13 b Fn(=)g Fg(r)p Fe(e)-25 b Fi(g)q Fn(,)15 b(still)i(in)f(the)f(sense)h(of)f(distributions.)21 b(In)16 b(addition,)g(since)g Fe(e)-25 b Fi(g)1267 2490 y Fv(\003)1309 2483 y Fn(satis\014es)564 2582 y Fg(\000)p Fn(\001)p Fe(e)g Fi(g)659 2589 y Fv(\003)698 2582 y Fn(=)13 b Fg(\000)p Fn(div)q(\()p Fg(r)p Fe(e)-25 b Fi(g)921 2589 y Fv(\003)947 2582 y Fn(\))12 b(=)h(4)p Fi(\031)1088 2545 y Fe(\002)1111 2582 y(e)-29 b Fi(\037)1135 2591 y Fv(\000\(\003\))1222 2582 y Fg(\000)13 b Fe(e)-28 b Fi(\032)1290 2589 y Fv(\003)1317 2545 y Fe(\003)1336 2582 y Fi(;)37 2681 y Fn(w)o(e)15 b(deduce)i(that)e Fe(e)-26 b Fi(g)17 b Fn(is)f(a)e(solution)i(to)645 2779 y Fg(\000)p Fn(div)r(\()p Fg(r)p Fe(e)-25 b Fi(g)q Fn(\))12 b(=)h Fg(\000)p Fn(\001)p Fe(e)-25 b Fi(g)14 b Fn(=)f(4)p Fi(\031)h Fn([1)9 b Fg(\000)14 b Fe(e)-28 b Fi(\032)p Fn(])p Fi(;)499 b Fn(\(4.32\))p eop %%Page: 38 38 38 37 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(38)37 199 y(in)17 b Fg(D)127 183 y Fu(0)139 199 y Fn(\()p Fj(R)196 183 y Fv(3)215 199 y Fn(\))e(\(see)g([11)o(]\).) k(With)d(\(4.31\),)d(w)o(e)i(obtain)478 322 y(lim)8 b(inf)493 353 y Fv(\003)p Fu(!1)633 292 y Fn(1)p 616 312 57 2 v 616 353 a Fg(j)p Fn(\003)p Fg(j)678 322 y Fi(D)q Fn(\()p Fi(\037)763 332 y Fv(\000\(\003\))849 322 y Fg(\000)j Fi(\032)918 329 y Fv(\003)945 322 y Fi(;)d(\037)994 332 y Fv(\000\(\003\))1080 322 y Fg(\000)i Fi(\032)1148 329 y Fv(\003)1175 322 y Fn(\))i Fg(\025)1253 260 y Fe(Z)1279 364 y Fh(Q)1316 322 y Fg(jr)p Fe(e)-25 b Fi(g)q Fg(j)1403 304 y Fv(2)1422 322 y Fi(:)37 454 y Fn(No)o(w,)15 b(w)o(e)g(notice)g (that)g(w)o(e)g(already)g(kno)o(w)g(another)g(solution)h(to)e (\(4.32\),)f(namely)384 577 y(\026)-24 b Fi(g)q Fn(\()p Fi(x)p Fn(\))12 b(=)528 515 y Fe(Z)553 618 y Fh(Q)591 577 y Fi(G)p Fn(\()p Fi(x)d Fg(\000)i Fi(y)r Fn(\)\(1)e Fg(\000)k Fe(e)-28 b Fi(\032)q Fn(\()p Fi(y)r Fn(\)\))11 b Fi(dy)j Fn(=)f Fg(\000)1127 515 y Fe(Z)1153 618 y Fh(Q)1190 577 y Fi(G)p Fn(\()p Fi(x)d Fg(\000)g Fi(y)r Fn(\))s Fe(e)-28 b Fi(\032)p Fn(\()p Fi(y)r Fn(\))12 b Fi(dy)r(;)236 b Fn(\(4.33\))37 709 y(thanks)15 b(to)g(the)g(normalization)h(\(2.14\)) e(on)h Fi(G)p Fn(.)k(The)d(function)h(\026)-24 b Fi(g)16 b Fn(is)g(p)q(erio)q(dic)h(and)f(satis\014es)609 770 y Fe(Z)634 873 y Fh(Q)671 832 y Fg(jr)r Fn(\026)-25 b Fi(g)q Fg(j)758 813 y Fv(2)790 832 y Fn(=)838 770 y Fe(Z)863 873 y Fh(Q)901 832 y Fg(\000)p Fn(\001)r(\026)g Fi(g)11 b Fg(\001)19 b Fn(\026)-24 b Fi(g)14 b Fn(=)f Fi(D)1160 839 y Fh(G)1189 832 y Fn(\()s Fe(e)-28 b Fi(\032;)10 b Fe(e)-27 b Fi(\032)p Fn(\))p Fi(;)37 966 y Fn(thanks)15 b(to)g(\(4.32\),)e(for)455 929 y Fe(R)476 982 y Fh(Q)515 966 y Fn(\026)-25 b Fi(g)14 b Fn(=)f(0.)20 b(W)l(e)15 b(are)g(going)g(to)g(sho)o(w)f(that)751 1038 y Fe(Z)776 1141 y Fh(Q)813 1100 y Fg(jr)p Fe(e)-25 b Fi(g)q Fg(j)900 1081 y Fv(2)932 1100 y Fg(\025)980 1038 y Fe(Z)1005 1141 y Fh(Q)1043 1100 y Fg(jr)r Fn(\026)g Fi(g)q Fg(j)1130 1081 y Fv(2)1149 1100 y Fi(:)597 b Fn(\(4.34\))37 1231 y(F)l(or)17 b(this)g(purp)q(ose,)g(w)o(e)f(\014rst)h(remark)f(that)f Fi(@)840 1238 y Fv(1)861 1231 y Fe(e)-26 b Fi(g)13 b Fg(\000)e Fi(@)965 1238 y Fv(1)986 1231 y Fn(\026)-24 b Fi(g)18 b Fn(\(where)e Fi(@)1199 1238 y Fv(1)1236 1231 y Fn(denotes)h(the)f(\014rst)h(deriv)m(ativ)o(e)h(with)37 1288 y(resp)q(ect)c(to)e(the)h(\014rst)f(co)q(ordinate)h Fi(x)658 1295 y Fv(1)690 1288 y Fn(of)g Fj(R)779 1271 y Fv(3)799 1288 y Fn(\))f(is)h(an)g(harmonic)g(function,)h(for)e Fe(e)-25 b Fi(g)14 b Fn(and)g(\026)-24 b Fi(g)13 b Fn(are)g(t)o(w)o(o)e (solutions)37 1344 y(to)k(the)g(same)g(Laplace)i(equation)e(\(4.32\).) 108 1401 y(No)o(w,)c(b)q(oth)h Fi(@)349 1408 y Fv(1)369 1401 y Fe(e)-25 b Fi(g)13 b Fn(\(as)e(the)h(limit)h(of)e Fi(@)726 1408 y Fv(1)746 1401 y Fe(e)-25 b Fi(g)768 1408 y Fv(\003)794 1401 y Fn(\))12 b(and)g Fi(@)933 1408 y Fv(1)954 1401 y Fn(\026)-24 b Fi(g)13 b Fn(\(b)o(y)e(construction,)h (see)g(\(4.33\)\))e(are)h(in)i Fi(L)1704 1384 y Fv(2)1704 1415 y(unif)1768 1401 y Fn(\()p Fj(R)1825 1384 y Fv(3)1845 1401 y Fn(\).)37 1457 y(Therefore,)24 b Fi(@)289 1464 y Fv(1)309 1457 y Fe(e)-25 b Fi(g)16 b Fg(\000)f Fi(@)421 1464 y Fv(1)442 1457 y Fn(\026)-24 b Fi(g)23 b Fn(is)g(an)f(harmonic)h (function)g(whic)o(h)g(b)q(elongs)g(to)f Fi(L)1402 1441 y Fv(2)1402 1471 y(unif)1466 1457 y Fn(\()p Fj(R)1523 1441 y Fv(3)1543 1457 y Fn(\),)h(and)g(th)o(us)f(is)g(a)37 1514 y(constan)o(t,)h(that)f(w)o(e)g(denote)h(b)o(y)f Fi(a)670 1521 y Fv(1)712 1514 y Fn(\(use)h(for)e(instance)j(the)e (mean-v)m(alue)i(inequalit)o(y\).)43 b(The)22 b(same)37 1570 y(argumen)o(t)13 b(applies)j(to)d(the)h(\014rst)f(deriv)m(ativ)o (es)i(with)f(resp)q(ect)g(to)f(the)h(co)q(ordinates)g Fi(x)1491 1577 y Fv(2)1525 1570 y Fn(and)g Fi(x)1638 1577 y Fv(3)1657 1570 y Fn(.)20 b(Hence)14 b(w)o(e)37 1627 y(obtain)790 1729 y Fe(e)-25 b Fi(g)11 b Fg(\000)h Fn(\026)-25 b Fi(g)14 b Fn(=)f Fi(a)e Fg(\001)e Fi(x)h Fn(+)h Fi(b;)37 1831 y Fn(where)16 b Fi(a)d Fn(=)g(\()p Fi(a)296 1838 y Fv(1)315 1831 y Fi(;)8 b(a)360 1838 y Fv(2)379 1831 y Fi(;)g(a)424 1838 y Fv(3)443 1831 y Fn(\))15 b(and)g Fi(b)h Fn(are)f(t)o(w)o(o)f(\014xed)h(v)o(ectors)g(of)g Fj(R)1116 1814 y Fv(3)1135 1831 y Fn(.)20 b(It)c(follo)o(ws)f(from)f (this)i(equalit)o(y)l(,)g(that)501 1892 y Fe(Z)527 1995 y Fh(Q)564 1954 y Fg(jr)p Fe(e)-25 b Fi(g)q Fg(j)651 1935 y Fv(2)712 1954 y Fn(=)789 1892 y Fe(Z)814 1995 y Fh(Q)851 1954 y Fg(jr)p Fn(\()r(\026)g Fi(g)11 b Fn(+)f Fi(a)g Fg(\001)g Fi(x)g Fn(+)h Fi(b)p Fn(\))p Fg(j)1187 1935 y Fv(2)712 2080 y Fn(=)789 2018 y Fe(Z)814 2121 y Fh(Q)851 2080 y Fg(jr)r Fn(\026)-25 b Fi(g)q Fg(j)938 2061 y Fv(2)967 2080 y Fn(+)1013 2018 y Fe(Z)1038 2121 y Fh(Q)1076 2080 y Fg(j)p Fi(a)p Fg(j)1126 2061 y Fv(2)1155 2080 y Fn(+)10 b(2)1231 2018 y Fe(Z)1256 2121 y Fh(Q)1293 2080 y Fi(a)g Fg(\001)g(r)r Fn(\026)-25 b Fi(g)712 2206 y Fn(=)789 2144 y Fe(Z)814 2247 y Fh(Q)851 2206 y Fg(jr)r Fn(\026)g Fi(g)q Fg(j)938 2187 y Fv(2)967 2206 y Fn(+)1013 2144 y Fe(Z)1038 2247 y Fh(Q)1076 2206 y Fg(j)p Fi(a)p Fg(j)1126 2187 y Fv(2)712 2332 y Fg(\025)789 2270 y Fe(Z)814 2373 y Fh(Q)851 2332 y Fg(jr)r Fn(\026)g Fi(g)q Fg(j)938 2313 y Fv(2)957 2332 y Fi(;)37 2466 y Fn(since)149 2430 y Fe(R)170 2482 y Fh(Q)208 2466 y Fi(a)9 b Fg(\001)h(r)r Fn(\026)-25 b Fi(g)14 b Fn(=)f Fi(a)d Fg(\001)442 2430 y Fe(R)463 2482 y Fh(Q)501 2466 y Fg(r)r Fn(\026)-25 b Fi(g)14 b Fn(=)f(0)h(b)q(ecause)i(of)f(the)g(p)q(erio)q(dicit)o(y)i (of)g(\026)-25 b Fi(g)q Fn(.)20 b(The)15 b(inequalit)o(y)i(\(4.34\))c (follo)o(ws,)37 2527 y(from)i(whic)o(h)h(w)o(e)f(deduce)h(easily)h (\(4.29\))c(and)i(\(4.30\).)p eop %%Page: 39 39 39 38 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(39)108 199 y(A)o(t)15 b(this)g(stage,)f(w)o(e)h (collect)i(\(4.9\))c(with)j(\(4.30\))d(to)i(obtain)179 328 y(lim)8 b(inf)194 358 y Fv(\003)p Fu(!1)318 297 y Fi(I)342 280 y Fh(RH)s(F)338 310 y Fv(\003)p 318 318 112 2 v 345 359 a Fg(j)p Fn(\003)p Fg(j)475 328 y Fn(=)42 b(lim)8 b(inf)568 358 y Fv(\003)p Fu(!1)708 297 y Fn(1)p 691 318 57 2 v 691 359 a Fg(j)p Fn(\003)p Fg(j)752 264 y Fe(\022)786 328 y Fn(T)l(r)833 338 y Fh(L)857 328 y Ft(2)874 338 y Fv(\()p Fd(R)918 328 y Ft(3)935 338 y Fv(\))951 291 y Fe(\002)970 328 y Fn(\()p Fg(\000)p Fn(\001)i Fg(\000)h Fi(V)1143 335 y Fv(\003)1170 328 y Fn(\))p Fi(K)1227 335 y Fv(\003)1253 291 y Fe(\003)1282 328 y Fn(+)1332 297 y(1)p 1332 318 23 2 v 1332 359 a(2)1360 328 y Fi(D)q Fn(\()p Fi(\032)1440 335 y Fv(\003)1466 328 y Fi(;)d(\032)1510 335 y Fv(\003)1537 328 y Fn(\))h(+)1615 297 y(1)p 1615 318 V 1615 359 a(2)1642 328 y Fi(U)1673 335 y Fv(\003)1700 264 y Fe(\023)475 450 y Fg(\025)552 388 y Fe(Z)577 491 y Fh(Q)605 482 y Fc(?)633 450 y Fn(T)l(r)680 462 y Fh(L)704 451 y Ft(2)704 474 y Fc(\030)721 462 y Fv(\()p Fh(Q)p Fv(\))778 450 y Fn([)p Fg(\000)p Fn(\001)864 457 y Fh(\030)906 439 y Fe(e)895 450 y Fi(K)934 457 y Fh(\030)953 450 y Fn(])1013 419 y Fi(d\030)p 983 440 106 2 v 983 481 a Fn(\(2)p Fi(\031)r Fn(\))1070 468 y Fv(3)1104 450 y Fg(\000)1149 388 y Fe(Z)1174 491 y Fh(Q)1212 450 y Fi(G)h Fe(e)-27 b Fi(\032)10 b Fn(+)1339 419 y(1)p 1339 440 23 2 v 1339 481 a(2)1367 450 y Fi(D)1405 457 y Fh(G)1434 450 y Fn(\()s Fe(e)-28 b Fi(\032;)10 b Fe(e)-27 b Fi(\032)p Fn(\))10 b(+)1597 419 y Fi(M)p 1597 440 50 2 v 1610 481 a Fn(2)475 576 y(=)42 b Fi(E)589 557 y Fh(RH)s(F)586 587 y(per)676 576 y Fn(\()704 565 y Fe(e)694 576 y Fi(K)r Fn(\))10 b(+)814 545 y Fi(M)p 814 566 V 827 607 a Fn(2)475 686 y Fg(\025)42 b Fi(I)576 667 y Fh(RH)s(F)572 697 y(per)673 686 y Fn(+)724 655 y Fi(M)p 724 675 V 737 717 a Fn(2)778 686 y Fi(:)968 b Fn(\(4.35\))37 796 y(This)16 b(concludes)h(the)e(pro)q (of)g(of)g(Prop)q(osition)h(4.1.)928 b Fg(})108 894 y Fn(Before)20 b(turning)h(to)e(the)h(study)h(of)e(the)i(upp)q(er)g (limit)g(of)f(the)g(energy)g(p)q(er)h(unit)g(v)o(olume,)g(w)o(e)f(can) 37 950 y(an)o(ticipate)c(a)f(little)i(bit)f(and)f(assume)g(that)f(w)o (e)h(ha)o(v)o(e)g(already)h(pro)o(v)o(en)641 1068 y Fi(I)665 1049 y Fh(RH)s(F)661 1079 y(per)762 1068 y Fn(+)812 1037 y Fi(M)p 812 1057 V 826 1099 a Fn(2)879 1068 y Fg(\025)d Fn(lim)8 b(sup)949 1107 y Fv(\003)p Fu(!1)1096 1037 y Fn(1)p 1079 1057 57 2 v 1079 1099 a Fg(j)p Fn(\003)p Fg(j)1140 1068 y Fi(I)1164 1049 y Fh(RH)s(F)1160 1079 y Fv(\003)1252 1068 y Fi(;)494 b Fn(\(4.36\))37 1190 y(whic)o(h)20 b(will)h(b)q(e)f(the)f(purp)q(ose)h(of)e(Prop)q(osition)i (4.2)e(b)q(elo)o(w.)32 b(It)19 b(will)i(follo)o(w)e(from)f(the)h (comparison)g(of)37 1246 y(\(4.35\))13 b(and)i(\(4.36\))e(that)g(all)j (equalities)g(in)g(these)e(strings)h(of)f(inequalities)j(\(4.9\),)c (\(4.30\),)f(\(4.35\),)h(\(4.36\))37 1303 y(are)i(indeed)i(equalities.) 22 b(In)16 b(particular,)f(w)o(e)g(shall)i(reco)o(v)o(er)412 1421 y(lim)396 1451 y Fv(\003)p Fu(!1)521 1390 y Fn(1)p 504 1411 V 504 1452 a Fg(j)p Fn(\003)p Fg(j)565 1421 y Fn(T)l(r)612 1431 y Fh(L)636 1421 y Ft(2)653 1431 y Fv(\()p Fd(R)697 1421 y Ft(3)715 1431 y Fv(\))731 1421 y Fn([)p Fg(\000)p Fn(\001)p Fi(K)856 1428 y Fv(\003)882 1421 y Fn(])12 b(=)955 1359 y Fe(Z)980 1462 y Fh(Q)1008 1453 y Fc(?)1036 1421 y Fn(T)l(r)1082 1433 y Fh(L)1106 1421 y Ft(2)1106 1445 y Fc(\030)1124 1433 y Fv(\()p Fh(Q)p Fv(\))1181 1421 y Fn([)p Fg(\000)p Fn(\001)1267 1428 y Fh(\030)1309 1410 y Fe(e)1298 1421 y Fi(K)1337 1428 y Fh(\030)1356 1421 y Fn(])1416 1390 y Fi(d\030)p 1386 1411 106 2 v 1386 1452 a Fn(\(2)p Fi(\031)r Fn(\))1473 1439 y Fv(3)1496 1421 y Fi(;)250 b Fn(\(4.37\))37 1560 y(and)16 b(w)o(e)f(shall)h(also)f(obtain)h(that)641 1549 y Fe(e)630 1560 y Fi(K)i Fn(is)e(a)f(minimizer)i(of)e Fi(I)1059 1544 y Fh(RH)s(F)1055 1571 y(per)1146 1560 y Fn(.)108 1641 y(In)c(order)g(to)f(pro)o(v)o(e)g(that)g(the)h(upp)q (er)h(limit)g(b)q(eha)o(v)o(es)f(in)h(the)e(exp)q(ected)i(w)o(a)o(y)l (,)f(w)o(e)f(shall)i(mak)o(e)e(use)i(of)e(the)37 1698 y(minimizer)17 b(of)d(\(2.17\))f(\(it)i(is)g(not)f Ff(stricto)i(sensu) 30 b Fn(necessary)l(,)15 b(as)f(w)o(e)g(migh)o(t)h(use)g(an)f(almost)h (minimizer\).)37 1754 y(Therefore,)f(w)o(e)f(dev)o(ote)g(the)h(next)g (section)g(to)f(the)g(study)h(of)f(problem)h(\(2.17\),)e(and)h(in)i (particular)f(to)f(the)37 1811 y(pro)q(of)k(of)g(Theorem)g(2.1.)25 b(W)l(e)17 b(shall)i(come)e(bac)o(k)g(to)f(the)i(pro)q(of)f(of)f(the)i (upp)q(er)g(limit)g(\(4.36\),)e(and)h(th)o(us)37 1867 y(conclude)g(the)f(pro)q(of)e(of)h(Theorem)g(2.2,)f(in)i(Subsection)h (4.3.)37 1989 y Fb(4.2)57 b(The)20 b(p)r(erio)r(dic)d(RHF)i(problem)37 2074 y Fn(W)l(e)d(b)q(egin)g(this)g(section)f(with)h(the)f(pro)q(of)g (of)g(Theorem)g(2.1,)e(that)i(w)o(e)g(recall)h(here)g(for)e(the)h(con)o (v)o(enience)37 2131 y(of)g(the)h(reader.)37 2229 y Fj(Theorem)i(2.1)h (\(W)l(ell-p)q(osedness)f(of)g(the)f(RHF)h(p)q(erio)q(dic)i(mo)q(del\)) 37 2285 y Ff(The)d(minimization)f(pr)n(oblem)g(de\014ne)n(d)g(by)g (\(2.17\){\(2.18\),)g(i.e.)624 2385 y Fi(I)648 2366 y Fh(RH)s(F)644 2396 y(per)748 2385 y Fn(=)d(inf)t Fg(f)p Fi(E)911 2366 y Fh(RH)s(F)908 2396 y(per)998 2385 y Fn(\()p Fi(K)s Fn(\))f(;)19 b Fi(K)d Fg(2)d(K)q(g)p Fi(;)333 2554 y(E)370 2536 y Fh(RH)s(F)367 2566 y(per)457 2554 y Fn(\()p Fi(K)s Fn(\))f(=)595 2493 y Fe(Z)620 2596 y Fh(Q)648 2586 y Fc(?)675 2554 y Fn(T)l(r)722 2566 y Fh(L)746 2555 y Ft(2)746 2578 y Fc(\030)764 2566 y Fv(\()p Fh(Q)p Fv(\))821 2518 y Fe(\002)840 2554 y Fg(\000)p Fn(\001)913 2561 y Fh(\030)932 2554 y Fi(K)971 2561 y Fh(\030)989 2518 y Fe(\003)1051 2524 y Fi(d\030)p 1021 2544 V 1021 2586 a Fn(\(2)p Fi(\031)r Fn(\))1108 2573 y Fv(3)1141 2554 y Fg(\000)1187 2493 y Fe(Z)1212 2596 y Fh(Q)1249 2554 y Fi(G\032)f Fn(+)1369 2524 y(1)p 1369 2544 23 2 v 1369 2586 a(2)1397 2554 y Fi(D)1435 2561 y Fh(G)1464 2554 y Fn(\()p Fi(\032;)d(\032)p Fn(\))p Fi(;)37 2683 y Ff(\(r)n(esp)n(e)n(ctively)j(in)i(the)g(sme)n(ar)n(e)n(d)f(nuclei)g (c)n(ase)h(by)f(e)n(quations)h(\(2.19\){\(2.20\))g(in)f(Se)n(ction)g (2\))h(admits)g(a)g(min-)37 2740 y(imum.)21 b(In)15 b(addition,)g(the)g (minimizing)e(density)g Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))14 b Ff(is)g(unique)g(and,)h(thus,)g(shar)n(es)f(the)g (symmetries)37 2796 y(of)j(the)f(unique)h(cub)n(e.)p eop %%Page: 40 40 40 39 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(40)108 199 y(Before)15 b(w)o(e)g(b)q(egin)h(with)g(the) 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(smeared)h(n)o(uclei)h(case)f(\(2.19\){\(2.20)o(\))d(is)j(straigh)o (tforw)o(ard.)28 b(Let)19 b(us)37 594 y(also)14 b(men)o(tion)g(at)f (this)h(stage)f(that)g(some)g(of)g(our)g(argumen)o(ts)g(are)g(similar)i (to)e(those)g(used)i(b)o(y)e(E.H.)g(Lieb,)37 651 y(J.)j(P)l(.)f(Solo)o (v)o(ej)g(and)g(J.)g(Yngv)m(ason)g(in)h([34],)e(where)h(a)g(close)h (problem)g(is)g(studied.)37 749 y Fj(Pro)q(of)j(of)e(Theorem)h(2.1)108 830 y Fn(Let)h(us)g(consider)h(a)f(minimizing)i(sequence)f Fi(K)950 814 y Fh(n)993 830 y Fn(for)e(the)h(minimization)i(problem)e (\(2.17\){\(2.18\))o(.)37 887 y(F)l(or)13 b(eac)o(h)g Fi(n)p Fn(,)g(the)g(op)q(erator)f Fi(K)569 870 y Fh(n)605 887 y Fn(ma)o(y)h(b)q(e)g(decomp)q(osed)h(in)o(to)f(op)q(erators)f Fi(K)1344 870 y Fh(n)1341 901 y(\030)1367 887 y Fn(.)19 b(W)l(e)13 b(denote)h(b)o(y)e Fi(\032)1703 894 y Fh(n)1727 887 y Fn(\()p Fi(\030)r(;)c(x;)g(y)r Fn(\))37 955 y(the)15 b(k)o(ernel)g(of)f Fi(K)341 938 y Fh(n)338 969 y(\030)365 955 y Fn(,)g(and)g(b)o(y)h Fi(\032)565 962 y Fh(n)589 955 y Fn(\()p Fi(x;)8 b(y)r Fn(\))j(=)755 918 y Fe(R)776 971 y Fh(Q)804 961 y Fc(?)832 955 y Fi(\032)855 962 y Fh(n)879 955 y Fn(\()p Fi(\030)r(;)d(x;)g(y)r Fn(\))1063 934 y Fh(d\030)p 1039 944 84 2 v 1039 972 a Fv(\(2)p Fh(\031)q Fv(\))1106 962 y Ft(3)1142 955 y Fn(the)14 b(Sc)o(h)o(w)o(artz)g(k)o(ernel)h(of)f Fi(K)1635 962 y Fh(n)1658 955 y Fn(,)g(according)37 1016 y(to)h(Lemma)g(3.4.)k(More)c (precisely)l(,)i(w)o(e)e(ha)o(v)o(e,)f(in)i(view)g(of)f(Prop)q(osition) g(3.2,)516 1122 y Fi(\032)539 1129 y Fh(n)563 1122 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))j(=)772 1079 y Fe(X)773 1177 y Fh(p)p Fu(\025)p Fv(1)845 1122 y Fi(\025)871 1103 y Fv(\()p Fh(n)p Fv(\))871 1133 y Fh(p)922 1122 y Fn(\()p Fi(\030)r Fn(\))h Fi(u)1018 1103 y Fv(\()p Fh(n)p Fv(\))1018 1133 y Fh(p)1069 1122 y Fn(\()p Fi(\030)r(;)c(x)p Fn(\))j Fi(u)1211 1103 y Fv(\()p Fh(n)p Fv(\))1211 1133 y Fh(p)1262 1122 y Fn(\()p Fi(\030)r(;)d(y)r Fn(\))1365 1103 y Fu(\003)1383 1122 y Fi(:)37 1269 y Fn(In)14 b(the)f(righ)o (t-hand)h(side,)g(the)f(index)h Fi(n)p Fn(,)f(referring)g(to)g(the)g (index)h(in)g(the)f(minimizing)j(sequence)e Fi(K)1760 1252 y Fh(n)1783 1269 y Fn(,)f(has)37 1325 y(b)q(een)k(put)f(in)o(to)f (paren)o(theses)h(in)g(order)f(to)g(a)o(v)o(oid)g(am)o(biguit)o(y)g (with)h(p)q(o)o(w)o(ers)f(of)g Fi(u)1460 1332 y Fh(p)1480 1325 y Fn(.)20 b(Since,)d(in)f(addition,)37 1382 y Fi(K)79 1365 y Fh(n)76 1396 y(\030)118 1382 y Fn(is)g(trace-class)f(on)g Fi(L)478 1365 y Fv(2)478 1396 y Fh(\030)498 1382 y Fn(\()p Fi(Q)p Fn(\),)f(w)o(e)h(ma)o(y)f(also)h(de\014ne)601 1499 y Fi(\032)624 1506 y Fh(n)648 1499 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))i(=)858 1456 y Fe(X)859 1554 y Fh(p)p Fu(\025)p Fv(1)931 1499 y Fi(\025)957 1481 y Fv(\()p Fh(n)p Fv(\))957 1511 y Fh(p)1009 1499 y Fn(\()p Fi(\030)r Fn(\))e Fg(j)p Fi(u)1114 1481 y Fv(\()p Fh(n)p Fv(\))1114 1511 y Fh(p)1163 1499 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)1281 1481 y Fv(2)1299 1499 y Fi(;)37 1653 y Fn(whic)o(h)16 b(is)e(a)g(non-negativ)o(e,)h(p)q(erio)q(dic)h(function)f(in)g Fi(L)961 1636 y Fv(1)981 1653 y Fn(\()p Fi(Q)p Fn(\),)e(suc)o(h)i(that) e(T)l(r)1325 1665 y Fh(L)1349 1653 y Ft(2)1349 1676 y Fc(\030)1366 1665 y Fv(\()p Fh(Q)p Fv(\))1424 1653 y Fi(K)1466 1636 y Fh(n)1463 1667 y(\030)1502 1653 y Fn(=)1550 1616 y Fe(R)1571 1669 y Fh(Q)1608 1653 y Fi(\032)1631 1660 y Fh(n)1655 1653 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))g Fi(dx)p Fn(.)37 1720 y(Moreo)o(v)o(er,)18 b(let)h(us)g(recall)g (from)f(Prop)q(osition)h(3.2,)f(that)g(the)g(densit)o(y)h Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))18 b(whic)o(h)h(app)q (ears)g(in)g(the)37 1776 y(de\014nition)i(\(2.18\))c(of)h(the)h(energy) g(functional)h(is)f(also)g(a)f(non-negativ)o(e)h(p)q(erio)q(dic)i (function)f(in)f Fi(L)1784 1759 y Fv(1)1804 1776 y Fn(\()p Fi(Q)p Fn(\))37 1832 y(\(at)g(least\))g(de\014ned)i(b)o(y)e Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))18 b(=)676 1796 y Fe(R)698 1848 y Fh(Q)726 1839 y Fc(?)753 1832 y Fi(\032)776 1839 y Fh(n)800 1832 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))987 1812 y Fh(d\030)p 962 1822 V 962 1849 a Fv(\(2)p Fh(\031)q Fv(\))1029 1840 y Ft(3)1051 1832 y Fn(.)32 b(Our)20 b(\014rst)f(step)g(consists)h(in)g(\014nding)g (some)37 1896 y(b)q(ounds,)c(indep)q(enden)o(t)i(of)d Fi(n)p Fn(,)g(on)g(the)g(op)q(erators)f Fi(K)957 1879 y Fh(n)996 1896 y Fn(and)h(on)g(the)h(functions)g Fi(\032)1445 1903 y Fh(n)1469 1896 y Fn(\()p Fg(\001)p Fi(;)8 b Fg(\001)p Fn(\))k(and)k Fi(\032)1676 1903 y Fh(n)1700 1896 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fi(;)g Fg(\001)p Fn(\).)108 1994 y(Step)16 b(1:)j(A)c(priori)h(estimates)f(on)h(the)f(minimizing)j (sequence)p 108 2001 1082 2 v 108 2075 a(First)d(of)g(all,)g(w)o(e)g (remark)g(that)f(the)i(follo)o(wing)g(b)q(ound)g(holds)g(ob)o(viously) 832 2177 y(0)d Fg(\024)g Fi(K)958 2158 y Fh(n)955 2189 y(\030)994 2177 y Fg(\024)g Fj(1)p Fi(;)37 2279 y Fn(for)19 b(almost)g(ev)o(ery)g Fi(\030)i Fn(in)f Fi(Q)520 2263 y Fh(?)540 2279 y Fn(,)g(whic)o(h)g(comes)f(straigh)o(tforw)o(ardly)f (from)g(Lemma)h(3.3.)31 b(In)20 b(view)g(of)f(the)37 2336 y(decomp)q(osition)h(of)d(the)i(op)q(erators)e Fi(K)722 2319 y Fh(n)719 2349 y(\030)763 2336 y Fn(along)h(their)h(eigen)o (basis,)g(let)g(us)f(rewrite)g(the)g(kinetic)i(energy)37 2392 y(term:)320 2442 y Fe(Z)345 2545 y Fh(Q)373 2535 y Fc(?)400 2504 y Fn(T)l(r)447 2515 y Fh(L)471 2504 y Ft(2)471 2527 y Fc(\030)489 2515 y Fv(\()p Fh(Q)p Fv(\))546 2467 y Fe(\002)565 2504 y Fg(\000)p Fn(\001)638 2511 y Fh(\030)657 2504 y Fi(K)699 2485 y Fh(n)696 2515 y(\030)722 2467 y Fe(\003)749 2504 y Fi(d\030)14 b Fn(=)855 2442 y Fe(Z)880 2545 y Fh(Q)908 2535 y Fc(?)936 2460 y Fe(X)937 2558 y Fh(p)p Fu(\025)p Fv(1)1009 2504 y Fi(\025)1035 2485 y Fv(\()p Fh(n)p Fv(\))1035 2515 y Fh(p)1086 2504 y Fn(\()p Fi(\030)r Fn(\))1152 2442 y Fe(Z)1176 2545 y Fh(Q)1214 2504 y Fg(jr)p Fi(u)1291 2485 y Fv(\()p Fh(n)p Fv(\))1291 2515 y Fh(p)1341 2504 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fg(j)1459 2485 y Fv(2)1485 2504 y Fi(dxd\030)r(:)165 b Fn(\(4.38\))37 2651 y(Next,)15 b(w)o(e)g(remark)g(that)f(the)h (constrain)o(t)g(of)g(c)o(harge)g(1,)f(namely)707 2709 y Fe(Z)733 2813 y Fh(Q)761 2803 y Fc(?)788 2771 y Fn(T)l(r)835 2783 y Fh(L)859 2772 y Ft(2)859 2795 y Fc(\030)876 2783 y Fv(\()p Fh(Q)p Fv(\))933 2771 y Fi(K)975 2753 y Fh(n)972 2783 y(\030)1041 2741 y Fi(d\030)p 1011 2761 106 2 v 1011 2803 a Fn(\(2)p Fi(\031)r Fn(\))1098 2789 y Fv(3)1134 2771 y Fn(=)f(1)p eop %%Page: 41 41 41 40 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(41)37 199 y(ma)o(y)15 b(also)g(b)q(e)h(written)f(as)430 234 y Fe(Z)455 337 y Fh(Q)483 328 y Fc(?)553 265 y Fi(d\030)p 523 286 106 2 v 523 327 a Fn(\(2)p Fi(\031)r Fn(\))610 314 y Fv(3)641 234 y Fe(Z)667 337 y Fh(Q)704 296 y Fi(\032)727 303 y Fh(n)751 296 y Fn(\()p Fi(\030)r Fn(;)8 b Fi(x)p Fn(;)g Fi(x)p Fn(\))g Fi(dx)h Fn(=)1018 234 y Fe(Z)1044 337 y Fh(Q)1072 328 y Fc(?)1099 253 y Fe(X)1101 351 y Fh(p)p Fu(\025)p Fv(1)1172 296 y Fi(\025)1198 277 y Fv(\()p Fh(n)p Fv(\))1198 307 y Fh(p)1250 296 y Fn(\()p Fi(\030)r Fn(\))1350 265 y Fi(d\030)p 1320 286 V 1320 327 a Fn(\(2)p Fi(\031)r Fn(\))1407 314 y Fv(3)944 440 y Fn(=)1021 379 y Fe(Z)1046 482 y Fh(Q)1084 440 y Fi(\032)1107 447 y Fh(n)1131 440 y Fn(\()p Fi(x)p Fn(;)f Fi(x)p Fn(\))g Fi(dx)55 b Fn(=)k(1)p Fi(:)276 b Fn(\(4.39\))37 546 y(Another)16 b(consequence)g(of)f(this)h(constrain)o(t)e(is)261 581 y Fe(Z)286 684 y Fh(Q)314 674 y Fc(?)376 612 y Fi(d\030)p 346 632 V 346 674 a Fn(\(2)p Fi(\031)r Fn(\))433 661 y Fv(3)464 581 y Fe(Z)5 b(Z)540 684 y Fh(Q)p Fu(\002)p Fh(Q)632 643 y Fg(j)p Fi(\032)668 650 y Fh(n)692 643 y Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fg(j)855 624 y Fv(2)880 643 y Fi(dxdy)420 769 y Fn(=)497 707 y Fe(Z)522 810 y Fh(Q)550 801 y Fc(?)578 726 y Fe(X)579 824 y Fh(p)p Fu(\025)p Fv(1)651 726 y Fe(X)653 824 y Fh(q)q Fu(\025)p Fv(1)724 769 y Fi(\025)750 750 y Fv(\()p Fh(n)p Fv(\))750 780 y Fh(p)802 769 y Fn(\()p Fi(\030)r Fn(\))p Fi(\025)886 750 y Fv(\()p Fh(n)p Fv(\))886 780 y Fh(q)936 769 y Fn(\()p Fi(\030)r Fn(\))994 730 y Fe(\014)994 758 y(\014)1009 707 y(Z)1034 810 y Fh(Q)1072 769 y Fi(u)1098 750 y Fv(\()p Fh(n)p Fv(\))1098 780 y Fh(p)1148 769 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fi(u)1279 750 y Fv(\()p Fh(n)p Fv(\))1279 780 y Fh(q)1329 769 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))1434 750 y Fu(\003)1452 769 y Fi(dx)1502 730 y Fe(\014)1502 758 y(\014)1517 744 y Fv(2)1571 738 y Fi(d\030)p 1542 759 V 1542 800 a Fn(\(2)p Fi(\031)r Fn(\))1629 787 y Fv(3)420 914 y Fn(=)497 852 y Fe(Z)522 955 y Fh(Q)550 946 y Fc(?)578 871 y Fe(X)579 969 y Fh(p)p Fu(\025)p Fv(1)651 871 y Fe(X)653 969 y Fh(q)q Fu(\025)p Fv(1)724 914 y Fi(\025)750 895 y Fv(\()p Fh(n)p Fv(\))750 925 y Fh(p)802 914 y Fn(\()p Fi(\030)r Fn(\))p Fi(\025)886 895 y Fv(\()p Fh(n)p Fv(\))886 925 y Fh(q)936 914 y Fn(\()p Fi(\030)r Fn(\))k Fi(\016)1026 921 y Fh(p;q)1121 883 y Fi(d\030)p 1091 903 V 1091 945 a Fn(\(2)p Fi(\031)r Fn(\))1178 932 y Fv(3)420 1059 y Fn(=)497 997 y Fe(Z)522 1100 y Fh(Q)550 1090 y Fc(?)578 1015 y Fe(X)579 1113 y Fh(p)p Fu(\025)p Fv(1)651 1059 y Fg(j)p Fi(\025)690 1040 y Fv(\()p Fh(n)p Fv(\))690 1070 y Fh(p)741 1059 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)812 1040 y Fv(2)865 1028 y Fi(d\030)p 836 1048 V 836 1090 a Fn(\(2)p Fi(\031)r Fn(\))923 1077 y Fv(3)420 1203 y Fg(\024)497 1141 y Fe(Z)522 1244 y Fh(Q)550 1235 y Fc(?)578 1160 y Fe(X)579 1258 y Fh(p)p Fu(\025)p Fv(1)651 1203 y Fi(\025)677 1184 y Fv(\()p Fh(n)p Fv(\))677 1214 y Fh(p)728 1203 y Fn(\()p Fi(\030)r Fn(\))821 1172 y Fi(d\030)p 791 1193 V 791 1234 a Fn(\(2)p Fi(\031)r Fn(\))878 1221 y Fv(3)959 1203 y Fn(=)59 b(1)p Fi(;)37 1364 y Fn(whence)537 1442 y Fi(\032)560 1449 y Fh(n)584 1442 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))13 b(is)i(b)q(ounded)i(in)f Fi(L)1062 1423 y Fv(2)1082 1442 y Fn(\()p Fi(Q)1136 1423 y Fh(?)1165 1442 y Fg(\002)10 b Fi(Q)g Fg(\002)h Fi(Q)p Fn(\))p Fi(;)390 b Fn(\(4.40\))37 1520 y(or,)15 b(equiv)m(alen)o(tly)l(,)i(b)q(ecause)f(of)f(\(3.26\))e (in)j(Lemma)g(3.4,)606 1598 y Fi(\032)629 1605 y Fh(n)653 1598 y Fn(\()p Fi(x;)8 b(y)r Fn(\))13 b(is)j(b)q(ounded)h(in)f Fi(L)1089 1579 y Fv(2)1108 1598 y Fn(\()p Fi(Q)10 b Fg(\002)h Fj(R)1257 1579 y Fv(3)1276 1598 y Fn(\))p Fi(:)452 b Fn(\(4.41\))37 1676 y(\(Note)18 b(that)g(actually)h Fi(\032)469 1683 y Fh(n)493 1676 y Fn(\()p Fi(x;)8 b(y)r Fn(\))17 b(is)i(b)q(ounded)g(in)h Fi(L)942 1660 y Fv(2)942 1690 y(unif)1006 1676 y Fn(\()p Fj(R)1063 1660 y Fv(3)1083 1676 y Fn(;)8 b Fi(L)1135 1660 y Fv(2)1154 1676 y Fn(\()p Fj(R)1211 1660 y Fv(3)1230 1676 y Fn(\)\),)18 b(thanks)g(to)g(the)h (translation)f(in-)37 1733 y(v)m(ariance.\))i(W)l(e)14 b(are)f(no)o(w)g(going)h(to)e(w)o(ork)h(on)g(the)h(energy)f (functional.)21 b(Owing)14 b(to)f(the)h(con)o(v)o(exit)o(y)f(of)g(the) 37 1789 y(function)k Fi(f)g Fg(7!)313 1752 y Fe(R)335 1805 y Fh(Q)372 1789 y Fg(jr)423 1755 y(p)p 461 1755 28 2 v 34 x Fi(f)5 b Fg(j)501 1773 y Fv(2)520 1789 y Fn(,)15 b(and)g(b)q(ecause)h(of)f(\(4.39\),)e(w)o(e)i(ha)o(v)o(e)233 1850 y Fe(Z)258 1953 y Fh(Q)295 1912 y Fg(jr)346 1881 y(p)p 384 1881 24 2 v 31 x Fi(\032)407 1927 y Fh(n)431 1912 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fg(j)553 1893 y Fv(2)578 1912 y Fi(dx)41 b Fn(=)746 1850 y Fe(Z)771 1953 y Fh(Q)801 1846 y Fe(\014)801 1873 y(\014)801 1900 y(\014)801 1927 y(\014)816 1912 y Fg(r)854 1875 y Fe(\000)875 1850 y(Z)900 1953 y Fh(Q)928 1943 y Fc(?)956 1868 y Fe(X)957 1966 y Fh(p)p Fu(\025)p Fv(1)1029 1912 y Fi(\025)1055 1893 y Fv(\()p Fh(n)p Fv(\))1055 1923 y Fh(p)1106 1912 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)1203 1893 y Fv(\()p Fh(n)p Fv(\))1203 1923 y Fh(p)1253 1912 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fg(j)1371 1893 y Fv(2)1432 1881 y Fi(d\030)p 1402 1901 106 2 v 1402 1943 a Fn(\(2)p Fi(\031)r Fn(\))1489 1930 y Fv(3)1512 1875 y Fe(\001)1533 1886 y Fv(1)p Fh(=)p Fv(2)1588 1846 y Fe(\014)1588 1873 y(\014)1588 1900 y(\014)1588 1927 y(\014)1603 1859 y Fv(2)1630 1912 y Fi(dx)669 2056 y Fg(\024)746 1994 y Fe(Z)d(Z)822 2097 y Fh(Q)p Fu(\002)p Fh(Q)905 2088 y Fc(?)932 2013 y Fe(X)934 2111 y Fh(p)p Fu(\025)p Fv(1)1006 2056 y Fi(\025)1032 2037 y Fv(\()p Fh(n)p Fv(\))1032 2067 y Fh(p)1083 2056 y Fn(\()p Fi(\030)r Fn(\))1141 2018 y Fe(\014)1141 2045 y(\014)1156 2056 y Fg(rj)p Fi(u)1233 2037 y Fv(\()p Fh(n)p Fv(\))1233 2067 y Fh(p)1283 2056 y Fn(\()p Fi(\030)r(;)j(x)p Fn(\))p Fg(j)1401 2018 y Fe(\014)1401 2045 y(\014)1414 2031 y Fv(2)1452 2025 y Fi(dxd\030)p 1447 2046 V 1447 2087 a Fn(\(2)p Fi(\031)r Fn(\))1534 2074 y Fv(3)669 2201 y Fg(\024)746 2139 y Fe(Z)d(Z)822 2242 y Fh(Q)p Fu(\002)p Fh(Q)905 2233 y Fc(?)932 2158 y Fe(X)934 2256 y Fh(p)p Fu(\025)p Fv(1)1006 2201 y Fi(\025)1032 2182 y Fv(\()p Fh(n)p Fv(\))1032 2212 y Fh(p)1083 2201 y Fn(\()p Fi(\030)r Fn(\))1141 2162 y Fe(\014)1141 2190 y(\014)1156 2201 y Fg(r)p Fi(u)1220 2182 y Fv(\()p Fh(n)p Fv(\))1220 2212 y Fh(p)1270 2201 y Fn(\()p Fi(\030)r(;)j(x)p Fn(\))1375 2162 y Fe(\014)1375 2190 y(\014)1389 2175 y Fv(2)1426 2170 y Fi(dxd\030)p 1421 2190 V 1421 2232 a Fn(\(2)p Fi(\031)r Fn(\))1508 2219 y Fv(3)669 2346 y Fn(=)746 2284 y Fe(Z)771 2387 y Fh(Q)799 2377 y Fc(?)827 2346 y Fn(T)l(r)874 2357 y Fh(L)898 2346 y Ft(2)898 2369 y Fc(\030)915 2357 y Fv(\()p Fh(Q)p Fv(\))972 2309 y Fe(\002)991 2346 y Fg(\000)p Fn(\001)1064 2353 y Fh(\030)1083 2346 y Fi(K)1125 2327 y Fh(n)1122 2357 y(\030)1149 2309 y Fe(\003)1210 2315 y Fi(d\030)p 1180 2335 V 1180 2377 a Fn(\(2)p Fi(\031)r Fn(\))1267 2364 y Fv(3)1290 2346 y Fi(;)456 b Fn(\(4.42\))37 2453 y(the)16 b(second)h(inequalit)o(y)g(b) q(eing)g(true)f(since)h Fg(jrj)p Fi(f)5 b Fg(jj)12 b(\024)i(jr)p Fi(f)5 b Fg(j)15 b Fn(for)g(an)o(y)g(complex-v)m(alued)j(function)f Fi(f)5 b Fn(.)22 b(Let)37 2510 y(us)16 b(no)o(w)f(observ)o(e)g(that)358 2559 y Fe(\014)358 2586 y(\014)373 2535 y(Z)399 2639 y Fh(Q)436 2597 y Fi(G)p Fn(\()p Fi(x)p Fn(\))p Fi(\032)557 2604 y Fh(n)580 2597 y Fn(\()p Fi(x;)8 b(x)p Fn(\))g Fi(dx)747 2559 y Fe(\014)747 2586 y(\014)801 2597 y Fg(\024)42 b Fi(C)17 b Fg(k)p Fi(G)p Fg(k)1009 2607 y Fh(L)1033 2598 y Ft(2)1049 2607 y Fv(\()p Fh(Q)p Fv(\))1119 2597 y Fg(k)p Fi(\032)1165 2604 y Fh(n)1189 2597 y Fg(k)1212 2573 y Fv(3)p Fh(=)p Fv(4)1212 2614 y Fh(L)1236 2605 y Ft(1)1253 2614 y Fv(\()p Fh(Q)p Fv(\))1323 2597 y Fg(k)p Fi(\032)1369 2604 y Fh(n)1392 2597 y Fg(k)1415 2573 y Fv(1)p Fh(=)p Fv(4)1415 2614 y Fh(L)1439 2605 y Ft(3)1456 2614 y Fv(\()p Fh(Q)p Fv(\))801 2710 y Fg(\024)42 b Fi(C)17 b Fg(k)p Fi(G)p Fg(k)1009 2720 y Fh(L)1033 2711 y Ft(2)1049 2720 y Fv(\()p Fh(Q)p Fv(\))1119 2710 y Fg(k)1142 2680 y(p)p 1180 2680 24 2 v 30 x Fi(\032)1203 2726 y Fh(n)1227 2710 y Fg(k)1250 2687 y Fv(1)p Fh(=)p Fv(2)1250 2728 y Fh(H)1282 2718 y Ft(1)1298 2728 y Fv(\()p Fh(Q)p Fv(\))801 2796 y Fg(\024)42 b Fi(C)17 b Fg(k)p Fi(G)p Fg(k)1009 2806 y Fh(L)1033 2797 y Ft(2)1049 2806 y Fv(\()p Fh(Q)p Fv(\))1119 2796 y Fn(\(1)9 b(+)i Fg(kr)1276 2766 y(p)p 1313 2766 V 1313 2796 a Fi(\032)1337 2811 y Fh(n)1360 2796 y Fg(k)1383 2777 y Fv(2)1383 2810 y Fh(L)1407 2801 y Ft(2)1424 2810 y Fv(\()p Fh(Q)p Fv(\))1482 2796 y Fn(\))1500 2777 y Fv(1)p Fh(=)p Fv(4)1759 2796 y Fn(\(4.43\))p eop %%Page: 42 42 42 41 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(42)37 199 y(since)15 b Fi(G)e Fn(is)h Fi(L)271 183 y Fv(2)304 199 y Fn(on)f Fi(Q)g Fn(\(it)g(has)h(only)f(a)g (singularit)o(y)i(lik)o(e)1009 181 y Fv(1)p 998 188 40 2 v 998 216 a Fu(j)p Fh(x)p Fu(j)1043 199 y Fn(\))e(and)g Fg(k)p Fi(\032)1206 206 y Fh(n)1230 199 y Fg(k)1253 209 y Fh(L)1277 200 y Ft(1)1294 209 y Fv(\()p Fh(Q)p Fv(\))1364 199 y Fn(=)g(1,)g(and)g(where)h Fi(C)j Fn(denotes)37 262 y(here)g(and)f(b)q(elo)o(w)h(some)f(p)q(ositiv)o(e)h(constan)o(t)f (that)f(is)i(indep)q(enden)o(t)h(of)e Fi(n)p Fn(.)23 b(Inserting)17 b(\(4.42\))d(and)j(\(4.43\))37 319 y(in)o(to)h(the)g (de\014nition)h(\(2.18\))d(of)h(the)h(energy)l(,)g(and)g(noticing)g (that)1216 301 y Fv(1)p 1216 308 18 2 v 1216 334 a(2)1239 319 y Fi(D)1277 326 y Fh(G)1306 319 y Fn(\()p Fg(\001)p Fi(;)8 b Fg(\001)p Fn(\))15 b Fg(\025)i Fn(0,)g(w)o(e)h(obtain)f(that)g (w)o(e)37 375 y(ha)o(v)o(e,)e(for)f(the)i(minimizing)i(sequence)e Fi(K)768 358 y Fh(n)807 375 y Fn(,)332 432 y Fe(p)p 378 432 155 2 v 378 473 a Fi(\032)401 480 y Fh(n)425 473 y Fn(\()p Fi(x;)8 b(x)p Fn(\))13 b(is)j(b)q(ounded)h(in)f Fi(H)874 454 y Fv(1)893 473 y Fn(\()p Fi(Q)p Fn(\))p Fi(;)21 b Fn(and)16 b(th)o(us)f(in)h Fi(L)1272 454 y Fh(p)1292 473 y Fn(\()p Fi(Q)p Fn(\))e(1)e Fg(\024)h Fi(p)g Fg(\024)g Fn(6)p Fi(:)178 b Fn(\(4.44\))37 571 y(This)16 b(can)g(also)f(b)q(e)h(expressed)g(b)o(y)f(stating)f(that)320 626 y Fe(Z)345 729 y Fh(Q)373 719 y Fc(?)400 688 y Fn(T)l(r)447 699 y Fh(L)471 688 y Ft(2)471 711 y Fc(\030)488 699 y Fv(\()p Fh(Q)p Fv(\))546 651 y Fe(\002)565 688 y Fg(\000)p Fn(\001)638 695 y Fh(\030)657 688 y Fi(K)699 669 y Fh(n)696 699 y(\030)722 651 y Fe(\003)748 688 y Fi(d\030)h Fn(=)855 626 y Fe(Z)5 b(Z)910 729 y Fh(Q)p Fu(\002)p Fh(Q)993 719 y Fc(?)1021 644 y Fe(X)1022 742 y Fh(p)p Fu(\025)p Fv(1)1094 688 y Fi(\025)1120 669 y Fv(\()p Fh(n)p Fv(\))1120 699 y Fh(p)1172 688 y Fn(\()p Fi(\030)r Fn(\))p Fg(jr)p Fi(u)1307 669 y Fv(\()p Fh(n)p Fv(\))1307 699 y Fh(p)1356 688 y Fn(\()p Fi(\030)r(;)j(y)r Fn(\))p Fg(j)1472 669 y Fv(2)1498 688 y Fi(dxd\030)167 b Fn(\(4.45\))37 831 y(is)16 b(b)q(ounded)h(indep)q(enden)o(tly)h(of)d Fi(n)p Fn(.)37 912 y(Using)h(the)g(Cauc)o(h)o(y-Sc)o(h)o(w)o(arz)e(inequalit)o (y)j(as)e(follo)o(ws)86 1032 y Fg(j)p Fi(\032)122 1039 y Fh(n)145 1032 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)40 b Fn(=)382 967 y Fe(\014)382 994 y(\014)382 1021 y(\014)382 1048 y(\014)397 971 y(Z)422 1074 y Fh(Q)450 1064 y Fc(?)478 989 y Fe(X)479 1087 y Fh(p)p Fu(\025)p Fv(1)551 1032 y Fi(\025)577 1014 y Fv(\()p Fh(n)p Fv(\))577 1044 y Fh(p)628 1032 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)712 1014 y Fv(\()p Fh(n)p Fv(\))712 1044 y Fh(p)763 1032 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fi(u)894 1014 y Fv(\()p Fh(n)p Fv(\))894 1044 y Fh(p)943 1032 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1046 1014 y Fu(\003)1099 1002 y Fi(d\030)p 1069 1022 106 2 v 1069 1064 a Fn(\(2)p Fi(\031)r Fn(\))1156 1051 y Fv(3)1179 967 y Fe(\014)1179 994 y(\014)1179 1021 y(\014)1179 1048 y(\014)305 1197 y Fg(\024)382 1133 y Fe(\022)415 1135 y(Z)440 1238 y Fh(Q)468 1229 y Fc(?)496 1154 y Fe(X)497 1252 y Fh(p)p Fu(\025)p Fv(1)569 1197 y Fi(\025)595 1178 y Fv(\()p Fh(n)p Fv(\))595 1208 y Fh(p)647 1197 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)744 1178 y Fv(\()p Fh(n)p Fv(\))744 1208 y Fh(p)793 1197 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)911 1178 y Fv(2)964 1167 y Fi(d\030)p 934 1187 V 934 1228 a Fn(\(2)p Fi(\031)r Fn(\))1021 1215 y Fv(3)1045 1133 y Fe(\023)1083 1131 y Ft(1)p 1083 1137 16 2 v 1083 1158 a(2)1105 1133 y Fe(\022)1139 1135 y(Z)1164 1238 y Fh(Q)1192 1229 y Fc(?)1219 1154 y Fe(X)1221 1252 y Fh(p)p Fu(\025)p Fv(1)1293 1197 y Fi(\025)1319 1178 y Fv(\()p Fh(n)p Fv(\))1319 1208 y Fh(p)1370 1197 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)1467 1178 y Fv(\()p Fh(n)p Fv(\))1467 1208 y Fh(p)1517 1197 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))p Fg(j)1633 1178 y Fv(2)1686 1167 y Fi(d\030)p 1656 1187 106 2 v 1656 1228 a Fn(\(2)p Fi(\031)r Fn(\))1743 1215 y Fv(3)1766 1133 y Fe(\023)1805 1131 y Ft(1)p 1805 1137 16 2 v 1805 1158 a(2)305 1324 y Fn(=)382 1283 y Fe(p)p 427 1283 155 2 v 41 x Fi(\032)450 1331 y Fh(n)474 1324 y Fn(\()p Fi(x;)g(x)p Fn(\))594 1283 y Fe(p)p 640 1283 151 2 v 640 1324 a Fi(\032)663 1331 y Fh(n)687 1324 y Fn(\()p Fi(y)r(;)g(y)r Fn(\))n Fi(;)956 b Fn(\(4.46\))37 1422 y(w)o(e)15 b(obtain)h(a)f(direct)h (corollary)f(of)g(the)g(b)q(ound)h(\(4.44\):)505 1520 y Fi(\032)528 1527 y Fh(n)552 1520 y Fn(\()p Fi(x;)8 b(y)r Fn(\))13 b(is)j(b)q(ounded)h(in)f Fi(L)988 1502 y Fh(p)1007 1520 y Fn(\()p Fi(Q)10 b Fg(\002)h Fi(Q)p Fn(\))p Fi(;)22 b Fn(1)12 b Fg(\024)h Fi(p)f Fg(\024)h Fn(6)p Fi(:)351 b Fn(\(4.47\))37 1619 y(Another)19 b(corollary)g(of)g (these)g(b)q(ounds)g(is)h(obtained)f(b)o(y)g(using)g(the)g(con)o(v)o (exit)o(y)g(of)f(the)h(function)h Fi(f)k Fg(7!)55 1638 y Fe(R)r(R)108 1691 y Fh(Q)p Fu(\002)p Fh(Q)191 1681 y Fc(?)219 1675 y Fg(jr)270 1641 y(p)p 307 1641 28 2 v 307 1675 a Fi(f)5 b Fg(j)347 1658 y Fv(2)374 1675 y Fi(dxd\030)r Fn(,)14 b(it)i(is)532 1747 y Fe(p)p 577 1747 197 2 v 41 x Fi(\032)600 1795 y Fh(n)624 1788 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))13 b(is)j(b)q(ounded)h(in)f Fi(L)1105 1769 y Fv(2)1124 1788 y Fn(\()p Fi(Q)1178 1769 y Fh(?)1198 1788 y Fn(;)8 b Fi(H)1261 1769 y Fv(1)1279 1788 y Fn(\()p Fi(Q)p Fn(\)\))p Fi(:)377 b Fn(\(4.48\))37 1886 y(Finally)l(,)17 b(a)e(v)o(ery)g(useful)i(b)q(ound)f(is)g (obtained)g(from)e(\(4.45\))g(b)o(y)h(using)h(the)f(Lieb-Thirring)j (inequalit)o(y)f(in)37 1942 y(this)g(setting.)22 b(\(This)16 b(is)g(an)g(easy)g(adaptation)f(of)g(the)h(Lieb-Thirring)j(inequalit)o (y)e(in)g(the)f(p)q(erio)q(dic)i(case)37 1999 y(giv)o(en)d(in)f(the)g (App)q(endix)i(of)d([54)o(])g(for)g(\014nite-rank)i(pro)s(jectors)d (together)h(with)h(the)g(results)g(of)f([39)o(])h(for)f(its)37 2055 y(extension)k(to)d(general)i(densit)o(y)g(matrices.\))j(W)l(e)c (ha)o(v)o(e,)g(for)f(almost)h(ev)o(ery)g Fi(\030)g Fg(2)e Fi(Q)1471 2039 y Fh(?)1490 2055 y Fn(,)496 2112 y Fe(Z)521 2215 y Fh(Q)558 2174 y Fi(\032)581 2181 y Fh(n)605 2174 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))757 2155 y Fv(5)p Fh(=)p Fv(3)810 2174 y Fi(dx)k Fg(\024)h Fi(C)952 2181 y Fv(0)985 2174 y Fn(T)l(r)1032 2186 y Fh(L)1056 2175 y Ft(2)1056 2198 y Fc(\030)1073 2186 y Fv(\()p Fh(Q)p Fv(\))1131 2137 y Fe(\002)1149 2174 y Fn(\(1)d Fg(\000)g Fn(\001)1283 2181 y Fh(\030)1302 2174 y Fn(\))p Fi(K)1362 2155 y Fh(n)1359 2186 y(\030)1385 2137 y Fe(\003)1404 2174 y Fi(;)37 2299 y Fn(for)19 b(some)h(constan)o(t)e Fi(C)449 2306 y Fv(0)469 2299 y Fn(,)j(whic)o(h)f(ma)o(y)f(b)q(e)h(c)o (hosen)g(indep)q(enden)o(tly)i(of)d Fi(\030)r Fn(,)h(since)h Fi(\030)g Fn(lies)g(in)f(a)g(b)q(ounded)37 2356 y(subset)c(of)f Fj(R)268 2339 y Fv(3)288 2356 y Fn(,)f(and)i(therefore)f(b)o(y)g(in)o (tegration)g(on)g Fi(Q)988 2339 y Fh(?)1023 2356 y Fn(\(since)h(the)f (left-hand)h(side)h(lies)f(in)g Fi(L)1677 2339 y Fv(1)1697 2356 y Fn(\()p Fi(Q)1751 2339 y Fh(?)1770 2356 y Fn(\)\),)299 2413 y Fe(Z)324 2516 y Fh(Q)352 2507 y Fc(?)415 2444 y Fi(d\030)p 385 2465 106 2 v 385 2506 a Fn(\(2)p Fi(\031)r Fn(\))472 2493 y Fv(3)503 2413 y Fe(Z)528 2516 y Fh(Q)565 2475 y Fi(\032)588 2482 y Fh(n)612 2475 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))764 2456 y Fv(5)p Fh(=)p Fv(3)825 2475 y Fi(dx)k Fg(\024)h Fi(C)978 2413 y Fe(Z)1003 2516 y Fh(Q)1031 2507 y Fc(?)1059 2475 y Fn(T)l(r)1106 2487 y Fh(L)1130 2475 y Ft(2)1130 2499 y Fc(\030)1147 2487 y Fv(\()p Fh(Q)p Fv(\))1204 2438 y Fe(\002)1223 2475 y Fn(\(1)d Fg(\000)g Fn(\001\))1375 2482 y Fh(\030)1394 2475 y Fi(K)1436 2456 y Fh(n)1433 2487 y(\030)1459 2438 y Fe(\003)1520 2444 y Fi(d\030)p 1490 2465 V 1490 2506 a Fn(\(2)p Fi(\031)r Fn(\))1577 2493 y Fv(3)1601 2475 y Fi(:)37 2600 y Fn(This)16 b(sho)o(ws)f(that)564 2698 y Fi(\032)587 2705 y Fh(n)611 2698 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))13 b(is)i(b)q(ounded)i(in)f Fi(L)1091 2679 y Fv(5)p Fh(=)p Fv(3)1146 2698 y Fn(\()p Fi(Q)1200 2679 y Fh(?)1229 2698 y Fg(\002)11 b Fi(Q)p Fn(\))p Fi(;)417 b Fn(\(4.49\))37 2796 y(and)16 b(concludes)h(our)e(\014rst)f(step,)h (dev)o(oted)g(to)g(the)g Ff(a)i(priori)f Fn(b)q(ounds)g(on)f(the)g (sequence)i Fi(K)1615 2780 y Fh(n)1638 2796 y Fn(.)p eop %%Page: 43 43 43 42 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(43)108 199 y(Step)16 b(2:)j(P)o(assing)c(to)g(the)g (limit)i(in)f(the)f(constrain)o(t)p 108 206 896 2 v 108 297 a(Let)20 b(us)h(\014rst)e(remark)h(that,)g(in)h(view)f(of)g(the)g (b)q(ound)h(\(4.40\),)e(w)o(e)h(ma)o(y)f(assume)h(without)g(loss)g(of) 37 354 y(generalit)o(y)12 b(that)e(the)h(sequence)h Fi(\032)618 361 y Fh(n)642 354 y Fn(\()p Fi(\030)r(;)c(x;)g(y)r Fn(\))g(con)o(v)o (erges)i(w)o(eakly)h(in)h Fi(L)1225 337 y Fv(2)1244 354 y Fn(\()p Fi(Q)1298 337 y Fh(?)1319 354 y Fg(\002)q Fi(Q)q Fg(\002)q Fi(Q)p Fn(\))f(to)f(some)h Fi(\032)1677 361 y Fu(1)1715 354 y Fn(\()p Fi(\030)r(;)d(x;)g(y)r Fn(\).)37 410 y(According)15 b(to)e(the)h(formal)f(decomp)q(osition)i(giv)o(en)f (in)g([45)o(],)g(and)f(recalled)j(in)e(Section)h(3.2,)d(w)o(e)i(ma)o(y) e(no)o(w)37 467 y(de\014ne)23 b(a)e(self-adjoin)o(t)h(op)q(erator)f Fi(K)691 450 y Fu(1)749 467 y Fn(on)h Fi(L)850 450 y Fv(2)869 467 y Fn(\()p Fj(R)926 450 y Fv(3)946 467 y Fn(\),)g(b)o(y)f Fi(K)1110 450 y Fu(1)1170 467 y Fn(=)1229 430 y Fe(R)1250 483 y Fh(Q)1278 473 y Fc(?)1306 467 y Fi(K)1348 450 y Fu(1)1345 480 y Fh(\030)1422 446 y(d\030)p 1397 456 84 2 v 1397 483 a Fv(\(2)p Fh(\031)q Fv(\))1464 474 y Ft(3)1486 467 y Fn(,)i(where)e Fi(K)1701 450 y Fu(1)1698 480 y Fh(\030)1760 467 y Fn(is)h(the)37 534 y(Hilb)q(ert-Sc)o(hmidt)g(op)q(erator)d(on)g Fi(L)662 517 y Fv(2)662 548 y Fh(\030)681 534 y Fn(\()p Fi(Q)p Fn(\))g(whose)g(k)o(ernel)h(is)g(the)f(function)h Fi(\032)1385 541 y Fu(1)1423 534 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\).)30 b(Another)19 b(w)o(a)o(y)37 590 y(to)h(state)e(the)i(w)o(eak) f(con)o(v)o(ergence)h(of)g Fi(\032)750 597 y Fh(n)793 590 y Fn(to)f Fi(\032)876 597 y Fu(1)934 590 y Fn(is)h(to)f(sa)o(y)g (that)g(for)g(almost)h Fi(\030)i Fg(2)e Fi(Q)1584 574 y Fh(?)1604 590 y Fn(,)g(and)g(for)f(an)o(y)37 647 y(self-adjoin)o(t)h (op)q(erator)d Fi(L)i Fn(on)f Fi(L)608 630 y Fv(2)628 647 y Fn(\()p Fj(R)685 630 y Fv(3)705 647 y Fn(\),)g(suc)o(h)h(that)f Fi(L)g Fn(=)1065 610 y Fe(R)1086 663 y Fh(Q)1114 653 y Fc(?)1142 647 y Fi(L)1173 654 y Fh(\030)1228 626 y(d\030)p 1204 636 V 1204 664 a Fv(\(2)p Fh(\031)q Fv(\))1271 654 y Ft(3)1293 647 y Fn(,)h(where)g(the)f(op)q(erators)g Fi(L)1777 654 y Fh(\030)1814 647 y Fn(are)37 714 y(Hilb)q(ert-Sc)o (hmidt)i(op)q(erators)c(on)h Fi(L)673 697 y Fv(2)673 728 y Fh(\030)693 714 y Fn(\()p Fi(Q)p Fn(\),)f(whose)h(k)o(ernels)h Fi(L)p Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))15 b(b)q(elong)j(to)f Fi(L)1516 697 y Fv(2)1535 714 y Fn(\()p Fi(Q)1589 697 y Fh(?)1609 714 y Fn(;)8 b Fi(L)1661 697 y Fv(2)1679 714 y Fn(\()p Fi(Q)j Fg(\002)h Fi(Q)p Fn(\)\),)37 770 y(w)o(e)j(ha)o(v)o(e)354 876 y(lim)315 904 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)464 814 y Fe(Z)489 917 y Fh(Q)517 908 y Fc(?)545 876 y Fn(T)l(r)591 888 y Fh(L)615 876 y Ft(2)615 900 y Fc(\030)633 888 y Fv(\()p Fh(Q)p Fv(\))690 876 y Fn([)p Fi(K)745 857 y Fh(n)742 887 y(\030)778 876 y Fg(\001)10 b Fi(L)832 857 y Fu(\003)832 887 y Fh(\030)851 876 y Fn(])907 845 y Fi(d\030)p 877 865 106 2 v 877 907 a Fn(\(2)p Fi(\031)r Fn(\))964 894 y Fv(3)1000 876 y Fn(=)1048 814 y Fe(Z)1073 917 y Fh(Q)1101 908 y Fc(?)1128 876 y Fn(T)l(r)1175 888 y Fh(L)1199 876 y Ft(2)1199 900 y Fc(\030)1216 888 y Fv(\()p Fh(Q)p Fv(\))1274 876 y Fn([)p Fi(K)1329 857 y Fu(1)1326 887 y Fh(\030)1376 876 y Fg(\001)f Fi(L)1429 857 y Fu(\003)1429 887 y Fh(\030)1449 876 y Fn(])1504 845 y Fi(d\030)p 1474 865 V 1474 907 a Fn(\(2)p Fi(\031)r Fn(\))1561 894 y Fv(3)1584 876 y Fi(:)162 b Fn(\(4.50\))37 998 y(Clearly)l(,)16 b(the)g(op)q(erators)e Fi(K)526 982 y Fu(1)523 1012 y Fh(\030)578 998 y Fn(satisfy)h(0)d Fg(\024)h Fi(K)845 982 y Fu(1)842 1012 y Fh(\030)895 998 y Fg(\024)g Fj(1)p Fn(,)i(and)h(th)o(us)f(0)d Fg(\024)h Fi(K)1311 982 y Fu(1)1361 998 y Fg(\024)g Fj(1)p Fn(.)108 1080 y(A)i(second)h(consequence)h(of)e(the)g(b)q(ounds)h(of)f(Step)h(1) f(comes)g(from)g(\(4.44\).)j(Again,)d(w)o(e)g(ma)o(y)g(alw)o(a)o(ys)37 1136 y(assume)e(that)f(the)h(sequence)550 1097 y Fe(p)p 595 1097 155 2 v 39 x Fi(\032)618 1143 y Fh(n)642 1136 y Fn(\()p Fi(x;)8 b(x)p Fn(\))j(con)o(v)o(erges)i(w)o(eakly)f(in)i Fi(H)1204 1120 y Fv(1)1200 1147 y(p)q(er)1252 1136 y Fn(\()p Fi(Q)p Fn(\),)e(strongly)g(in)i Fi(L)1604 1120 y Fh(p)1624 1136 y Fn(\()p Fi(Q)p Fn(\),)e(1)g Fg(\024)h Fi(p)g(<)37 1200 y Fn(6,)i(and)g(almost)g(ev)o(erywhere)g(on)h Fj(R)662 1184 y Fv(3)682 1200 y Fn(,)e(to)h(some)g(function)1057 1162 y Fe(p)p 1102 1162 169 2 v 1106 1200 a Fn(\026)-27 b Fi(\032)1125 1207 y Fu(1)1163 1200 y Fn(\()p Fi(x;)8 b(x)p Fn(\))j Fg(2)i Fi(H)1368 1184 y Fv(1)1364 1212 y(p)q(er)1416 1200 y Fn(\()p Fi(Q)p Fn(\).)108 1282 y(A)g(third)h (consequence)g(of)f(the)g(b)q(ounds)h(of)f(Step)g(1)g(is)h(deduced)g (from)f(\(4.49\):)k(w)o(e)12 b(ma)o(y)h(supp)q(ose)h(that)37 1338 y(the)f(sequence)h(of)f(\(non-negativ)o(e\))f(functions)i Fi(\032)866 1345 y Fh(n)890 1338 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))i(con)o(v)o(erges)i(w)o(eakly)h(in)h Fi(L)1483 1322 y Fv(5)p Fh(=)p Fv(3)1538 1338 y Fn(\()p Fi(Q)1592 1322 y Fh(?)1616 1338 y Fg(\002)5 b Fi(Q)p Fn(\))13 b(to)f(some)37 1395 y(\(non-negativ)o(e\))k(function)g(that)e (w)o(e)h(denote)g(for)g(the)g(momen)o(t)g(b)o(y)k(\026)-27 b Fi(\032)1240 1402 y Fu(1)1278 1395 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\).)108 1451 y(Let)14 b(us)f(\014rst)g(pro)o(v)o(e)g (that)553 1415 y Fe(R)575 1467 y Fh(Q)603 1458 y Fc(?)634 1451 y Fn(\026)-27 b Fi(\032)653 1458 y Fu(1)691 1451 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))877 1431 y Fh(d\030)p 853 1441 84 2 v 853 1468 a Fv(\(2)p Fh(\031)q Fv(\))920 1459 y Ft(3)954 1451 y Fn(=)18 b(\026)-28 b Fi(\032)1025 1458 y Fu(1)1063 1451 y Fn(\()p Fi(x;)8 b(x)p Fn(\).)18 b(F)l(or)13 b(this)g(purp)q(ose,)h(w)o(e)f(note)g(that)g(the)37 1522 y(w)o(eak)i(con)o(v)o(ergence)h(in)g Fi(L)487 1505 y Fv(5)p Fh(=)p Fv(3)541 1522 y Fn(\()p Fi(Q)595 1505 y Fh(?)625 1522 y Fg(\002)10 b Fi(Q)p Fn(\))15 b(implies)i(in)f (particular)g(that,)e(for)h(an)o(y)g(function)h Fi(v)e Fg(2)f Fi(L)1711 1505 y Fv(5)p Fh(=)p Fv(2)1766 1522 y Fn(\()p Fi(Q)p Fn(\),)272 1639 y(lim)233 1666 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)382 1577 y Fe(Z)407 1680 y Fh(Q)435 1671 y Fc(?)497 1608 y Fi(d\030)p 468 1628 106 2 v 468 1670 a Fn(\(2)p Fi(\031)r Fn(\))555 1657 y Fv(3)585 1577 y Fe(Z)611 1680 y Fh(Q)648 1639 y Fi(\032)671 1646 y Fh(n)695 1639 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))p Fi(v)r Fn(\()p Fi(x)p Fn(\))f Fi(dx)j Fn(=)1048 1577 y Fe(Z)1073 1680 y Fh(Q)1101 1671 y Fc(?)1163 1608 y Fi(d\030)p 1133 1628 V 1133 1670 a Fn(\(2)p Fi(\031)r Fn(\))1220 1657 y Fv(3)1251 1577 y Fe(Z)1276 1680 y Fh(Q)1318 1639 y Fn(\026)-27 b Fi(\032)1337 1646 y Fu(1)1375 1639 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))p Fi(v)r Fn(\()p Fi(x)p Fn(\))f Fi(dx:)37 1761 y Fn(No)o(w,)15 b(the)g(left-hand)h(side) g(is)g(also)f(giv)o(en)h(b)o(y)311 1876 y(lim)272 1903 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)420 1814 y Fe(Z)446 1917 y Fh(Q)475 1839 y Fe(\000)496 1814 y(Z)522 1917 y Fh(Q)550 1907 y Fc(?)577 1876 y Fi(\032)600 1883 y Fh(n)624 1876 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))816 1845 y Fi(d\030)p 786 1865 V 786 1907 a Fn(\(2)p Fi(\031)r Fn(\))873 1894 y Fv(3)896 1839 y Fe(\001)917 1876 y Fi(v)r Fn(\()p Fi(x)p Fn(\))p Fi(dx)k Fn(=)51 b(lim)1113 1903 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)1261 1814 y Fe(Z)1286 1917 y Fh(Q)1324 1876 y Fi(\032)1347 1883 y Fh(n)1371 1876 y Fn(\()p Fi(x;)8 b(x)p Fn(\))g Fi(v)r Fn(\()p Fi(x)p Fn(\))f Fi(dx;)37 1998 y Fn(and)16 b(th)o(us)f(b)o(y)518 2112 y(lim)479 2140 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)628 2050 y Fe(Z)653 2153 y Fh(Q)690 2112 y Fi(\032)713 2119 y Fh(n)737 2112 y Fn(\()p Fi(x;)8 b(x)p Fn(\))g Fi(v)r Fn(\()p Fi(x)p Fn(\))g Fi(dx)h Fn(=)1055 2050 y Fe(Z)1080 2153 y Fh(Q)1122 2112 y Fn(\026)-27 b Fi(\032)1141 2119 y Fu(1)1179 2112 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fi(v)r Fn(\()p Fi(x)p Fn(\))p Fi(dx:)37 2234 y Fn(Therefore,)15 b(w)o(e)g(ha)o(v)o(e)632 2287 y Fe(Z)657 2390 y Fh(Q)685 2380 y Fc(?)717 2349 y Fn(\026)-27 b Fi(\032)736 2356 y Fu(1)774 2349 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))958 2318 y Fi(d\030)p 931 2338 V 931 2380 a Fn(\(2)p Fi(\031)q Fn(\))1017 2367 y Fv(3)1051 2349 y Fn(=)17 b(\026)-27 b Fi(\032)1122 2356 y Fu(1)1160 2349 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fi(:)477 b Fn(\(4.51\))108 2474 y(A)o(t)16 b(this)g(stage,)f(w)o(e)h(do)g(not)f(kno)o(w)h Ff(a)h(priori)g Fn(that)e Fi(\032)1018 2481 y Fu(1)1056 2474 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))k(=)18 b(\026)-27 b Fi(\032)1292 2481 y Fu(1)1330 2474 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\),)13 b(but)k(w)o(e)e(shall)j(pro)o(v)o(e)37 2530 y(this)e(claim)g(b)q(elo)o(w)g(in)g(Step)g(3.)108 2587 y(Let)i(us)h(no)o(w)e(turn)i(to)e(the)h(pro)q(of)g(of)g(the)g(fact)g (that)f(the)h(op)q(erator)g Fi(K)1357 2570 y Fu(1)1412 2587 y Fn(necessarily)h(satis\014es)g(the)37 2643 y(constrain)o(t)c(:) 692 2687 y Fe(Z)717 2790 y Fh(Q)745 2780 y Fc(?)772 2749 y Fn(T)l(r)819 2760 y Fh(L)843 2749 y Ft(2)843 2772 y Fc(\030)860 2760 y Fv(\()p Fh(Q)p Fv(\))918 2749 y Fi(K)960 2730 y Fu(1)957 2760 y Fh(\030)1044 2718 y Fi(d\030)p 1015 2738 V 1015 2780 a Fn(\(2)p Fi(\031)r Fn(\))1102 2767 y Fv(3)1137 2749 y Fn(=)e(1)p Fi(:)538 b Fn(\(4.52\))p eop %%Page: 44 44 44 43 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(44)37 199 y(The)22 b(di\016cult)o(y)h(to)e(deduce)h (\(4.52\),)f(from)g(the)g(con)o(v)o(ergence)h(\(4.50\))d(and)j(the)f (fact)g(that)g(the)g(ab)q(o)o(v)o(e)37 256 y(constrain)o(t)h(is)g (satis\014ed)g(for)f(all)h Fi(n)p Fn(,)i(is)e(of)f(course)g(that)g(w)o (e)h(cannot)f(tak)o(e)g Fi(L)g Fn(as)h(b)q(eing)h(the)e(iden)o(tit)o(y) 37 312 y(op)q(erator)16 b(in)i(\(4.50\),)d(for)h(the)h(iden)o(tit)o(y)g (is)h(not)e(a)h(Hilb)q(ert-Sc)o(hmidt)i(op)q(erator.)24 b(In)17 b(order)g(to)f(conclude,)37 369 y(w)o(e)f(shall)i(need)f(to)e (use)i(the)f(b)q(ound)h(on)f(the)h(kinetic)g(energy)g(term.)j(W)l(e)c (argue)g(as)g(follo)o(ws.)108 425 y(F)l(or)22 b(all)h Fi(n)p Fn(,)h(and)e(for)g(almost)f(all)i Fi(\030)k Fg(2)d Fi(Q)863 409 y Fh(?)883 425 y Fn(,)f(w)o(e)f(kno)o(w)g(that)f Fi(K)1266 409 y Fh(n)1263 439 y(\030)1312 425 y Fn(and)h Fg(\000)p Fn(\001)1480 432 y Fh(\030)1499 425 y Fi(K)1541 409 y Fh(n)1538 439 y(\030)1587 425 y Fn(are)g(trace-class)37 498 y(op)q(erators)d(on)g Fi(L)341 482 y Fv(2)341 512 y Fh(\030)360 498 y Fn(\()p Fi(Q)p Fn(\).)31 b(In)20 b(particular,)g(this)f(implies)j(that)c(the)h(op)q(erator)f(\(1)12 b Fg(\000)h Fn(\001\))1543 474 y Fv(1)p Fh(=)p Fv(2)1543 513 y Fh(\030)1598 498 y Fi(K)1640 482 y Fh(n)1637 512 y(\030)1663 498 y Fn(\(1)f Fg(\000)h Fn(\001\))1820 474 y Fv(1)p Fh(=)p Fv(2)1820 513 y Fh(\030)37 555 y Fn(is)18 b(also)e(trace-class)h(\(th)o(us)f(in)i(particular)f(Hilb)q(ert-Sc)o (hmidt\).)27 b(In)17 b(addition,)h(since)f(w)o(e)g(ha)o(v)o(e)f(a)g(b)q (ound,)37 611 y(deriv)o(ed)h(from)d(\(4.45\))f(and)j(\(4.38\),)192 672 y Fe(Z)217 776 y Fh(Q)245 766 y Fc(?)273 734 y Fn(T)l(r)320 746 y Fh(L)344 735 y Ft(2)344 758 y Fc(\030)361 746 y Fv(\()p Fh(Q)p Fv(\))418 698 y Fe(\002)437 734 y Fn(\(1)10 b Fg(\000)g Fn(\001\))589 710 y Fv(1)p Fh(=)p Fv(2)589 749 y Fh(\030)644 734 y Fi(K)686 716 y Fh(n)683 746 y(\030)709 734 y Fn(\(1)g Fg(\000)g Fn(\001\))861 710 y Fv(1)p Fh(=)p Fv(2)861 749 y Fh(\030)916 698 y Fe(\003)942 734 y Fi(d\030)k Fn(=)1048 672 y Fe(Z)1074 776 y Fh(Q)1102 766 y Fc(?)1129 734 y Fn(T)l(r)1176 746 y Fh(L)1200 735 y Ft(2)1200 758 y Fc(\030)1217 746 y Fv(\()p Fh(Q)p Fv(\))1274 734 y Fn([\(1)c Fg(\000)g Fn(\001\))1439 741 y Fh(\030)1457 734 y Fi(K)1499 716 y Fh(n)1496 746 y(\030)1523 734 y Fn(])e Fi(d\030)26 b Fg(\024)f Fi(C)q(;)37 868 y Fn(w)o(e)13 b(ma)o(y)f(assume,)h(extracting)g(a)f(subsequence)j(if)e(necessary)l(,) h(that)e(the)h(sequence)h(of)e(op)q(erators)1724 832 y Fe(R)1746 884 y Fh(Q)1774 875 y Fc(?)1794 868 y Fn(\(1)6 b Fg(\000)37 944 y Fn(\001\))93 920 y Fv(1)p Fh(=)p Fv(2)93 959 y Fh(\030)148 944 y Fi(K)190 927 y Fh(n)187 958 y(\030)213 944 y Fn(\(1)k Fg(\000)h Fn(\001\))366 920 y Fv(1)p Fh(=)p Fv(2)366 959 y Fh(\030)457 923 y(d\030)p 433 933 84 2 v 433 961 a Fv(\(2)p Fh(\031)q Fv(\))500 951 y Ft(3)537 944 y Fn(con)o(v)o(erges)k(in)h(the)g(sense)g(of)f(\(4.50\),)e(and)i (its)h(limit)h(is)f(necessarily)1720 907 y Fe(R)1741 960 y Fh(Q)1769 950 y Fc(?)1789 944 y Fn(\(1)10 b Fg(\000)37 1022 y Fn(\001\))93 998 y Fv(1)p Fh(=)p Fv(2)93 1037 y Fh(\030)148 1022 y Fi(K)190 1006 y Fu(1)187 1036 y Fh(\030)227 1022 y Fn(\(1)e Fg(\000)h Fn(\001\))376 998 y Fv(1)p Fh(=)p Fv(2)376 1037 y Fh(\030)468 1002 y(d\030)p 443 1012 V 443 1039 a Fv(\(2)p Fh(\031)q Fv(\))510 1029 y Ft(3)532 1022 y Fn(.)20 b(T)l(esting)14 b(this)h(w)o(eak)f(con)o(v)o (ergence)h(with)f(the)h(op)q(erators)e Fi(L)1587 1029 y Fh(\030)1619 1022 y Fn(=)g(\(1)8 b Fg(\000)h Fn(\001\))1816 1003 y Fu(\000)p Fv(1)1816 1037 y Fh(\030)1862 1022 y Fn(,)37 1089 y(whic)o(h)17 b(are)d(Hilb)q(ert-Sc)o(hmidt)k(on)d Fi(L)672 1073 y Fv(2)672 1103 y Fh(\030)692 1089 y Fn(\()p Fi(Q)p Fn(\),)f(w)o(e)h(obtain)302 1225 y(lim)264 1253 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)412 1163 y Fe(Z)437 1267 y Fh(Q)465 1257 y Fc(?)493 1225 y Fn(T)l(r)540 1237 y Fh(L)564 1226 y Ft(2)564 1249 y Fc(\030)581 1237 y Fv(\()p Fh(Q)p Fv(\))638 1161 y Fe(\024)662 1225 y Fn(\(1)10 b Fg(\000)g Fn(\001\))814 1201 y Fv(1)p Fh(=)p Fv(2)814 1240 y Fh(\030)869 1225 y Fi(K)911 1207 y Fh(n)908 1237 y(\030)934 1225 y Fn(\(1)g Fg(\000)g Fn(\001\))1086 1201 y Fv(1)p Fh(=)p Fv(2)1086 1240 y Fh(\030)1141 1225 y Fn(\(1)f Fg(\000)h Fn(\001\))1292 1206 y Fu(\000)p Fv(1)1292 1240 y Fh(\030)1339 1161 y Fe(\025)1371 1225 y Fi(d\030)541 1355 y Fn(=)618 1294 y Fe(Z)643 1397 y Fh(Q)671 1387 y Fc(?)699 1355 y Fn(T)l(r)746 1367 y Fh(L)770 1356 y Ft(2)770 1379 y Fc(\030)787 1367 y Fv(\()p Fh(Q)p Fv(\))844 1291 y Fe(\024)868 1355 y Fn(\(1)g Fg(\000)g Fn(\001\))1020 1332 y Fv(1)p Fh(=)p Fv(2)1020 1370 y Fh(\030)1075 1355 y Fi(K)1117 1337 y Fu(1)1114 1367 y Fh(\030)1154 1355 y Fn(\(1)f Fg(\000)i Fn(\001\))1306 1332 y Fv(1)p Fh(=)p Fv(2)1306 1370 y Fh(\030)1360 1355 y Fn(\(1)f Fg(\000)g Fn(\001\))1512 1336 y Fu(\000)p Fv(1)1512 1370 y Fh(\030)1559 1291 y Fe(\025)1591 1355 y Fi(d\030)r Fn(;)37 1485 y(that)15 b(is)528 1596 y(lim)489 1624 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)637 1534 y Fe(Z)663 1637 y Fh(Q)691 1628 y Fc(?)718 1596 y Fn(T)l(r)765 1608 y Fh(L)789 1597 y Ft(2)789 1620 y Fc(\030)806 1608 y Fv(\()p Fh(Q)p Fv(\))863 1596 y Fi(K)905 1577 y Fh(n)902 1608 y(\030)941 1596 y Fi(d\030)15 b Fn(=)1048 1534 y Fe(Z)1073 1637 y Fh(Q)1101 1628 y Fc(?)1128 1596 y Fn(T)l(r)1175 1608 y Fh(L)1199 1597 y Ft(2)1199 1620 y Fc(\030)1216 1608 y Fv(\()p Fh(Q)p Fv(\))1274 1596 y Fi(K)1316 1577 y Fu(1)1313 1608 y Fh(\030)1365 1596 y Fi(d\030)r(:)37 1737 y Fn(Therefore,)f(as)309 1701 y Fe(R)330 1753 y Fh(Q)358 1744 y Fc(?)386 1737 y Fn(T)l(r)433 1749 y Fh(L)457 1738 y Ft(2)457 1761 y Fc(\030)474 1749 y Fv(\()p Fh(Q)p Fv(\))531 1737 y Fi(K)573 1721 y Fh(n)570 1751 y(\030)639 1717 y(d\030)p 614 1727 V 614 1754 a Fv(\(2)p Fh(\031)q Fv(\))681 1745 y Ft(3)716 1737 y Fn(=)f(1)g(for)g(all)h Fi(n)p Fn(,)g(w)o(e)f(deduce)i(that)e (the)h(op)q(erator)e Fi(K)1595 1721 y Fu(1)1646 1737 y Fn(satis\014es)i(the)37 1800 y(constrain)o(t.)108 1898 y(Step)i(3:)j(P)o(assing)c(to)g(the)g(limit)i(in)f(the)f(energy)p 108 1905 828 2 v 108 1996 a(A)j(simple)h(argumen)o(t,)d(using)j(the)e (op)q(erator)g(\(1)11 b Fg(\000)h Fn(\001\))17 b(and)h(F)l(atou's)e (lemma)i(allo)o(ws)f(one)h(to)f(sho)o(w,)37 2053 y(arguing)f(as)f(in)h (the)f(pro)q(of)g(of)f(Prop)q(osition)i(4.1,)e(and)h(making)h(use)f(of) g(Step)h(2,)e(that)141 2177 y(lim)8 b(inf)134 2205 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)275 2113 y Fe(\024)299 2177 y Fn(1)9 b(+)377 2115 y Fe(Z)402 2219 y Fh(Q)430 2209 y Fc(?)458 2177 y Fn(T)l(r)504 2189 y Fh(L)528 2178 y Ft(2)528 2201 y Fc(\030)546 2189 y Fv(\()p Fh(Q)p Fv(\))603 2140 y Fe(\002)622 2177 y Fg(\000)p Fn(\001)695 2184 y Fh(\030)714 2177 y Fi(K)756 2159 y Fh(n)753 2189 y(\030)779 2140 y Fe(\003)806 2177 y Fi(d\030)852 2113 y Fe(\025)917 2177 y Fn(=)49 b(lim)8 b(inf)994 2205 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)1142 2115 y Fe(Z)1168 2219 y Fh(Q)1196 2209 y Fc(?)1223 2177 y Fn(T)l(r)1270 2189 y Fh(L)1294 2178 y Ft(2)1294 2201 y Fc(\030)1311 2189 y Fv(\()p Fh(Q)p Fv(\))1368 2140 y Fe(\002)1387 2177 y Fn(\(1)10 b Fg(\000)g Fn(\001)1521 2184 y Fh(\030)1540 2177 y Fn(\))p Fi(K)1600 2159 y Fh(n)1597 2189 y(\030)1623 2140 y Fe(\003)1650 2177 y Fi(d\030)917 2304 y Fg(\025)81 b Fn(lim)994 2332 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)1142 2242 y Fe(Z)1168 2345 y Fh(Q)1196 2336 y Fc(?)1223 2304 y Fn(T)l(r)1270 2316 y Fh(L)1294 2304 y Ft(2)1294 2328 y Fc(\030)1311 2316 y Fv(\()p Fh(Q)p Fv(\))1368 2267 y Fe(\002)1387 2304 y Fn(\(1)10 b Fg(\000)g Fn(\001)1521 2311 y Fh(\030)1540 2304 y Fn(\))p Fi(K)1600 2285 y Fu(1)1597 2315 y Fh(\030)1637 2267 y Fe(\003)1698 2273 y Fi(d\030)p 1669 2293 106 2 v 1669 2335 a Fn(\(2)p Fi(\031)r Fn(\))1756 2322 y Fv(3)917 2430 y Fn(=)42 b(1)10 b(+)1072 2369 y Fe(Z)1097 2472 y Fh(Q)1125 2462 y Fc(?)1153 2430 y Fn(T)l(r)1200 2442 y Fh(L)1224 2431 y Ft(2)1224 2454 y Fc(\030)1241 2442 y Fv(\()p Fh(Q)p Fv(\))1298 2394 y Fe(\002)1317 2430 y Fg(\000)p Fn(\001)1390 2437 y Fh(\030)1409 2430 y Fi(K)1451 2412 y Fu(1)1448 2442 y Fh(\030)1488 2394 y Fe(\003)1550 2400 y Fi(d\030)p 1520 2420 V 1520 2462 a Fn(\(2)p Fi(\031)r Fn(\))1607 2449 y Fv(3)1630 2430 y Fi(;)37 2560 y Fn(and,)16 b(therefore,)e(that)302 2681 y(lim)8 b(inf)294 2708 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)443 2619 y Fe(Z)468 2722 y Fh(Q)496 2712 y Fc(?)523 2681 y Fn(T)l(r)570 2692 y Fh(L)594 2681 y Ft(2)594 2704 y Fc(\030)611 2692 y Fv(\()p Fh(Q)p Fv(\))669 2644 y Fe(\002)688 2681 y Fg(\000)p Fn(\001)761 2688 y Fh(\030)780 2681 y Fi(K)822 2662 y Fh(n)819 2692 y(\030)845 2644 y Fe(\003)907 2650 y Fi(d\030)p 877 2670 V 877 2712 a Fn(\(2)p Fi(\031)r Fn(\))964 2699 y Fv(3)1000 2681 y Fg(\025)1048 2619 y Fe(Z)1073 2722 y Fh(Q)1101 2712 y Fc(?)1128 2681 y Fn(T)l(r)1175 2692 y Fh(L)1199 2681 y Ft(2)1199 2704 y Fc(\030)1216 2692 y Fv(\()p Fh(Q)p Fv(\))1274 2644 y Fe(\002)1293 2681 y Fg(\000)p Fn(\001)1366 2688 y Fh(\030)1385 2681 y Fi(K)1427 2662 y Fu(1)1424 2692 y Fh(\030)1464 2644 y Fe(\003)1525 2650 y Fi(d\030)p 1495 2670 V 1495 2712 a Fn(\(2)p Fi(\031)r Fn(\))1582 2699 y Fv(3)1606 2681 y Fi(:)140 b Fn(\(4.53\))p eop %%Page: 45 45 45 44 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(45)37 199 y(Next,)13 b(a)f(standard)g(argumen)o(t)g(on) g(the)h(sequence)h Fi(\032)927 206 y Fh(n)951 199 y Fn(\()p Fi(x;)8 b(x)p Fn(\),)j(whose)h(square)g(ro)q(ot)g(con)o(v)o(erges)g(in) h Fi(H)1743 183 y Fv(1)1739 210 y(p)q(er)1791 199 y Fn(\()p Fi(Q)p Fn(\),)37 256 y(sho)o(ws)i(that)327 359 y(lim)288 386 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)429 295 y Fe(\024)453 359 y Fg(\000)496 297 y Fe(Z)522 400 y Fh(Q)559 359 y Fi(G)p Fn(\()p Fi(x)p Fn(\))p Fi(\032)680 366 y Fh(n)703 359 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fi(dx)g Fn(+)921 328 y(1)p 921 348 23 2 v 921 390 a(2)948 359 y Fi(D)986 366 y Fh(G)1016 359 y Fn(\()p Fi(\032)1057 366 y Fh(n)1080 359 y Fn(\()p Fi(x;)g(x)p Fn(\))p Fi(;)g(\032)1233 366 y Fh(n)1255 359 y Fn(\()p Fi(x;)g(x)p Fn(\)\))1382 295 y Fe(\025)566 485 y Fn(=)42 b Fg(\000)686 423 y Fe(Z)711 526 y Fh(Q)749 485 y Fi(G)p Fn(\()p Fi(x)p Fn(\))t(\026)-27 b Fi(\032)870 492 y Fu(1)907 485 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fi(dx)g Fn(+)1124 454 y(1)p 1124 474 V 1124 516 a(2)1152 485 y Fi(D)1190 492 y Fh(G)1219 485 y Fn(\()t(\026)-27 b Fi(\032)1260 492 y Fu(1)1298 485 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fi(;)i Fn(\026)-25 b Fi(\032)1451 492 y Fu(1)1486 485 y Fn(\()p Fi(x;)8 b(x)p Fn(\)\))p Fi(:)133 b Fn(\(4.54\))37 604 y(It)11 b(is)g(then)g(clear)g(that)f(\(4.53\))e (along)j(with)g(\(4.54\))e(will)j(su\016ce)f(to)f(establish)h(the)g (existence)h(of)e(a)g(minimizer)37 660 y(for)15 b(the)f(p)q(erio)q(dic) j(RHF)e(mo)q(del,)g(pro)o(vided)h(w)o(e)e(are)h(able)g(to)f(sho)o(w)g (that,)g(for)g(almost)g(ev)o(ery)h Fi(\030)g Fg(2)d Fi(Q)1767 644 y Fh(?)1802 660 y Fn(and)37 717 y Fi(x)h Fg(2)g Fi(Q)p Fn(,)713 806 y(\026)-27 b Fi(\032)732 813 y Fu(1)770 806 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))i(=)j Fi(\032)1003 813 y Fu(1)1041 806 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))p Fi(:)553 b Fn(\(4.55\))37 896 y(W)l(e)11 b(\014nally)h(pro)o(v) o(e)e(this)h(fact.)18 b(F)l(or)10 b(this)h(purp)q(ose,)h(w)o(e)e(c)o (ho)q(ose)h(an)f(arbitrary)g(function)h Fi(\022)q Fn(\()p Fi(\030)r(;)d(x)p Fn(\))k Fg(2)h Fi(L)1728 879 y Fu(1)1765 896 y Fn(\()p Fi(Q)1819 879 y Fh(?)1840 896 y Fg(\002)37 952 y Fi(Q)p Fn(\),)i(and)g(de\014ne)i(the)e(Hilb)q(ert-Sc)o(hmidt)j (op)q(erators)604 1050 y Fi(L)635 1057 y Fh(\030)667 1050 y Fn(=)13 b(\(1)c Fg(\000)i Fn(\001\))867 1026 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)867 1064 y Fh(\030)949 1050 y Fi(\022)q Fn(\()p Fi(\030)r(;)d Fg(\001)p Fn(\)\(1)g Fg(\000)i Fn(\001\))1213 1026 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1213 1064 y Fh(\030)1295 1050 y Fi(:)37 1157 y Fn(Using)16 b(the)f(con)o(v)o(ergence)g(\(4.50\))f(for)g(the)h(sequence)h(of)f(op)q (erators)f(\(1)9 b Fg(\000)h Fn(\001\))1365 1133 y Fv(1)p Fh(=)p Fv(2)1365 1171 y Fh(\030)1420 1157 y Fi(K)1462 1140 y Fh(n)1459 1171 y(\030)1485 1157 y Fn(\(1)f Fg(\000)h Fn(\001\))1636 1133 y Fv(1)p Fh(=)p Fv(2)1636 1171 y Fh(\030)1691 1157 y Fn(,)15 b(w)o(e)f(ha)o(v)o(e)117 1277 y(lim)78 1305 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)226 1215 y Fe(Z)251 1318 y Fh(Q)279 1309 y Fc(?)307 1277 y Fn(T)l(r)354 1289 y Fh(L)378 1277 y Ft(2)378 1301 y Fc(\030)395 1289 y Fv(\()p Fh(Q)p Fv(\))452 1240 y Fe(\002)471 1277 y Fn(\(1)10 b Fg(\000)g Fn(\001\))623 1253 y Fv(1)p Fh(=)p Fv(2)623 1292 y Fh(\030)678 1277 y Fi(K)720 1258 y Fh(n)717 1288 y(\030)743 1277 y Fn(\(1)g Fg(\000)g Fn(\001\))895 1253 y Fv(1)p Fh(=)p Fv(2)895 1292 y Fh(\030)950 1277 y Fn(\(1)f Fg(\000)h Fn(\001\))1101 1253 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1101 1292 y Fh(\030)1183 1277 y Fi(\022)q Fn(\()p Fi(\030)r(;)e Fg(\001)p Fn(\)\(1)h Fg(\000)h Fn(\001\))1448 1253 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1448 1292 y Fh(\030)1530 1240 y Fe(\003)1557 1277 y Fi(d\030)355 1403 y Fn(=)432 1341 y Fe(Z)457 1444 y Fh(Q)485 1435 y Fc(?)513 1403 y Fn(T)l(r)560 1415 y Fh(L)584 1403 y Ft(2)584 1427 y Fc(\030)601 1415 y Fv(\()p Fh(Q)p Fv(\))658 1366 y Fe(\002)677 1403 y Fn(\(1)g Fg(\000)g Fn(\001\))829 1379 y Fv(1)p Fh(=)p Fv(2)829 1418 y Fh(\030)884 1403 y Fi(K)926 1384 y Fu(1)923 1415 y Fh(\030)963 1403 y Fn(\(1)f Fg(\000)i Fn(\001\))1115 1379 y Fv(1)p Fh(=)p Fv(2)1115 1418 y Fh(\030)1169 1403 y Fn(\(1)f Fg(\000)g Fn(\001\))1321 1379 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1321 1418 y Fh(\030)1403 1403 y Fi(\022)q Fn(\()p Fi(\030)r(;)e Fg(\001)p Fn(\)\(1)h Fg(\000)h Fn(\001\))1668 1379 y Fu(\000)p Fv(1)p Fh(=)p Fv(2)1668 1418 y Fh(\030)1750 1366 y Fe(\003)1777 1403 y Fi(d\030)r(:)37 1519 y Fn(This)16 b(ma)o(y)f(also)g(read)265 1627 y(lim)226 1655 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)375 1565 y Fe(Z)5 b(Z)451 1669 y Fh(Q)479 1659 y Fc(?)496 1669 y Fu(\002)p Fh(Q)561 1627 y Fi(\032)584 1634 y Fh(n)608 1627 y Fn(\()p Fi(\030)r(;)j(x;)g(x) p Fn(\))p Fi(\022)q Fn(\()p Fi(\030)r(;)g(x)p Fn(\))g Fi(d)o(xd\030)k Fn(=)1048 1565 y Fe(Z)5 b(Z)1123 1669 y Fh(Q)1151 1659 y Fc(?)1169 1669 y Fu(\002)p Fh(Q)1234 1627 y Fi(\032)1257 1634 y Fu(1)1295 1627 y Fn(\()p Fi(\030)r(;)j(x;)g (x)p Fn(\))p Fi(\022)q Fn(\()p Fi(\030)r(;)g(x)p Fn(\))f Fi(dxd)o(\030)r(:)37 1746 y Fn(But)17 b(on)f(the)g(other)f(hand,)i(w)o (e)e(kno)o(w)h(that)f(the)h(sequence)h(of)f(functions)h Fi(\032)1346 1753 y Fh(n)1370 1746 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))13 b(con)o(v)o(erges)j(w)o(eakly)37 1803 y(in)h Fi(L)122 1786 y Fv(5)p Fh(=)p Fv(3)176 1803 y Fn(\()p Fi(Q)230 1786 y Fh(?)260 1803 y Fg(\002)10 b Fi(Q)p Fn(\))15 b(to)k(\026)-27 b Fi(\032)453 1810 y Fu(1)491 1803 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\),)k(th)o(us)j (w)o(e)g(also)g(ha)o(v)o(e)265 1913 y(lim)226 1941 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)375 1851 y Fe(Z)5 b(Z)451 1954 y Fh(Q)479 1945 y Fc(?)496 1954 y Fu(\002)p Fh(Q)561 1913 y Fi(\032)584 1920 y Fh(n)608 1913 y Fn(\()p Fi(\030)r(;)j(x;)g(x)p Fn(\))p Fi(\022)q Fn(\()p Fi(\030)r(;)g(x)p Fn(\))g Fi(d)o(xd\030)k Fn(=)1048 1851 y Fe(Z)5 b(Z)1123 1954 y Fh(Q)1151 1945 y Fc(?)1169 1954 y Fu(\002)p Fh(Q)1238 1913 y Fn(\026)-27 b Fi(\032)1257 1920 y Fu(1)1295 1913 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))p Fi(\022)q Fn(\()p Fi(\030)r(;)g(x)p Fn(\))f Fi(dxd)o(\030)r(:)37 2032 y Fn(This)17 b(sho)o(ws)f(the)h(equalit)o(y)g(\(4.55\),)d(and)j (concludes)h(the)e(pro)q(of)g(of)g(the)h(existence)g(of)f(a)g(minim)o (um.)25 b(W)l(e)37 2089 y(no)o(w)14 b(sho)o(w)g(the)h(uniqueness)h(of)e (the)g(p)q(erio)q(dic)j(densit)o(y)e Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\).)18 b(The)d(argumen)o(t)e(is)i(an)f (adaptation)g(of)g(a)37 2145 y(similar)i(claim)g(in)f([34)o(].)k (Assume)c(b)o(y)g(con)o(tradiction)g(that)f(there)g(exist)h(t)o(w)o(o)e (minimizers)k Fi(K)1643 2152 y Fv(1)1677 2145 y Fn(and)d Fi(K)1803 2152 y Fv(2)1837 2145 y Fn(in)37 2201 y Fg(K)i Fn(of)f Fi(I)164 2185 y Fh(RH)s(F)160 2213 y(per)251 2201 y Fn(.)20 b(Denoting)15 b(b)o(y)g Fi(\032)566 2208 y Fv(1)602 2201 y Fn(and)g Fi(\032)713 2208 y Fv(2)748 2201 y Fn(their)h(resp)q(ectiv)o(e)g(densit)o(y)l(,)g(it)f(is)h(easily) g(c)o(hec)o(k)o(ed)g(that)271 2316 y Fi(E)308 2298 y Fh(RH)s(F)305 2328 y(per)395 2316 y Fn(\()p Fi(K)s Fn(\))41 b(=)595 2286 y(1)p 595 2306 V 595 2348 a(2)623 2316 y Fi(E)660 2298 y Fh(RH)s(F)657 2328 y(per)747 2316 y Fn(\()p Fi(K)804 2323 y Fv(1)823 2316 y Fn(\))10 b(+)901 2286 y(1)p 901 2306 V 901 2348 a(2)929 2316 y Fi(E)966 2298 y Fh(RH)s(F)963 2328 y(per)1053 2316 y Fn(\()p Fi(K)1110 2323 y Fv(2)1129 2316 y Fn(\))f Fg(\000)1207 2286 y Fn(1)p 1207 2306 V 1207 2348 a(8)1235 2316 y Fi(D)1273 2323 y Fh(G)1302 2316 y Fn(\()p Fi(\032)1343 2323 y Fv(1)1373 2316 y Fg(\000)h Fi(\032)1441 2323 y Fv(2)1462 2316 y Fi(;)e(\032)1506 2323 y Fv(1)1535 2316 y Fg(\000)i Fi(\032)1603 2323 y Fv(2)1624 2316 y Fn(\))514 2424 y(=)42 b Fi(I)615 2405 y Fh(RH)s(F)611 2435 y(per)712 2424 y Fg(\000)762 2393 y Fn(1)p 762 2414 V 762 2455 a(8)790 2424 y Fi(D)828 2431 y Fh(G)857 2424 y Fn(\()p Fi(\032)898 2431 y Fv(1)928 2424 y Fg(\000)11 b Fi(\032)997 2431 y Fv(2)1017 2424 y Fi(;)d(\032)1061 2431 y Fv(1)1090 2424 y Fg(\000)j Fi(\032)1159 2431 y Fv(2)1179 2424 y Fn(\))p Fi(:)37 2527 y Fn(W)l(e)j(th)o(us)g(ha)o(v)o(e)f Fi(D)354 2534 y Fh(G)383 2527 y Fn(\()p Fi(\032)424 2534 y Fv(1)451 2527 y Fg(\000)7 b Fi(\032)516 2534 y Fv(2)537 2527 y Fi(;)h(\032)581 2534 y Fv(1)607 2527 y Fg(\000)f Fi(\032)672 2534 y Fv(2)693 2527 y Fn(\))12 b(=)h(0.)19 b(No)o(w,)13 b(since)i Fi(G)p Fn(,)e Fi(\032)1136 2534 y Fv(1)1170 2527 y Fn(and)h Fi(\032)1280 2534 y Fv(2)1313 2527 y 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b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(46)37 199 y(w)o(e)13 b(observ)o(e)f(that)g Fi(D)395 206 y Fh(G)424 199 y Fn(\()p Fi(\032)465 206 y Fv(1)490 199 y Fg(\000)5 b Fi(\032)553 206 y Fv(2)574 199 y Fi(;)j(\032)618 206 y Fv(1)642 199 y Fg(\000)d Fi(\032)705 206 y Fv(2)726 199 y Fn(\))12 b(=)h(0)f(if)h(and)g(only)g (if)g Fi(\032)1123 206 y Fv(1)1148 199 y Fg(\000)5 b Fi(\032)1211 206 y Fv(2)1244 199 y Fn(is)14 b(constan)o(t.)k(But,)1589 163 y Fe(R)1610 215 y Fh(Q)1647 199 y Fi(\032)1670 206 y Fv(1)1696 199 y Fg(\000)5 b Fi(\032)1759 206 y Fv(2)1792 199 y Fn(=)13 b(0,)37 268 y(hence)k Fi(\032)187 275 y Fv(1)220 268 y Fn(=)d Fi(\032)292 275 y Fv(2)312 268 y Fn(.)21 b(In)16 b(particular,)g(in)g(the)g(ab)q(o)o(v)o(e)f(pro)q (of,)g(the)g(whole)i(sequence)1415 229 y Fe(p)p 1461 229 155 2 v 1461 268 a Fi(\032)1484 275 y Fh(n)1508 268 y Fn(\()p Fi(x)p Fn(;)8 b Fi(x)p Fn(\))14 b(con)o(v)o(erges)h(to)37 286 y Fe(p)p 83 286 169 2 v 83 325 a Fi(\032)106 332 y Fu(1)144 325 y Fn(\()p Fi(x)p 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Fi(\030)r(;)e Fg(\001)p Fn(\))762 1927 y(=)815 1884 y Fe(X)810 1982 y Fh(m)p Fu(\025)p Fv(1)894 1927 y Fi(")915 1934 y Fh(nm)970 1927 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)1054 1934 y Fh(m)1086 1927 y Fn(\()p Fi(\030)r(;)g Fg(\001)p Fn(\))p Fi(;)66 b Fn(a.e.)20 b(on)15 b Fi(Q;)159 2049 y Fe(Z)184 2152 y Fh(Q)214 2074 y Fe(\000)235 2111 y Fg(jr)8 b Fi(u)320 2118 y Fh(n)342 2111 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)460 2092 y Fv(2)488 2111 y Fg(\000)j Fi(G)p Fn(\()p Fi(x)p Fn(\))p Fg(j)p Fi(u)671 2118 y Fh(n)693 2111 y Fn(\()p Fi(\030)r(;)d(x)p Fn(\))p Fg(j)811 2092 y Fv(2)829 2074 y Fe(\001)857 2111 y Fi(dx)i Fn(+)968 2067 y Fe(X)962 2165 y Fh(m)p Fu(\025)p Fv(1)1046 2111 y Fi(\025)1072 2118 y Fh(m)1106 2111 y Fn(\()p Fi(\030)r Fn(\))p Fi(D)1202 2118 y Fh(G)1231 2111 y Fn(\()p Fg(j)p Fi(u)1288 2118 y Fh(n)1310 2111 y Fn(\()p Fi(\030)r(;)e Fg(\001)p Fn(\))p Fg(j)1415 2092 y Fv(2)1433 2111 y Fi(;)g Fg(j)p Fi(u)1493 2118 y Fh(m)1525 2111 y Fn(\()p Fi(\030)r(;)g Fg(\001)p Fn(\))p Fg(j)1630 2092 y Fv(2)1648 2111 y Fn(\))762 2215 y(=)13 b Fi(\026)837 2199 y Fv(0)837 2226 y Fh(n)861 2215 y Fn(\()p Fi(\030)r Fn(\))c(+)i Fi(\026)1001 2199 y Fv(1)1001 2226 y Fh(n)1025 2215 y Fn(\()p Fi(\030)r Fn(\))e(+)i Fi(\031)r(:)1759 2011 y Fn(\(4.57\))108 2320 y(Since)22 b Fi(E)269 2303 y Fh(RH)s(F)266 2331 y(per)356 2320 y Fn(\()p Fi(K)s Fn(\))d(is)i(indep)q(enden)o(t)i(of)d(the)h(c)o (hoice)g(of)f(an)h(eigen)o(basis)g(for)f Fi(K)1504 2327 y Fh(\030)1523 2320 y Fn(,)h(w)o(e)f(ma)o(y)g(assume)37 2376 y(without)f(loss)h(of)e(generalit)o(y)h(that)g(the)g(matrix)f(of)h Fi(")982 2383 y Fh(nm)1037 2376 y Fn(\()p Fi(\030)r Fn(\))f(is)h (diagonal,)i(for)d(almost)g(ev)o(ery)h Fi(\030)i Fn(in)f Fi(Q)1843 2360 y Fh(?)1862 2376 y Fn(;)37 2433 y(in)d(other)f(w)o (ords,)f(the)h(righ)o(t-hand)g(side)h(term)860 2399 y Fe(P)908 2446 y Fh(m)p Fu(\025)p Fv(1)994 2433 y Fi(")1015 2440 y Fh(nm)1070 2433 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)1154 2440 y Fh(m)1186 2433 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fn(\))14 b(in)j(the)f(\014rst)g(equation)g(of)g(\(4.57\))37 2489 y(ma)o(y)g(b)q(e)g(replaced)h(b)o(y)f Fi(")464 2496 y Fh(n)487 2489 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)571 2496 y Fh(n)594 2489 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fn(\).)20 b(Moreo)o(v)o(er,)14 b(o)o(wing)h(to)h(the)f(fact)h(that)f (the)g(Lagrange)h(m)o(ultipliers)37 2546 y Fi(\026)64 2529 y Fv(0)64 2557 y Fh(n)88 2546 y Fn(\()p Fi(\030)r Fn(\))f(and)g Fi(\026)276 2529 y Fv(1)276 2557 y Fh(n)300 2546 y Fn(\()p Fi(\030)r Fn(\))f(are)g(resp)q(ectiv)o(ely)j(asso)q (ciated)e(to)f(the)i(constrain)o(ts)e(0)e Fg(\024)h Fi(\025)1385 2553 y Fh(n)1409 2546 y Fn(\()p Fi(\030)r Fn(\))i(and)g Fi(\025)1596 2553 y Fh(n)1620 2546 y Fn(\()p Fi(\030)r Fn(\))d Fg(\024)h Fn(1,)h(they)37 2602 y(satisfy)l(,)h(for)g(all)h Fi(n)d Fg(\025)g Fn(1)i(and)g(for)g(almost)f(ev)o(ery)h Fi(\030)j Fn(in)e Fi(Q)998 2586 y Fh(?)1017 2602 y Fn(,)679 2739 y Fi(\026)706 2721 y Fv(0)706 2751 y Fh(n)730 2739 y Fn(\()p Fi(\030)r Fn(\))796 2662 y Fe(\()832 2708 y Fn(=)d(0)p Fi(;)60 b Fn(if)15 b Fi(\025)1043 2715 y Fh(n)1067 2708 y Fn(\()p Fi(\030)r Fn(\))d Fi(>)h Fn(0)p Fi(;)832 2775 y Fg(\025)g Fn(0)p Fi(;)60 b Fn(if)15 b Fi(\025)1043 2782 y Fh(n)1067 2775 y Fn(\()p Fi(\030)r Fn(\))d(=)h(0)p Fi(;)1736 2739 y Fn(\(4.58a\))p eop %%Page: 47 47 47 46 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(47)679 233 y Fi(\026)706 214 y Fv(1)706 244 y Fh(n)730 233 y Fn(\()p Fi(\030)r Fn(\))796 155 y Fe(\()832 201 y Fn(=)13 b(0)p Fi(;)60 b Fn(if)15 b Fi(\025)1043 208 y Fh(n)1067 201 y Fn(\()p Fi(\030)r Fn(\))d Fi(<)h Fn(1)p Fi(;)832 269 y Fg(\024)g Fn(0)p Fi(;)60 b Fn(if)15 b Fi(\025)1043 276 y Fh(n)1067 269 y Fn(\()p Fi(\030)r Fn(\))d(=)h(1)p Fi(:)1734 233 y Fn(\(4.58b\))37 368 y(W)l(e)g(no)o(w)g(apply)g Fi(u)354 375 y Fh(n)378 368 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fn(\))i(to)j(the)f(\014rst)h (equation)g(of)f(\(4.57\),)f(next)i(in)o(tegrate)f(o)o(v)o(er)g Fi(Q)p Fn(,)h(and,)g(\014nally)l(,)i(insert)37 424 y(the)h(result)f(in) o(to)h(the)f(second)h(equation)f(of)g(\(4.57\),)e(to)h(obtain,)i(using) f(\(4.58a\))e(and)j(\(4.58b\),)618 486 y Fe(8)618 527 y(>)618 540 y(<)618 622 y(>)618 636 y(:)659 527 y Fi(\025)685 534 y Fh(n)709 527 y Fn(\()p Fi(\030)r Fn(\))c(=)g(0)129 b(=)-8 b Fg(\))29 b Fi(")1100 534 y Fh(n)1124 527 y Fn(\()p Fi(\030)r Fn(\))12 b Fg(\025)h Fi(\031)r(;)659 595 y Fn(0)f Fi(<)h(\025)768 602 y Fh(n)792 595 y Fn(\()p Fi(\030)r Fn(\))f Fi(<)h Fn(1)45 b(=)-8 b Fg(\))29 b Fi(")1100 602 y Fh(n)1124 595 y Fn(\()p Fi(\030)r Fn(\))12 b(=)h Fi(\031)r(;)659 662 y(\025)685 669 y Fh(n)709 662 y Fn(\()p Fi(\030)r Fn(\))f(=)g(1)129 b(=)-8 b Fg(\))29 b Fi(")1100 669 y Fh(n)1124 662 y Fn(\()p Fi(\030)r Fn(\))12 b Fg(\024)h Fi(\031)r(:)1759 593 y Fn(\(4.59\))37 784 y Fb(4.3)57 b(Upp)r(er)19 b(limit)f(of)h(the)g(energy)h(p)r(er)f(unit)g(v)n(olume)h (and)f(conclusion)37 870 y Fn(In)d(order)f(to)g(conclude)i(the)e(pro)q (of)g(of)g(Theorem)g(2.2,)e(w)o(e)i(no)o(w)g(pro)o(v)o(e)g(the)37 959 y Fj(Prop)q(osition)20 b(4.2)k Ff(We)c(assume)h(that)g(the)f(V)m (an)g(Hove)g(se)n(quenc)n(e)f Fn(\003)g Ff(satis\014es)g(\(2.21\),)i (and)g(that)g(the)37 1015 y(unit)c(c)n(el)r(l)e Fi(Q)h Ff(is)g(a)h(cub)n(e.)j(Then,)669 1138 y Fn(lim)8 b(sup)691 1178 y Fv(\003)p Fu(!1)821 1107 y Fi(I)845 1091 y Fh(RH)s(F)841 1121 y Fv(\003)p 821 1128 112 2 v 848 1170 a Fg(j)p Fn(\003)p Fg(j)950 1138 y(\024)13 b Fi(I)1022 1120 y Fh(RH)s(F)1018 1150 y(per)1119 1138 y Fn(+)1169 1108 y Fi(M)p 1169 1128 50 2 v 1182 1170 a Fn(2)1223 1138 y Fi(;)523 b Fn(\(4.60\))37 1265 y Ff(wher)n(e)17 b Fi(I)190 1249 y Fh(RH)s(F)186 1276 y(per)293 1265 y Ff(is)f(de\014ne)n(d)f(by)i(\(2.17\){\(2.18\).)37 1365 y Fj(Remark)h(4.1)24 b Ff(As)12 b(state)n(d)h(in)g(The)n(or)n(em)g (2.2,)h(the)f(same)g(r)n(esult)g(holds)g(true)g(in)g(the)g(sme)n(ar)n (e)n(d)f(nuclei)h(c)n(ase,)37 1422 y(if)k(we)g(assume)g(mor)n(e)n(over) g(that)h Fi(m)f Ff(shar)n(es)f(the)h(symmetries)g(of)g(the)g(unit)g (cub)n(e)f Fi(Q)p Ff(,)h(and)g(de\014ne)f Fi(M)22 b Ff(in)17 b(a)37 1478 y(c)n(onvenient)e(way.)108 1579 y Fn(As)g(a)g(corollary)g (of)f(Prop)q(osition)i(4.1)d(and)j(Prop)q(osition)f(4.2)f(\(and)h(the)g (sligh)o(t)g(mo)q(di\014cations)h(whic)o(h)37 1635 y(are)f(necessary)h (to)e(treat)h(the)g(smeared)g(n)o(uclei)i(case\),)e(w)o(e)f(shall)j (obtain)e(Theorem)g(2.2.)37 1733 y Fj(Pro)q(of)j(of)f(Prop)q(osition)i (4.2)c Fn(:)k(Let)c(us)g(denote)f(b)o(y)h Fi(K)i Fn(a)d(minimizer)j(of) d(the)g(p)q(erio)q(dic)j(RHF)e(problem.)37 1790 y(As)h(usual,)g(w)o(e)g (ma)o(y)e(decomp)q(ose)j Fi(K)h Fn(in)o(to)e(op)q(erators)e Fi(K)1023 1797 y Fh(\030)1058 1790 y Fn(\()p Fi(\030)h Fg(2)e Fi(Q)1190 1773 y Fh(?)1210 1790 y Fn(\),)i(whose)g(k)o(ernels)i Fi(\032)p Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))13 b(ma)o(y)i(b)q(e)37 1846 y(written)h(as)582 1942 y Fi(\032)p Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))i(=)814 1899 y Fe(X)813 1997 y Fh(n)p Fu(\025)p Fv(1)887 1942 y Fi(\025)913 1949 y Fh(n)937 1942 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)1021 1949 y Fh(n)1044 1942 y Fn(\()p Fi(\030)r(;)e(x)p Fn(\))p Fi(u)1175 1949 y Fh(n)1197 1942 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1300 1923 y Fu(\003)1318 1942 y Fi(:)37 2083 y Fn(W)l(e)16 b(denote)668 2189 y Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))j(=)857 2127 y Fe(Z)883 2230 y Fh(Q)911 2220 y Fc(?)938 2189 y Fi(\032)p Fn(\()p Fi(\030)r(;)d(x;)g(y)r Fn(\))1152 2158 y Fi(d\030)p 1122 2178 106 2 v 1122 2220 a Fn(\(2)p Fi(\031)r Fn(\))1209 2207 y Fv(3)1232 2189 y Fi(:)37 2318 y Fn(Let)16 b(no)o(w)e(\003)h(b)q(e)h(\014xed.)k(W)l(e)15 b(build)i(a)e(cut-o\013)g(function)h Fi(\037)1029 2325 y Fv(\003)1069 2318 y Fg(2)d(D)q Fn(\()p Fj(R)1205 2302 y Fv(3)1225 2318 y Fn(\))h(satisfying)i(the)f(follo)o(wing)h(prop)q (er-)37 2375 y(ties)g(:)514 2421 y Fe(8)514 2462 y(<)514 2544 y(:)575 2458 y Fn(0)d Fg(\024)g Fi(\037)687 2465 y Fv(\003)726 2458 y Fg(\024)g Fn(1)8 b(;)575 2514 y Fi(\037)603 2521 y Fv(\003)643 2514 y Fg(\021)13 b Fn(1)i(on)g Fg(f)p Fi(x)d Fg(2)h Fn(\000\(\003\);)h Fi(d)p Fn(\()p Fi(x;)8 b(@)s Fn(\000\(\003\)\))i Fg(\025)j Fn(2)p Fg(g)8 b Fn(;)575 2571 y Fi(\037)603 2578 y Fv(\003)643 2571 y Fg(\021)13 b Fn(0)i(on)g(\000\(\003\))888 2554 y Fh(c)912 2571 y Fi(:)37 2663 y Fn(In)h(addition,)g(w)o(e)f(c)o(ho)q(ose)g Fi(\037)525 2670 y Fv(\003)567 2663 y Fn(in)i(suc)o(h)e(a)g(w)o(a)o(y)f (that)h(it)g(also)g(satis\014es)617 2716 y Fe(Z)642 2819 y Fd(R)672 2810 y Ft(3)699 2778 y Fi(\037)727 2759 y Fv(2)727 2789 y(\003)754 2778 y Fn(\()p Fi(x)p Fn(\))p Fi(\032)p Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fi(dx)j Fn(=)i Fg(j)p Fn(\003)p Fg(j)c Fn(+)h Fi(o)p Fn(\()p Fg(j)p Fn(\003)p Fg(j)p Fn(\))p Fi(:)p eop %%Page: 48 48 48 47 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(48)37 199 y(W)l(e)16 b(next)f(consider)h(the)f(op)q (erator)g Fi(K)696 206 y Fv(\003)737 199 y Fn(on)g Fi(L)831 183 y Fv(2)851 199 y Fn(\()p Fj(R)908 183 y Fv(3)928 199 y Fn(\))f(whose)h(k)o(ernel)h(is)662 301 y Fi(\032)685 308 y Fv(\003)712 301 y Fn(\()p Fi(x;)8 b(y)r Fn(\))j(=)i Fi(\037)906 308 y Fv(\003)933 301 y Fn(\()p Fi(x)p Fn(\))p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fi(\037)1153 308 y Fv(\003)1178 301 y Fn(\()p Fi(y)r Fn(\))p Fi(:)508 b Fn(\(4.61\))37 403 y(A)16 b(simple)g(computation)g(sho)o(ws)e(that) 539 506 y(\()p Fi(K)596 513 y Fv(\003)622 506 y Fi( )r(;)8 b( )r Fn(\))725 515 y Fh(L)749 506 y Ft(2)764 515 y Fv(\()p Fd(R)808 506 y Ft(3)825 515 y Fv(\))854 506 y Fn(=)13 b(\()p Fi(K)s Fn(\()p Fi(\037)1008 513 y Fv(\003)1034 506 y Fi( )r Fn(\))p Fi(;)8 b Fn(\()p Fi(\037)1151 513 y Fv(\003)1176 506 y Fi( )r Fn(\)\))1244 515 y Fh(L)1268 506 y Ft(2)1283 515 y Fv(\()p Fd(R)1327 506 y Ft(3)1345 515 y Fv(\))1361 506 y Fi(;)37 608 y Fn(and)22 b(therefore)f(w)o(e)h (ha)o(v)o(e)f(0)h Fg(\024)i Fi(K)657 615 y Fv(\003)706 608 y Fg(\024)g Fj(1)p Fn(.)38 b(The)22 b(c)o(hoice)g(of)f Fi(\037)1169 615 y Fv(\003)1218 608 y Fn(ensures)h(also)f(that)g(T)l(r) 1633 618 y Fh(L)1657 608 y Ft(2)1674 618 y Fv(\()p Fd(R)1718 608 y Ft(3)1736 618 y Fv(\))1751 608 y Fi(K)1790 615 y Fv(\003)1840 608 y Fn(=)37 664 y Fg(j)p Fn(\003)p Fg(j)9 b Fn(+)i Fi(o)p Fn(\()p Fg(j)p Fn(\003)p Fg(j)p Fn(\))g Fg(\024)i(j)p Fn(\003)p Fg(j)p Fn(.)108 721 y(W)l(e)i(no)o(w)g(compute) g(the)h(RHF)f(energy)g(of)g Fi(K)892 728 y Fv(\003)918 721 y Fn(.)108 777 y(Since)22 b Fi(\032)255 784 y Fv(\003)282 777 y Fn(\()p Fi(x;)8 b(x)p Fn(\))21 b(=)h Fi(\037)497 761 y Fv(2)497 791 y(\003)524 777 y Fn(\()p Fi(x)p Fn(\))p Fi(\032)p Fn(\()p Fi(x;)8 b(x)p Fn(\),)20 b(it)h(is)g(a)g(simple)h (matter,)e(arguing)h(as)g(in)g([11)o(])f(and)h(using)h(the)37 834 y(p)q(erio)q(dicit)o(y)c(of)d Fi(\032)p Fn(\()p Fi(x;)8 b(x)p Fn(\),)13 b(to)i(sho)o(w)g(that)f(the)h(electrostatic)h(terms)628 957 y Fg(\000)671 895 y Fe(Z)696 998 y Fd(R)726 988 y Ft(3)753 957 y Fi(V)779 964 y Fv(\003)806 957 y Fi(\032)829 964 y Fv(\003)866 957 y Fn(+)917 926 y(1)p 917 946 23 2 v 917 988 a(2)945 957 y Fi(D)q Fn(\()p Fi(\032)1025 964 y Fv(\003)1051 957 y Fi(;)8 b(\032)1095 964 y Fv(\003)1121 957 y Fn(\))i(+)1200 926 y(1)p 1200 946 V 1200 988 a(2)1227 957 y Fi(U)1258 964 y Fv(\003)37 1079 y Fn(b)q(eha)o(v)o(e)16 b(lik)o(e)629 1195 y Fg(j)p Fn(\003)p Fg(j)687 1131 y Fe(\024)710 1195 y Fg(\000)753 1133 y Fe(Z)778 1236 y Fh(Q)815 1195 y Fi(G\032)11 b Fn(+)935 1164 y(1)p 935 1184 V 935 1226 a(2)963 1195 y Fi(D)1001 1202 y Fh(G)1030 1195 y Fn(\()p Fi(\032;)d(\032)p Fn(\))i(+)1193 1164 y Fi(M)p 1193 1184 50 2 v 1206 1226 a Fn(2)1247 1131 y Fe(\025)1271 1195 y Fi(;)37 1326 y Fn(as)j(\003)f(go)q(es)h(to)f (in\014nit)o(y)l(.)21 b(This)13 b(is)g(precisely)i(where)e(w)o(e)f (need)i(the)f(assumption)g(\(2.21\),)e(the)i(fact)f(that)g Fi(Q)g Fn(is)37 1383 y(a)f(cub)q(e,)i(and)e(that)f Fi(\032)p Fn(\()p Fi(x)p Fn(;)e Fi(x)p Fn(\))i(shares)h(the)g(symmetries)g(of)g (the)g(unit)h(cub)q(e.)19 b(Both)11 b(facts)g(pla)o(y)g(a)g(fundamen)o (tal)37 1439 y(role)19 b(|)f(see)g(the)g(details)h(in)g([11)o(].)27 b(Therefore,)19 b(w)o(e)e(concen)o(trate)h(ourselv)o(es)g(on)g(the)g(b) q(eha)o(viour)g(of)g(the)37 1496 y(kinetic)f(energy)f(term.)j(W)l(e)c (in)o(tend)h(to)f(pro)o(v)o(e)g(that)431 1617 y(lim)404 1647 y Fv(\003)p Fu(\000)-5 b(!1)550 1586 y Fn(1)p 533 1606 57 2 v 533 1648 a Fg(j)p Fn(\003)p Fg(j)595 1617 y Fn(T)l(r)642 1627 y Fh(L)666 1617 y Ft(2)683 1627 y Fv(\()p Fd(R)727 1617 y Ft(3)745 1627 y Fv(\))760 1617 y Fn([)p Fg(\000)p Fn(\001)p Fi(K)885 1624 y Fv(\003)911 1617 y Fn(])13 b(=)985 1555 y Fe(Z)1010 1658 y Fh(Q)1038 1649 y Fc(?)1065 1617 y Fn(T)l(r)1112 1629 y Fh(L)1136 1617 y Ft(2)1136 1640 y Fc(\030)1153 1629 y Fv(\()p Fh(Q)p Fv(\))1211 1617 y Fn([)p Fg(\000)p Fn(\001)1297 1624 y Fh(\030)1315 1617 y Fi(K)1354 1624 y Fh(\030)1373 1617 y Fn(])1428 1586 y Fi(d\030)p 1398 1606 106 2 v 1398 1648 a Fn(\(2)p Fi(\031)r Fn(\))1485 1635 y Fv(3)1759 1617 y Fn(\(4.62\))37 1746 y(whic)o(h)k(will)f(of)f(course)g(conclude)i (the)f(pro)q(of)e(of)h(Prop)q(osition)h(4.2.)108 1802 y(Let)g(us)f(denote)g(b)o(y)g(\()p Fi(')505 1809 y Fh(m)538 1802 y Fn(\))556 1809 y Fh(m)p Fu(\025)p Fv(1)649 1802 y Fn(an)g(Hilb)q(ertian)j(basis)d(of)g Fi(L)1124 1786 y Fv(2)1144 1802 y Fn(\()p Fj(R)1201 1786 y Fv(3)1220 1802 y Fn(\).)20 b(W)l(e)15 b(b)q(egin)i(with)142 1930 y Fg(\000)p Fn(\001)p Fi(K)254 1937 y Fv(\003)280 1930 y Fi(')310 1937 y Fh(m)384 1930 y Fn(=)42 b Fg(\000)p Fn(\001)534 1866 y Fe(\022)568 1869 y(Z)593 1972 y Fd(R)623 1962 y Ft(3)651 1930 y Fi(\032)674 1937 y Fv(\003)701 1930 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fi(')838 1937 y Fh(m)869 1930 y Fn(\()p Fi(y)r Fn(\))g Fi(dy)985 1866 y Fe(\023)384 2052 y Fn(=)461 1990 y Fe(Z)487 2093 y Fd(R)517 2084 y Ft(3)544 2052 y Fg(\000)p Fn(\001)617 2015 y Fe(\000)638 2052 y Fi(\032)p Fn(\()p Fi(x;)g(y)r Fn(\))p Fi(\037)796 2059 y Fv(\003)822 2052 y Fn(\()p Fi(x)p Fn(\))884 2015 y Fe(\001)904 2052 y Fi(\037)932 2059 y Fv(\003)959 2052 y Fn(\()p Fi(y)r Fn(\))p Fi(')1049 2059 y Fh(m)1081 2052 y Fn(\()p Fi(y)r Fn(\))g Fi(dy)384 2172 y Fn(=)42 b Fg(\000)504 2110 y Fe(Z)6 b(Z)580 2213 y Fd(R)610 2204 y Ft(3)628 2213 y Fu(\002)p Fh(Q)683 2204 y Fc(?)711 2129 y Fe(X)711 2227 y Fh(n)p Fu(\025)p Fv(1)785 2172 y Fi(\025)811 2179 y Fh(n)835 2172 y Fn(\()p Fi(\030)r Fn(\)\001)931 2135 y Fe(\000)951 2172 y Fi(u)977 2179 y Fh(n)1000 2172 y Fn(\()p Fi(\030)r(;)i(x)p Fn(\))p Fi(\037)1133 2179 y Fv(\003)1159 2172 y Fn(\()p Fi(x)p Fn(\))1221 2135 y Fe(\001)1241 2172 y Fi(u)1267 2179 y Fh(n)1290 2172 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1393 2153 y Fu(\003)1412 2172 y Fi(\037)1440 2179 y Fv(\003)1466 2172 y Fn(\()p Fi(y)r Fn(\))p Fi(')1556 2179 y Fh(m)1589 2172 y Fn(\()p Fi(y)r Fn(\))1667 2142 y Fi(dy)r(d\030)p 1661 2162 V 1661 2203 a Fn(\(2)p Fi(\031)r Fn(\))1748 2190 y Fv(3)37 2318 y Fn(Hence,)79 2433 y(\()p Fg(\000)p Fn(\001)p Fi(K)209 2440 y Fv(\003)235 2433 y Fi(')265 2440 y Fh(m)298 2433 y Fi(;)g(')349 2440 y Fh(m)381 2433 y Fn(\))399 2442 y Fh(L)423 2433 y Ft(2)440 2442 y Fv(\()p Fd(R)484 2433 y Ft(3)502 2442 y Fv(\))159 2533 y Fn(=)207 2471 y Fe(Z)d(Z)282 2575 y Fd(R)312 2565 y Ft(3)330 2575 y Fu(\002)p Fh(Q)385 2565 y Fc(?)413 2490 y Fe(X)413 2588 y Fh(n)p Fu(\025)p Fv(1)487 2533 y Fi(\025)513 2540 y Fh(n)537 2533 y Fn(\()p Fi(\030)r Fn(\))595 2469 y Fe(\024)618 2471 y(Z)644 2575 y Fd(R)674 2565 y Ft(3)701 2533 y Fg(\000)p Fn(\001)774 2496 y Fe(\000)795 2533 y Fi(u)821 2540 y Fh(n)845 2533 y Fn(\()p Fi(\030)r(;)j(x)p Fn(\))p Fi(\037)978 2540 y Fv(\003)1003 2533 y Fn(\()p Fi(x)p Fn(\))1065 2496 y Fe(\001)1085 2533 y Fi(')1115 2515 y Fu(\003)1115 2545 y Fh(m)1148 2533 y Fn(\()p Fi(x)p Fn(\))p Fi(dx)1260 2469 y Fe(\025)1291 2533 y Fi(u)1317 2540 y Fh(n)1340 2533 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1443 2515 y Fu(\003)1461 2533 y Fi(\037)1489 2540 y Fv(\003)1516 2533 y Fn(\()p Fi(y)r Fn(\))p Fi(')1606 2540 y Fh(m)1639 2533 y Fn(\()p Fi(y)r Fn(\))1717 2503 y Fi(dy)r(d\030)p 1711 2523 V 1711 2564 a Fn(\(2)p Fi(\031)r Fn(\))1798 2551 y Fv(3)1821 2533 y Fi(:)p eop %%Page: 49 49 49 48 bop 53 50 a Fm(4.)23 b Fl(The)16 b(Reduced)g(Har)m(tree-F)o(ock)f (model)980 b Fn(49)37 199 y(Summing)17 b(up)e(next)g(in)h Fi(m)p Fn(,)f(w)o(e)g(obtain)43 274 y Fe(X)37 372 y Fh(m)p Fu(\025)p Fv(1)114 317 y Fn(\()p Fg(\000)p Fn(\001)p Fi(K)244 324 y Fv(\003)270 317 y Fi(')300 324 y Fh(m)333 317 y Fi(;)8 b(')384 324 y Fh(m)416 317 y Fn(\))434 327 y Fh(L)458 317 y Ft(2)475 327 y Fv(\()p Fd(R)519 317 y Ft(3)536 327 y Fv(\))594 317 y Fn(=)671 255 y Fe(Z)696 358 y Fd(R)726 348 y Ft(3)753 317 y Fi(dy)808 255 y Fe(Z)834 358 y Fh(Q)862 348 y Fc(?)889 274 y Fe(X)889 372 y Fh(n)p Fu(\025)p Fv(1)963 317 y Fi(\025)989 324 y Fh(n)1013 317 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)1097 324 y Fh(n)1120 317 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1223 298 y Fu(\003)1241 317 y Fi(\037)1269 324 y Fv(\003)1296 317 y Fn(\()p Fi(y)r Fn(\))858 421 y Fe(X)852 519 y Fh(m)p Fu(\025)p Fv(1)929 400 y Fe(\024)953 402 y(Z)978 505 y Fd(R)1008 496 y Ft(3)1035 464 y Fg(\000)p Fn(\001)1108 427 y Fe(\000)1129 464 y Fi(u)1155 471 y Fh(n)1179 464 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fi(\037)1312 471 y Fv(\003)1337 464 y Fn(\()p Fi(x)p Fn(\))1399 427 y Fe(\001)1420 464 y Fi(')1450 445 y Fu(\003)1450 475 y Fh(m)1483 464 y Fn(\()p Fi(x)p Fn(\))g Fi(dx)1603 400 y Fe(\025)1625 464 y Fi(')1655 471 y Fh(m)1688 464 y Fn(\()p Fi(y)r Fn(\))1790 433 y Fi(d\030)p 1760 454 106 2 v 1760 495 a Fn(\(2)p Fi(\031)r Fn(\))1847 482 y Fv(3)594 608 y Fn(=)671 546 y Fe(Z)696 649 y Fd(R)726 640 y Ft(3)753 608 y Fi(dy)808 546 y Fe(Z)834 649 y Fh(Q)862 640 y Fc(?)889 565 y Fe(X)889 663 y Fh(n)p Fu(\025)p Fv(1)963 608 y Fi(\025)989 615 y Fh(n)1013 608 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)1097 615 y Fh(n)1120 608 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1223 589 y Fu(\003)1241 608 y Fi(\037)1269 615 y Fv(\003)1296 608 y Fn(\()p Fi(y)r Fn(\))h Fg(\000)i Fn(\001)1449 571 y Fe(\000)1470 608 y Fi(u)1496 615 y Fh(n)1519 608 y Fn(\()p Fi(\030)r(;)d(y)r Fn(\))p Fi(\037)1650 615 y Fv(\003)1676 608 y Fn(\()p Fi(y)r Fn(\))1736 571 y Fe(\001)1798 577 y Fi(d\030)p 1768 598 V 1768 639 a Fn(\(2)p Fi(\031)r Fn(\))1855 626 y Fv(3)594 752 y Fn(=)671 690 y Fe(Z)696 793 y Fh(Q)724 784 y Fc(?)786 721 y Fi(d\030)p 756 741 V 756 783 a Fn(\(2)p Fi(\031)r Fn(\))843 770 y Fv(3)875 709 y Fe(X)874 807 y Fh(n)p Fu(\025)p Fv(1)948 752 y Fi(\025)974 759 y Fh(n)998 752 y Fn(\()p Fi(\030)r Fn(\))1064 690 y Fe(Z)1088 793 y Fd(R)1118 784 y Ft(3)1146 752 y Fi(u)1172 759 y Fh(n)1195 752 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1298 733 y Fu(\003)1316 752 y Fi(\037)1344 759 y Fv(\003)1371 752 y Fn(\()p Fi(y)r Fn(\))i Fg(\000)g Fn(\001)1524 715 y Fe(\000)1545 752 y Fi(u)1571 759 y Fh(n)1594 752 y Fn(\()p Fi(\030)r(;)e(y)r Fn(\))p Fi(\037)1725 759 y Fv(\003)1751 752 y Fn(\()p Fi(y)r Fn(\))1811 715 y Fe(\001)1839 752 y Fi(dy)594 896 y Fn(=)671 834 y Fe(Z)696 937 y Fh(Q)724 927 y Fc(?)786 865 y Fi(d\030)p 756 885 V 756 927 a Fn(\(2)p Fi(\031)r Fn(\))843 914 y Fv(3)875 853 y Fe(X)874 950 y Fh(n)p Fu(\025)p Fv(1)948 896 y Fi(\025)974 903 y Fh(n)998 896 y Fn(\()p Fi(\030)r Fn(\))1064 834 y Fe(Z)1088 937 y Fd(R)1118 927 y Ft(3)1146 896 y Fg(jr)p Fn(\()p Fi(u)1241 903 y Fh(n)1263 896 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))p Fi(\037)1394 903 y Fv(\003)1420 896 y Fn(\()p Fi(y)r Fn(\)\))p Fg(j)1511 877 y Fv(2)1537 896 y Fi(dy)r(:)37 1039 y Fn(W)l(e)16 b(ha)o(v)o(e)f(therefore)f(obtained)295 1148 y(T)l(r)342 1158 y Fh(L)366 1148 y Ft(2)383 1158 y Fv(\()p Fd(R)427 1148 y Ft(3)445 1158 y Fv(\))461 1148 y Fn([)p Fg(\000)p Fn(\001)p Fi(K)586 1155 y Fv(\003)612 1148 y Fn(])e(=)685 1086 y Fe(Z)710 1189 y Fh(Q)738 1180 y Fc(?)800 1117 y Fi(d\030)p 770 1138 V 770 1179 a Fn(\(2)p Fi(\031)r Fn(\))857 1166 y Fv(3)889 1105 y Fe(X)888 1203 y Fh(n)p Fu(\025)p Fv(1)962 1148 y Fi(\025)988 1155 y Fh(n)1012 1148 y Fn(\()p Fi(\030)r Fn(\))1078 1086 y Fe(Z)1103 1189 y Fd(R)1133 1180 y Ft(3)1160 1148 y Fg(jr)p Fn(\()p Fi(u)1255 1155 y Fh(n)1278 1148 y Fn(\()p Fi(\030)r(;)c(x)p Fn(\))p Fi(\037)1411 1155 y Fv(\003)1436 1148 y Fn(\()p Fi(x)p Fn(\)\))p Fg(j)1529 1129 y Fv(2)1555 1148 y Fi(dx:)141 b Fn(\(4.63\))37 1291 y(W)l(e)16 b(no)o(w)e(remark)h(that)80 1338 y Fe(Z)105 1441 y Fd(R)135 1432 y Ft(3)162 1400 y Fg(jr)p Fn(\()p Fi(u)257 1407 y Fh(n)280 1400 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fi(\037)413 1407 y Fv(\003)438 1400 y Fn(\()p Fi(x)p Fn(\)\))p Fg(j)531 1381 y Fv(2)557 1400 y Fi(dx)41 b Fn(=)725 1338 y Fe(Z)751 1441 y Fd(R)781 1432 y Ft(3)808 1400 y Fg(j)p Fi(\037)849 1407 y Fv(\003)876 1400 y Fn(\()p Fi(x)p Fn(\))p Fg(r)p Fn(\()p Fi(u)1020 1407 y Fh(n)1042 1400 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\)\))g(+)j Fi(u)1245 1407 y Fh(n)1268 1400 y Fn(\()p Fi(\030)r(;)d(x)p Fn(\))p Fg(r)p Fi(\037)1439 1407 y Fv(\003)1465 1400 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1540 1381 y Fv(2)1566 1400 y Fi(dx)648 1520 y Fn(=)725 1458 y Fe(Z)751 1561 y Fd(R)781 1551 y Ft(3)808 1520 y Fg(j)p Fi(\037)849 1527 y Fv(\003)876 1520 y Fn(\()p Fi(x)p Fn(\))p Fg(j)951 1501 y Fv(2)969 1520 y Fg(jr)p Fn(\()p Fi(u)1064 1527 y Fh(n)1087 1520 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\)\))p Fg(j)1223 1501 y Fv(2)1251 1520 y Fn(+)1296 1458 y Fe(Z)1321 1561 y Fd(R)1351 1551 y Ft(3)1379 1520 y Fg(j)p Fi(u)1418 1527 y Fh(n)1441 1520 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)1559 1501 y Fv(2)1577 1520 y Fg(jr)p Fi(\037)1656 1527 y Fv(\003)1682 1520 y Fn(\()p Fi(x)p Fn(\))p Fg(j)1757 1501 y Fv(2)1783 1520 y Fi(dx)771 1639 y Fn(+2)844 1577 y Fe(Z)869 1681 y Fd(R)899 1671 y Ft(3)927 1639 y Fi(\037)955 1646 y Fv(\003)982 1639 y Fn(\()p Fi(x)p Fn(\))p Fi(u)1070 1621 y Fu(\003)1070 1651 y Fh(n)1092 1639 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(r)p Fi(u)1261 1646 y Fh(n)1283 1639 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))h Fg(\001)h(r)p Fi(\037)1486 1646 y Fv(\003)1513 1639 y Fn(\()p Fi(x)p Fn(\))e Fi(dx)37 1761 y Fn(Eac)o(h)15 b(of)g(the)g(three)g(in)o(tegrals)g(in)h(the)f (righ)o(t-hand)g(side)h(ma)o(y)f(b)q(e)g(restricted)h(on)e (\000\(\003\),)g(for)h Fi(\037)1670 1768 y Fv(\003)1712 1761 y Fn(v)m(anishes)37 1818 y(outside)21 b(this)f(domain.)33 b(In)20 b(addition,)i(the)e(third)g(in)o(tegral)g(ma)o(y)f(b)q(e)h (restricted)g(on)f(the)h(\\b)q(oundary")37 1874 y Fi(@)s Fn(\003)13 b(=)g Fg(f)p Fi(x)g Fg(2)g Fn(\000\(\003\);)20 b Fi(d)p Fn(\()p Fi(x)p Fn(;)8 b Fi(@)s Fn(\000\(\003\)\))i Fg(\024)k Fn(2)p Fg(g)h Fn(of)g(\000\(\003\),)f(as)h Fg(r)p Fi(\037)1038 1881 y Fv(\003)1078 1874 y Fg(\021)e Fn(0)j(in)g(the)f(\\in)o(terior")h(\000\(\003\))9 b Fg(n)h Fi(@)s Fn(\003)15 b(of)g(\000\(\003\).)37 1931 y(Therefore,)g(w)o(e)g (ha)o(v)o(e)315 1987 y Fe(Z)341 2090 y Fh(Q)369 2080 y Fc(?)431 2018 y Fi(d\030)p 401 2038 V 401 2080 a Fn(\(2)p Fi(\031)r Fn(\))488 2067 y Fv(3)519 2006 y Fe(X)519 2103 y Fh(n)p Fu(\025)p Fv(1)593 2049 y Fi(\025)619 2056 y Fh(n)643 2049 y Fn(\()p Fi(\030)r Fn(\))709 1987 y Fe(Z)733 2090 y Fd(R)763 2080 y Ft(3)791 2049 y Fg(jr)p Fn(\()p Fi(u)886 2056 y Fh(n)908 2049 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fi(\037)1041 2056 y Fv(\003)1067 2049 y Fn(\()p Fi(x)p Fn(\)\))p Fg(j)1160 2030 y Fv(2)1186 2049 y Fi(dx)k Fn(=)h Fi(I)1320 2030 y Fv(1)1316 2060 y(\003)1353 2049 y Fn(+)d Fi(I)1422 2030 y Fv(2)1418 2060 y(\003)1455 2049 y Fn(+)g(2)e Fi(I)1555 2030 y Fv(3)1551 2060 y(\003)1577 2049 y Fi(;)169 b Fn(\(4.64\))37 2198 y(where)16 b Fi(I)193 2182 y Fv(1)189 2212 y(\003)215 2198 y Fn(,)f Fi(I)267 2182 y Fv(2)263 2212 y(\003)290 2198 y Fn(,)g Fi(I)342 2182 y Fv(3)338 2212 y(\003)379 2198 y Fn(denote)h(resp)q(ectiv)o(ely) 312 2321 y Fi(I)336 2302 y Fv(1)332 2332 y(\003)371 2321 y Fn(=)419 2259 y Fe(Z)445 2362 y Fh(Q)473 2353 y Fc(?)535 2290 y Fi(d\030)p 505 2310 V 505 2352 a Fn(\(2)p Fi(\031)r Fn(\))592 2339 y Fv(3)623 2278 y Fe(X)623 2376 y Fh(n)p Fu(\025)p Fv(1)697 2321 y Fi(\025)723 2328 y Fh(n)747 2321 y Fn(\()p Fi(\030)r Fn(\))813 2278 y Fe(X)813 2376 y Fh(k)q Fu(2)p Fv(\003)887 2259 y Fe(Z)912 2362 y Fh(Q)949 2321 y Fg(j)p Fi(\037)990 2328 y Fv(\003)1017 2321 y Fn(\()p Fi(x)10 b Fn(+)g Fi(k)q Fn(\))p Fg(j)1172 2302 y Fv(2)1191 2321 y Fg(jr)p Fn(\()p Fi(u)1286 2328 y Fh(n)1309 2321 y Fn(\()p Fi(\030)r(;)e(x)h Fn(+)h Fi(k)q Fn(\)\))p Fg(j)1524 2302 y Fv(2)1551 2321 y Fi(dx)330 2537 y(I)354 2519 y Fv(2)350 2549 y(\003)389 2537 y Fn(=)437 2476 y Fe(Z)462 2579 y Fh(Q)490 2569 y Fc(?)552 2507 y Fi(d\030)p 523 2527 V 523 2569 a Fn(\(2)p Fi(\031)r Fn(\))610 2556 y Fv(3)641 2494 y Fe(X)641 2592 y Fh(n)p Fu(\025)p Fv(1)715 2537 y Fi(\025)741 2544 y Fh(n)765 2537 y Fn(\()p Fi(\030)r Fn(\))830 2494 y Fe(X)831 2593 y Fh(k)q Fu(2)p Fv(\003)904 2476 y Fe(Z)930 2579 y Fh(Q)967 2537 y Fg(j)p Fi(u)1006 2544 y Fh(n)1029 2537 y Fn(\()p Fi(\030)r(;)e(x)h Fn(+)h Fi(k)q Fn(\))p Fg(j)1226 2519 y Fv(2)1246 2537 y Fg(jr)p Fi(\037)1325 2544 y Fv(\003)1351 2537 y Fn(\()p Fi(x)g Fn(+)g Fi(k)q Fn(\))p Fg(j)1506 2519 y Fv(2)1533 2537 y Fi(dx)308 2754 y(I)332 2735 y Fv(3)328 2765 y(\003)367 2754 y Fn(=)415 2692 y Fe(Z)440 2795 y Fh(Q)468 2786 y Fc(?)531 2723 y Fi(d\030)p 501 2744 V 501 2785 a Fn(\(2)p Fi(\031)r Fn(\))588 2772 y Fv(3)619 2711 y Fe(X)619 2809 y Fh(n)p Fu(\025)p Fv(1)693 2754 y Fi(\025)719 2761 y Fh(n)743 2754 y Fn(\()p Fi(\030)r Fn(\))809 2692 y Fe(Z)833 2795 y Fh(@)r Fv(\003)887 2754 y Fi(\037)915 2761 y Fv(\003)942 2754 y Fn(\()p Fi(x)p Fn(\))p Fi(u)1030 2735 y Fu(\003)1030 2765 y Fh(n)1053 2754 y Fn(\()p Fi(\030)r(;)e(x)p Fn(\))p Fg(r)p Fi(u)1222 2761 y Fh(n)1244 2754 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))h Fg(\001)h(r)p Fi(\037)1447 2761 y Fv(\003)1473 2754 y Fn(\()p Fi(x)p Fn(\))e Fi(dx:)p eop %%Page: 50 50 50 49 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(50)37 199 y(As)16 b(\003)g(is)g(a)g(V)l(an)g(Ho)o(v)o(e)f (sequence,)i(w)o(e)f(exp)q(ect)g(that)g(the)g(in)o(tegrals)g(o)o(v)o (er)f(the)h(\\b)q(oundary")g(of)f(\000\(\003\))g(are)37 256 y(negligible)20 b(with)d(resp)q(ect)h(to)e Fg(j)p Fn(\003)p Fg(j)p Fn(,)g(or,)g(in)i(other)e(w)o(ords,)g(that)h(only)g (the)g(in)o(tegrals)g(o)o(v)o(er)f(the)h(\\in)o(terior")37 312 y(of)h(\000\(\003\))f(pla)o(y)h(a)f(role)i(in)f(iden)o(tifying)i (the)e(limit)h(p)q(er)f(unit)h(v)o(olume.)28 b(Indeed,)19 b(it)f(is)g(easy)g(to)f(see,)i(b)o(y)e(a)37 369 y(standard)g(argumen)o (t)e(that)h(w)o(e)g(ha)o(v)o(e)h(already)f(used)h(in)h([11)o(])e(and)h (that)f(is)h(based)f(up)q(on)i(the)e(prop)q(erties)37 425 y(of)f Fi(\037)117 432 y Fv(\003)159 425 y Fn(and)h(of)f Fi(u)326 432 y Fh(n)349 425 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fn(\),)13 b(that)362 549 y Fi(I)386 530 y Fv(1)382 560 y(\003)450 549 y Fn(=)42 b Fg(j)p Fn(\003)p Fg(j)593 487 y Fe(Z)616 590 y Fh(Q)644 580 y Fc(?)707 518 y Fi(d\030)p 677 538 106 2 v 677 580 a Fn(\(2)p Fi(\031)r Fn(\))764 567 y Fv(3)795 505 y Fe(X)795 603 y Fh(n)p Fu(\025)p Fv(1)869 549 y Fi(\025)895 556 y Fh(n)919 549 y Fn(\()p Fi(\030)r Fn(\))985 487 y Fe(Z)1009 590 y Fh(Q)1046 549 y Fg(jr)p Fn(\()p Fi(u)1141 556 y Fh(n)1164 549 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\)\))p Fg(j)1300 530 y Fv(2)1325 549 y Fi(dx)i Fn(+)g Fi(o)1452 512 y Fe(\000)1473 549 y Fg(j)p Fn(\003)p Fg(j)1531 512 y Fe(\001)450 692 y Fn(=)42 b Fg(j)p Fn(\003)p Fg(j)593 631 y Fe(Z)616 734 y Fh(Q)644 724 y Fc(?)672 692 y Fn(T)l(r)719 704 y Fh(L)743 693 y Ft(2)743 716 y Fc(\030)760 704 y Fv(\()p Fh(Q)p Fv(\))817 692 y Fn([)p Fg(\000)p Fn(\001)903 699 y Fh(\030)922 692 y Fi(K)961 699 y Fh(\030)979 692 y Fn(])1034 662 y Fi(d\030)p 1005 682 V 1005 724 a Fn(\(2)p Fi(\031)r Fn(\))1092 710 y Fv(3)1125 692 y Fn(+)10 b Fi(o)1192 656 y Fe(\000)1213 692 y Fg(j)p Fn(\003)p Fg(j)1271 656 y Fe(\001)1291 692 y Fi(;)455 b Fn(\(4.65\))37 821 y(and)570 934 y Fi(I)594 915 y Fv(2)590 945 y(\003)630 934 y Fn(=)13 b Fi(o)700 897 y Fe(\000)720 934 y Fg(j)p Fn(\003)p Fg(j)778 897 y Fe(\001)806 872 y(Z)831 975 y Fh(Q)859 965 y Fc(?)886 934 y Fn(T)l(r)933 945 y Fh(L)957 934 y Ft(2)957 957 y Fc(\030)974 945 y Fv(\()p Fh(Q)p Fv(\))1032 934 y Fn([)p Fg(\000)p Fn(\001)1118 941 y Fh(\030)1136 934 y Fi(K)1175 941 y Fh(\030)1194 934 y Fn(])1249 903 y Fi(d\030)p 1219 923 V 1219 965 a Fn(\(2)p Fi(\031)r Fn(\))1306 952 y Fv(3)1329 934 y Fi(:)417 b Fn(\(4.66\))37 1069 y(Finally)l(,)17 b(w)o(e)e(ma)o(y)f(b)q(ound)j Fi(I)531 1052 y Fv(3)527 1082 y(\003)568 1069 y Fn(as)e(follo)o(ws)151 1215 y Fg(j)p Fi(I)188 1196 y Fv(3)184 1226 y(\003)210 1215 y Fg(j)42 b(\024)19 b Fi(o)341 1178 y Fe(\000)362 1215 y Fg(j)p Fn(\003)p Fg(j)420 1178 y Fe(\001)447 1153 y(Z)472 1256 y Fh(Q)500 1246 y Fc(?)528 1171 y Fe(X)527 1269 y Fh(n)p Fu(\025)p Fv(1)601 1215 y Fi(\025)627 1222 y Fh(n)651 1215 y Fn(\()p Fi(\030)r Fn(\))709 1151 y Fe(\022)742 1153 y(Z)767 1256 y Fh(Q)805 1215 y Fg(jr)p Fn(\()p Fi(u)900 1222 y Fh(n)923 1215 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\)\))p Fg(j)1059 1196 y Fv(2)1076 1151 y Fe(\023)1115 1148 y Ft(1)p 1115 1154 16 2 v 1115 1175 a(2)1137 1151 y Fe(\022)1170 1153 y(Z)1196 1256 y Fh(Q)1233 1215 y Fg(j)p Fi(u)1272 1222 y Fh(n)1295 1215 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)1413 1196 y Fv(2)1431 1151 y Fe(\023)1469 1148 y Ft(1)p 1469 1154 V 1469 1175 a(2)1499 1215 y Fi(d\030)265 1379 y Fg(\024)19 b Fi(o)341 1342 y Fe(\000)362 1379 y Fg(j)p Fn(\003)p Fg(j)420 1342 y Fe(\001)447 1317 y(Z)472 1420 y Fh(Q)500 1410 y Fc(?)520 1314 y Fe(\022)554 1335 y(X)553 1433 y Fh(n)p Fu(\025)p Fv(1)627 1379 y Fi(\025)653 1386 y Fh(n)677 1379 y Fn(\()p Fi(\030)r Fn(\))743 1317 y Fe(Z)767 1420 y Fh(Q)805 1379 y Fg(jr)p Fn(\()p Fi(u)900 1386 y Fh(n)923 1379 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\)\))p Fg(j)1059 1360 y Fv(2)1076 1314 y Fe(\023)1115 1312 y Ft(1)p 1115 1318 V 1115 1339 a(2)1137 1314 y Fe(\022)1171 1335 y(X)1170 1433 y Fh(n)p Fu(\025)p Fv(1)1244 1379 y Fi(\025)1270 1386 y Fh(n)1294 1379 y Fn(\()p Fi(\030)r Fn(\))1360 1317 y Fe(Z)1385 1420 y Fh(Q)1422 1379 y Fg(j)p Fi(u)1461 1386 y Fh(n)1484 1379 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)1602 1360 y Fv(2)1620 1314 y Fe(\023)1658 1312 y Ft(1)p 1658 1318 V 1658 1339 a(2)1688 1379 y Fi(d\030)265 1542 y Fg(\024)19 b Fi(o)341 1506 y Fe(\000)362 1542 y Fg(j)p Fn(\003)p Fg(j)420 1506 y Fe(\001)439 1478 y(\022)473 1481 y(Z)498 1584 y Fh(Q)526 1574 y Fc(?)554 1499 y Fe(X)553 1597 y Fh(n)p Fu(\025)p Fv(1)627 1542 y Fi(\025)653 1549 y Fh(n)677 1542 y Fn(\()p Fi(\030)r Fn(\))743 1481 y Fe(Z)767 1584 y Fh(Q)805 1542 y Fg(jr)p Fn(\()p Fi(u)900 1549 y Fh(n)923 1542 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\)\))p Fg(j)1059 1524 y Fv(2)1076 1478 y Fe(\023)1115 1476 y Ft(1)p 1115 1482 V 1115 1503 a(2)1137 1478 y Fe(\022)1170 1481 y(Z)1196 1584 y Fh(Q)1224 1574 y Fc(?)1251 1499 y Fe(X)1251 1597 y Fh(n)p Fu(\025)p Fv(1)1325 1542 y Fi(\025)1351 1549 y Fh(n)1375 1542 y Fn(\()p Fi(\030)r Fn(\))1441 1481 y Fe(Z)1465 1584 y Fh(Q)1503 1542 y Fg(j)p Fi(u)1542 1549 y Fh(n)1565 1542 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)1683 1524 y Fv(2)1701 1478 y Fe(\023)1739 1476 y Ft(1)p 1739 1482 V 1739 1503 a(2)265 1706 y Fn(=)19 b Fi(o)341 1670 y Fe(\000)362 1706 y Fg(j)p Fn(\003)p Fg(j)420 1670 y Fe(\001)439 1642 y(\022)473 1645 y(Z)498 1748 y Fh(Q)526 1738 y Fc(?)553 1706 y Fn(T)l(r)600 1718 y Fh(L)624 1707 y Ft(2)624 1730 y Fc(\030)641 1718 y Fv(\()p Fh(Q)p Fv(\))699 1706 y Fn([)p Fg(\000)p Fn(\001)785 1713 y Fh(\030)803 1706 y Fi(K)842 1713 y Fh(\030)861 1706 y Fn(])916 1676 y Fi(d\030)p 886 1696 106 2 v 886 1738 a Fn(\(2)p Fi(\031)r Fn(\))973 1724 y Fv(3)996 1642 y Fe(\023)1035 1640 y Ft(1)p 1035 1646 16 2 v 1035 1667 a(2)1057 1706 y Fi(;)689 b Fn(\(4.67\))37 1838 y(b)o(y)22 b(a)f(rep)q(eated)h(use)g(of)f(the)h(Cauc)o(h)o(y-Sc)o(h)o(w)o(arz)f (inequalit)o(y)l(.)40 b(Inserting)23 b(\(4.65\),)e(\(4.66\),)f (\(4.67\))g(in)o(to)37 1894 y(\(4.64\),)13 b(next)j(in)g(\(4.63\),)d(w) o(e)i(obtain)373 2018 y(T)l(r)420 2028 y Fh(L)444 2018 y Ft(2)461 2028 y Fv(\()p Fd(R)505 2018 y Ft(3)523 2028 y Fv(\))538 2018 y Fn([)p Fg(\000)p Fn(\001)p Fi(K)663 2025 y Fv(\003)689 2018 y Fn(])e(=)g Fg(j)p Fn(\003)p Fg(j)829 1956 y Fe(Z)852 2059 y Fh(Q)880 2050 y Fc(?)908 2018 y Fn(T)l(r)954 2030 y Fh(L)978 2018 y Ft(2)978 2042 y Fc(\030)996 2030 y Fv(\()p Fh(Q)p Fv(\))1053 2018 y Fn([)p Fg(\000)p Fn(\001)1139 2025 y Fh(\030)1158 2018 y Fi(K)1197 2025 y Fh(\030)1215 2018 y Fn(])1270 1987 y Fi(d\030)p 1240 2007 106 2 v 1240 2049 a Fn(\(2)p Fi(\031)r Fn(\))1327 2036 y Fv(3)1361 2018 y Fn(+)d Fi(o)1428 1981 y Fe(\000)1449 2018 y Fg(j)p Fn(\003)p Fg(j)1507 1981 y Fe(\001)1527 2018 y Fi(;)37 2150 y Fn(whic)o(h)17 b(sho)o(ws)d (\(4.62\))f(and)j(concludes)h(the)e(pro)q(of)g(of)f(the)i(prop)q (osition.)538 b Fg(})37 2293 y Fk(5)67 b(The)23 b(Hartree-F)-6 b(o)r(c)n(k)23 b(mo)r(del)37 2394 y Fn(Let)16 b(us)f(\014rst)g(of)g (all)h(recall)g(the)g(Hartree-F)l(o)q(c)o(k)e(mo)q(del)i (\(2.3\){\(2.4\){\()o(2.5)o(\))c(in)o(tro)q(duced)17 b(in)f(Section)g(2)f(:)565 2515 y Fi(I)589 2497 y Fh(H)s(F)585 2527 y Fv(\003)662 2515 y Fn(=)e(inf)t Fg(f)p Fi(E)825 2497 y Fh(H)s(F)822 2527 y Fv(\003)885 2515 y Fn(\()p Fi(K)s Fn(\))c(+)1023 2485 y(1)p 1023 2505 23 2 v 1023 2547 a(2)1051 2515 y Fi(U)1082 2522 y Fv(\003)1121 2515 y Fn(;)20 b Fi(K)15 b Fg(2)e(K)1286 2522 y Fv(\003)1312 2515 y Fg(g)p Fi(;)37 2628 y Fn(where)j(the)f(set)g(of)g(minimization)i (is)372 2730 y Fg(K)407 2737 y Fv(\003)446 2730 y Fn(=)c Fg(f)p Fn(0)f Fg(\024)h Fi(K)i Fg(\024)e Fj(1)p Fi(;)23 b Fn(T)l(r)15 b Fi(K)g Fn(=)e Fg(j)p Fn(\003)p Fg(j)p Fi(;)21 b Fn(T)l(r)1067 2693 y Fe(\002)1086 2730 y Fn(\()p Fg(\000)p Fn(\001)10 b Fg(\000)g Fi(V)1258 2737 y Fv(\003)1285 2730 y Fn(\))p Fi(K)1345 2693 y Fe(\003)1377 2730 y Fi(<)j Fn(+)p Fg(1g)p Fi(;)p eop %%Page: 51 51 51 50 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(51)37 199 y(and)16 b(the)f(energy)g(functional)i(writes)76 318 y Fi(E)113 299 y Fh(H)s(F)110 329 y Fv(\003)173 318 y Fn(\()p Fi(K)s Fn(\))12 b(=)h(T)l(r)358 281 y Fe(\002)377 318 y Fn(\()p Fg(\000)p Fn(\001)d Fg(\000)g Fi(V)549 325 y Fv(\003)576 318 y Fn(\))p Fi(K)636 281 y Fe(\003)665 318 y Fn(+)715 287 y(1)p 715 307 23 2 v 715 349 a(2)763 256 y Fe(Z)5 b(Z)728 390 y Fd(R)758 380 y Ft(3)775 390 y Fu(\002)p Fd(R)832 380 y Ft(3)863 287 y Fi(\032)p Fn(\()p Fi(x;)j(x)p Fn(\))p Fi(\032)p Fn(\()p Fi(y)r(;)g(y)r Fn(\))p 863 307 258 2 v 927 349 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1134 318 y Fi(dxdy)g Fg(\000)1291 287 y Fn(1)p 1291 307 23 2 v 1291 349 a(2)1339 256 y Fe(Z)5 b(Z)1304 390 y Fd(R)1334 380 y Ft(3)1352 390 y Fu(\002)p Fd(R)1409 380 y Ft(3)1439 287 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1595 270 y Fv(2)p 1439 307 174 2 v 1461 349 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1626 318 y Fi(dxdy)r(;)37 473 y Fn(where)20 b Fi(\032)p Fn(\()p Fi(x)p Fn(;)8 b Fi(y)r Fn(\))18 b(is)i(the)g(Hilb)q(ert-Sc)o(hmidt)i(k) o(ernel)e(of)f Fi(K)s Fn(.)32 b(As)19 b(is)h(w)o(ell-kno)o(wn,)h(this)f (functional)g(is)g(not)37 529 y(con)o(v)o(ex,)15 b(and)f(therefore)h(w) o(e)f(exp)q(ect)h(the)g(thermo)q(dynamic)g(limit)h(problem)g(to)d(b)q (e)j(m)o(uc)o(h)e(more)g(di\016cult)37 586 y(than)g(in)h(the)f(Reduced) i(Hartree-F)l(o)q(c)o(k)d(setting.)20 b(In)14 b(this)h(latter)e(case,)h (w)o(e)g(ha)o(v)o(e)g(used)g(in)h(a)f(fundamen)o(tal)37 642 y(w)o(a)o(y)g(the)g(con)o(v)o(exit)o(y)g(of)g(the)h(energy)f (functional)i(through)e(the)g(use)h(of)f(the)g Fg(\030)p Fn(-transform.)19 b(Here,)14 b(for)g(the)37 699 y(Hartree-F)l(o)q(c)o (k)k(case,)i(this)f(is)g(not)f(p)q(ossible)i(an)o(y)e(more.)30 b(Let)19 b(us)f(at)g(once)h(sa)o(y)f(that)g(this)h(is)g(the)g(main)37 755 y(reason)d(wh)o(y)g(w)o(e)f(are)h(not)f(able)i(to)e(pro)o(v)o(e)g (a)h(result)g(on)g(the)g(con)o(v)o(ergence)g(of)g(the)g(Hartree-F)l(o)q (c)o(k)f(energy)37 812 y(p)q(er)k(unit)f(v)o(olume)g(in)h(the)e(thermo) q(dynamic)i(limit,)g(and)f(wh)o(y)f(w)o(e)h(cannot)f(establish)i(the)f (analogue)f(of)37 868 y(Theorem)i(2.2.)30 b(Nev)o(ertheless,)20 b(in)g(this)f(section,)h(w)o(e)f(shall)h(\(a\))d(giv)o(e)i(some)g (formal)f(computations)h(in)37 925 y(order)i(to)g(justify)g(our)g (guess)g(on)f(the)i(p)q(erio)q(dic)h(Hartree-F)l(o)q(c)o(k)d(problem)i (\(2.23\){\(2.2)o(4\){\()o(2.2)o(5\))c(that)37 981 y(should)f(b)q(e)f (obtained)g(in)g(the)f(thermo)q(dynamic)h(limit,)h(\(b\))d(sho)o(w)h (that)g(this)h(p)q(erio)q(dic)h(problem)f(is)g(w)o(ell-)37 1038 y(p)q(osed)f(mathematically)h(\(Theorem)d(2.3)h(in)h(Subsection)h (5.1)d(b)q(elo)o(w\),)i(and,)f(\014nally)l(,)i(\(c\))e(c)o(hec)o(k)g (that)g(the)37 1094 y(upp)q(er)g(limit)h(of)d(the)i(Hartree-F)l(o)q(c)o (k)e(energy)h(p)q(er)h(unit)f(v)o(olume)h(ma)o(y)e(at)g(least)h(b)q(e)h (compared)f(from)f(ab)q(o)o(v)o(e)37 1150 y(b)o(y)k(the)f(HF)g(p)q (erio)q(dic)i(problem)f(\(Prop)q(osition)f(2.1)g(in)h(Subsection)g (5.2\).)108 1232 y(Let)h(us)f(b)q(egin)i(with)e(some)g(formal)g (computations)g(on)h(the)f(Hartree-F)l(o)q(c)o(k)g(energy)g(of)g(a)g (minimizer)37 1288 y Fi(K)76 1295 y Fv(\003)118 1288 y Fn(of)f Fi(I)194 1272 y Fh(H)s(F)190 1302 y Fv(\003)269 1288 y Fn(when)h(\003)f(go)q(es)g(to)f(in\014nit)o(y)l(.)108 1345 y(W)l(e)j Ff(p)n(ostulate)k Fn(that)16 b(the)i(sequence)g(of)f(op) q(erators)f Fi(K)1049 1352 y Fv(\003)1092 1345 y Fn(con)o(v)o(erges)h (to)f(some)h(self-adjoin)o(t)h(op)q(erator)37 1401 y Fi(K)g Fn(that)c(comm)o(utes)h(with)g(the)g(translation)g(of)g Fj(Z)901 1385 y Fv(3)920 1401 y Fn(,)g(and)g(that)f(b)q(elongs)i(to)e Fg(K)q Fn(.)19 b(By)c(sa)o(ying)g(so,)f(the)i(main)37 1458 y(assumption)d(w)o(e)f(do)g(is)h(the)g(follo)o(wing)g(one)f(:)19 b(W)l(e)12 b(p)q(ostulate)h(that)e(the)i(densit)o(y)g Fi(\032)1438 1465 y Fv(\003)1465 1458 y Fn(\()p Fi(x;)8 b(x)p Fn(\))i(asymptotically)37 1514 y(b)q(eha)o(v)o(es)17 b(lik)o(e)h(a)d Fi(Q)p Fn(-p)q(erio)q(dic)k(densit)o(y)e Fi(\032)p Fn(\()p Fi(x;)8 b(x)p Fn(\).)22 b(W)l(e)17 b(emphasize)g(this)g(is)g(an)f Ff(assumption)p Fn(,)g(and)h(that)e(w)o (e)37 1571 y(only)i(ha)o(v)o(e)f(the)h(in)o(tuition)g(that)f(it)h(is)f (true.)24 b(The)16 b(lac)o(k)h(of)f(con)o(v)o(exit)o(y)g(of)g(the)g (Hartree-F)l(o)q(c)o(k)g(mo)q(del)h(has)37 1627 y(prev)o(en)o(ted)f(us) f(so)g(far)g(to)f Ff(pr)n(ove)19 b Fn(this)d(p)q(ostulate.)108 1683 y(In)j(view)g(of)e(the)h(results)h(w)o(e)f(ha)o(v)o(e)g(obtained)g (on)g(the)h(Reduced)h(Hartree-F)l(o)q(c)o(k)d(mo)q(del,)j(it)e(is)h (then)37 1740 y(reasonable)d(to)f(b)q(eliev)o(e)i(that)d(in)i(the)g (energy)f(\(2.5\))f(of)g(a)h(minimizer)j Fi(K)1291 1747 y Fv(\003)1332 1740 y Fn(of)d(\(2.3\){\(2.4\))o({\(2)o(.5\))57 1861 y Fi(E)94 1842 y Fh(H)s(F)91 1872 y Fv(\003)154 1861 y Fn(\()p Fi(K)211 1868 y Fv(\003)237 1861 y Fn(\))d(=)h(T)l(r)362 1824 y Fe(\002)381 1861 y Fn(\()p Fg(\000)p Fn(\001)d Fg(\000)g Fi(V)553 1868 y Fv(\003)580 1861 y Fn(\))p Fi(K)637 1868 y Fv(\003)663 1824 y Fe(\003)692 1861 y Fn(+)743 1830 y(1)p 743 1850 23 2 v 743 1892 a(2)790 1799 y Fe(Z)c(Z)755 1933 y Fd(R)785 1924 y Ft(3)803 1933 y Fu(\002)p Fd(R)860 1924 y Ft(3)891 1830 y Fi(\032)914 1837 y Fv(\003)941 1830 y Fn(\()p Fi(x;)i(x)p Fn(\))p Fi(\032)1073 1837 y Fv(\003)1098 1830 y Fn(\()p Fi(y)r(;)g(y)r Fn(\))p 891 1850 311 2 v 981 1892 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1214 1861 y Fi(dxdy)i Fg(\000)1385 1830 y Fn(1)p 1385 1850 23 2 v 1385 1892 a(2)1432 1799 y Fe(Z)5 b(Z)1397 1933 y Fd(R)1427 1924 y Ft(3)1445 1933 y Fu(\002)p Fd(R)1502 1924 y Ft(3)1533 1830 y Fg(j)p Fi(\032)1569 1837 y Fv(\003)1595 1830 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1715 1814 y Fv(2)p 1533 1850 201 2 v 1567 1892 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1746 1861 y Fi(dxdy)r(;)37 2044 y Fn(the)16 b(\014rst)f(three)g(terms)g(globally)h(b)q(eha)o(v)o (e)f(lik)o(e)i Fg(j)p Fn(\003)p Fg(j)916 2007 y Fe(\000)935 2044 y Fi(E)972 2025 y Fh(RH)s(F)969 2055 y(per)1059 2044 y Fn(\()p Fi(K)s Fn(\))10 b(+)1197 2013 y Fi(M)p 1197 2033 50 2 v 1210 2075 a Fn(2)1251 2007 y Fe(\001)1272 2044 y Fi(:)108 2111 y Fn(Therefore,)23 b(it)g(remains)f(to)g (understand,)i(at)d(least)i(formally)l(,)h(the)e(b)q(eha)o(viour)h(of)e (the)i(so-called)37 2192 y(exc)o(hange)d(term)f Fg(\000)389 2162 y Fn(1)p 389 2182 23 2 v 389 2224 a(2)425 2130 y Fe(Z)5 b(Z)500 2234 y Fd(R)530 2224 y Ft(3)548 2234 y Fu(\002)p Fd(R)605 2224 y Ft(3)638 2162 y Fg(j)p Fi(\032)674 2169 y Fv(\003)700 2162 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)820 2145 y Fv(2)p 638 2182 201 2 v 673 2224 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)851 2192 y Fi(dxdy)q Fn(.)32 b(F)l(or)19 b(this)h(purp)q(ose,)g(let)g(us)g(replace)g(the)f(densit)o (y)37 2275 y(matrix)c Fi(\032)208 2282 y Fv(\003)235 2275 y Fn(\()p Fi(x;)8 b(y)r Fn(\))14 b(b)o(y)h(a)g(matrix)g(of)g(the)g (form)f Fi(\037)870 2282 y Fv(\003)897 2275 y Fn(\()p Fi(x)p Fn(\))p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fi(\037)1117 2282 y Fv(\003)1143 2275 y Fn(\()p Fi(y)r Fn(\))14 b(mimic)o(king)j(the)e(argumen)o(t)f(w)o(e)h(ha)o(v)o(e)37 2332 y(made)k(ab)q(o)o(v)o(e)f(to)f(determine)i(the)g(upp)q(er)g(limit) h(of)d(the)i(RHF)f(mo)q(del.)30 b(The)19 b(function)g Fi(\037)1613 2339 y Fv(\003)1658 2332 y Fn(is)g(a)e(cut-o\013)37 2388 y(function,)e(whic)o(h)f(has)f(all)h(the)f(go)q(o)q(d)h(prop)q (erties)g(the)f(reader)g(ma)o(y)g(think)h(of,)f(and)g(whic)o(h)h(are)f (recalled)i(in)37 2445 y(the)h(pro)q(of)g(of)f(Prop)q(osition)i(2.1)e (b)q(elo)o(w.)22 b(Then,)17 b(w)o(e)e(establish,)i(still)g(in)g(the)f (course)g(of)g(Prop)q(osition)g(2.1)37 2501 y(in)h(Subsection)f(5.2)e (b)q(elo)o(w,)i(that)351 2620 y(lim)335 2650 y Fv(\003)p Fu(!1)459 2589 y Fn(1)p 442 2609 57 2 v 442 2651 a Fg(j)p Fn(\003)p Fg(j)512 2558 y Fe(Z)5 b(Z)587 2661 y Fd(R)617 2651 y Ft(3)635 2661 y Fu(\002)p Fd(R)692 2651 y Ft(3)725 2589 y Fg(j)p Fi(\032)761 2596 y Fv(\003)788 2589 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)908 2572 y Fv(2)p 725 2609 201 2 v 760 2651 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)938 2620 y Fi(dxdy)k Fn(=)1096 2558 y Fe(Z)5 b(Z)1172 2661 y Fh(Q)p Fu(\002)p Fd(R)1257 2651 y Ft(3)1289 2589 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1445 2572 y Fv(2)p 1289 2609 174 2 v 1311 2651 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1468 2620 y Fi(dxdy)r(:)37 2740 y Fn(Moreo)o(v)o(er,)20 b(the)h(quan)o(tit)o(y)e(whic)o(h)j(app)q(ears) e(in)h(the)f(righ)o(t-hand)h(side)g(of)f(the)h(ab)q(o)o(v)o(e)e (equalit)o(y)j(mak)o(es)37 2796 y(sense)16 b(thanks)f(to)g(the)g(follo) o(wing)p eop %%Page: 52 52 52 51 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(52)37 199 y Fj(Lemma)18 b(5.1)23 b Ff(F)m(or)16 b(any)g Fi(K)j Ff(in)d Fg(K)q Ff(,)g(we)g(have)658 247 y Fe(Z)684 350 y Fh(Q)721 308 y Fi(dx)779 247 y Fe(Z)804 350 y Fd(R)834 340 y Ft(3)866 278 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1022 261 y Fv(2)p 866 298 174 2 v 888 340 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1052 308 y Fi(dy)k(<)e Fn(+)p Fg(1)p Fi(;)37 419 y Ff(and)340 458 y Fe(Z)365 561 y Fh(Q)403 520 y Fi(dx)461 458 y Fe(Z)485 561 y Fd(R)515 552 y Ft(3)547 489 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)703 473 y Fv(2)p 547 509 V 569 551 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)734 520 y Fi(dy)979 b Fn(\(5.1a\))381 650 y(=)460 588 y Fe(Z)6 b(Z)f(Z)g(Z)436 722 y Fv(\()p Fh(Q)478 713 y Fc(?)495 722 y Fv(\))509 713 y Ft(2)526 722 y Fu(\002)p Fh(Q)581 713 y Ft(2)606 650 y Fi(\032)p Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fi(W)822 657 y Fu(1)858 650 y Fn(\()p Fi(\030)j Fg(\000)g Fi(\030)975 631 y Fu(0)986 650 y Fi(;)d(x)i Fg(\000)g Fi(y)r Fn(\))p Fi(\032)1153 631 y Fu(\003)1173 650 y Fn(\()p Fi(\030)1213 631 y Fu(0)1224 650 y Fi(;)e(x;)g(y)r Fn(\))g Fi(dxdy)1451 619 y(d\030)r(d\030)1543 603 y Fu(0)p 1450 639 106 2 v 1450 681 a Fn(\(2)p Fi(\031)r Fn(\))1537 668 y Fv(6)1560 650 y Fi(;)183 b Fn(\(5.1b\))37 801 y Ff(wher)n(e)17 b(the)f(p)n(otential)g Fi(W)470 808 y Fu(1)524 801 y Ff(is)g(given)f(by)h(\(2.25\))g(in)g(Se)n(ction)f(2)i(;) f(that)h(is)712 914 y Fi(W)755 921 y Fu(1)793 914 y Fn(\()p Fi(\021)r(;)8 b(z)r Fn(\))i(=)966 871 y Fe(X)956 972 y Fh(k)q Fu(2)p Fd(Z)1024 963 y Ft(3)1076 883 y Fi(e)1097 867 y Fh(ik)q Fu(\001)p Fh(\021)p 1054 904 130 2 v 1054 945 a Fg(j)p Fi(z)h Fn(+)g Fi(k)q Fg(j)1188 914 y Fi(:)37 1042 y Fn(The)16 b(pro)q(of)f(of)g(this)g(lemma)h(is)f(giv)o(en)h(b)q (elo)o(w.)108 1124 y(The)g(ab)q(o)o(v)o(e)g(argumen)o(t)f(justi\014es,) h(at)f(least)h(formally)l(,)g(the)g(in)o(tro)q(duction)h(of)e(the)h(p)q (erio)q(dic)i(problem)37 1180 y(\(2.23\){\(2.24\){)o(\(2.)o(25\))o(,)12 b(that)j(w)o(e)g(recall)h(no)o(w)f(:)651 1264 y Fi(I)675 1245 y Fh(H)s(F)671 1275 y(per)748 1264 y Fn(=)e(inf)t Fg(f)p Fi(E)911 1245 y Fh(H)s(F)908 1275 y(per)971 1264 y Fn(\()p Fi(K)s Fn(\))f(;)20 b Fi(K)15 b Fg(2)e(K)q(g)p Fi(;)220 1404 y(E)257 1385 y Fh(H)s(F)254 1415 y(per)317 1404 y Fn(\()p Fi(K)s Fn(\))f(=)455 1342 y Fe(Z)480 1445 y Fh(Q)508 1435 y Fc(?)536 1404 y Fn(T)l(r)583 1415 y Fh(L)607 1404 y Ft(2)607 1427 y Fc(\030)624 1415 y Fv(\()p Fh(Q)p Fv(\))681 1367 y Fe(\002)700 1404 y Fg(\000)p Fn(\001)773 1411 y Fh(\030)792 1404 y Fi(K)831 1411 y Fh(\030)850 1367 y Fe(\003)911 1373 y Fi(d\030)p 881 1393 106 2 v 881 1435 a Fn(\(2)p Fi(\031)r Fn(\))968 1422 y Fv(3)1002 1404 y Fg(\000)1047 1342 y Fe(Z)1072 1445 y Fh(Q)1110 1404 y Fi(G\032)e Fn(+)1230 1373 y(1)p 1230 1393 23 2 v 1230 1435 a(2)1257 1404 y Fi(D)1295 1411 y Fh(G)1324 1404 y Fn(\()p Fi(\032;)e(\032)p Fn(\))i Fg(\000)1488 1373 y Fn(1)p 1488 1393 V 1488 1435 a(2)1515 1404 y Fi(E)1549 1411 y Fh(exc)1602 1404 y Fn(\()p Fi(K)s Fn(\))p Fi(;)37 1517 y Fn(where)16 b(w)o(e)f(denote)h(b)o(y)f Fi(E)480 1524 y Fh(exc)533 1517 y Fn(\()p Fi(K)s Fn(\))f(an)o(y)h(of)g (the)g(t)o(w)o(o)f(equiv)m(alen)o(t)j(form)o(ulations)e(\(5.1a\))f(and) h(\(5.1b\))f(of)h(the)37 1574 y(p)q(erio)q(dic)k(exc)o(hange)e(term.)22 b(Before)17 b(pro)o(ving)f(in)h(the)f(forthcoming)g(subsection)i(that)d (the)i(HF)f(p)q(erio)q(dic)37 1630 y(problem)g(is)g(w)o(ell-p)q(osed,)h (w)o(e)e(giv)o(e)g(no)o(w)g(the)g(pro)q(of)g(of)g(Lemma)g(5.1.)37 1711 y Fj(Pro)q(of)j(of)f(Lemma)g(5.1:)k Fn(W)l(e)15 b(decomp)q(ose)g(the)g(exc)o(hange)g(term)f(in)i(t)o(w)o(o)d(terms)h (in)i(the)f(follo)o(wing)g(w)o(a)o(y)339 1759 y Fe(Z)364 1862 y Fh(Q)401 1821 y Fi(dx)459 1759 y Fe(Z)484 1862 y Fd(R)514 1852 y Ft(3)546 1790 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)702 1773 y Fv(2)p 546 1810 174 2 v 568 1852 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)732 1821 y Fi(dy)380 1954 y Fn(=)457 1892 y Fe(Z)482 1995 y Fh(Q)520 1954 y Fi(dx)578 1892 y Fe(Z)602 1995 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\024)p Fv(1)747 1923 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)d(y)r Fn(\))p Fg(j)903 1906 y Fv(2)p 747 1943 V 769 1985 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)934 1954 y Fi(dy)h Fn(+)1037 1892 y Fe(Z)1062 1995 y Fh(Q)1100 1954 y Fi(dx)1158 1892 y Fe(Z)1182 1995 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\025)p Fv(1)1327 1923 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)d(y)r Fn(\))p Fg(j)1483 1906 y Fv(2)p 1327 1943 V 1349 1985 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1514 1954 y Fi(dy)r(:)37 2066 y Fn(F)l(or)15 b(the)g(second)h(term,)e(w)o(e)h(clearly)i(ha)o(v)o(e)329 2114 y Fe(Z)355 2217 y Fh(Q)392 2176 y Fi(dx)450 2114 y Fe(Z)475 2217 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\025)p Fv(1)620 2145 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)776 2128 y Fv(2)p 620 2165 V 641 2207 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)806 2176 y Fi(dy)j Fg(\024)914 2114 y Fe(Z)940 2217 y Fh(Q)977 2176 y Fi(dx)1035 2114 y Fe(Z)1060 2217 y Fd(R)1090 2207 y Ft(3)1117 2176 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1273 2157 y Fv(2)1298 2176 y Fi(dy)15 b(<)e Fn(+)p Fg(1)p Fi(:)37 2295 y Fn(Let)j(us)g(concen)o(trate)f(no)o(w)f(on)i(the)f (\014rst)g(term.)20 b(Since)d(w)o(e)e(ma)o(y)f(pro)o(v)o(e)h(lik)o(e)i (in)f(\(4.46\))d(that)i Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1809 2278 y Fv(2)1840 2295 y Fg(\024)37 2351 y Fi(\032)p Fn(\()p Fi(x)p Fn(\))g Fi(\032)p Fn(\()p Fi(y)r Fn(\),)14 b(almost)h(ev)o(erywhere)h(on)f Fj(R)726 2335 y Fv(3)756 2351 y Fg(\002)10 b Fj(R)840 2335 y Fv(3)860 2351 y Fn(,)15 b(w)o(e)g(ma)o(y)f(write)307 2401 y Fe(Z)332 2504 y Fh(Q)370 2463 y Fi(dx)428 2401 y Fe(Z)452 2504 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\024)p Fv(1)597 2432 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)753 2416 y Fv(2)p 597 2453 V 619 2494 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)784 2463 y Fi(dy)43 b Fg(\024)950 2401 y Fe(Z)975 2504 y Fh(Q)1012 2463 y Fi(dx)1070 2401 y Fe(Z)1095 2504 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\024)p Fv(1)1240 2432 y Fi(\032)p Fn(\()p Fi(x)p Fn(\))8 b Fi(\032)p Fn(\()p Fi(y)r Fn(\))p 1240 2453 176 2 v 1262 2494 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1428 2463 y Fi(dy)873 2594 y Fg(\024)950 2532 y Fe(Z)5 b(Z)1025 2635 y Fv(\()p Fh(Q)p Fv(+)p Fh(B)1121 2640 y Ft(1)1138 2635 y Fv(\))p Fu(\002)p Fv(\()p Fh(Q)p Fv(+)p Fh(B)1275 2640 y Ft(1)1292 2635 y Fv(\))1321 2563 y Fi(\032)p Fn(\()p Fi(x)p Fn(\))j Fi(\032)p Fn(\()p Fi(y)r Fn(\))p 1321 2584 V 1343 2625 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1508 2594 y Fi(dxdy)873 2700 y Fg(\024)42 b Fi(C)11 b Fg(k)p Fi(\032)p Fg(k)1062 2681 y Fv(2)1062 2717 y Fh(L)1086 2707 y Ft(6)p Fc(=)p Ft(5)1134 2717 y Fv(\()p Fh(Q)p Fv(+)p Fh(B)1230 2722 y Ft(1)1247 2717 y Fv(\))873 2782 y Fg(\024)42 b Fi(C)11 b Fg(k)p Fi(\032)p Fg(k)1062 2764 y Fv(2)1062 2807 y Fh(L)1086 2790 y Ft(6)p Fc(=)p Ft(5)1086 2819 y(unif)1142 2807 y Fv(\()p Fd(R)1186 2798 y Ft(3)1203 2807 y Fv(\))1277 2782 y Fi(<)59 b Fn(+)p Fg(1)p Fi(;)p eop %%Page: 53 53 53 52 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(53)37 199 y(since)19 b Fi(\032)e Fn(lies)h(in)g Fi(H)368 183 y Fv(1)364 213 y(unif)428 199 y Fn(\()p Fj(R)485 183 y Fv(3)505 199 y Fn(\).)25 b(W)l(e)17 b(c)o(hec)o(k)g(no)o (w)f(the)h(equiv)m(alence)j(of)c(the)h(t)o(w)o(o)f(form)o(ulations)h (\(5.1a\))e(and)37 256 y(\(5.1b\),)f(whic)o(h)i(follo)o(ws)f(from)g (the)g(follo)o(wing)h(string)f(of)g(equalities)i(:)103 324 y Fe(Z)5 b(Z)78 458 y Fh(Q)p Fu(\002)p Fd(R)163 448 y Ft(3)193 355 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)349 338 y Fv(2)p 193 375 174 2 v 215 417 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)380 386 y Fi(dxdy)42 b Fn(=)598 324 y Fe(Z)5 b(Z)573 458 y Fh(Q)p Fu(\002)p Fd(R)658 448 y Ft(3)742 355 y Fn(1)p 688 375 131 2 v 688 417 a Fg(j)p Fi(x)10 b Fg(\000)g Fi(y)r Fg(j)831 386 y Fi(dxdy)946 324 y Fe(Z)c(Z)913 455 y Fh(Q)941 445 y Fc(?)959 455 y Fu(\002)p Fh(Q)1014 445 y Fc(?)1040 386 y Fi(\032)p Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))p Fi(\032)1236 367 y Fu(\003)1254 386 y Fn(\()p Fi(\030)1294 367 y Fu(0)1305 386 y Fi(;)g(x;)g(y)r Fn(\))1427 355 y Fi(d\030)r(d\030)1519 338 y Fu(0)p 1426 375 106 2 v 1426 417 a Fn(\(2)p Fi(\031)r Fn(\))1513 404 y Fv(6)518 551 y Fn(=)597 489 y Fe(Z)e(Z)f(Z)g(Z)573 623 y Fh(Q)601 614 y Ft(2)618 623 y Fu(\002)p Fv(\()p Fh(Q)687 614 y Fc(?)705 623 y Fv(\))719 614 y Ft(2)753 508 y Fe(X)743 609 y Fh(k)q Fu(2)p Fd(Z)811 600 y Ft(3)935 520 y Fn(1)p 841 541 212 2 v 841 582 a Fg(j)p Fi(x)k Fg(\000)i Fi(y)h Fg(\000)e Fi(k)q Fg(j)1057 551 y Fi(\032)p Fn(\()p Fi(\030)r(;)e(x;)g (y)j Fn(+)f Fi(k)q Fn(\))p Fi(\032)1332 532 y Fu(\003)1352 551 y Fn(\()p Fi(\030)1392 532 y Fu(0)1403 551 y Fi(;)e(x;)g(y)j Fn(+)f Fi(k)q Fn(\))e Fi(dxdy)1703 520 y(d\030)r(d\030)1795 504 y Fu(0)p 1702 541 106 2 v 1702 582 a Fn(\(2)p Fi(\031)r Fn(\))1789 569 y Fv(6)518 726 y Fn(=)597 664 y Fe(Z)e(Z)f(Z)g(Z)573 798 y Fh(Q)601 788 y Ft(2)618 798 y Fu(\002)p Fv(\()p Fh(Q)687 788 y Fc(?)705 798 y Fv(\))719 788 y Ft(2)743 726 y Fi(\032)p Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fi(\032)939 707 y Fu(\003)957 726 y Fn(\()p Fi(\030)997 707 y Fu(0)1009 726 y Fi(;)g(x;)g(y)r Fn(\))1134 682 y Fe(X)1127 784 y Fh(k)q Fu(2)p Fd(Z)1195 774 y Ft(3)1232 695 y Fi(e)1253 678 y Fu(\000)p Fh(i)p Fv(\()p Fh(\030)q Fu(\000)p Fh(\030)1367 667 y Fp(0)1378 678 y Fv(\))p Fu(\001)p Fh(k)p 1222 715 212 2 v 1222 757 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)i Fg(\000)e Fi(k)q Fg(j)1446 726 y Fi(dxdy)1549 695 y(d\030)r(d\030)1641 678 y Fu(0)p 1548 715 106 2 v 1548 757 a Fn(\(2)p Fi(\031)r Fn(\))1635 744 y Fv(6)518 893 y Fn(=)597 831 y Fe(Z)c(Z)f(Z)g(Z)573 965 y Fh(Q)601 956 y Ft(2)618 965 y Fu(\002)p Fv(\()p Fh(Q)687 956 y Fc(?)705 965 y Fv(\))719 956 y Ft(2)743 893 y Fi(\032)p Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fi(W)965 874 y Fu(1)1001 893 y Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)1118 874 y Fu(0)1130 893 y Fi(;)e(x)h Fg(\000)h Fi(y)r Fn(\))p Fi(\032)1296 874 y Fu(\003)1316 893 y Fn(\()p Fi(\030)1356 874 y Fu(0)1368 893 y Fi(;)e(x;)g(y)r Fn(\))g Fi(dxd)o(y)1587 862 y(d\030)r(d\030)1679 846 y Fu(0)p 1585 882 V 1585 924 a Fn(\(2)p Fi(\031)r Fn(\))1672 911 y Fv(6)1696 893 y Fi(;)37 1062 y Fn(where)16 b(w)o(e)e(ha)o(v)o(e)h(used)g(the)g(prop)q (erties)h(of)e(p)q(erio)q(dicit)o(y)j(of)e(the)g(functions)h Fi(\032)p Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))13 b(with)i(resp)q (ect)g(to)f Fi(y)r Fn(,)37 1118 y(and)i(the)f(de\014nition)i(\(2.25\))d (of)g Fi(W)638 1102 y Fu(1)676 1118 y Fn(.)108 1200 y(And)i(the)f(pro)q (of)g(of)g(Lemma)g(5.1)f(is)i(complete.)913 b Fg(})108 1298 y Fn(This)14 b(section)g(is)g(organized)g(as)f(follo)o(ws.)19 b(W)l(e)14 b(\014rst)f(pro)o(v)o(e)g(in)h(Subsection)h(5.1)d(that)h (the)h(p)q(erio)q(dic)h(HF)37 1354 y(problem)j(is)g(w)o(ell-p)q(osed.) 27 b(Next,)17 b(in)h(Subsection)g(5.2,)f(w)o(e)f(c)o(hec)o(k)i(that)e (w)o(e)h(ma)o(y)f(compare)h(from)f(ab)q(o)o(v)o(e)37 1411 y(the)f(Hartree-F)l(o)q(c)o(k)e(energy)h(p)q(er)h(unit)f(v)o (olume)h(with)f(this)g(p)q(erio)q(dic)j(problem,)d(b)o(y)g(using)h(a)e (minimizer)j(of)37 1467 y Fi(I)61 1451 y Fh(H)s(F)57 1478 y(per)122 1467 y Fn(.)37 1589 y Fb(5.1)57 b(The)20 b(p)r(erio)r(dic)d(HF)i(problem)37 1675 y Fn(This)e(section)g(is)g(dev) o(oted)f(to)g(the)g(pro)q(of)g(of)g(Theorem)g(2.3,)g(that)f(w)o(e)h (recall)i(here)e(for)g(the)g(con)o(v)o(enience)37 1731 y(of)f(the)h(reader.)37 1812 y Fj(Theorem)e(2.3)g(\(W)l(ell-p)q (osedness)g(of)f(the)g(HF)g(p)q(erio)q(dic)i(problem\))e Ff(The)g(minimization)f(pr)n(oblem)37 1869 y(de\014ne)n(d)k(by)g (\(2.23\))g(and)g(\(2.24\))g(\(r)n(esp)n(e)n(ctively)f(by)h(\(2.26\))g (and)g(\(2.27\)\))g(admits)g(a)h(minimum.)108 1950 y Fn(W)l(e)f(shall)h(pro)o(vide)f(t)o(w)o(o)f(di\013eren)o(t)i(pro)q(ofs) e(of)g(this)h(claim.)23 b(The)16 b(\014rst)f(one)h(mak)o(es)f(use)h(of) g(regularit)o(y)37 2007 y(prop)q(erties)i(of)e(the)h(p)q(oten)o(tial)h Fi(W)6 b Fn(\()p Fi(\021)r(;)i(z)r Fn(\))15 b(giv)o(en)i(b)o(y)g (\(2.25\),)d(and)j(whic)o(h)h(app)q(ears)f(when)g(one)g(writes)g(the)37 2063 y(exc)o(hange)g(terms)f(according)g(to)g(the)g(Blo)q(c)o(h)h(w)o (a)o(v)o(es)e(decomp)q(osition.)25 b(The)16 b(second)h(one,)f(whic)o(h) h(is)g(also)37 2120 y(the)f(shortest)e(one,)h(is)h(based)f(up)q(on)h (the)f(form)o(ulation)h(\(5.1a\))d(of)i(the)g(exc)o(hange)g(term.)37 2201 y Fj(First)22 b(pro)q(of)h(of)e(Theorem)h(2.3)g(:)27 b Fn(In)19 b(order)f(to)g(c)o(hec)o(k)g(that)g(the)h(minimization)h (problem)f(\(2.23\){)37 2257 y(\(2.24\){\(2.25\))13 b(is)k(w)o(ell-p)q (osed,)h(w)o(e)e(no)o(w)f(pro)o(v)o(e)h(that,)g(giv)o(en)g(an)h (arbitrary)e(sequence)j(of)e(op)q(erators)f Fi(K)1852 2264 y Fh(n)37 2314 y Fn(\()p Fg(2)f(K)q Fn(\))h(suc)o(h)h(that)f Fi(E)407 2297 y Fh(H)s(F)404 2325 y(per)467 2314 y Fn(\()p Fi(K)524 2321 y Fh(n)547 2314 y Fn(\))g(go)q(es)h(to)f Fi(I)761 2297 y Fh(H)s(F)757 2325 y(per)822 2314 y Fn(,)g(as)h Fi(n)g Fn(go)q(es)f(to)h(in\014nit)o(y)l(,)h(this)f(sequence)h(con)o(v) o(erges,)e(up)i(to)37 2375 y(an)d(extraction)f(of)g(a)g(subsequence,)h (to)f(some)g(op)q(erator)f Fi(K)k Fn(\()p Fg(2)d(K)q Fn(\))f(that)h(satis\014es)g Fi(E)1494 2358 y Fh(H)s(F)1491 2386 y(per)1554 2375 y Fn(\()p Fi(K)s Fn(\))f(=)h Fi(I)1716 2358 y Fh(H)s(F)1712 2386 y(per)1776 2375 y Fn(.)20 b(F)l(or)37 2431 y(this)c(purp)q(ose,)g(w)o(e)f(shall)h(hea)o(vily)h(rely)f(up)q (on)f(the)h(pro)q(of)f(of)g(Theorem)g(2.1.)k(But,)c(\014rst)g(of)g (all,)h(w)o(e)f(b)q(egin)37 2488 y(our)g(pro)q(of)f(with)h(a)f(careful) h(study)f(of)h(the)f(prop)q(erties)h(of)f(regularit)o(y)h(of)f(the)h (in)o(teraction)g(p)q(oten)o(tial)g Fi(W)1838 2495 y Fu(1)37 2544 y Fn(and)h(consequen)o(tly)g(of)f(the)g(prop)q(erties)h (of)f(the)g(exc)o(hange)h(term.)108 2642 y(Step)g(1:)j(Decomp)q (osition)d(of)f(the)g(exc)o(hange)h(p)q(oten)o(tial)p 108 2649 970 2 v 108 2740 a(W)l(e)e(\014rst)g(remark)f(that)h(the)g (function)h Fi(e)805 2724 y Fh(i\021)q Fu(\001)p Fh(x)868 2740 y Fi(W)911 2747 y Fu(1)948 2740 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))k(is)j Fi(Q)p Fn(-p)q(erio)q(dic)h(with)e(resp)q(ect)h(to)f Fi(x)g Fn(when)g Fi(\021)h Fn(is)p eop %%Page: 54 54 54 53 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(54)37 199 y(\014xed.)21 b(Indeed,)16 b Fi(j)f Fg(2)e Fj(Z)434 183 y Fv(3)469 199 y Fn(b)q(eing)k(\014xed,)e(w)o(e)g (ha)o(v)o(e)295 321 y Fi(e)316 302 y Fh(i\021)q Fu(\001)p Fv(\()p Fh(x)p Fv(+)p Fh(j)r Fv(\))450 321 y Fi(W)493 328 y Fu(1)530 321 y Fn(\()p Fi(\021)r(;)8 b(x)g Fn(+)j Fi(j)s Fn(\))40 b(=)i Fi(e)852 302 y Fh(i\021)q Fu(\001)p Fh(x)915 321 y Fi(e)936 302 y Fh(i\021)q Fu(\001)p Fh(j)1012 278 y Fe(X)1003 379 y Fh(k)q Fu(2)p Fd(Z)1071 370 y Ft(3)1163 290 y Fi(e)1184 274 y Fh(ik)q Fu(\001)p Fh(\021)p 1100 311 209 2 v 1100 352 a Fg(j)p Fi(x)10 b Fn(+)g Fi(j)j Fn(+)d Fi(k)q Fg(j)754 471 y Fn(=)42 b Fi(e)852 453 y Fh(i\021)q Fu(\001)p Fh(x)938 428 y Fe(X)922 530 y Fh(m)p Fu(2)p Fd(Z)1002 520 y Ft(3)1058 441 y Fi(e)1079 424 y Fh(im)p Fu(\001)p Fh(\021)p 1032 461 147 2 v 1032 503 a Fg(j)p Fi(x)10 b Fn(+)g Fi(m)p Fg(j)1242 471 y Fn(=)58 b Fi(e)1356 453 y Fh(i\021)q Fu(\001)p Fh(x)1419 471 y Fi(W)1462 478 y Fu(1)1499 471 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))p Fi(;)37 617 y Fn(for)17 b(almost)g(all)h(\()p Fi(\021)r(;)8 b(x)p Fn(\))14 b Fg(2)j Fi(Q)528 601 y Fh(?)559 617 y Fg(\002)12 b Fi(Q)p Fn(.)26 b(W)l(e)17 b(ma)o(y)g(therefore)g(decomp)q(ose)h Fi(e)1307 601 y Fh(i\021)q Fu(\001)p Fh(x)1370 617 y Fi(W)1413 624 y Fu(1)1450 617 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))15 b(in)o(to)i(its)h(F)l(ourier)37 674 y(series:)606 769 y Fi(e)627 750 y Fh(i\021)q Fu(\001)p Fh(x)690 769 y Fi(W)733 776 y Fu(1)771 769 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))i(=)952 725 y Fe(X)937 827 y Fh(m)p Fu(2)p Fd(Z)1017 817 y Ft(3)1041 769 y Fi(a)1065 776 y Fh(m)1099 769 y Fn(\()p Fi(\021)r Fn(\))p Fi(e)1181 750 y Fv(2)p Fh(i\031)q(m)p Fu(\001)p Fh(x)1293 769 y Fi(;)37 910 y Fn(with)16 b(co)q(e\016cien)o (ts)g Fi(a)395 917 y Fh(m)428 910 y Fn(\()p Fi(\021)r Fn(\))e(giv)o(en)i(b)o(y)234 1034 y Fi(a)258 1041 y Fh(m)291 1034 y Fn(\()p Fi(\021)r Fn(\))40 b(=)469 972 y Fe(Z)495 1075 y Fh(Q)532 1034 y Fi(e)553 1015 y Fh(i\021)q Fu(\001)p Fh(x)616 1034 y Fi(W)659 1041 y Fu(1)696 1034 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))p Fi(e)825 1015 y Fu(\000)p Fv(2)p Fh(i\031)q(m)p Fu(\001)p Fh(x)972 1034 y Fi(dx)25 b Fn(=)1108 972 y Fe(Z)1133 1075 y Fh(Q)1170 1034 y Fi(e)1191 1015 y Fh(i)p Fv(\()p Fh(\021)q Fu(\000)p Fv(2)p Fh(\031)q(m)p Fv(\))p Fu(\001)p Fh(x)1396 991 y Fe(X)1387 1092 y Fh(k)q Fu(2)p Fd(Z)1455 1083 y Ft(3)1509 1003 y Fi(e)1530 987 y Fh(ik)q Fu(\001)p Fh(\021)p 1484 1024 132 2 v 1484 1065 a Fg(j)p Fi(x)10 b Fn(+)g Fi(k)q Fg(j)1629 1034 y Fi(dx)392 1177 y Fn(=)479 1134 y Fe(X)469 1235 y Fh(k)q Fu(2)p Fd(Z)537 1226 y Ft(3)562 1115 y Fe(Z)587 1219 y Fh(Q)625 1177 y Fi(e)646 1159 y Fh(i)p Fv(\()p Fh(\021)q Fu(\000)p Fv(2)p Fh(\031)q(m)p Fv(\))p Fu(\001)p Fv(\()p Fh(x)p Fv(+)p Fh(k)q Fv(\))967 1147 y Fn(1)p 913 1167 V 913 1209 a Fg(j)p Fi(x)f Fn(+)i Fi(k)q Fg(j)1057 1177 y Fi(dx)392 1321 y Fn(=)469 1259 y Fe(Z)495 1362 y Fd(R)525 1352 y Ft(3)552 1321 y Fi(e)573 1302 y Fh(i)p Fv(\()p Fh(\021)q Fu(\000)p Fv(2)p Fh(\031)q(m)p Fv(\))p Fu(\001)p Fh(y)778 1290 y Fn(1)p 765 1310 50 2 v 765 1352 a Fg(j)p Fi(y)r Fg(j)826 1321 y Fi(dy)27 b Fn(=)1047 1290 y(4)p Fi(\031)p 965 1310 215 2 v 965 1352 a Fg(j)p Fi(\021)11 b Fg(\000)f Fn(2)p Fi(\031)r(m)p Fg(j)1161 1339 y Fv(2)1185 1321 y Fi(;)37 1438 y Fn(for)15 b(almost)g(all)h Fi(\021)e Fg(2)f Fi(Q)433 1421 y Fh(?)452 1438 y Fn(.)20 b(Hence,)c(w)o(e)f(ha)o (v)o(e)574 1558 y Fi(W)617 1565 y Fu(1)654 1558 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))i(=)j(4)p Fi(\031)c(e)899 1539 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)1013 1515 y Fe(X)997 1616 y Fh(m)p Fu(2)p Fd(Z)1077 1607 y Ft(3)1146 1527 y Fi(e)1167 1511 y Fv(2)p Fh(i\031)q(m)p Fu(\001)p Fh(x)p 1106 1548 V 1106 1589 a Fg(j)p Fi(\021)i Fg(\000)g Fn(2)p Fi(\031)r(m)p Fg(j)1303 1576 y Fv(2)1326 1558 y Fi(:)443 b Fn(\(5.2\))37 1697 y(It)13 b(is)h(easily)f(deduced)i(from)d(this)h (expression)h(that,)e(with)h Fi(x)g Fn(b)q(eing)h(\014xed)f(in)h Fi(Q)p Fn(,)e(the)h(function)h(de\014ned)g(b)o(y)37 1779 y Fi(W)80 1786 y Fu(1)118 1779 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))g Fg(\000)i Fn(4)p Fi(\031)342 1748 y(e)363 1732 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)p 342 1769 112 2 v 363 1810 a Fg(j)p Fi(\021)r Fg(j)414 1797 y Fv(2)473 1779 y Fn(is)16 b(con)o(tin)o(uous)g(with)g(resp)q(ect)f(to)g Fi(\021)f Fg(2)f Fi(Q)1178 1763 y Fh(?)1198 1779 y Fn(,)i(and)g(ev)o (en)h(to)e Fi(\021)g Fg(2)g Fn(\(1)9 b(+)i Fi(")p Fn(\))p Fi(Q)k Fn(\()p Fi(")d(>)i Fn(0,)37 1860 y(small)i(enough\),)f(and)h (satis\014es)405 1983 y(lim)390 2012 y Fh(\021)q Fu(\000)-5 b(!)p Fv(0)484 1919 y Fe(\022)517 1983 y Fi(W)560 1990 y Fu(1)597 1983 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))g Fg(\000)j Fn(4)p Fi(\031)821 1952 y(e)842 1936 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)p 821 1973 V 843 2014 a Fg(j)p Fi(\021)r Fg(j)894 2001 y Fv(2)938 1919 y Fe(\023)984 1983 y Fn(=)1083 1940 y Fe(X)1032 2041 y Fh(m)p Fu(2)p Fd(Z)1112 2032 y Ft(3)1129 2041 y Fu(nf)p Fv(0)p Fu(g)1212 1952 y Fi(e)1233 1936 y Fv(2)p Fh(i\031)q(m)p Fu(\001)p Fh(x)p 1212 1973 136 2 v 1223 2014 a Fi(\031)r Fg(j)p Fi(m)p Fg(j)1317 2001 y Fv(2)1365 1983 y Fn(=)i Fi(G)p Fn(\()p Fi(x)p Fn(\))p Fi(:)37 2128 y Fn(Since)19 b(w)o(e)e(ha)o(v)o(e) f(isolated)i(the)f(singularit)o(y)h(in)g Fi(\021)r Fn(,)e(let)i(us)f (no)o(w)f(examine)i(the)f(singularit)o(y)h(in)g Fi(x)p Fn(.)26 b(Let)17 b(us)37 2185 y(then)f(consider)511 2296 y Fi(f)5 b Fn(\()p Fi(\021)r(;)j(x)p Fn(\))i(=)j Fi(W)747 2303 y Fu(1)785 2296 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))g Fg(\000)i Fi(e)967 2278 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)1057 2296 y Fi(G)p Fn(\()p Fi(x)p Fn(\))f Fg(\000)i Fn(4)p Fi(\031)1273 2266 y(e)1294 2249 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)p 1273 2286 112 2 v 1294 2328 a Fg(j)p Fi(\021)r Fg(j)1345 2314 y Fv(2)1389 2296 y Fi(:)380 b Fn(\(5.3\))37 2416 y(W)l(e)17 b(no)o(w)e(c)o(hec)o(k)h(that)g Fi(f)5 b Fn(\()p Fi(\021)r(;)j(x)p Fn(\))13 b(is)k(in)g Fi(L)715 2399 y Fu(1)752 2416 y Fn(\()p Fi(Q)806 2399 y Fh(?)836 2416 y Fg(\002)11 b Fi(Q)p Fn(\),)k(and)i(ev)o(en)f(in)h Fi(L)1243 2399 y Fu(1)1280 2416 y Fn(\(\(1)10 b(+)h Fi(")p Fn(\))p Fi(Q)1470 2399 y Fh(?)1500 2416 y Fg(\002)g Fn(\(1)f(+)h Fi(")p Fn(\))p Fi(Q)p Fn(\).)22 b(F)l(rom)37 2472 y(\(5.2\))14 b(and)h(the)h(F)l(ourier)f(series)h(expansion)g(of)f Fi(G)p Fn(,)f(w)o(e)h(obtain)331 2592 y Fi(f)5 b Fn(\()p Fi(\021)r(;)j(x)p Fn(\))39 b(=)j(4)p Fi(\031)14 b(e)666 2573 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)815 2549 y Fe(X)764 2650 y Fh(m)p Fu(2)p Fd(Z)844 2641 y Ft(3)861 2650 y Fu(nf)p Fv(0)p Fu(g)939 2592 y Fi(e)960 2573 y Fv(2)p Fh(i\031)q(m)p Fu(\001)p Fh(x)1082 2528 y Fe(\022)1216 2561 y Fn(1)p 1120 2581 215 2 v 1120 2623 a Fg(j)p Fi(\021)d Fg(\000)f Fn(2)p Fi(\031)r(m)p Fg(j)1316 2610 y Fv(2)1350 2592 y Fg(\000)1457 2561 y Fn(1)p 1401 2581 136 2 v 1401 2623 a Fg(j)p Fn(2)p Fi(\031)r(m)p Fg(j)1518 2610 y Fv(2)1541 2528 y Fe(\023)505 2748 y Fn(=)42 b(4)p Fi(\031)14 b(e)666 2729 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)815 2705 y Fe(X)764 2806 y Fh(m)p Fu(2)p Fd(Z)844 2797 y Ft(3)861 2806 y Fu(nf)p Fv(0)p Fu(g)939 2748 y Fi(e)960 2729 y Fv(2)p Fh(i\031)q(m)p Fu(\001)p Fh(x)1118 2717 y Fn(4)p Fi(\031)g(\021)d Fg(\001)f Fi(m)g Fg(\000)g(j)p Fi(\021)r Fg(j)1384 2701 y Fv(2)p 1079 2738 363 2 v 1079 2779 a Fg(j)p Fi(\021)h Fg(\000)f Fn(2)p Fi(\031)r(m)p Fg(j)1275 2766 y Fv(2)1307 2779 y Fg(j)p Fn(2)p Fi(\031)r(m)p Fg(j)1424 2766 y Fv(2)1447 2748 y Fi(:)322 b Fn(\(5.4\))p eop %%Page: 55 55 55 54 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(55)37 199 y(It)16 b(is)g(ob)o(vious)f(from)g(the)g(last)g (equalit)o(y)i(that,)d(at)h(\014xed)h Fi(\021)r Fn(,)e Fi(e)1093 183 y Fh(i\021)q Fu(\001)p Fh(x)1156 199 y Fi(f)5 b Fn(\()p Fi(\021)r(;)j(x)p Fn(\))13 b(is)j Fi(Q)p Fn(-p)q(erio)q(dic)h(with)f(resp)q(ect)g(to)37 256 y Fi(x)p Fn(,)f(and)h(that)486 383 y Fg(k)p Fi(f)5 b Fn(\()p Fi(\021)r(;)j Fg(\001)p Fn(\))p Fg(k)654 364 y Fv(2)654 397 y Fh(L)678 388 y Ft(2)692 397 y Fv(\()p Fh(Q)p Fv(\))762 383 y Fg(\024)13 b Fi(C)917 340 y Fe(X)866 441 y Fh(m)p Fu(2)p Fd(Z)946 432 y Ft(3)963 441 y Fu(nf)p Fv(0)p Fu(g)1146 352 y Fn(1)d(+)g Fg(j)p Fi(m)p Fg(j)1290 336 y Fv(2)p 1046 373 363 2 v 1046 414 a Fg(j)p Fi(\021)h Fg(\000)g Fn(2)p Fi(\031)r(m)p Fg(j)1243 401 y Fv(4)1274 414 y Fg(j)p Fn(2)p Fi(\031)r(m)p Fg(j)1391 401 y Fv(4)1414 383 y Fi(;)37 538 y Fn(for)19 b(some)f(p)q(ositiv)o(e)i(constan)o(t)d (whic)o(h)j(is)f(indep)q(enden)o(t)i(of)d Fi(\021)i Fg(2)f Fn(\(1)12 b(+)g Fi(")p Fn(\))p Fi(Q)1352 521 y Fh(?)1372 538 y Fn(;)20 b(in)g(other)e(w)o(ords,)g(for)g(an)o(y)37 594 y Fi(")13 b(>)g Fn(0)i(small)h(enough,)643 696 y Fg(k)p Fi(f)5 b Fn(\()p Fi(\021)r(;)j Fg(\001)p Fn(\))p Fg(k)811 678 y Fv(2)811 710 y Fh(L)835 701 y Ft(2)850 710 y Fv(\()p Fh(Q)p Fv(\))919 696 y Fg(2)13 b Fi(L)993 678 y Fu(1)1031 696 y Fn(\(\(1)c(+)h Fi(")p Fn(\))p Fi(Q)1219 678 y Fh(?)1239 696 y Fn(\))p Fi(:)512 b Fn(\(5.5\))37 799 y(With)16 b Fi(\021)g Fn(b)q(eing)h(still)f(\014xed)g(in)g(\(1)10 b(+)g Fi(")p Fn(\))p Fi(Q)739 782 y Fh(?)759 799 y Fn(,)15 b(it)g(is)h(clear)f(from)g(\(5.3\))e(that)532 907 y Fg(\000)p Fn(\001)605 914 y Fh(x)627 907 y Fi(f)5 b Fn(\()p Fi(\021)r(;)j(x)p Fn(\))j(=)i(4)p Fi(\031)924 864 y Fe(X)879 966 y Fh(k)q Fu(2)p Fd(Z)947 956 y Ft(3)964 966 y Fu(nf)p Fv(0)p Fu(g)1034 871 y Fe(\000)1055 907 y Fi(e)1076 889 y Fu(\000)p Fh(ik)q Fu(\001)p Fh(\021)1176 907 y Fg(\000)d Fn(1)1244 871 y Fe(\001)1265 907 y Fi(\016)1285 914 y Fh(k)1306 907 y Fn(\()p Fi(x)p Fn(\))p Fi(;)37 1062 y Fn(and)j(th)o(us)g Fi(f)5 b Fn(\()p Fi(\021)r(;)j(x)p Fn(\))i(is)j(harmonic)g(in)g(\(1)5 b(+)g Fi(")p Fn(\))p Fi(Q)p Fn(.)19 b(With)12 b(the)h(help)h(of)e(the)h (mean-v)m(alue)h(prop)q(ert)o(y)l(,)e(w)o(e)h(\014nally)37 1119 y(obtain)j(for)e(ev)o(ery)i Fi(x)f Fn(in)h Fi(Q)p Fn(,)488 1239 y Fg(j)p Fi(f)5 b Fn(\()p Fi(\021)r(;)j(x)p Fn(\))p Fg(j)i(\024)707 1177 y Fe(Z)732 1280 y Fh(x)p Fv(+)p Fh("Q)833 1239 y Fg(j)p Fi(f)5 b Fn(\()p Fi(\021)r(;)j(y)r Fn(\))p Fg(j)g Fi(d)o(y)k Fg(\024)h Fi(C)e Fg(k)p Fi(f)5 b Fn(\()p Fi(\021)r(;)j Fg(\001)p Fn(\))p Fg(k)1316 1215 y Fv(1)p Fh(=)o Fv(2)1316 1256 y Fh(L)1340 1247 y Ft(2)1355 1256 y Fv(\()p Fh(Q)p Fv(\))1412 1239 y Fi(;)37 1371 y Fn(and)18 b(w)o(e)e(conclude)j(since)f(the)f(righ)o(t-hand)g(side)h (in)g(the)f(ab)q(o)o(v)o(e)f(inequalit)o(y)j(lies)f(in)g Fi(L)1540 1354 y Fu(1)1577 1371 y Fn(\()p Fi(Q)1631 1354 y Fh(?)1650 1371 y Fn(\))f(thanks)g(to)37 1427 y(\(5.5\).)108 1484 y(Next,)g(w)o(e)f(remark)g(that)g(the)g(exc)o(hange)h(term)f(in)o (v)o(olv)o(es)h(the)g(function)g Fi(W)1418 1491 y Fu(1)1456 1484 y Fn(\()p Fi(\030)12 b Fg(\000)g Fi(\030)1575 1467 y Fu(0)1586 1484 y Fi(;)c(x)j Fg(\000)g Fi(y)r Fn(\))16 b(with)h Fi(\030)37 1540 y Fn(and)i Fi(\030)151 1524 y Fu(0)181 1540 y Fn(v)m(arying)g(in)g Fi(Q)438 1524 y Fh(?)458 1540 y Fn(,)f(and)h Fi(x)f Fn(and)h Fi(y)h Fn(v)m(arying)f(in)g Fi(Q)p Fn(.)29 b(Therefore,)19 b(w)o(e)f(need)h (some)g(information)f(on)37 1597 y Fi(W)80 1604 y Fu(1)118 1597 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))16 b(on)j(2)p Fi(Q)368 1580 y Fh(?)399 1597 y Fg(\002)13 b Fn(2)p Fi(Q)18 b Fn(=)h([)p Fg(\000)p Fn(2)p Fi(\031)r(;)8 b Fn(+2)p Fi(\031)r Fn([)797 1580 y Fv(3)814 1597 y Fg(\002)p Fn([)p Fg(\000)p Fn(1)p Fi(;)g Fn(+1[)1012 1580 y Fv(3)1031 1597 y Fn(.)30 b(In)19 b(a)f(straigh)o(tforw)o(ard)f(w)o(a)o(y)l(,)h (for)g(almost)g(all)37 1653 y Fi(\021)c Fg(2)f Fn([)p Fg(\000)p Fn(2)p Fi(\031)r(;)8 b Fn(+2)p Fi(\031)r Fn([)336 1637 y Fv(3)354 1653 y Fn(,)14 b Fi(x)f Fg(2)g Fn([)p Fg(\000)p Fn(1)p Fi(;)8 b Fn(+1[)626 1637 y Fv(3)644 1653 y Fn(,)15 b(w)o(e)g(obtain)g(from)g(\(5.3\))e(the)j(decomp)q (osition)118 1783 y Fi(W)161 1790 y Fu(1)198 1783 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))j(=)p 365 1747 50 2 v 13 w Fi(W)414 1790 y Fu(1)451 1783 y Fn(\()p Fi(\021)r(;)d(x)p Fn(\))g Fg(\000)685 1740 y Fe(X)613 1842 y Fh(k)q Fu(2)p Fd(Z)681 1832 y Ft(3)698 1842 y Fh(;)p Fu(j)p Fh(k)q Fu(j)747 1846 y Fp(1)779 1842 y Fu(\024)p Fv(1)891 1753 y Fn(1)p 836 1773 132 2 v 836 1815 a Fg(j)p Fi(x)i Fn(+)h Fi(k)q Fg(j)983 1783 y(\000)g Fn(4)p Fi(\031)e(e)1108 1765 y Fu(\000)p Fh(i\021)q Fu(\001)p Fh(x)1290 1740 y Fe(X)1206 1842 y Fh(m)p Fu(2)p Fd(Z)1286 1832 y Ft(3)1302 1842 y Fh(;)p Fu(j)p Fh(m)p Fu(j)1363 1846 y Fp(1)1396 1842 y Fu(\024)p Fv(1)1493 1753 y Fi(e)1514 1736 y Fv(2)p Fh(i\031)q(m)p Fu(\001)p Fh(x)p 1453 1773 215 2 v 1453 1815 a Fg(j)p Fi(\021)i Fg(\000)g Fn(2)p Fi(\031)r(m)p Fg(j)1650 1802 y Fv(2)1681 1783 y Fi(;)88 b Fn(\(5.6\))37 1944 y(with)19 b Fg(j)p Fi(x)p Fg(j)196 1951 y Fu(1)251 1944 y Fn(=)f(max\()p Fg(j)p Fi(x)446 1951 y Fv(1)465 1944 y Fg(j)p Fi(;)8 b Fg(j)p Fi(x)538 1951 y Fv(2)556 1944 y Fg(j)p Fi(;)g Fg(j)p Fi(x)629 1951 y Fv(3)647 1944 y Fg(j)p Fn(\),)18 b(and)h(where)p 936 1908 50 2 v 19 w Fi(W)985 1951 y Fu(1)1022 1944 y Fn(\()p Fi(\021)r(;)8 b(x)p Fn(\))16 b(b)q(elongs)k(to)d Fi(L)1402 1928 y Fu(1)1440 1944 y Fn(\(2)p Fi(Q)1517 1928 y Fh(?)1548 1944 y Fg(\002)c Fn(2)p Fi(Q)p Fn(\),)18 b(o)o(wing)g(in)37 2001 y(particular)e(to)f (the)g(fact)g(that)f Fi(G)p Fn(\()p Fi(x)p Fn(\))9 b Fg(\000)722 1967 y Fe(P)770 2014 y Fh(k)q Fu(2)p Fd(Z)838 2005 y Ft(3)855 2014 y Fh(;)p Fu(j)p Fh(k)q Fu(j)904 2018 y Fp(1)936 2014 y Fu(\024)p Fv(1)1030 1983 y(1)p 996 1990 87 2 v 996 2017 a Fu(j)p Fh(x)p Fv(+)p Fh(k)q Fu(j)1102 2001 y Fn(is)16 b(b)q(ounded)g(in)g(2)p Fi(Q)1445 1984 y Fh(?)1475 2001 y Fg(\002)10 b Fn(2)p Fi(Q)p Fn(.)108 2099 y(Step)16 b(2:)j(A)c(priori)h(estimates)p 108 2106 518 2 v 108 2197 a(Using)e(the)f(decomp)q(osition)h(\(5.6\),)e(let)h (us)h(no)o(w)e(sho)o(w)h(that)f(the)h Ff(a)i(priori)f Fn(estimates)f(that)f(ha)o(v)o(e)h(b)q(een)37 2253 y(established)k(in)f (Step)g(1)f(of)g(the)g(pro)q(of)g(of)f(Theorem)h(2.1)g(also)g(hold)h (true)f(here.)108 2310 y(Since)h(some)e(estimates)h(only)g(dep)q(end)h (on)f(the)f(constrain)o(ts)g(and)h(not)f(on)h(the)f(energy)h (functional,)h(it)37 2366 y(is)g(easy)f(to)g(see)g(that)g(the)g(follo)o (wing)h(b)q(ounds)g(of)f(the)g(RHF)g(setting)h(are)f(also)g(v)m(alid)i (here:)493 2428 y Fe(8)493 2469 y(<)493 2550 y(:)554 2464 y Fi(\032)577 2471 y Fh(n)601 2464 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))13 b(is)j(b)q(ounded)h(in)f Fi(L)1080 2448 y Fv(2)1099 2464 y Fn(\()p Fi(Q)1153 2448 y Fh(?)1183 2464 y Fg(\002)10 b Fi(Q)g Fg(\002)h Fi(Q)p Fn(\))p Fi(;)554 2521 y(\032)577 2528 y Fh(n)601 2521 y Fn(\()p Fi(x;)d(y)r Fn(\))14 b(is)i(b)q(ounded)g(in)g Fi(L)1037 2504 y Fv(2)1057 2521 y Fn(\()p Fi(Q)10 b Fg(\002)g Fj(R)1205 2504 y Fv(3)1225 2521 y Fn(\))p Fi(;)554 2577 y(\032)577 2584 y Fh(n)601 2577 y Fn(\()p Fi(x;)e(x)p Fn(\))14 b(is)i(b)q(ounded)g(in)g Fi(L)1039 2561 y Fv(1)1059 2577 y Fn(\()p Fi(Q)p Fn(\))p Fi(:)1782 2521 y Fn(\(5.7\))p eop %%Page: 56 56 56 55 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(56)37 199 y(Next,)17 b(w)o(e)f(are)g(going)g(to)g(estimate)g (the)h(exc)o(hange)f(term)g(b)o(y)g(splitting)i(it)f(in)o(to)f(three)h (terms)e(according)37 256 y(to)g(the)g(decomp)q(osition)i(\(5.6\).)h (First,)d(w)o(e)g(ha)o(v)o(e)388 317 y Fe(\014)388 344 y(\014)388 371 y(\014)388 399 y(\014)405 321 y(Z)5 b(Z)g(Z)h(Z)380 455 y Fh(Q)408 446 y Ft(2)425 455 y Fu(\002)p Fv(\()p Fh(Q)494 446 y Fc(?)512 455 y Fv(\))526 446 y Ft(2)551 383 y Fi(\032)574 390 y Fh(n)598 383 y Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))p 748 346 81 2 v Fi(W)791 390 y Fu(1)825 383 y Fn(\()p Fi(\030)k Fg(\000)f Fi(\030)943 364 y Fu(0)954 383 y Fi(;)d(x)i Fg(\000)g Fi(y)r Fn(\))p Fi(\032)1121 364 y Fu(\003)1121 394 y Fh(n)1145 383 y Fn(\()p Fi(\030)1185 364 y Fu(0)1196 383 y Fi(;)e(x;)g(y)r Fn(\))g Fi(dxdy)e(d\030)r(d\030) 1508 364 y Fu(0)1519 317 y Fe(\014)1519 344 y(\014)1519 371 y(\014)1519 399 y(\014)429 546 y Fg(\024)42 b(k)p 529 510 V Fi(W)572 553 y Fu(1)609 546 y Fg(k)632 553 y Fh(L)656 544 y Fp(1)700 484 y Fe(Z)5 b(Z)g(Z)h(Z)675 619 y Fh(Q)703 609 y Ft(2)720 619 y Fu(\002)p Fv(\()p Fh(Q)789 609 y Fc(?)807 619 y Fv(\))821 609 y Ft(2)846 546 y Fg(j)p Fi(\032)882 553 y Fh(n)905 546 y Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))p Fg(j)i(j)p Fi(\032)1114 528 y Fu(\003)1114 558 y Fh(n)1138 546 y Fn(\()p Fi(\030)1178 528 y Fu(0)1189 546 y Fi(;)e(x;)g(y)r Fn(\))p Fg(j)g Fi(dxdy)e(d\030)r(d\030)1514 528 y Fu(0)429 710 y Fg(\024)42 b(k)p 529 673 V Fi(W)572 717 y Fu(1)609 710 y Fg(k)632 717 y Fh(L)656 707 y Fp(1)690 710 y Fg(j)p Fi(Q)739 691 y Fh(?)759 710 y Fg(j)780 648 y Fe(Z)t(Z)5 b(Z)762 782 y Fh(Q)790 773 y Ft(2)806 782 y Fu(\002)p Fh(Q)861 773 y Fc(?)892 710 y Fg(j)p Fi(\032)928 717 y Fh(n)952 710 y Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fg(j)1115 691 y Fv(2)1140 710 y Fi(dxdy)r(d\030)26 b Fg(\024)f Fi(C)q(;)368 b Fn(\(5.8\))37 877 y(b)o(y)13 b(the)g(Cauc)o(h)o(y-Sc)o(h)o(w)o(arz)e (inequalit)o(y)k(and)d(\(5.7\),)f(where)i Fi(C)j Fn(denotes)d(here)g (and)g(b)q(elo)o(w)g(v)m(arious)g(p)q(ositiv)o(e)37 933 y(constan)o(ts)i(that)f(are)h(indep)q(enden)o(t)j(of)c Fi(n)p Fn(.)37 990 y(Next,)h(w)o(e)g(treat)f(for)h(instance)h(the)f (term)g(:)314 1049 y Fe(\014)314 1076 y(\014)314 1103 y(\014)314 1130 y(\014)331 1053 y(Z)5 b(Z)g(Z)h(Z)306 1187 y Fh(Q)334 1177 y Ft(2)351 1187 y Fu(\002)p Fv(\()p Fh(Q)420 1177 y Fc(?)438 1187 y Fv(\))452 1177 y Ft(2)477 1114 y Fi(\032)500 1121 y Fh(n)524 1114 y Fn(\()p Fi(\030)r(;)i(x;)g(y) r Fn(\))730 1084 y(1)p 679 1104 131 2 v 679 1146 a Fg(j)p Fi(x)f Fg(\000)j Fi(y)r Fg(j)812 1114 y Fi(\032)835 1096 y Fh(?)835 1126 y(n)859 1114 y Fn(\()p Fi(\030)899 1096 y Fu(0)910 1114 y Fi(;)e(x;)g(y)r Fn(\))1032 1084 y Fi(d\030)r(d\030) 1124 1067 y Fu(0)p 1031 1104 106 2 v 1031 1146 a Fn(\(2)p Fi(\031)r Fn(\))1118 1133 y Fv(6)1149 1114 y Fi(dxdy)1247 1049 y Fe(\014)1247 1076 y(\014)1247 1103 y(\014)1247 1130 y(\014)355 1285 y Fn(=)432 1223 y Fe(Z)d(Z)508 1326 y Fh(Q)p Fu(\002)p Fh(Q)605 1254 y Fg(j)p Fi(\032)641 1261 y Fh(n)665 1254 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)785 1238 y Fv(2)p 605 1274 198 2 v 639 1316 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)815 1285 y Fi(dxdy)27 b Fg(\024)998 1223 y Fe(Z)6 b(Z)1074 1326 y Fh(Q)p Fu(\002)p Fh(Q)1172 1254 y Fi(\032)1195 1261 y Fh(n)1219 1254 y Fn(\()p Fi(x;)i(x)p Fn(\))p Fi(\032)1351 1261 y Fh(n)1373 1254 y Fn(\()p Fi(y)r(;)g(y)r Fn(\))p 1172 1274 305 2 v 1259 1316 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1489 1285 y Fi(dxdy)r(;)182 b Fn(\(5.9\))37 1414 y(since)18 b(w)o(e)e(recall)h(that,)e(b)o(y)h(the) g(discrete)h(and)f(next)g(the)h(con)o(tin)o(uous)f(Cauc)o(h)o(y-Sc)o(h) o(w)o(arz)f(inequalit)o(y)l(,)j(w)o(e)37 1470 y(ha)o(v)o(e)140 1586 y Fg(j)p Fi(\032)176 1593 y Fh(n)199 1586 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)40 b Fn(=)436 1520 y Fe(\014)436 1547 y(\014)436 1575 y(\014)436 1602 y(\014)451 1524 y(Z)476 1627 y Fh(Q)504 1618 y Fc(?)532 1543 y Fe(X)532 1641 y Fh(n)p Fu(\025)p Fv(1)606 1586 y Fi(\025)632 1593 y Fh(n)656 1586 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)740 1593 y Fh(n)763 1586 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fi(u)894 1567 y Fu(\003)894 1597 y Fh(n)916 1586 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1052 1555 y Fi(d\030)p 1024 1576 106 2 v 1024 1617 a Fn(\(2)p Fi(\031)r Fn(\))1111 1604 y Fv(3)1132 1520 y Fe(\014)1132 1547 y(\014)1132 1575 y(\014)1132 1602 y(\014)359 1730 y Fg(\024)436 1668 y Fe(Z)461 1771 y Fh(Q)489 1762 y Fc(?)509 1693 y Fe(\000)530 1687 y(X)530 1785 y Fh(n)p Fu(\025)p Fv(1)604 1730 y Fi(\025)630 1737 y Fh(n)654 1730 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)751 1737 y Fh(n)773 1730 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)891 1711 y Fv(2)909 1693 y Fe(\001)935 1691 y Ft(1)p 935 1697 16 2 v 935 1717 a(2)957 1693 y Fe(\000)978 1687 y(X)978 1785 y Fh(n)p Fu(\025)p Fv(1)1052 1730 y Fi(\025)1078 1737 y Fh(n)1102 1730 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)1199 1737 y Fh(n)1222 1730 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))p Fg(j)1338 1711 y Fv(2)1355 1693 y Fe(\001)1381 1691 y Ft(1)p 1381 1697 V 1381 1717 a(2)1438 1699 y Fi(d\030)p 1408 1719 106 2 v 1408 1761 a Fn(\(2)p Fi(\031)r Fn(\))1495 1748 y Fv(3)359 1894 y Fg(\024)436 1830 y Fe(\022)469 1832 y(Z)495 1935 y Fh(Q)523 1926 y Fc(?)550 1851 y Fe(X)550 1949 y Fh(n)p Fu(\025)p Fv(1)624 1894 y Fi(\025)650 1901 y Fh(n)674 1894 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)771 1901 y Fh(n)793 1894 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)911 1875 y Fv(2)964 1863 y Fi(d\030)p 934 1883 V 934 1925 a Fn(\(2)p Fi(\031)r Fn(\))1021 1912 y Fv(3)1045 1830 y Fe(\023)1083 1827 y Ft(1)p 1083 1833 16 2 v 1083 1854 a(2)1105 1830 y Fe(\022)1139 1832 y(Z)1164 1935 y Fh(Q)1192 1926 y Fc(?)1220 1851 y Fe(X)1219 1949 y Fh(n)p Fu(\025)p Fv(1)1293 1894 y Fi(\025)1319 1901 y Fh(n)1343 1894 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)1440 1901 y Fh(n)1463 1894 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))p Fg(j)1579 1875 y Fv(2)1632 1863 y Fi(d\030)p 1602 1883 106 2 v 1602 1925 a Fn(\(2)p Fi(\031)r Fn(\))1689 1912 y Fv(3)1712 1830 y Fe(\023)1750 1827 y Ft(1)p 1750 1833 16 2 v 1750 1854 a(2)359 2020 y Fg(\024)436 1979 y Fe(p)p 481 1979 155 2 v 41 x Fi(\032)504 2027 y Fh(n)528 2020 y Fn(\()p Fi(x;)g(x)p Fn(\))648 1979 y Fe(p)p 694 1979 151 2 v 694 2020 a Fi(\032)717 2027 y Fh(n)741 2020 y Fn(\()p Fi(y)r(;)g(y)r Fn(\))n Fi(:)37 2122 y Fn(Th)o(us,)15 b(w)o(e)g(ha)o(v)o(e)g(in)h(\(5.9\))349 2183 y Fe(\014)349 2211 y(\014)349 2238 y(\014)349 2265 y(\014)366 2187 y(Z)5 b(Z)g(Z)h(Z)341 2321 y Fh(Q)369 2312 y Ft(2)386 2321 y Fu(\002)p Fv(\()p Fh(Q)455 2312 y Fc(?)473 2321 y Fv(\))487 2312 y Ft(2)512 2249 y Fi(\032)535 2256 y Fh(n)559 2249 y Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))765 2219 y(1)p 714 2239 131 2 v 714 2280 a Fg(j)p Fi(x)f Fg(\000)j Fi(y)r Fg(j)847 2249 y Fi(\032)870 2230 y Fu(\003)870 2260 y Fh(n)894 2249 y Fn(\()p Fi(\030)934 2230 y Fu(0)945 2249 y Fi(;)e(x;)g(y)r Fn(\))g Fi(dxdy)1164 2219 y(d\030)r(d\030)1256 2202 y Fu(0)p 1163 2239 106 2 v 1163 2280 a Fn(\(2)p Fi(\031)r Fn(\))1250 2267 y Fv(6)1273 2183 y Fe(\014)1273 2211 y(\014)1273 2238 y(\014)1273 2265 y(\014)626 2413 y Fg(\024)703 2351 y Fe(Z)729 2454 y Fh(Q)766 2413 y Fi(\032)789 2420 y Fh(n)813 2413 y Fn(\()p Fi(x;)g(x)p Fn(\)\()p Fi(\032)963 2420 y Fh(n)985 2413 y Fn(\()p Fg(\001)p Fi(;)g Fg(\001)p Fn(\))p Fi(\037)1096 2420 y Fh(Q)1134 2413 y Fi(?)1186 2382 y Fn(1)p 1172 2402 52 2 v 1172 2444 a Fg(j)p Fi(x)p Fg(j)1228 2413 y Fn(\))g Fi(dx)626 2537 y Fg(\024)42 b Fi(C)747 2498 y Fe(\015)747 2526 y(\015)791 2506 y Fn(1)p 777 2527 V 777 2568 a Fg(j)p Fi(x)p Fg(j)833 2498 y Fe(\015)833 2526 y(\015)858 2555 y Fh(L)882 2546 y Ft(3)p Fc(;)p Fp(1)954 2537 y Fg(k)p Fi(\032)1000 2544 y Fh(n)1024 2537 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fg(k)1156 2518 y Fv(2)1156 2554 y Fh(L)1180 2544 y Ft(6)p Fc(=)p Ft(5)1227 2554 y Fv(\()p Fh(Q)p Fv(\))626 2656 y Fg(\024)42 b Fi(C)747 2618 y Fe(\015)747 2645 y(\015)791 2626 y Fn(1)p 777 2646 V 777 2687 a Fg(j)p Fi(x)p Fg(j)833 2618 y Fe(\015)833 2645 y(\015)858 2674 y Fh(L)882 2665 y Ft(3)p Fc(;)p Fp(1)942 2656 y Fg(k)p Fi(\032)988 2663 y Fh(n)1011 2656 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fg(k)1143 2632 y Fv(3)p Fh(=)p Fv(2)1143 2673 y Fh(L)1167 2664 y Ft(1)1183 2673 y Fv(\()p Fh(Q)p Fv(\))1253 2656 y Fg(k)p Fi(\032)1299 2663 y Fh(n)1322 2656 y Fn(\()p Fi(x;)g(x)p Fn(\))p Fg(k)1454 2632 y Fv(1)p Fh(=)p Fv(2)1454 2673 y Fh(L)1478 2664 y Ft(3)1494 2673 y Fv(\()p Fh(Q)p Fv(\))1551 2656 y Fi(:)p eop %%Page: 57 57 57 56 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(57)37 199 y(using)18 b(Y)l(oung's)e(and)h(then)g(H\177)-23 b(older's)16 b(inequalit)o(y)l(.)26 b(Next,)17 b(using)g(the)g(Sob)q (olev)g(em)o(b)q(eddings)h(and)f(\(5.7\),)37 256 y(w)o(e)e(obtain)272 303 y Fe(\014)272 330 y(\014)272 358 y(\014)272 385 y(\014)289 307 y(Z)6 b(Z)f(Z)g(Z)265 441 y Fh(Q)293 432 y Ft(2)310 441 y Fu(\002)p Fv(\()p Fh(Q)379 432 y Fc(?)397 441 y Fv(\))411 432 y Ft(2)435 369 y Fi(\032)458 376 y Fh(n)482 369 y Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))689 338 y(1)p 637 359 131 2 v 637 400 a Fg(j)p Fi(x)g Fg(\000)i Fi(y)r Fg(j)771 369 y Fi(\032)794 350 y Fu(\003)794 380 y Fh(n)818 369 y Fn(\()p Fi(\030)858 350 y Fu(0)869 369 y Fi(;)e(x;)g(y)r Fn(\))991 338 y Fi(d\030)r(d\030)1083 322 y Fu(0)p 989 359 106 2 v 989 400 a Fn(\(2)p Fi(\031)r Fn(\))1076 387 y Fv(6)1100 369 y Fi(dxdy)1198 303 y Fe(\014)1198 330 y(\014)1198 358 y(\014)1198 385 y(\014)432 531 y Fg(\024)42 b Fi(C)552 492 y Fe(\015)552 519 y(\015)597 500 y Fn(1)p 582 520 52 2 v 582 562 a Fg(j)p Fi(x)p Fg(j)639 492 y Fe(\015)639 519 y(\015)664 549 y Fh(L)688 539 y Ft(3)p Fc(;)p Fp(1)760 531 y Fg(k)p Fi(\032)806 538 y Fh(n)829 531 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fg(k)961 507 y Fv(3)p Fh(=)p Fv(2)961 548 y Fh(L)985 539 y Ft(1)1001 548 y Fv(\()p Fh(Q)p Fv(\))1071 531 y Fg(k)1094 490 y Fe(p)p 1139 490 155 2 v 41 x Fi(\032)1162 538 y Fh(n)1186 531 y Fn(\()p Fi(x;)g(x)p Fn(\))n Fg(k)1316 541 y Fh(H)1348 531 y Ft(1)1365 541 y Fv(\()p Fh(Q)p Fv(\))432 650 y Fg(\024)42 b Fi(C)552 611 y Fe(\015)552 639 y(\015)597 619 y Fn(1)p 582 640 52 2 v 582 681 a Fg(j)p Fi(x)p Fg(j)639 611 y Fe(\015)639 639 y(\015)664 668 y Fh(L)688 659 y Ft(3)p Fc(;)p Fp(1)760 650 y Fg(k)p Fi(\032)806 657 y Fh(n)829 650 y Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fg(k)961 626 y Fv(3)p Fh(=)p Fv(2)961 667 y Fh(L)985 658 y Ft(1)1001 667 y Fv(\()p Fh(Q)p Fv(\))1071 613 y Fe(\000)1092 650 y Fn(1)h(+)i Fg(kr)1231 609 y Fe(p)p 1276 609 155 2 v 41 x Fi(\032)1299 657 y Fh(n)1323 650 y Fn(\()p Fi(x;)d(x)p Fn(\))n Fg(k)1453 660 y Fh(L)1477 650 y Ft(2)1494 660 y Fv(\()p Fh(Q)p Fv(\))1552 613 y Fe(\001)1573 625 y Fv(1)p Fh(=)p Fv(2)1628 650 y Fi(:)118 b Fn(\(5.10\))37 774 y(Finally)l(,)17 b(w)o(e)e(treat)f(the)h(con)o(tribution)h(of)f (the)g(third)h(term)f(in)h(\(5.6\),)d(namely)340 840 y Fe(\014)340 867 y(\014)340 894 y(\014)340 922 y(\014)357 844 y(Z)5 b(Z)g(Z)h(Z)332 978 y Fh(Q)360 969 y Ft(2)377 978 y Fu(\002)p Fv(\()p Fh(Q)446 969 y Fc(?)464 978 y Fv(\))478 969 y Ft(2)503 906 y Fi(\032)526 913 y Fh(n)550 906 y Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))705 875 y Fi(e)726 859 y Fu(\000)p Fh(i)p Fv(\()p Fh(\030)q Fu(\000)p Fh(\030)840 847 y Fp(0)849 859 y Fv(\))p Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 705 895 266 2 v 757 937 a Fg(j)p Fi(\030)k Fg(\000)e Fi(\030)869 924 y Fu(0)881 937 y Fg(j)894 924 y Fv(2)973 906 y Fi(\032)996 887 y Fu(\003)996 917 y Fh(n)1020 906 y Fn(\()p Fi(\030)1060 887 y Fu(0)1071 906 y Fi(;)e(x;)g(y)r Fn(\))1193 875 y Fi(d\030)r(d\030)1285 859 y Fu(0)p 1191 895 106 2 v 1191 937 a Fn(\(2)p Fi(\031)r Fn(\))1278 924 y Fv(6)1302 906 y Fi(dxdy)1400 840 y Fe(\014)1400 867 y(\014)1400 894 y(\014)1400 922 y(\014)499 1069 y Fg(\024)576 1008 y Fe(Z)d(Z)561 1139 y Fh(Q)p Fu(\002)p Fh(Q)652 1031 y Fe(\015)652 1058 y(\015)677 1069 y Fn(\()p Fg(j)p Fi(\032)731 1076 y Fh(n)754 1069 y Fn(\()p Fg(\001)p Fi(;)j(x;)g(y)r Fn(\))p Fg(j)f Fi(?)938 1076 y Fh(Q)966 1067 y Fc(?)1023 1039 y Fn(1)p 1001 1059 67 2 v 1001 1101 a Fg(j)p Fi(\030)r Fg(j)1049 1087 y Fv(2)1073 1069 y Fn(\))12 b Fg(j)p Fi(\032)1139 1076 y Fh(n)1163 1069 y Fn(\()p Fg(\001)p Fi(;)c(x;)g(y)r Fn(\))p Fg(j)1317 1031 y Fe(\015)1317 1058 y(\015)1339 1087 y Fh(L)1363 1078 y Ft(1)1380 1087 y Fv(\()p Fh(Q)1422 1078 y Fc(?)1440 1087 y Fv(\))1463 1069 y Fi(dxdy)r(;)37 1243 y Fn(where,)14 b(here)h(and)f(b)q(elo)o(w,)g(w)o(e)f(shall)i(use)g (the)e(notation)h Fi(f)e(?)1067 1250 y Fh(Q)1095 1241 y Fc(?)1122 1243 y Fi(g)r Fn(\()p Fi(\030)r Fn(\))f(=)1263 1207 y Fe(R)1285 1259 y Fh(Q)1313 1250 y Fc(?)1340 1243 y Fi(f)5 b Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)1478 1227 y Fu(0)1491 1243 y Fn(\))h Fi(g)r Fn(\()p Fi(\030)1581 1227 y Fu(0)1591 1243 y Fn(\))1640 1223 y Fh(d\030)1675 1211 y Fp(0)p 1621 1233 84 2 v 1621 1260 a Fv(\(2)p Fh(\031)q Fv(\))1688 1251 y Ft(3)1710 1243 y Fn(.)19 b(By)14 b(the)37 1304 y(Y)l(oung)i(and)f(the)h(H\177)-23 b(older)15 b(inequalities,)j(w) o(e)d(ha)o(v)o(e,)f(with)i Fi(x)f Fn(and)g Fi(y)i Fn(b)q(eing)g (\014xed,)377 1382 y Fe(\015)377 1409 y(\015)414 1421 y Fn(\()p Fg(j)p Fi(\032)468 1428 y Fh(n)492 1421 y Fn(\()p Fg(\001)p Fi(;)8 b(x;)g(y)r Fn(\))p Fg(j)f Fi(?)676 1428 y Fh(Q)704 1418 y Fc(?)761 1390 y Fn(1)p 738 1410 67 2 v 738 1452 a Fg(j)p Fi(\030)r Fg(j)786 1439 y Fv(2)810 1421 y Fn(\))h Fg(j)p Fi(\032)872 1428 y Fh(n)895 1421 y Fn(\()p Fg(\001)p Fi(;)g(x;)g(y)r Fn(\))p Fg(j)1059 1382 y Fe(\015)1059 1409 y(\015)1084 1439 y Fh(L)1108 1429 y Ft(1)1125 1439 y Fv(\()p Fh(Q)1167 1429 y Fc(?)1185 1439 y Fv(\))536 1540 y Fg(\024)613 1501 y Fe(\015)613 1529 y(\015)651 1540 y Fg(j)p Fi(\032)687 1547 y Fh(n)711 1540 y Fn(\()p Fg(\001)p Fi(;)g(x;)g(y)r Fn(\))p Fg(j)f Fi(?)895 1547 y Fh(Q)923 1537 y Fc(?)979 1509 y Fn(1)p 957 1529 V 957 1571 a Fg(j)p Fi(\030)r Fg(j)1005 1558 y Fv(2)1029 1501 y Fe(\015)1029 1529 y(\015)1054 1558 y Fh(L)1078 1548 y Ft(2)1096 1558 y Fv(\()p Fh(Q)1138 1548 y Fc(?)1155 1558 y Fv(\))1184 1501 y Fe(\015)1184 1529 y(\015)1209 1540 y Fi(\032)1232 1547 y Fh(n)1256 1540 y Fn(\()p Fg(\001)p Fi(;)h(x;)g(y)r Fn(\))1397 1501 y Fe(\015)1397 1529 y(\015)1419 1558 y Fh(L)1443 1548 y Ft(2)1461 1558 y Fv(\()p Fh(Q)1503 1548 y Fc(?)1520 1558 y Fv(\))536 1659 y Fg(\024)613 1620 y Fe(\015)613 1648 y(\015)665 1628 y Fn(1)p 643 1649 V 643 1690 a Fg(j)p Fi(\030)r Fg(j)691 1677 y Fv(2)715 1620 y Fe(\015)715 1648 y(\015)741 1677 y Fh(L)765 1668 y Ft(1)782 1677 y Fv(\()p Fh(Q)824 1668 y Fc(?)841 1677 y Fv(\))870 1620 y Fe(\015)870 1648 y(\015)895 1659 y Fi(\032)918 1666 y Fh(n)942 1659 y Fn(\()p Fg(\001)p Fi(;)g(x;)g(y)r Fn(\))1083 1620 y Fe(\015)1083 1648 y(\015)1105 1634 y Fv(2)1105 1677 y Fh(L)1129 1668 y Ft(2)1147 1677 y Fv(\()p Fh(Q)1189 1668 y Fc(?)1206 1677 y Fv(\))1222 1659 y Fi(:)37 1781 y Fn(Therefore,)394 1844 y Fe(\014)394 1871 y(\014)394 1899 y(\014)394 1926 y(\014)411 1848 y(Z)d(Z)h(Z)f(Z)386 1982 y Fh(Q)414 1973 y Ft(2)431 1982 y Fu(\002)p Fv(\()p Fh(Q)500 1973 y Fc(?)518 1982 y Fv(\))532 1973 y Ft(2)557 1910 y Fi(\032)580 1917 y Fh(n)604 1910 y Fn(\()p Fi(\030)r(;)j(x;)g(y) r Fn(\))759 1879 y Fi(e)780 1863 y Fu(\000)p Fh(i)p Fv(\()p Fh(\030)q Fu(\000)p Fh(\030)894 1851 y Fp(0)903 1863 y Fv(\))p Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 759 1899 266 2 v 811 1941 a Fg(j)p Fi(\030)k Fg(\000)e Fi(\030)923 1928 y Fu(0)935 1941 y Fg(j)948 1928 y Fv(2)1027 1910 y Fi(\032)1050 1891 y Fu(\003)1050 1921 y Fh(n)1074 1910 y Fn(\()p Fi(\030)1114 1891 y Fu(0)1125 1910 y Fi(;)e(x;)g(y)r Fn(\))1247 1879 y Fi(d\030)r(d\030)1339 1863 y Fu(0)p 1246 1899 106 2 v 1246 1941 a Fn(\(2)p Fi(\031)r Fn(\))1333 1928 y Fv(6)1356 1910 y Fi(dxdy)1454 1844 y Fe(\014)1454 1871 y(\014)1454 1899 y(\014)1454 1926 y(\014)554 2073 y Fg(\024)630 2035 y Fe(\015)630 2062 y(\015)683 2043 y Fn(1)p 661 2063 67 2 v 661 2105 a Fg(j)p Fi(\030)r Fg(j)709 2091 y Fv(2)733 2035 y Fe(\015)733 2062 y(\015)758 2091 y Fh(L)782 2082 y Ft(1)799 2091 y Fv(\()p Fh(Q)841 2082 y Fc(?)859 2091 y Fv(\))882 2012 y Fe(Z)d(Z)958 2115 y Fh(Q)p Fu(\002)p Fh(Q)1050 2073 y Fg(k)p Fi(\032)1096 2080 y Fh(n)1120 2073 y Fn(\()p Fg(\001)p Fi(;)j(x;)g(y)r Fn(\))p Fg(k)1284 2055 y Fv(2)1284 2087 y Fh(L)1308 2078 y Ft(2)1322 2087 y Fv(\()p Fh(Q)1364 2078 y Fc(?)1382 2087 y Fv(\))1405 2073 y Fi(dxdy)554 2198 y Fn(=)630 2159 y Fe(\015)630 2186 y(\015)683 2167 y Fn(1)p 661 2187 V 661 2229 a Fg(j)p Fi(\030)r Fg(j)709 2216 y Fv(2)733 2159 y Fe(\015)733 2186 y(\015)758 2216 y Fh(L)782 2206 y Ft(1)799 2216 y Fv(\()p Fh(Q)841 2206 y Fc(?)859 2216 y Fv(\))887 2159 y Fe(\015)887 2186 y(\015)912 2198 y Fi(\032)935 2205 y Fh(n)959 2198 y Fn(\()p Fi(\030)r(;)g(x;)g(y)r Fn(\))1109 2159 y Fe(\015)1109 2186 y(\015)1132 2172 y Fv(2)1132 2216 y Fh(L)1156 2206 y Ft(2)1173 2216 y Fv(\()p Fh(Q)1215 2206 y Fc(?)1233 2216 y Fu(\002)p Fh(Q)1288 2206 y Ft(2)1305 2216 y Fv(\))1379 2198 y Fg(\024)59 b Fi(C)q(;)240 b Fn(\(5.11\))37 2322 y(in)16 b(view)g(of)f(\(5.7\).)j (W)l(e)d(no)o(w)f(collect)j(\(5.8\),)c(\(5.10\),)f(and)k(\(5.11\),)c (and)k(estimate)e(the)i(exc)o(hange)f(term)f(as)37 2378 y(follo)o(ws)394 2426 y Fe(\014)394 2453 y(\014)394 2480 y(\014)394 2508 y(\014)411 2430 y(Z)5 b(Z)g(Z)h(Z)386 2564 y Fh(Q)414 2555 y Ft(2)431 2564 y Fu(\002)p Fv(\()p Fh(Q)500 2555 y Fc(?)518 2564 y Fv(\))532 2555 y Ft(2)557 2492 y Fi(\032)580 2499 y Fh(n)604 2492 y Fn(\()p Fi(\030)r(;)i(x;)g(y) r Fn(\))p Fi(W)797 2499 y Fu(1)832 2492 y Fn(\()p Fi(\030)j Fg(\000)g Fi(\030)949 2473 y Fu(0)960 2492 y Fi(;)d(x)i Fg(\000)g Fi(y)r Fn(\))p Fi(\032)1127 2473 y Fu(\003)1127 2503 y Fh(n)1151 2492 y Fn(\()p Fi(\030)1191 2473 y Fu(0)1202 2492 y Fi(;)e(x;)g(y)r Fn(\))1324 2461 y Fi(d\030)r(d\030)1416 2445 y Fu(0)p 1323 2481 106 2 v 1323 2523 a Fn(\(2)p Fi(\031)r Fn(\))1410 2510 y Fv(6)1433 2492 y Fi(dxdy)1531 2426 y Fe(\014)1531 2453 y(\014)1531 2480 y(\014)1531 2508 y(\014)908 2638 y Fg(\024)42 b Fi(C)13 b Fn(+)e Fi(C)16 b Fg(kr)1185 2597 y Fe(p)p 1230 2597 155 2 v 41 x Fi(\032)1253 2645 y Fh(n)1277 2638 y Fn(\()p Fi(x;)8 b(x)p Fn(\))o Fg(k)1408 2648 y Fh(L)1432 2639 y Ft(2)1449 2648 y Fv(\()p Fh(Q)p Fv(\))1506 2638 y Fi(:)240 b Fn(\(5.12\))37 2738 y(It)22 b(remains)g(no)o(w)e(to)h(cop)o(y)g(the)h(pro)q(of)f(of)f (Theorem)i(2.1)e(:)32 b(The)22 b(kinetic)g(energy)g(term)f(is)h(b)q (ounded)37 2794 y(from)16 b(b)q(elo)o(w)g(b)o(y)g Fg(kr)400 2756 y Fe(p)p 445 2756 V 38 x Fi(\032)468 2801 y Fh(n)492 2794 y Fn(\()p Fi(x;)8 b(x)p Fn(\))o Fg(k)623 2778 y Fv(2)623 2811 y Fh(L)647 2801 y Ft(2)664 2811 y Fv(\()p Fh(Q)p Fv(\))721 2794 y Fn(,)16 b(and)g(therefore)f(the)h(fact)g(that)f (the)h(energy)g(of)f(the)h(minimizing)p eop %%Page: 58 58 58 57 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(58)37 199 y(sequence)13 b(con)o(v)o(erges)d(to)g(the)h (in\014m)o(um)h(implies)h(that)d Fg(kr)1025 160 y Fe(p)p 1070 160 155 2 v 39 x Fi(\032)1093 206 y Fh(n)1117 199 y Fn(\()p Fi(x;)e(x)p Fn(\))n Fg(k)1247 209 y Fh(L)1271 200 y Ft(2)1288 209 y Fv(\()p Fh(Q)p Fv(\))1356 199 y Fn(is)k(b)q(ounded.)19 b(Consequen)o(tly)l(,)37 256 y(all)e(the)e(b)q (ounds)h(sho)o(wn)f(in)h(the)f(RHF)h(setting)f(still)h(hold)g(here)g(:) 329 320 y Fe(8)329 361 y(>)329 374 y(>)329 388 y(<)329 470 y(>)329 484 y(>)329 497 y(:)390 316 y(p)p 436 316 V 436 355 a Fi(\032)459 362 y Fh(n)483 355 y Fn(\()p Fi(x;)8 b(x)p Fn(\))13 b(is)j(b)q(ounded)h(in)f Fi(H)932 338 y Fv(1)951 355 y Fn(\()p Fi(Q)p Fn(\))p Fi(;)22 b Fn(th)o(us)15 b(in)h Fi(L)1242 338 y Fh(p)1261 355 y Fn(\()p Fi(Q)p Fn(\))f(1)d Fg(\024)h Fi(p)g Fg(\024)g Fn(6)p Fi(;)390 411 y(\032)413 418 y Fh(n)437 411 y Fn(\()p Fi(x;)8 b(y)r Fn(\))14 b(is)i(b)q(ounded)g(in)g Fi(L)873 395 y Fh(p)893 411 y Fn(\()p Fi(Q)10 b Fg(\002)g Fi(Q)p Fn(\))p Fi(;)22 b Fn(1)13 b Fg(\024)f Fi(p)h Fg(\024)g Fn(6)p Fi(;)390 432 y Fe(p)p 436 432 197 2 v 436 471 a Fi(\032)459 478 y Fh(n)483 471 y Fn(\()p Fi(\030)r(;)8 b(x;)g(x)p Fn(\))13 b(is)i(b)q(ounded)i(in)f Fi(L)963 454 y Fv(2)983 471 y Fn(\()p Fi(Q)1037 454 y Fh(?)1056 471 y Fn(;)8 b Fi(H)1119 454 y Fv(1)1137 471 y Fn(\()p Fi(Q)p Fn(\)\))p Fi(;)390 529 y(\032)413 536 y Fh(n)437 529 y Fn(\()p Fi(\030)r(;)g(x;)g(x)p Fn(\))13 b(is)j(b)q(ounded)g(in)h Fi(L)918 512 y Fv(5)p Fh(=)p Fv(3)972 529 y Fn(\()p Fi(Q)1026 512 y Fh(?)1056 529 y Fg(\002)10 b Fi(Q)p Fn(\))p Fi(;)1759 440 y Fn(\(5.13\))37 627 y(and)17 b(eac)o(h)g(of)f(the)g(four)h(terms)e (of)i(the)f(energy)h(\(2.24\))d(is)j(b)q(ounded)h(indep)q(enden)o(tly)i (of)c Fi(n)p Fn(.)24 b(As)16 b(a)h(conse-)37 684 y(quence)g(of)e (\(5.13\),)d(w)o(e)j(sho)o(w)g(the)g(follo)o(wing)h(b)q(ound)g(that)f (will)i(b)q(e)f(useful)g(in)g(Step)f(3:)612 784 y Fi(\032)635 791 y Fh(n)659 784 y Fn(\()p Fi(x;)8 b(y)r Fn(\))14 b(is)h(b)q(ounded)i (in)f Fi(H)1106 765 y Fv(1)1125 784 y Fn(\()p Fi(Q)10 b Fg(\002)g Fi(Q)p Fn(\))p Fi(:)458 b Fn(\(5.14\))37 884 y(Indeed,)17 b(w)o(e)e(ha)o(v)o(e)132 1006 y Fg(jr)183 1013 y Fh(x)204 1006 y Fi(\032)227 1013 y Fh(n)251 1006 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)40 b Fn(=)488 940 y Fe(\014)488 967 y(\014)488 995 y(\014)488 1022 y(\014)503 944 y(Z)528 1047 y Fh(Q)556 1038 y Fc(?)589 963 y Fe(X)583 1061 y Fh(m)p Fu(\025)p Fv(1)667 1006 y Fi(\025)693 1013 y Fh(m)727 1006 y Fn(\()p Fi(\030)r Fn(\))p Fg(r)823 1013 y Fh(x)844 1006 y Fi(u)870 1013 y Fh(m)904 1006 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fi(u)1035 987 y Fu(\003)1035 1017 y Fh(m)1066 1006 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))1210 975 y Fi(d\030)p 1180 996 106 2 v 1180 1037 a Fn(\(2)p Fi(\031)r Fn(\))1267 1024 y Fv(3)1291 940 y Fe(\014)1291 967 y(\014)1291 995 y(\014)1291 1022 y(\014)411 1165 y Fg(\024)488 1103 y Fe(Z)513 1207 y Fh(Q)541 1197 y Fc(?)561 1101 y Fe(\022)599 1122 y(X)594 1220 y Fh(m)p Fu(\025)p Fv(1)678 1165 y Fi(\025)704 1172 y Fh(m)738 1165 y Fn(\()p Fi(\030)r Fn(\))p Fg(jr)847 1172 y Fh(x)868 1165 y Fi(u)894 1172 y Fh(m)927 1165 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)1045 1147 y Fv(2)1063 1101 y Fe(\023)1096 1113 y Fv(1)p Fh(=)p Fv(2)1151 1101 y Fe(\022)1190 1122 y(X)1185 1220 y Fh(m)p Fu(\025)p Fv(1)1269 1165 y Fi(\025)1295 1172 y Fh(m)1328 1165 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)1425 1172 y Fh(m)1458 1165 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))p Fg(j)1574 1147 y Fv(2)1591 1101 y Fe(\023)1625 1113 y Fv(1)p Fh(=)p Fv(2)411 1324 y Fg(\024)488 1260 y Fe(\022)521 1263 y(Z)546 1366 y Fh(Q)574 1356 y Fc(?)607 1281 y Fe(X)602 1379 y Fh(m)p Fu(\025)p Fv(1)686 1324 y Fi(\025)712 1331 y Fh(m)745 1324 y Fn(\()p Fi(\030)r Fn(\))p Fg(jr)854 1331 y Fh(x)875 1324 y Fi(u)901 1331 y Fh(m)934 1324 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)1052 1306 y Fv(2)1070 1260 y Fe(\023)1104 1272 y Fv(1)p Fh(=)p Fv(2)1159 1260 y Fe(\022)1192 1263 y(Z)1218 1366 y Fh(Q)1246 1356 y Fc(?)1278 1281 y Fe(X)1273 1379 y Fh(m)p Fu(\025)p Fv(1)1357 1324 y Fi(\025)1383 1331 y Fh(m)1417 1324 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)1514 1331 y Fh(m)1546 1324 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))p Fg(j)1662 1306 y Fv(2)1680 1260 y Fe(\023)1713 1272 y Fv(1)p Fh(=)p Fv(2)1768 1324 y Fi(:)37 1468 y Fn(Hence,)130 1524 y Fe(Z)d(Z)206 1628 y Fh(Q)p Fu(\002)p Fh(Q)298 1586 y Fg(jr)349 1593 y Fh(x)371 1586 y Fi(\032)394 1593 y Fh(n)418 1586 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)538 1568 y Fv(2)563 1586 y Fi(dxdy)408 1713 y Fg(\024)486 1651 y Fe(Z)d(Z)462 1782 y Fh(Q)p Fu(\002)p Fh(Q)545 1773 y Fc(?)576 1670 y Fe(X)570 1768 y Fh(m)p Fu(\025)p Fv(1)654 1713 y Fi(\025)680 1720 y Fh(m)714 1713 y Fn(\()p Fi(\030)r Fn(\))p Fg(jr)823 1720 y Fh(x)844 1713 y Fi(u)870 1720 y Fh(m)903 1713 y Fn(\()p Fi(\030)r(;)j(x)p Fn(\))p Fg(j)1021 1694 y Fv(2)1049 1682 y Fi(dxd\030)p 1044 1703 V 1044 1744 a Fn(\(2)p Fi(\031)r Fn(\))1131 1731 y Fv(3)1176 1651 y Fe(Z)d(Z)1152 1782 y Fh(Q)p Fu(\002)p Fh(Q)1235 1773 y Fc(?)1266 1670 y Fe(X)1260 1768 y Fh(m)p Fu(\025)p Fv(1)1344 1713 y Fi(\025)1370 1720 y Fh(m)1404 1713 y Fn(\()p Fi(\030)r Fn(\))p Fg(j)p Fi(u)1501 1720 y Fh(m)1533 1713 y Fn(\()p Fi(\030)r(;)j(y)r Fn(\))p Fg(j)1649 1694 y Fv(2)1678 1682 y Fi(dy)r(d\030)p 1672 1703 V 1672 1744 a Fn(\(2)p Fi(\031)r Fn(\))1759 1731 y Fv(3)408 1872 y Fn(=)485 1810 y Fe(Z)510 1914 y Fh(Q)538 1904 y Fc(?)565 1872 y Fn(T)l(r)612 1884 y Fh(L)636 1873 y Ft(2)636 1896 y Fc(\030)653 1884 y Fv(\()p Fh(Q)p Fv(\))711 1872 y Fn([)p Fg(\000)p Fn(\001)797 1879 y Fh(\030)815 1872 y Fi(K)854 1879 y Fh(\030)873 1872 y Fn(])928 1842 y Fi(d\030)p 898 1862 V 898 1903 a Fn(\(2)p Fi(\031)r Fn(\))985 1890 y Fv(3)1034 1872 y Fg(k)p Fi(\032)1080 1879 y Fh(n)1103 1872 y Fn(\()p Fi(x;)g(x)p Fn(\))p Fg(k)1235 1882 y Fh(L)1259 1873 y Ft(1)1275 1882 y Fv(\()p Fh(Q)p Fv(\))1436 1872 y Fg(\024)104 b Fi(C)q(:)37 2002 y Fn(This)16 b(yields)h(\(5.14\).)108 2100 y(Step)f(3:)j(Con)o(v)o(ergence)c(of)g(the)g(exc)o(hange)h(term)p 108 2107 842 2 v 108 2198 a(In)21 b(view)h(of)e(the)h(b)q(ounds)h (\(5.7\),)e(\(5.13\),)g(\(5.14\),)f(w)o(e)i(ma)o(y)f(c)o(ho)q(ose)h(a)f (subsequence)i(of)f Fi(K)1744 2205 y Fh(n)1767 2198 y Fn(,)h(still)37 2254 y(denoted)16 b(b)o(y)f Fi(K)311 2261 y Fh(n)334 2254 y Fn(,)g(suc)o(h)h(that)e(the)i(follo)o(wing)f (con)o(v)o(ergences)h(hold:)387 2314 y Fe(8)387 2355 y(<)387 2437 y(:)448 2350 y Fi(\032)471 2357 y Fh(n)495 2350 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))13 b(con)o(v)o(erges)i(w) o(eakly)g(in)h Fi(L)1096 2333 y Fv(2)1116 2350 y Fn(\()p Fi(Q)1170 2333 y Fh(?)1199 2350 y Fg(\002)11 b Fi(Q)f Fg(\002)g Fi(Q)p Fn(\))p Fi(;)448 2408 y(\032)471 2415 y Fh(n)495 2408 y Fn(\()p Fi(\030)r(;)e(x;)g(x)p Fn(\))13 b(con)o(v)o(erges)i(w)o(eakly)g(in)h Fi(L)1098 2392 y Fv(5)p Fh(=)p Fv(3)1153 2408 y Fn(\()p Fi(Q)1207 2392 y Fh(?)1236 2408 y Fg(\002)11 b Fi(Q)p Fn(\))p Fi(;)448 2465 y(\032)471 2472 y Fh(n)495 2465 y Fn(\()p Fi(x;)d(y)r Fn(\))14 b(con)o(v)o(erges)h(strongly)f(in)j Fi(L)1080 2448 y Fh(p)1099 2465 y Fn(\()p Fi(Q)10 b Fg(\002)g Fi(Q)p Fn(\))p Fi(;)15 b Fn(1)d Fg(\024)h Fi(p)g(<)g Fn(6)p Fi(:)1759 2407 y Fn(\(5.15\))37 2563 y(The)g(last)e(con)o(v)o(ergence)h (holds,)h(b)q(ecause)g(b)o(y)e(Rellic)o(h's)j(theorem,)e(\(5.14\))e (implies)k(that)d Fi(\032)1573 2570 y Fh(n)1597 2563 y Fn(\()p Fi(x;)d(y)r Fn(\))i(strongly)37 2620 y(con)o(v)o(erges)19 b(\(up)g(to)f(the)h(extraction)g(of)g(a)f(subsequence\))j(in)e Fi(L)1156 2603 y Fh(p)1176 2620 y Fn(\()p Fi(Q)1230 2603 y Fv(2)1249 2620 y Fn(\))g(for)f(all)i(1)f Fg(\024)g Fi(p)g(<)g Fn(3.)31 b(But)19 b(since)37 2676 y(w)o(e)d(ha)o(v)o(e)f(in) h(addition)h(the)f(second)g(b)q(ound)g(of)g(\(5.7\),)d(the)j(in)o(terp) q(olation)h(inequalit)o(y)g(yields)g(the)f(strong)37 2733 y(con)o(v)o(ergence)e(in)g Fi(L)369 2716 y Fh(p)388 2733 y Fn(\()p Fi(Q)442 2716 y Fv(2)462 2733 y Fn(\))e(for)h(all)h(1)e Fg(\024)h Fi(p)g(<)g Fn(6.)18 b(Let)c(us)f(denote)g(b)o(y)g Fi(K)j Fn(=)1286 2696 y Fe(R)1307 2749 y Fh(Q)1335 2739 y Fc(?)1362 2733 y Fi(K)1401 2740 y Fh(\030)1457 2712 y(d\030)p 1433 2722 84 2 v 1433 2750 a Fv(\(2)p Fh(\031)q Fv(\))1500 2740 y Ft(3)1534 2733 y Fn(the)d(op)q(erator)g(that)37 2796 y(is)k(the)g(limit)h(of)d Fi(K)365 2803 y Fh(n)405 2796 y Fn(in)i(the)g(sense)f(of)g(the)h(op)q(erators)e(w)o(eak)h(top)q (ology)l(,)g(and)h(in)g(the)f(sense)h(of)f(\(4.50\))f(|)p eop %%Page: 59 59 59 58 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(59)37 199 y(the)15 b(latter)f(b)q(eing)h(equiv)m(alen)o(t)h (to)e(the)g(\014rst)g(con)o(v)o(ergence)h(of)f(\(5.15\).)j(Let)e Fi(\032)p Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))k(denotes)j(the)f(k)o (ernel)37 256 y(of)f(the)g(asso)q(ciated)g Fi(K)416 263 y Fh(\030)448 256 y Fn(:)18 b(It)13 b(is)h(indeed)g(clear)g(that)e Fi(K)k Fg(2)d(K)g Fn(for)f(the)h(same)g(reasons)f(as)h(those)f (indicated)j(in)37 312 y(the)g(pro)q(of)f(of)g(Theorem)h(2.1.)j(In)e (particular,)e Fi(K)k Fn(still)e(satis\014es)f(the)f(constrain)o(t)g (of)g(c)o(harge)g(1.)20 b(Lik)o(ewise,)37 369 y(w)o(e)15 b(ha)o(v)o(e)313 481 y(lim)8 b(inf)305 508 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)454 419 y Fe(Z)479 522 y Fh(Q)507 512 y Fc(?)534 481 y Fn(T)l(r)581 492 y Fh(L)605 481 y Ft(2)605 504 y Fc(\030)622 492 y Fv(\()p Fh(Q)p Fv(\))680 444 y Fe(\002)699 481 y Fg(\000)p Fn(\001)772 488 y Fh(\030)791 481 y Fi(K)833 462 y Fh(n)830 492 y(\030)856 444 y Fe(\003)917 450 y Fi(d\030)p 888 470 106 2 v 888 512 a Fn(\(2)p Fi(\031)r Fn(\))975 499 y Fv(3)1010 481 y Fg(\025)1058 419 y Fe(Z)1084 522 y Fh(Q)1112 512 y Fc(?)1139 481 y Fn(T)l(r)1186 492 y Fh(L)1210 481 y Ft(2)1210 504 y Fc(\030)1227 492 y Fv(\()p Fh(Q)p Fv(\))1284 444 y Fe(\002)1303 481 y Fg(\000)p Fn(\001)1376 488 y Fh(\030)1396 481 y Fi(K)1435 488 y Fh(\030)1453 444 y Fe(\003)1514 450 y Fi(d\030)p 1485 470 V 1485 512 a Fn(\(2)p Fi(\031)r Fn(\))1572 499 y Fv(3)1595 481 y Fi(;)37 610 y Fn(and)19 b(the)f(electrostatic)h(terms)f (con)o(v)o(erge)f(as)h(they)h(do)f(in)h(the)f(RHF)h(setting.)29 b(Concluding)20 b(the)e(pro)q(of)37 666 y(of)e(Theorem)g(2.3)f(amoun)o (ts)f(therefore)i(to)f(pro)o(ving)h(that)f(w)o(e)h(ma)o(y)f(pass)h(to)f (the)h(limit)h(in)g(the)f(exc)o(hange)37 722 y(term,)f(or)g(in)h(other) e(w)o(ords)h(that)126 855 y(lim)87 883 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)237 793 y Fe(Z)5 b(Z)h(Z)f(Z)212 928 y Fh(Q)240 918 y Ft(2)257 928 y Fu(\002)p Fv(\()p Fh(Q)326 918 y Fc(?)344 928 y Fv(\))358 918 y Ft(2)383 855 y Fi(\032)406 862 y Fh(n)430 855 y Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fi(W)623 862 y Fu(1)658 855 y Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)775 837 y Fu(0)787 855 y Fi(;)e(x)h Fg(\000)h Fi(y)r Fn(\))p Fi(\032)953 837 y Fu(\003)953 867 y Fh(n)977 855 y Fn(\()p Fi(\030)1017 837 y Fu(0)1028 855 y Fi(;)e(x;)g(y)r Fn(\))g Fi(d\030)r(d\030)1238 837 y Fu(0)1247 855 y Fi(dxdy)575 1019 y Fn(=)625 957 y Fe(Z)d(Z)h(Z)f(Z)600 1091 y Fh(Q)628 1082 y Ft(2)645 1091 y Fu(\002)p Fv(\()p Fh(Q)714 1082 y Fc(?)732 1091 y Fv(\))746 1082 y Ft(2)771 1019 y Fi(\032)p Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fi(W)987 1026 y Fu(1)1022 1019 y Fn(\()p Fi(\030)k Fg(\000)f Fi(\030)1140 1000 y Fu(0)1151 1019 y Fi(;)d(x)h Fg(\000)i Fi(y)r Fn(\))p Fi(\032)1318 1000 y Fu(\003)1338 1019 y Fn(\()p Fi(\030)1378 1000 y Fu(0)1389 1019 y Fi(;)d(x;)g(y)r Fn(\))g Fi(d\030)r(d\030)1599 1000 y Fu(0)1607 1019 y Fi(dxdy)r(:)41 b Fn(\(5.16\))37 1188 y(In)18 b(order)f(to)f(pro)o(v)o(e)g(\(5.16\),)g(w)o(e)g(again)h (decomp)q(ose)h(the)f(in)o(teraction)g(p)q(oten)o(tial)h Fi(W)1514 1195 y Fu(1)1551 1188 y Fn(\()p Fg(\001)p Fi(;)8 b Fg(\001)p Fn(\))15 b(accordingly)37 1244 y(to)i(\(5.6\).)25 b(W)l(e)17 b(treat)f(eac)o(h)i(of)e(the)i(three)f(categories)g(of)g (terms)g(separately)l(,)g(pro)o(ving)h(the)f(analogue)g(of)37 1301 y(\(5.16\))d(for)g(eac)o(h)i(of)e(them.)20 b(The)c(assertion)f (\(5.16\))e(will)k(then)e(follo)o(w)h(b)o(y)f(addition.)108 1357 y(W)l(e)g(b)q(egin)i(with)e(pro)o(ving)126 1490 y(lim)87 1518 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)237 1428 y Fe(Z)5 b(Z)h(Z)f(Z)212 1562 y Fh(Q)240 1553 y Ft(2)257 1562 y Fu(\002)p Fv(\()p Fh(Q)326 1553 y Fc(?)344 1562 y Fv(\))358 1553 y Ft(2)383 1490 y Fi(\032)406 1497 y Fh(n)430 1490 y Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p 588 1453 81 2 v 8 w Fi(W)631 1497 y Fu(1)665 1490 y Fn(\()p Fi(\030)k Fg(\000)f Fi(\030)783 1471 y Fu(0)794 1490 y Fi(;)d(x)i Fg(\000)g Fi(y)r Fn(\))e Fi(\032)969 1471 y Fu(\003)969 1501 y Fh(n)992 1490 y Fn(\()p Fi(\030)1032 1471 y Fu(0)1043 1490 y Fi(;)g(x;)g(y)r Fn(\))g Fi(d\030)r(d\030)1253 1471 y Fu(0)1262 1490 y Fi(dxdy)676 1653 y Fn(=)726 1592 y Fe(Z)d(Z)h(Z)f(Z)701 1726 y Fh(Q)729 1716 y Ft(2)746 1726 y Fu(\002)p Fv(\()p Fh(Q)815 1716 y Fc(?)833 1726 y Fv(\))847 1716 y Ft(2)872 1653 y Fi(\032)p Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p 1053 1617 V 8 w Fi(W)1096 1660 y Fu(1)1131 1653 y Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)1248 1635 y Fu(0)1260 1653 y Fi(;)e(x)h Fg(\000)h Fi(y)r Fn(\))e Fi(\032)1434 1635 y Fu(\003)1454 1653 y Fn(\()p Fi(\030)1494 1635 y Fu(0)1505 1653 y Fi(;)g(x;)g(y)r Fn(\))g Fi(d\030)r(d\030)1715 1635 y Fu(0)1724 1653 y Fi(dxdy)r(:)37 1820 y Fn(Clearly)l(,)16 b(it)g(su\016ces)f(to)g(pro)o(v)o(e)138 1940 y(lim)99 1968 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)250 1878 y Fe(Z)5 b(Z)g(Z)h(Z)225 2012 y Fh(Q)253 2003 y Ft(2)270 2012 y Fu(\002)p Fv(\()p Fh(Q)339 2003 y Fc(?)357 2012 y Fv(\))371 2003 y Ft(2)388 1940 y Fn([)p Fi(\032)424 1947 y Fh(n)448 1940 y Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))f Fg(\000)k Fi(\032)p Fn(\()p Fi(\030)r(;)d(x;)g(y)r Fn(\)])p 837 1904 V Fi(W)880 1947 y Fu(1)915 1940 y Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)1032 1921 y Fu(0)1044 1940 y Fi(;)e(x)h Fg(\000)i Fi(y)r Fn(\))p Fi(\032)1211 1921 y Fu(\003)1211 1951 y Fh(n)1234 1940 y Fn(\()p Fi(\030)1274 1921 y Fu(0)1285 1940 y Fi(;)d(x;)g(y)r Fn(\))g Fi(d\030)r(d\030)1495 1921 y Fu(0)1504 1940 y Fi(dxdy)14 b Fn(=)f(0)p Fi(:)61 b Fn(\(5.17\))37 2107 y(F)l(or)15 b(this)h(purp)q(ose,)f(w)o(e)g (observ)o(e)g(that)37 2192 y Fe(\014)37 2219 y(\014)55 2169 y(Z)5 b(Z)g(Z)h(Z)30 2303 y Fh(Q)58 2294 y Ft(2)75 2303 y Fu(\002)p Fv(\()p Fh(Q)144 2294 y Fc(?)162 2303 y Fv(\))176 2294 y Ft(2)193 2231 y Fn([)p Fi(\032)229 2238 y Fh(n)253 2231 y Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))f Fg(\000)k Fi(\032)p Fn(\()p Fi(\030)r(;)d(x;)g(y)r Fn(\)])p 650 2194 V 8 w Fi(W)693 2238 y Fu(1)727 2231 y Fn(\()p Fi(\030)k Fg(\000)f Fi(\030)845 2212 y Fu(0)856 2231 y Fi(;)d(x)h Fg(\000)i Fi(y)r Fn(\))d Fi(\032)1031 2212 y Fu(\003)1031 2242 y Fh(n)1054 2231 y Fn(\()p Fi(\030)1094 2212 y Fu(0)1105 2231 y Fi(;)g(x;)g(y)r Fn(\))1227 2200 y Fi(d\030)r(d\030)1319 2184 y Fu(0)p 1226 2220 106 2 v 1226 2262 a Fn(\(2)p Fi(\031)r Fn(\))1313 2249 y Fv(6)1336 2231 y Fi(dxdy)1434 2192 y Fe(\014)1434 2219 y(\014)79 2398 y Fg(\024)156 2336 y Fe(Z)d(Z)141 2468 y Fh(Q)p Fu(\002)p Fh(Q)239 2359 y Fe(\014)239 2387 y(\014)255 2336 y(Z)244 2468 y Fh(Q)272 2458 y Fc(?)308 2398 y Fi(\032)331 2405 y Fh(n)355 2398 y Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))537 2367 y Fi(d\030)p 510 2388 V 510 2429 a Fn(\(2)p Fi(\031)q Fn(\))596 2416 y Fv(3)628 2398 y Fg(\000)673 2336 y Fe(Z)663 2468 y Fh(Q)691 2458 y Fc(?)726 2398 y Fi(\032)p Fn(\()p Fi(\030)r(;)g(x;)g(y)r Fn(\))932 2367 y Fi(d\030)p 904 2388 V 904 2429 a Fn(\(2)p Fi(\031)r Fn(\))991 2416 y Fv(3)1013 2359 y Fe(\014)1013 2387 y(\014)1028 2398 y Fg(j)p 1041 2362 81 2 v Fi(W)1084 2405 y Fu(1)1121 2398 y Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)1238 2379 y Fu(0)1250 2398 y Fi(;)e(x)h Fg(\000)i Fi(y)r Fn(\))p Fg(j)1415 2359 y Fe(\014)1415 2387 y(\014)1429 2336 y(Z)1418 2468 y Fh(Q)1446 2458 y Fc(?)1482 2398 y Fi(\032)1505 2379 y Fu(\003)1505 2409 y Fh(n)1529 2398 y Fn(\()p Fi(\030)1569 2379 y Fu(0)1580 2398 y Fi(;)d(x;)g(y)r Fn(\))1717 2367 y Fi(d\030)1763 2351 y Fu(0)p 1695 2388 106 2 v 1695 2429 a Fn(\(2)p Fi(\031)r Fn(\))1782 2416 y Fv(3)1803 2359 y Fe(\014)1803 2387 y(\014)1826 2398 y Fi(dxdy)79 2557 y Fg(\024)42 b(k)p 179 2521 81 2 v Fi(W)222 2564 y Fu(1)259 2557 y Fg(k)282 2564 y Fh(L)306 2555 y Fp(1)348 2495 y Fe(Z)5 b(Z)333 2626 y Fh(Q)p Fu(\002)p Fh(Q)431 2557 y Fg(j)p Fi(\032)467 2564 y Fh(n)491 2557 y Fn(\()p Fi(x;)j(y)r Fn(\))g Fg(\000)j Fi(\032)p Fn(\()p Fi(x;)d(y)r Fn(\))p Fg(j)j(j)p Fi(\032)842 2564 y Fh(n)866 2557 y Fn(\()p Fi(x;)d(y)r Fn(\))p Fg(j)g Fi(dxd)o(y)79 2694 y Fg(\024)42 b(k)p 179 2658 V Fi(W)222 2701 y Fu(1)259 2694 y Fg(k)282 2701 y Fh(L)306 2692 y Fp(1)353 2694 y Fg(k)p Fi(\032)399 2701 y Fh(n)423 2694 y Fn(\()p Fi(x;)8 b(y)r Fn(\))g Fg(\000)j Fi(\032)p Fn(\()p Fi(x;)d(y)r Fn(\))p Fg(k)737 2704 y Fh(L)761 2695 y Ft(2)777 2704 y Fv(\()p Fh(Q)819 2695 y Ft(2)835 2704 y Fv(\))864 2694 y Fg(k)p Fi(\032)910 2701 y Fh(n)933 2694 y Fn(\()p Fi(x;)g(y)r Fn(\))p Fg(k)1063 2704 y Fh(L)1087 2695 y Ft(2)1103 2704 y Fv(\()p Fh(Q)1145 2695 y Ft(2)1162 2704 y Fv(\))1177 2694 y Fi(;)37 2796 y Fn(and)16 b(\(5.17\))d(follo)o(ws)i(from)g(the)g (strong)f(con)o(v)o(ergence)i(of)f Fi(\032)1058 2803 y Fh(n)1082 2796 y Fn(\()p Fi(x;)8 b(y)r Fn(\))13 b(to)i Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))14 b(in)i Fi(L)1486 2780 y Fv(2)1506 2796 y Fn(\()p Fi(Q)1560 2780 y Fv(2)1579 2796 y Fn(\).)p eop %%Page: 60 60 60 59 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(60)37 199 y(F)l(or)15 b(the)g(second)h(term,)e(w)o(e)h (remark)g(that)301 257 y Fe(\014)301 285 y(\014)301 312 y(\014)301 339 y(\014)318 261 y(Z)5 b(Z)h(Z)f(Z)293 395 y Fh(Q)321 386 y Ft(2)338 395 y Fu(\002)p Fv(\()p Fh(Q)407 386 y Fc(?)425 395 y Fv(\))439 386 y Ft(2)456 286 y Fe(\002)475 323 y Fg(j)p Fi(\032)511 330 y Fh(n)535 323 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)655 304 y Fv(2)683 323 y Fg(\000)i(j)p Fi(\032)p Fn(\()p Fi(x;)e(y)r Fn(\))p Fg(j)884 304 y Fv(2)902 286 y Fe(\003)980 292 y Fn(1)p 926 313 131 2 v 926 354 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1069 323 y Fi(dxdy)1167 257 y Fe(\014)1167 285 y(\014)1167 312 y(\014)1167 339 y(\014)579 491 y Fg(\024)41 b Fi(C)699 425 y Fe(\015)699 452 y(\015)699 479 y(\015)699 507 y(\015)783 460 y Fn(1)p 729 480 V 729 522 a Fg(j)p Fi(x)10 b Fg(\000)g Fi(y)r Fg(j)865 425 y Fe(\015)865 452 y(\015)865 479 y(\015)865 507 y(\015)890 536 y Fh(L)914 527 y Ft(2)931 536 y Fv(\()p Fh(Q)973 527 y Ft(2)990 536 y Fv(\))1018 425 y Fe(\015)1018 452 y(\015)1018 479 y(\015)1018 507 y(\015)1044 491 y Fg(j)p Fi(\032)1080 498 y Fh(n)1103 491 y Fn(\()p Fi(x;)e(y)r Fn(\))p Fg(j)1223 472 y Fv(2)1251 491 y Fg(\000)j(j)p Fi(\032)p Fn(\()p Fi(x;)d(y)r Fn(\))p Fg(j)1453 472 y Fv(2)1471 425 y Fe(\015)1471 452 y(\015)1471 479 y(\015)1471 507 y(\015)1496 536 y Fh(L)1520 527 y Ft(2)1537 536 y Fv(\()p Fh(Q)1579 527 y Ft(2)1596 536 y Fv(\))37 628 y Fn(where)15 b(the)g(righ)o(t-hand)g(side)h(con)o(v)o (erges)e(to)g(zero)g(b)q(ecause)i Fi(\032)1104 635 y Fh(n)1128 628 y Fn(\()p Fi(x;)8 b(y)r Fn(\))13 b(con)o(v)o(erges)h(to)g Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\),)13 b(strongly)h(in)37 684 y Fi(L)68 668 y Fv(4)88 684 y Fn(\()p Fi(Q)142 668 y Fv(2)161 684 y Fn(\),)h(th)o(us)g Fg(j)p Fi(\032)343 691 y Fh(n)367 684 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)487 668 y Fv(2)519 684 y Fn(con)o(v)o(erges)15 b(to)g Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)935 668 y Fv(2)953 684 y Fn(,)14 b(strongly)h(in)h Fi(L)1240 668 y Fv(2)1260 684 y Fn(\()p Fi(Q)1314 668 y Fv(2)1333 684 y Fn(\))f(.)108 741 y(It)g(no)o(w)g(remains)h(to)e(treat)g(the)i (third)f(term)g(of)g(\(5.6\),)e(namely)j(for)f(instance)h(to)e(pro)o(v) o(e)339 874 y(lim)300 902 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)450 812 y Fe(Z)6 b(Z)f(Z)g(Z)426 946 y Fh(Q)454 937 y Ft(2)471 946 y Fu(\002)p Fv(\()p Fh(Q)540 937 y Fc(?)558 946 y Fv(\))572 937 y Ft(2)596 874 y Fi(\032)619 881 y Fh(n)643 874 y Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fi(\032)816 855 y Fu(\003)816 885 y Fh(n)838 874 y Fn(\()p Fi(\030)878 855 y Fu(0)889 874 y Fi(;)g(x;)g(y)r Fn(\))1004 843 y Fi(e)1025 827 y Fu(\000)p Fh(i)p Fv(\()p Fh(\030)q Fu(\000)p Fh(\030)1139 815 y Fp(0)1149 827 y Fv(\))p Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 1004 864 266 2 v 1057 905 a Fg(j)p Fi(\030)j Fg(\000)g Fi(\030)1169 892 y Fu(0)1181 905 y Fg(j)1194 892 y Fv(2)1280 874 y Fi(d\030)r(d\030)1372 855 y Fu(0)1383 874 y Fi(dxdy)519 1049 y Fn(=)597 987 y Fe(Z)6 b(Z)f(Z)g(Z)573 1121 y Fh(Q)601 1111 y Ft(2)618 1121 y Fu(\002)p Fv(\()p Fh(Q)687 1111 y Fc(?)705 1121 y Fv(\))719 1111 y Ft(2)743 1049 y Fi(\032)p Fn(\()p Fi(\030)r(;)j(x;)g(y)r Fn(\))p Fi(\032)939 1030 y Fu(\003)958 1049 y Fn(\()p Fi(\030)998 1030 y Fu(0)1009 1049 y Fi(;)g(x;)g(y)r Fn(\))1124 1018 y Fi(e)1145 1001 y Fu(\000)p Fh(i)p Fv(\()p Fh(\030)q Fu(\000)p Fh(\030)1259 990 y Fp(0)1268 1001 y Fv(\))p Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 1124 1038 V 1177 1080 a Fg(j)p Fi(\030)j Fg(\000)g Fi(\030)1289 1067 y Fu(0)1300 1080 y Fg(j)1313 1067 y Fv(2)1400 1049 y Fi(d\030)r(d\030)1492 1030 y Fu(0)1503 1049 y Fi(dxdy)r(:)37 1214 y Fn(W)l(e)16 b(in)o(tro)q(duce)g(the)f(function)470 1336 y Fi(F)499 1343 y Fh(n)523 1336 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))i(=)731 1275 y Fe(Z)757 1378 y Fh(Q)785 1368 y Fc(?)812 1336 y Fi(\032)835 1318 y Fu(\003)835 1348 y Fh(n)859 1336 y Fn(\()p Fi(\030)899 1318 y Fu(0)910 1336 y Fi(;)e(x;)g(y)r Fn(\))1025 1306 y Fi(e)1046 1289 y Fu(\000)p Fh(i)p Fv(\()p Fh(\030)q Fu(\000)p Fh(\030)1160 1277 y Fp(0)1170 1289 y Fv(\))p Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 1025 1326 V 1078 1368 a Fg(j)p Fi(\030)k Fg(\000)e Fi(\030)1190 1354 y Fu(0)1202 1368 y Fg(j)1215 1354 y Fv(2)1336 1306 y Fi(d\030)1382 1289 y Fu(0)p 1311 1326 106 2 v 1311 1368 a Fn(\(2)p Fi(\031)r Fn(\))1398 1354 y Fv(3)1422 1336 y Fi(;)324 b Fn(\(5.18\))37 1465 y(and)16 b(the)f(analogous)g(function)h Fi(F)22 b Fn(when)15 b Fi(\032)784 1472 y Fh(n)824 1465 y Fn(is)g(replaced)i(b) o(y)e Fi(\032)p Fn(.)20 b(What)15 b(w)o(e)g(ha)o(v)o(e)f(to)h(pro)o(v)o (e)g(is)g(that)323 1585 y(lim)284 1613 y Fh(n)p Fu(\000)-5 b(!)p Fv(+)p Fu(1)433 1523 y Fe(Z)5 b(Z)h(Z)416 1657 y Fh(Q)444 1648 y Ft(2)461 1657 y Fu(\002)p Fh(Q)516 1648 y Fc(?)539 1585 y Fn([)p Fi(\032)575 1592 y Fh(n)599 1585 y Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))p Fi(F)778 1592 y Fh(n)799 1585 y Fn(\()p Fi(\030)r(;)g(x;)g(y)r Fn(\))g Fg(\000)i Fi(\032)p Fn(\()p Fi(\030)r(;)e(x;)g(y)r Fn(\))p Fi(F)e Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))o(])i Fi(d\030)r(dxdy)j Fn(=)g(0)p Fi(;)138 b Fn(\(5.19\))37 1755 y(and)11 b(for)g(this)g(purp)q(ose)g(it)g(su\016ces)h(to)e(sho)o (w)g(that)g Fi(\032)905 1762 y Fh(n)929 1755 y Fn(\()p Fi(\030)r(;)e(x;)g(y)r Fn(\))g(con)o(v)o(erges)j(w)o(eakly)g(to)f Fi(\032)p Fn(\()p Fi(\030)r(;)e(x;)g(y)r Fn(\))h(in)i Fi(L)1745 1739 y Fv(2)1765 1755 y Fn(\()p Fi(Q)1819 1739 y Fh(?)1840 1755 y Fg(\002)37 1812 y Fi(Q)73 1795 y Fv(2)93 1812 y Fn(\),)i(whic)o(h)i(w)o(e)e(already)h(kno)o(w)g(b)o(y)f (\(5.15\),)f(and)i(that)f Fi(F)1025 1819 y Fh(n)1048 1812 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))k(con)o(v)o(erges)h (strongly)g(to)g Fi(F)6 b Fn(\()p Fi(\030)r(;)i(x;)g(y)r Fn(\))k(in)37 1868 y Fi(L)68 1852 y Fv(2)88 1868 y Fn(\()p Fi(Q)142 1852 y Fh(?)172 1868 y Fg(\002)e Fi(Q)253 1852 y Fv(2)273 1868 y Fn(\),)k(whic)o(h)i(w)o(e)f(no)o(w)g(establish)h (\(up)g(to)e(the)h(extraction)g(of)g(a)g(subsequence\).)108 1932 y(Since)229 1914 y Fh(e)245 1902 y Fp(\000)p Fc(i\030)q Fp(\001)p Ft(\()p Fc(x)p Fp(\000)p Fc(y)q Ft(\))p 229 1921 160 2 v 282 1949 a Fu(j)p Fh(\030)q Fu(j)319 1939 y Ft(2)407 1932 y Fn(b)q(elongs)f(to)e Fi(L)652 1915 y Fu(1)689 1932 y Fn(\()p Fi(Q)5 b Fg(\002)g Fi(Q)p Fn(;)j Fi(L)876 1915 y Fv(1)896 1932 y Fn(\()p Fi(Q)950 1915 y Fh(?)969 1932 y Fn(\)\),)13 b(and)g(since)h Fi(\032)1249 1939 y Fh(n)1273 1932 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))i(is)j(b)q(ounded)i(in)e Fi(L)1741 1915 y Fv(2)1761 1932 y Fn(\()p Fi(Q)1815 1915 y Fh(?)1840 1932 y Fg(\002)37 1999 y Fi(Q)73 1983 y Fv(2)93 1999 y Fn(\),)i(w)o(e)g(easily)h(deduce)g (from)f(Y)l(oung's)g(inequalit)o(y)i(that)538 2101 y Fi(F)567 2108 y Fh(n)590 2101 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))13 b(is)j(b)q(ounded)h(in)f Fi(L)1069 2082 y Fv(2)1088 2101 y Fn(\()p Fi(Q)1142 2082 y Fh(?)1172 2101 y Fg(\002)10 b Fi(Q)g Fg(\002)h Fi(Q)p Fn(\))p Fi(:)37 2202 y Fn(Lik)o(ewise,)17 b(the)e(generalized)h(Y)l(oung)f(inequalit)o (y)i(for)d(the)h(Marcinkiewicz)i(spaces)e(\(or)f(w)o(eak)g Fi(L)1698 2186 y Fh(p)1733 2202 y Fn(spaces\))37 2259 y(implies)534 2360 y Fi(F)563 2367 y Fh(n)587 2360 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))k(is)k(b)q(ounded)h(in)f Fi(L)1065 2341 y Fv(6)1084 2360 y Fn(\()p Fi(Q)1138 2341 y Fh(?)1168 2360 y Fg(\002)10 b Fi(Q)g Fg(\002)h Fi(Q)p Fn(\))p Fi(;)387 b Fn(\(5.20\))37 2475 y(using)16 b(this)g(time)g(that) 452 2457 y Fh(e)468 2445 y Fp(\000)p Fc(i\030)q Fp(\001)p Ft(\()p Fc(x)p Fp(\000)p Fc(y)q Ft(\))p 452 2464 V 505 2492 a Fu(j)p Fh(\030)q Fu(j)542 2482 y Ft(2)632 2475 y Fn(b)q(elongs)g(to)f Fi(L)882 2458 y Fv(3)p Fh(=)p Fv(2)p Fh(;)p Fu(1)982 2475 y Fn(\()p Fi(Q)1036 2458 y Fh(?)1055 2475 y Fn(\).)37 2570 y(Let)20 b(us)g(no)o(w)e(pro)o(v)o(e) h(that)g Fi(F)543 2577 y Fh(n)567 2570 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))16 b(is)k(b)q(ounded)h(in)f Fi(W)1079 2554 y Fv(1)p Fh(=)p Fv(2)p Fh(;)p Fv(1)1162 2570 y Fn(\()p Fi(Q)1216 2554 y Fh(?)1248 2570 y Fg(\002)13 b Fi(Q)1332 2554 y Fv(2)1352 2570 y Fn(\).)32 b(This)20 b(will)h(imply)g(b)o(y)e (the)37 2627 y(Sob)q(olev)i(em)o(b)q(eddings)g(theorem)e(that)f Fi(F)775 2634 y Fh(n)799 2627 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))17 b(is)j(relativ)o(ely)h(compact)e(in)h Fi(L)1493 2610 y Fh(p)1513 2627 y Fn(\()p Fi(Q)1567 2610 y Fh(?)1599 2627 y Fg(\002)13 b Fi(Q)1683 2610 y Fv(2)1703 2627 y Fn(\),)19 b(for)g(all)37 2683 y(1)13 b Fg(\024)g Fi(p)f(<)209 2665 y Fv(19)p 209 2672 36 2 v 209 2699 a(18)250 2683 y Fn(.)19 b(In)d(view)f(of)g(\(5.20\),)d(H\177)-23 b(older's)15 b(inequalit)o(y)i(yields)f(the)f(compactness)g(in)h(all)g Fi(L)1642 2667 y Fh(p)1661 2683 y Fn(\()p Fi(Q)1715 2667 y Fh(?)1744 2683 y Fg(\002)10 b Fi(Q)1825 2667 y Fv(2)1845 2683 y Fn(\),)37 2740 y(for)19 b(all)g(1)g Fg(\024)g Fi(p)f(<)h Fn(6,)g(and)g(th)o(us)g(in)h(particular)f(the)g(desired)h Fi(L)1160 2723 y Fv(2)1180 2740 y Fn(\()p Fi(Q)1234 2723 y Fh(?)1265 2740 y Fg(\002)13 b Fi(Q)1349 2723 y Fv(2)1369 2740 y Fn(\))18 b(compactness.)31 b(T)l(o)19 b(pro)o(v)o(e)37 2796 y(that)g Fi(F)169 2803 y Fh(n)192 2796 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))17 b(is)i(b)q(ounded)i(in)e Fi(W)703 2780 y Fv(1)p Fh(=)p Fv(2)p Fh(;)p Fv(1)786 2796 y Fn(\()p Fi(Q)840 2780 y Fh(?)872 2796 y Fg(\002)13 b Fi(Q)956 2780 y Fv(2)976 2796 y Fn(\),)19 b(w)o(e)f(pro)o(v)o(e)h (that)f Fi(F)1354 2803 y Fh(n)1378 2796 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))16 b(is)k(b)q(ounded)g(b)q(oth)p eop %%Page: 61 61 61 60 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(61)37 199 y(in)16 b Fi(L)121 183 y Fv(1)141 199 y Fn(\()p Fi(Q)195 183 y Fv(2)214 199 y Fn(;)8 b Fi(W)284 183 y Fv(1)p Fh(=)p Fv(2)p Fh(;)p Fv(1)366 199 y Fn(\()p Fi(Q)420 183 y Fh(?)439 199 y Fn(\)\))14 b(and)h(in)g Fi(L)660 183 y Fv(1)680 199 y Fn(\()p Fi(Q)734 183 y Fh(?)753 199 y Fn(;)8 b Fi(W)823 183 y Fv(1)p Fh(=)p Fv(2)p Fh(;)p Fv(1)905 199 y Fn(\()p Fi(Q)959 183 y Fv(2)978 199 y Fn(\)\).)19 b(The)c(second)g(b)q(ound)g(is)h(an)e(immediate)i (conse-)37 256 y(quence)d(of)d(the)h(fact)f(that)h Fi(F)514 263 y Fh(n)537 256 y Fn(\()p Fi(\030)r(;)d(x;)g(y)r Fn(\))h(is)i(b)q (ounded)i(in)f Fi(L)999 239 y Fv(2)1018 256 y Fn(\()p Fi(Q)1072 239 y Fh(?)1092 256 y Fn(;)c Fi(H)1155 239 y Fv(1)1173 256 y Fn(\()p Fi(Q)1227 239 y Fv(2)1246 256 y Fn(\)\).)18 b(The)11 b(latter)g(fact)f(holds)i(b)q(ecause)37 320 y(b)q(oth)k Fi(\032)168 327 y Fh(n)192 320 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))14 b(and)i Fg(r)483 327 y Fh(x)505 320 y Fi(\032)528 327 y Fh(n)552 320 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))13 b(are)j(b)q(ounded)h(in)f Fi(L)1062 303 y Fv(2)1082 320 y Fn(\()p Fi(Q)1136 303 y Fh(?)1166 320 y Fg(\002)11 b Fi(Q)1248 303 y Fv(2)1267 320 y Fn(\),)1319 302 y Fh(e)1335 290 y Fp(\000)p Fc(i\030)q Fp(\001)p Ft(\()p Fc(x)p Fp(\000)p Fc(y)q Ft(\))p 1319 309 160 2 v 1372 336 a Fu(j)p Fh(\030)q Fu(j)1409 327 y Ft(2)1497 320 y Fg(2)j Fi(L)1572 303 y Fu(1)1610 320 y Fn(\()p Fi(Q)1664 303 y Fv(2)1683 320 y Fn(;)8 b Fi(L)1735 303 y Fv(1)1754 320 y Fn(\()p Fi(Q)1808 303 y Fh(?)1827 320 y Fn(\)\),)37 380 y(and)224 494 y Fg(r)262 501 y Fh(x)284 494 y Fi(F)313 501 y Fh(n)336 494 y Fn(\()p Fi(\030)r(;)g(x;)g(y)r Fn(\))j(=)i Fg(r)583 501 y Fh(x)605 494 y Fi(\032)628 475 y Fu(\003)628 505 y Fh(n)652 494 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))f Fi(?)832 501 y Fh(Q)860 491 y Fc(?)895 463 y Fi(e)916 446 y Fu(\000)p Fh(i\030)q Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 895 483 183 2 v 953 525 a Fg(j)p Fi(\030)r Fg(j)1001 512 y Fv(2)1093 494 y Fn(+)j Fi(\032)1161 475 y Fu(\003)1161 505 y Fh(n)1185 494 y Fn(\()p Fi(\030)r(;)e(x;)g(y)r Fn(\))g Fi(?)1366 501 y Fh(Q)1394 491 y Fc(?)1428 463 y Fg(r)1466 470 y Fh(x)1488 463 y Fi(e)1509 446 y Fu(\000)p Fh(i\030)q Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 1428 483 243 2 v 1516 525 a Fg(j)p Fi(\030)r Fg(j)1564 512 y Fv(2)1676 494 y Fi(:)37 618 y Fn(In)23 b(order)e(to)g(sho)o(w)g(that)f Fi(F)540 625 y Fh(n)564 618 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))19 b(is)j(b)q(ounded)h(in) f Fi(L)1067 602 y Fv(1)1087 618 y Fn(\()p Fi(Q)1141 602 y Fv(2)1160 618 y Fn(;)8 b Fi(W)1230 602 y Fv(1)p Fh(=)p Fv(2)p Fh(;)p Fv(1)1312 618 y Fn(\()p Fi(Q)1366 602 y Fh(?)1386 618 y Fn(\)\),)22 b(w)o(e)f(remark)g(\014rst)g(that)61 657 y Fv(1)p 42 664 54 2 v 42 692 a Fu(j)p Fh(\030)q Fu(j)79 682 y Ft(2)114 675 y Fg(2)13 b Fi(L)188 658 y Fu(1)225 675 y Fn(\()p Fi(Q)279 658 y Fv(2)298 675 y Fi(;)8 b(W)368 658 y Fv(1)p Fh(=)p Fv(2)p Fh(;)p Fv(1)450 675 y Fn(\()p Fi(Q)504 658 y Fh(?)524 675 y Fn(\),)14 b(th)o(us)321 775 y Fe(\015)321 802 y(\015)346 813 y Fi(F)375 820 y Fh(n)399 813 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))549 775 y Fe(\015)549 802 y(\015)572 831 y Fh(W)610 821 y Ft(1)p Fc(=)p Ft(2)p Fc(;)p Ft(1)683 831 y Fv(\()p Fh(Q)725 822 y Fc(?)742 831 y Fv(\))800 813 y Fn(=)877 775 y Fe(\015)877 802 y(\015)902 813 y Fi(\032)925 820 y Fh(n)949 813 y Fn(\()p Fi(\030)r(;)g(x;)g(y)r Fn(\))g Fi(?)1130 820 y Fh(Q)1158 811 y Fc(?)1192 783 y Fi(e)1213 766 y Fu(\000)p Fh(i\030)q Fu(\001)p Fv(\()p Fh(x)p Fu(\000)p Fh(y)q Fv(\))p 1192 803 183 2 v 1250 844 a Fg(j)p Fi(\030)r Fg(j)1298 831 y Fv(2)1380 775 y Fe(\015)1380 802 y(\015)1405 831 y Fh(W)1443 821 y Ft(1)p Fc(=)p Ft(2)p Fc(;)p Ft(1)1516 831 y Fv(\()p Fh(Q)1558 822 y Fc(?)1576 831 y Fv(\))800 932 y Fg(\024)877 894 y Fe(\015)877 921 y(\015)902 932 y Fi(\032)925 939 y Fh(n)949 932 y Fn(\()p Fi(\030)r(;)g(x;)g(y)r Fn(\))1099 894 y Fe(\015)1099 921 y(\015)1122 950 y Fh(L)1146 941 y Ft(1)1163 950 y Fv(\()p Fh(Q)1205 941 y Fc(?)1222 950 y Fv(\))1251 894 y Fe(\015)1251 921 y(\015)1303 902 y Fn(1)p 1281 922 67 2 v 1281 964 a Fg(j)p Fi(\030)r Fg(j)1329 951 y Fv(2)1353 894 y Fe(\015)1353 921 y(\015)1378 950 y Fh(W)1416 941 y Ft(1)p Fc(=)p Ft(2)p Fc(;)p Ft(1)1490 950 y Fv(\()p Fh(Q)1532 941 y Fc(?)1549 950 y Fv(\))37 1048 y Fn(whence)17 b(w)o(e)e(deduce)259 1110 y Fe(\015)259 1137 y(\015)284 1149 y Fi(F)313 1156 y Fh(n)337 1149 y Fn(\()p Fi(\030)r(;)8 b(x;)g(y)r Fn(\))487 1110 y Fe(\015)487 1137 y(\015)510 1167 y Fh(L)534 1157 y Ft(1)551 1167 y Fv(\()p Fh(Q)593 1157 y Ft(2)610 1167 y Fv(;)p Fh(W)658 1157 y Ft(1)p Fc(=)p Ft(2)p Fc(;)p Ft(1)731 1167 y Fv(\()p Fh(Q)773 1157 y Fc(?)790 1167 y Fv(\)\))832 1149 y Fg(\024)880 1110 y Fe(\015)880 1137 y(\015)906 1149 y Fi(\032)929 1156 y Fh(n)953 1149 y Fn(\()p Fi(\030)r(;)g(x;)g(y)r Fn(\))1103 1110 y Fe(\015)1103 1137 y(\015)1125 1167 y Fh(L)1149 1157 y Ft(1)1167 1167 y Fv(\()p Fh(Q)1209 1157 y Fc(?)1226 1167 y Fu(\002)p Fh(Q)1281 1157 y Ft(2)1299 1167 y Fv(\))1327 1110 y Fe(\015)1327 1137 y(\015)1379 1118 y Fn(1)p 1357 1138 V 1357 1180 a Fg(j)p Fi(\030)r Fg(j)1405 1167 y Fv(2)1429 1110 y Fe(\015)1429 1137 y(\015)1455 1167 y Fh(W)1493 1157 y Ft(1)p Fc(=)p Ft(2)p Fc(;)p Ft(1)1566 1167 y Fv(\()p Fh(Q)1608 1157 y Fc(?)1625 1167 y Fv(\))1641 1149 y Fi(:)37 1266 y Fn(This)16 b(concludes)h(the)e(\014rst)g(pro)q (of)g(of)g(\(5.16\),)e(and)i(th)o(us)g(the)g(pro)q(of)g(of)g(Theorem)g (2.3.)1840 1323 y Fg(})37 1404 y Fj(Second)h(pro)q(of)f(of)g(Theorem)g (2.3)h(:)41 b Fn(W)l(e)13 b(ma)o(y)f(observ)o(e)h(that)f(w)o(e)g(can)h (mo)q(dify)g(the)g(argumen)o(t)f(whic)o(h)37 1461 y(is)k(used)g(in)g (Step)g(3)e(ab)q(o)o(v)o(e)h(b)o(y)g(pro)o(ving)h(the)f(follo)o(wing.) 37 1547 y Fj(Lemma)j(5.2)23 b Ff(F)m(or)16 b(any)g(minimizing)f(se)n (quenc)n(e)g Fi(K)945 1554 y Fh(n)981 1547 y Fg(2)e(K)j Ff(of)h Fi(I)1153 1530 y Fh(H)s(F)1149 1558 y(per)1213 1547 y Ff(,)f(we)h(have)431 1674 y Fn(lim)403 1702 y Fh(n)p Fu(!)p Fv(+)p Fu(1)530 1612 y Fe(Z)555 1716 y Fh(Q)593 1674 y Fi(dx)651 1612 y Fe(Z)675 1716 y Fd(R)705 1706 y Ft(3)737 1644 y Fg(j)p Fi(\032)773 1651 y Fh(n)797 1644 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)917 1627 y Fv(2)p 737 1664 198 2 v 771 1705 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)947 1674 y Fi(dy)k Fn(=)1055 1612 y Fe(Z)1081 1716 y Fh(Q)1118 1674 y Fi(dx)1176 1612 y Fe(Z)1201 1716 y Fd(R)1231 1706 y Ft(3)1263 1644 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1419 1627 y Fv(2)p 1263 1664 174 2 v 1285 1705 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1449 1674 y Fi(dy)r(:)249 b Fn(\(5.21\))37 1819 y Fj(Pro)q(of)25 b(of)f(Lemma)f(5.2:)32 b Fn(The)21 b(pro)q(of)f(of)g(the)h(ab)q(o)o(v)o(e)f(Lemma)h(5.1)e(yields)j(in)g (particular)f(that)f(the)37 1883 y(exc)o(hange)d(term)343 1846 y Fe(R)364 1899 y Fh(Q)402 1883 y Fi(dx)460 1846 y Fe(R)480 1899 y Fd(R)510 1889 y Ft(3)543 1860 y Fu(j)p Fh(\032)571 1864 y Fc(n)592 1860 y Fv(\()p Fh(x;y)q Fv(\))p Fu(j)678 1848 y Ft(2)p 543 1872 152 2 v 576 1899 a Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j)707 1883 y Fi(dy)h Fn(is)f(b)q(ounded)h (indep)q(enden)o(tly)h(of)d Fi(n)p Fn(,)h(with)f(the)h(help)g(of)f (\(5.13\).)37 1949 y(Moreo)o(v)o(er,)d(in)h(view)h(of)e(\(5.15\),)f Fg(j)p Fi(\032)633 1956 y Fh(n)657 1949 y Fn(\()p Fi(x;)c(y)r Fn(\))p Fg(j)777 1933 y Fv(2)808 1949 y Fn(con)o(v)o(erges)14 b(to)f Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1221 1933 y Fv(2)1252 1949 y Fn(almost)14 b(ev)o(erywhere)g(on)f Fj(R)1733 1933 y Fv(3)1761 1949 y Fg(\002)7 b Fj(R)1842 1933 y Fv(3)1862 1949 y Fn(.)37 2006 y(Then,)16 b(b)o(y)f(F)l(atou's)f (lemma,)h(w)o(e)g(deduce)h(that)400 2124 y(lim)8 b(inf)403 2152 y Fh(n)p Fu(!)p Fv(+)p Fu(1)533 2062 y Fe(Z)558 2165 y Fh(Q)596 2124 y Fi(dx)654 2062 y Fe(Z)678 2165 y Fd(R)708 2156 y Ft(3)741 2093 y Fg(j)p Fi(\032)777 2100 y Fh(n)800 2093 y Fn(\()p Fi(x;)g(y)r Fn(\))p Fg(j)920 2077 y Fv(2)p 741 2114 198 2 v 774 2155 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)951 2124 y Fi(dy)k Fg(\025)1059 2062 y Fe(Z)1084 2165 y Fh(Q)1121 2124 y Fi(dx)1179 2062 y Fe(Z)1204 2165 y Fd(R)1234 2156 y Ft(3)1266 2093 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1422 2077 y Fv(2)p 1266 2114 174 2 v 1288 2155 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1453 2124 y Fi(dy)r(:)37 2244 y Fn(In)18 b(order)e(to)g(pro)o(v)o(e)g(the)h(rev)o(erse)f(inequalit)o(y) i(for)e(the)h(upp)q(er)h(limit,)f(w)o(e)g(argue)f(as)g(follo)o(ws.)24 b(Let)17 b Fi(R)f(>)f Fn(0)37 2301 y(b)q(e)h(\014xed,)g(w)o(e)f(ma)o(y) f(write)102 2357 y Fe(Z)128 2461 y Fh(Q)165 2419 y Fi(dx)223 2357 y Fe(Z)247 2461 y Fd(R)277 2451 y Ft(3)310 2389 y Fg(j)p Fi(\032)346 2396 y Fh(n)369 2389 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)489 2372 y Fv(2)p 310 2409 198 2 v 343 2451 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)520 2419 y Fi(dy)k Fn(=)628 2357 y Fe(Z)653 2461 y Fh(Q)691 2419 y Fi(dx)749 2357 y Fe(Z)773 2461 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\024)p Fh(R)927 2389 y Fg(j)p Fi(\032)963 2396 y Fh(n)987 2389 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1107 2372 y Fv(2)p 927 2409 V 961 2451 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1137 2419 y Fi(dy)h Fn(+)1241 2357 y Fe(Z)1266 2461 y Fh(Q)1303 2419 y Fi(dx)1361 2357 y Fe(Z)1386 2461 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\025)p Fh(R)1540 2389 y Fg(j)p Fi(\032)1576 2396 y Fh(n)1600 2389 y Fn(\()p Fi(x;)c(y)r Fn(\))p Fg(j)1720 2372 y Fv(2)p 1540 2409 V 1573 2451 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1750 2419 y Fi(dy)r(:)37 2541 y Fn(On)16 b(the)g(one)f(hand,)g(w)o (e)g(obtain)206 2598 y Fe(Z)231 2701 y Fh(Q)269 2660 y Fi(dx)327 2598 y Fe(Z)351 2701 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\025)p Fh(R)505 2629 y Fg(j)p Fi(\032)541 2636 y Fh(n)565 2629 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)685 2613 y Fv(2)p 505 2649 V 539 2691 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)715 2660 y Fi(dy)43 b Fg(\024)892 2629 y Fn(1)p 886 2649 35 2 v 886 2691 a Fi(R)946 2598 y Fe(Z)972 2701 y Fh(Q)1009 2660 y Fi(dx)1067 2598 y Fe(Z)1092 2701 y Fd(R)1122 2692 y Ft(3)1149 2660 y Fg(j)p Fi(\032)1185 2667 y Fh(n)1208 2660 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1328 2641 y Fv(2)1354 2660 y Fi(dy)804 2788 y Fn(=)892 2757 y(1)p 886 2777 V 886 2819 a Fi(R)939 2788 y Fg(k)p Fi(\032)985 2795 y Fh(n)1009 2788 y Fn(\()p Fi(\030)r Fn(;)g Fi(x;)g(y)r Fn(\))p Fg(k)1182 2769 y Fv(2)1182 2802 y Fh(L)1206 2792 y Ft(2)1220 2802 y Fv(\()p Fh(Q)1262 2792 y Fc(?)1280 2802 y Fu(\002)p Fh(Q)p Fu(\002)p Fh(Q)p Fv(\))1509 2788 y Fg(\024)1653 2757 y Fi(C)p 1653 2777 36 2 v 1653 2819 a(R)1694 2788 y(;)52 b Fn(\(5.22\))p eop %%Page: 62 62 62 61 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(62)37 199 y(for)17 b(some)h(p)q(ositiv)o(e)g(constan)o(t)f Fi(C)k Fn(that)c(is)g(indep)q(enden)o(t)j(of)d Fi(n)p Fn(,)h(and)g(b)q(ecause)g(of)f(the)h(b)q(ound)g(\(5.13\))e(on)37 256 y Fi(\032)60 263 y Fh(n)84 256 y Fn(.)108 312 y(On)j(the)g(other)f (hand,)h(w)o(e)f(kno)o(w)g(from)g(\(5.15\),)f(that)h Fi(\032)1097 319 y Fh(n)1121 312 y Fn(\()p Fi(x;)8 b(y)r Fn(\))17 b(con)o(v)o(erges)h(to)g Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))17 b(strongly)h(in)37 369 y Fi(L)68 352 y Fv(4)68 382 y(lo)q(c)115 369 y Fn(\()p Fj(R)172 352 y Fv(3)205 369 y Fg(\002)13 b Fj(R)292 352 y Fv(3)312 369 y Fn(\).)35 b(Th)o(us,)21 b Fg(j)p Fi(\032)548 376 y Fh(n)571 369 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)691 352 y Fv(2)729 369 y Fn(con)o(v)o(erges)20 b(to)f Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1154 352 y Fv(2)1192 369 y Fn(strongly)20 b(in)h Fi(L)1462 352 y Fv(2)1462 382 y(lo)q(c)1508 369 y Fn(\()p Fj(R)1565 352 y Fv(3)1598 369 y Fg(\002)14 b Fj(R)1686 352 y Fv(3)1706 369 y Fn(\).)34 b(Since)76 407 y Fv(1)p 42 414 86 2 v 42 441 a Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j)133 425 y Fi(\037)161 434 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\024)p Fh(R)303 425 y Fn(\()p Fi(x;)8 b(y)r Fn(\))14 b(b)q(elongs)i(to)f Fi(L)674 409 y Fv(2)674 439 y(lo)q(c)720 425 y Fn(\()p Fj(R)777 409 y Fv(3)806 425 y Fg(\002)c Fj(R)891 409 y Fv(3)911 425 y Fn(\),)j(w)o(e)h(clearly)h(obtain,)g(for)e(an)o(y)h (\014xed)h Fi(R)d(>)g Fn(0,)339 551 y(lim)311 578 y Fh(n)p Fu(!)p Fv(+)p Fu(1)438 489 y Fe(Z)463 592 y Fh(Q)500 551 y Fi(dx)558 489 y Fe(Z)583 592 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\024)p Fh(R)737 520 y Fg(j)p Fi(\032)773 527 y Fh(n)797 520 y Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)917 503 y Fv(2)p 737 540 198 2 v 771 582 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)947 551 y Fi(dy)k Fn(=)1055 489 y Fe(Z)1081 592 y Fh(Q)1118 551 y Fi(dx)1176 489 y Fe(Z)1201 592 y Fu(j)p Fh(x)p Fu(\000)p Fh(y)q Fu(j\024)p Fh(R)1355 520 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1511 503 y Fv(2)p 1355 540 174 2 v 1377 582 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1541 551 y Fi(dy)r(:)157 b Fn(\(5.23\))37 666 y(Collecting)17 b(\(5.22\))c(and)j(\(5.23\),)d (and)i(letting)h Fi(n)p Fn(,)f(and)h(then)f Fi(R)p Fn(,)h(go)e(to)h (in\014nit)o(y)l(,)h(w)o(e)f(\014nally)i(obtain)393 779 y(lim)8 b(sup)403 815 y Fh(n)p Fu(!)p Fv(+)p Fu(1)540 717 y Fe(Z)565 820 y Fh(Q)602 779 y Fi(dx)660 717 y Fe(Z)685 820 y Fd(R)715 810 y Ft(3)747 748 y Fg(j)p Fi(\032)783 755 y Fh(n)807 748 y Fn(\()p Fi(x;)g(y)r Fn(\))p Fg(j)927 731 y Fv(2)p 747 768 198 2 v 781 810 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)957 779 y Fi(dy)j Fg(\024)1065 717 y Fe(Z)1091 820 y Fh(Q)1128 779 y Fi(dx)1186 717 y Fe(Z)1210 820 y Fd(R)1240 810 y Ft(3)1273 748 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1429 731 y Fv(2)p 1273 768 174 2 v 1294 810 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1459 779 y Fi(dy)r(:)37 890 y Fn(The)16 b(pro)q(of)f(is)g (no)o(w)g(complete.)1261 b Fg(})108 988 y Fn(W)l(e)17 b(shall)h(no)o(w)e(write)h(do)o(wn)f(the)h(system)f(of)h (Euler-Lagrange)g(equations)g(satis\014ed)g(b)o(y)g(an)o(y)f(mini-)37 1045 y(mizer)j Fi(K)h Fn(of)d Fi(I)301 1028 y Fh(H)s(F)297 1056 y(per)362 1045 y Fn(.)26 b(These)18 b(equations)g(lo)q(ok)g(lik)o (e)h(v)o(ery)e(m)o(uc)o(h)g(the)h(usual)g(Hartree-F)l(o)q(c)o(k)f (equations)h(in)37 1101 y(Molecular)h(Chemistry)l(,)e(and)h(are)f(lik)o (ely)i(to)e(b)q(e)g(the)h(analogues)f(in)h(the)g(p)q(erio)q(dic)h (setting)f(of)e(the)i(w)o(ell-)37 1157 y(kno)o(wn)h(Hartree-F)l(o)q(c)o (k)g(equations.)32 b(The)19 b(form)f(of)h(these)g(equations)h(is)f(v)o (ery)g(similar)h(to)f(the)g(one)g(w)o(e)37 1214 y(ha)o(v)o(e)g(deriv)o (ed)g(for)f(the)h(p)q(erio)q(dic)i(RHF)d(mo)q(del,)i(namely)f(\(4.57\)) e({\(4.59\),)g(except)i(for)f(an)h(extra)e(term)37 1270 y(whic)o(h)f(comes)e(from)g(the)h(exc)o(hange)g(term.)k(Arguing)c(b)o (y)g(analogy)f(with)h(the)f(p)q(erio)q(dic)j(RHF)e(mo)q(del,)g(w)o(e)37 1327 y(obtain)j(the)g(follo)o(wing)g(system)f(of)h(Euler-Lagrange)g (equations,)g(for)f(almost)g(ev)o(ery)g Fi(\030)j Fn(in)e Fi(Q)1679 1310 y Fh(?)1699 1327 y Fn(,)g(and)g(for)37 1383 y(ev)o(ery)e Fi(n)c Fg(\025)h Fn(1,)106 1424 y Fe(8)106 1465 y(>)106 1479 y(>)106 1492 y(>)106 1506 y(>)106 1520 y(>)106 1533 y(>)106 1547 y(>)106 1561 y(>)106 1574 y(>)106 1588 y(>)106 1602 y(>)106 1615 y(>)106 1629 y(>)106 1642 y(>)106 1656 y(>)106 1670 y(>)106 1683 y(>)106 1697 y(>)106 1711 y(>)106 1724 y(>)106 1738 y(>)106 1752 y(>)106 1765 y(>)106 1779 y(<)106 1861 y(>)106 1874 y(>)106 1888 y(>)106 1902 y(>)106 1915 y(>)106 1929 y(>)106 1942 y(>)106 1956 y(>)106 1970 y(>)106 1983 y(>)106 1997 y(>)106 2011 y(>)106 2024 y(>)106 2038 y(>)106 2052 y(>)106 2065 y(>)106 2079 y(>)106 2092 y(>)106 2106 y(>)106 2120 y(>)106 2133 y(>)106 2147 y(>)106 2161 y(>)106 2174 y(:)286 1493 y Fg(\000)p Fn(\001)p Fi(u)385 1500 y Fh(n)409 1493 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fn(\))g Fg(\000)j Fi(G)d(u)625 1500 y Fh(n)647 1493 y Fn(\()p Fi(\030)r(;)g Fg(\001)p Fn(\))h(+)798 1450 y Fe(X)793 1548 y Fh(m)p Fu(\025)p Fv(1)877 1493 y Fi(\025)903 1500 y Fh(m)937 1493 y Fn(\()p Fi(\030)r Fn(\))1003 1429 y Fe(\022)1035 1431 y(Z)1060 1534 y Fh(Q)1098 1493 y Fi(u)1124 1500 y Fh(m)1157 1493 y Fn(\()p Fi(\030)r(;)f(y)r Fn(\))1260 1474 y Fv(2)1278 1493 y Fi(G)p Fn(\()p Fg(\001)h(\000)i Fi(y)r Fn(\))d Fi(dy)1498 1429 y Fe(\023)1538 1493 y Fi(u)1564 1500 y Fh(n)1587 1493 y Fn(\()p Fi(\030)r(;)g Fg(\001)p Fn(\))570 1640 y Fg(\000)613 1578 y Fe(Z)d(Z)689 1682 y Fh(Q)717 1672 y Fc(?)734 1682 y Fu(\002)p Fh(Q)799 1640 y Fi(\032)p Fn(\()p Fi(\030)862 1622 y Fu(0)874 1640 y Fn(;)j Fi(x)p Fn(;)g Fi(y)r Fn(\))g Fi(W)1041 1622 y Fu(1)1076 1640 y Fn(\()p Fi(\030)k Fg(\000)e Fi(\030)1193 1622 y Fu(0)1205 1640 y Fn(;)e Fi(x)h Fg(\000)i Fi(y)r Fn(\))d Fi(u)1383 1647 y Fh(n)1405 1640 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))g Fi(dy)1591 1610 y(d\030)1637 1593 y Fu(0)p 1567 1630 106 2 v 1567 1672 a Fn(\(2)p Fi(\031)r Fn(\))1654 1658 y Fv(3)982 1752 y Fn(=)1035 1709 y Fe(X)1030 1807 y Fh(m)p Fu(\025)p Fv(1)1114 1752 y Fi(")1135 1759 y Fh(nm)1190 1752 y Fn(\()p Fi(\030)r Fn(\))g Fi(u)1282 1759 y Fh(m)1314 1752 y Fn(\()p Fi(\030)r(;)g Fg(\001)p Fn(\))p Fi(;)51 b Fn(a.e.)19 b(on)c Fi(Q;)162 1834 y Fe(Z)187 1937 y Fh(Q)217 1896 y Fn(\()p Fg(jr)8 b Fi(u)320 1903 y Fh(n)342 1896 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)460 1877 y Fv(2)488 1896 y Fg(\000)j Fi(G)p Fn(\()p Fi(x)p Fn(\))d Fg(j)p Fi(u)679 1903 y Fh(n)701 1896 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fg(j)819 1877 y Fv(2)836 1896 y Fn(\))g Fi(dx)h Fn(+)972 1853 y Fe(X)967 1951 y Fh(m)p Fu(\025)p Fv(1)1051 1896 y Fi(\025)1077 1903 y Fh(m)1111 1896 y Fn(\()p Fi(\030)r Fn(\))f Fi(D)1215 1903 y Fh(G)1243 1896 y Fn(\()p Fg(j)p Fi(u)1300 1903 y Fh(n)1322 1896 y Fn(\()p Fi(\030)r(;)g Fg(\001)p Fn(\))p Fg(j)1427 1877 y Fv(2)1445 1896 y Fi(;)g Fg(j)p Fi(u)1505 1903 y Fh(m)1537 1896 y Fn(\()p Fi(\030)r(;)g Fg(\001)p Fn(\))p Fg(j)1642 1877 y Fv(2)1660 1896 y Fn(\))405 2043 y Fg(\000)477 1981 y Fe(Z)d(Z)g(Z)461 2122 y Fh(Q)489 2110 y Fc(?)506 2122 y Fu(\002)p Fh(Q)561 2110 y Ft(2)597 2043 y Fi(\032)p Fn(\()p Fi(\030)660 2024 y Fu(0)671 2043 y Fn(;)j Fi(x)p Fn(;)g Fi(y)r Fn(\))g Fi(W)838 2024 y Fu(1)873 2043 y Fn(\()p Fi(\030)k Fg(\000)f Fi(\030)991 2024 y Fu(0)1002 2043 y Fn(;)d Fi(x)i Fg(\000)g Fi(y)r Fn(\))e Fi(u)1180 2024 y Fu(\003)1180 2054 y Fh(n)1203 2043 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))p Fi(u)1334 2050 y Fh(n)1356 2043 y Fn(\()p Fi(\030)r(;)g(y)r Fn(\))g Fi(dxdy)1591 2012 y(d\030)1637 1996 y Fu(0)p 1567 2033 V 1567 2074 a Fn(\(2)p Fi(\031)r Fn(\))1654 2061 y Fv(3)1262 2195 y Fn(=)13 b Fi(\026)1337 2177 y Fv(0)1337 2207 y Fh(n)1360 2195 y Fn(\()p Fi(\030)r Fn(\))d(+)g Fi(\026)1500 2177 y Fv(1)1500 2207 y Fh(n)1524 2195 y Fn(\()p Fi(\030)r Fn(\))g(+)g Fi(\031)r(;)1759 1831 y Fn(\(5.24\))37 2282 y(with)25 b Fi(\032)p Fn(\()p Fi(\030)213 2266 y Fu(0)224 2282 y Fn(;)8 b Fi(x)p Fn(;)g Fi(y)r Fn(\))25 b(=)422 2248 y Fe(P)470 2296 y Fh(m)p Fu(\025)p Fv(1)556 2282 y Fi(\025)582 2289 y Fh(m)615 2282 y Fn(\()p Fi(\030)655 2266 y Fu(0)667 2282 y Fn(\))p Fi(u)711 2289 y Fh(m)744 2282 y Fn(\()p Fi(\030)784 2266 y Fu(0)795 2282 y Fi(;)8 b(x)p Fn(\))p Fi(u)886 2266 y Fu(\003)886 2294 y Fh(m)918 2282 y Fn(\()p Fi(\030)958 2266 y Fu(0)969 2282 y Fi(;)g(y)r Fn(\),)25 b(and)f(where)g(the)g(Lagrange)f(m)o(ultipliers)j Fi(\031)r Fn(,)37 2343 y Fi(\026)64 2327 y Fv(0)64 2355 y Fh(n)88 2343 y Fn(\()p Fi(\030)r Fn(\),)14 b Fi(\026)200 2327 y Fv(1)200 2355 y Fh(n)224 2343 y Fn(\()p Fi(\030)r Fn(\),)g Fi(")330 2350 y Fh(nm)385 2343 y Fn(\()p Fi(\030)r Fn(\))h(are)g(resp)q(ectiv)o(ely)i(asso)q(ciated)e(to)g(the)g(set)g(of) g(constrain)o(ts)366 2387 y Fe(8)366 2428 y(>)366 2442 y(>)366 2455 y(>)366 2469 y(>)366 2483 y(>)366 2496 y(>)366 2510 y(>)366 2523 y(>)366 2537 y(>)366 2551 y(<)366 2633 y(>)366 2646 y(>)366 2660 y(>)366 2673 y(>)366 2687 y(>)366 2701 y(>)366 2714 y(>)366 2728 y(>)366 2742 y(>)366 2755 y(:)427 2389 y(Z)452 2492 y Fh(Q)480 2483 y Fc(?)508 2408 y Fe(X)508 2506 y Fh(n)p Fu(\025)p Fv(1)582 2451 y Fi(\025)608 2458 y Fh(n)632 2451 y Fn(\()p Fi(\030)r Fn(\))731 2420 y Fi(d\030)p 702 2440 V 702 2482 a Fn(\(2)p Fi(\031)r Fn(\))789 2469 y Fv(3)824 2451 y Fn(=)e(1)p Fi(;)427 2612 y Fn(0)f Fg(\024)h Fi(\025)536 2619 y Fh(n)560 2612 y Fn(\()p Fi(\030)r Fn(\))f Fg(\024)h Fn(1)p Fi(;)i Fn(for)f(all)24 b Fi(n)13 b Fg(\025)g Fn(1)p Fi(;)h Fn(and)i(for)e (almost)h(all)9 b Fi(\030)14 b Fg(2)f Fi(Q)1481 2595 y Fh(?)1501 2612 y Fi(;)427 2685 y Fe(Z)452 2788 y Fh(Q)490 2747 y Fi(u)516 2754 y Fh(n)539 2747 y Fn(\()p Fi(\030)r(;)8 b(x)p Fn(\))p Fi(u)670 2728 y Fu(\003)670 2758 y Fh(m)702 2747 y Fn(\()p Fi(\030)r(;)g(x)p Fn(\))g Fi(dx)i Fn(=)j Fi(\016)943 2754 y Fh(n;m)1008 2747 y Fi(;)i Fn(for)f(almost)h(all)9 b Fi(\030)14 b Fg(2)f Fi(Q)1421 2728 y Fh(?)1441 2747 y Fi(:)1759 2603 y Fn(\(5.25\))p eop %%Page: 63 63 63 62 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(63)108 199 y(Since,)18 b(once)e(more,)g Fi(E)510 183 y Fh(H)s(F)507 210 y(per)570 199 y Fn(\()p Fi(K)s Fn(\))f(is)i(indep)q(enden)o(t)i(of)d(the)g(c)o(hoice)h(of)f(an)g (eigen)o(basis)i(for)d Fi(K)1676 206 y Fh(\030)1695 199 y Fn(,)h(w)o(e)g(ma)o(y)37 256 y(assume)i(without)g(loss)g(of)f (generalit)o(y)i(that)e(the)h(matrix)f(of)h Fi(")1133 263 y Fh(nm)1188 256 y Fn(\()p Fi(\030)r Fn(\))f(is)h(diagonal,)h(for)e (almost)g(ev)o(ery)h Fi(\030)37 312 y Fn(in)j Fi(Q)131 296 y Fh(?)151 312 y Fn(;)h(under)e(this)g(assumption,)h(the)f(righ)o (t-hand)h(side)f(term,)1205 278 y Fe(P)1253 326 y Fh(m)p Fu(\025)p Fv(1)1339 312 y Fi(")1360 319 y Fh(nm)1415 312 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)1499 319 y Fh(m)1532 312 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fn(\),)19 b(in)h(the)g(\014rst) 37 369 y(equation)f(of)f(\(5.24\))f(b)q(ecomes)i(simply)h Fi(")770 376 y Fh(n)794 369 y Fn(\()p Fi(\030)r Fn(\))p Fi(u)878 376 y Fh(n)900 369 y Fn(\()p Fi(\030)r(;)8 b Fg(\001)p Fn(\).)28 b(In)19 b(addition,)h(the)f(Lagrange)f(m)o (ultipliers)j(still)37 425 y(satisfy)c(the)f(prop)q(erties)h (\(4.58a\),)d(\(4.58b\),)h(and)h(\(4.59\),)f(that)g(w)o(e)h(recall)i (here)e(for)g(the)h(con)o(v)o(enience)g(of)37 482 y(the)f(reader;)f (namely)l(,)g(for)g(all)h Fi(n)d Fg(\025)g Fn(1)i(and)g(for)g(almost)f (ev)o(ery)h Fi(\030)j Fn(in)e Fi(Q)1244 465 y Fh(?)1263 482 y Fn(,)683 612 y Fi(\026)710 593 y Fv(0)710 623 y Fh(n)734 612 y Fn(\()p Fi(\030)r Fn(\))800 534 y Fe(\()836 581 y Fn(=)c(0)p Fi(;)60 b Fn(if)16 b Fi(\025)1047 588 y Fh(n)1071 581 y Fn(\()p Fi(\030)r Fn(\))c Fi(>)h Fn(0)p Fi(;)836 648 y Fg(\025)f Fn(0)p Fi(;)60 b Fn(if)16 b Fi(\025)1047 655 y Fh(n)1071 648 y Fn(\()p Fi(\030)r Fn(\))c(=)h(0)p Fi(;)683 829 y(\026)710 810 y Fv(1)710 840 y Fh(n)734 829 y Fn(\()p Fi(\030)r Fn(\))800 751 y Fe(\()836 797 y Fn(=)f(0)p Fi(;)60 b Fn(if)16 b Fi(\025)1047 804 y Fh(n)1071 797 y Fn(\()p Fi(\030)r Fn(\))c Fi(<)h Fn(1,)836 865 y Fg(\024)f Fn(0)p Fi(;)60 b Fn(if)16 b Fi(\025)1047 872 y Fh(n)1071 865 y Fn(\()p Fi(\030)r Fn(\))c(=)h(1,)37 960 y(and)622 1008 y Fe(8)622 1049 y(>)622 1063 y(<)622 1144 y(>)622 1158 y(:)662 1049 y Fi(\025)688 1056 y Fh(n)712 1049 y Fn(\()p Fi(\030)r Fn(\))f(=)h(0)129 b(=)-8 b Fg(\))29 b Fi(")1104 1056 y Fh(n)1127 1049 y Fn(\()p Fi(\030)r Fn(\))12 b Fg(\025)h Fi(\031)r(;)662 1117 y Fn(0)g Fi(<)g(\025)772 1124 y Fh(n)796 1117 y Fn(\()p Fi(\030)r Fn(\))f Fi(<)h Fn(1)45 b(=)-8 b Fg(\))29 b Fi(")1104 1124 y Fh(n)1127 1117 y Fn(\()p Fi(\030)r Fn(\))12 b(=)h Fi(\031)r(;)662 1185 y(\025)688 1192 y Fh(n)712 1185 y Fn(\()p Fi(\030)r Fn(\))f(=)h(1)129 b(=)-8 b Fg(\))29 b Fi(")1104 1192 y Fh(n)1127 1185 y Fn(\()p Fi(\030)r Fn(\))12 b Fg(\024)h Fi(\031)r(:)108 1321 y Fn(Once)f(w)o(e)f(ha)o(v)o(e)f(established)j(the)e(existence)h (of)e(a)h(minimizer)i(for)d(the)h(p)q(erio)q(dic)i(Hartree-F)l(o)q(c)o (k)d(mo)q(del,)37 1377 y(w)o(e)i(are)f(able)h(to)f(compare)g(from)g(ab) q(o)o(v)o(e)g(the)h(Hartree-F)l(o)q(c)o(k)e(energy)i(p)q(er)g(unit)g(v) o(olume)g(with)g(this)g(p)q(erio)q(dic)37 1433 y(problem.)37 1554 y Fb(5.2)57 b(Upp)r(er)19 b(limit)f(of)h(the)g(energy)h(p)r(er)f (unit)g(v)n(olume)37 1640 y Fn(In)d(this)g(section,)f(w)o(e)g(pro)o(v)o (e)g(the)g(follo)o(wing.)37 1721 y Fj(Prop)q(osition)23 b(2.1:)k Ff(We)19 b(assume)f(that)i(the)f(V)m(an)e(Hove)i(se)n(quenc)n (e)e Fn(\003)h Ff(satis\014es)f(\(2.21\).)28 b(In)19 b(addition,)37 1778 y(we)g(assume)f(that)i(the)e(unit)h(c)n(el)r(l)e Fi(Q)i Ff(is)f(a)h(cub)n(e,)f(and)h(that)g(ther)n(e)f(exists)g(a)h (minimizer)f Fi(K)h Fg(2)e(K)i Ff(of)g Fi(I)1815 1761 y Fh(H)s(F)1811 1789 y(per)37 1834 y Ff(whose)e(density)f Fi(\032)g Ff(shar)n(es)g(the)g(symmetries)g(of)h(the)f(unit)h(cub)n(e.) j(Then,)700 1956 y Fn(lim)8 b(sup)722 1995 y Fv(\003)p Fu(!1)851 1924 y Fi(I)875 1908 y Fh(H)s(F)871 1938 y Fv(\003)p 851 1945 85 2 v 865 1987 a Fg(j)p Fn(\003)p Fg(j)953 1956 y(\024)13 b Fi(I)1025 1937 y Fh(H)s(F)1021 1967 y(per)1096 1956 y Fn(+)1146 1925 y Fi(M)p 1146 1945 50 2 v 1159 1987 a Fn(2)1200 1956 y Fi(;)37 2081 y Ff(wher)n(e)k Fi(I)190 2064 y Fh(H)s(F)186 2092 y(per)267 2081 y Ff(is)e(de\014ne)n (d)h(by)g(\(2.23\){\(2.24\).)37 2204 y Fj(Remark)i(5.1)24 b(\(1\))f Ff(The)16 b(same)e(r)n(esult)h(holds)f(true)i(in)e(the)h(sme) n(ar)n(e)n(d)f(nuclei)g(c)n(ase,)g(if)h(we)g(assume)g(mor)n(e-)151 2261 y(over)i(that)g Fi(m)f Ff(shar)n(es)f(the)i(symmetries)f(of)g(the) h(cub)n(e)f Fi(Q)p Ff(,)g(and)g(de\014ne)g Fi(M)21 b Ff(ac)n(c)n(or)n(ding)15 b(to)i(\(2.22\).)37 2352 y Fj(\(2\))24 b Ff(In)16 b(the)g(HF)f(setting,)g(sinc)n(e)f(we)i(do)g(not)g(know)f (whether)i Fi(\032)f Ff(is)f(unique,)h(the)g(assertion)f(that)h Fi(\032)h Ff(shar)n(es)151 2409 y(the)g(symmetries)f(of)g(the)g(cub)n (e)h(ne)n(e)n(ds)d(to)j(b)n(e)f(assume)n(d.)37 2532 y Fj(Pro)q(of)27 b(of)e(Prop)q(osition)i(2.1:)36 b Fn(The)22 b(b)q(eginning)i(of)d(the)h(pro)q(of)g(is)h(the)f(same)f(as)h(in)h(the) f(pro)q(of)f(of)37 2589 y(Prop)q(osition)d(4.2)f(for)g(the)g(RHF)h (problem.)27 b(Let)18 b(us)f(denote)h(b)o(y)f Fi(K)k Fn(a)c(minimizer)i(of)e(the)h(p)q(erio)q(dic)h(HF)37 2645 y(problem)f(suc)o(h)f(that)f(the)h(corresp)q(onding)h(densit)o(y)f Fi(\032)g Fn(shares)g(the)g(symmetries)f(of)h(the)g(unit)g(cub)q(e.)26 b(W)l(e)37 2702 y(set)643 2796 y Fi(@)s Fn(\003)12 b(=)h Fg(f)p Fi(k)g Fg(2)g Fn(\003;)i Fi(d)p Fn(\()p Fi(k)q(;)8 b(@)s Fn(\000\(\003\)\))h Fg(\024)k Fn(2)p Fg(g)p Fi(;)p eop %%Page: 64 64 64 63 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(64)753 201 y(\000\()p Fi(@)s Fn(\003\))11 b(=)954 158 y Fe([)935 257 y Fh(k)q Fu(2)p Fh(@)r Fv(\003)1031 201 y Fi(k)g Fn(+)f Fi(Q;)844 349 y Fu(\016)837 389 y Fn(\003)882 385 y(=)j(\003)c Fg(n)h Fi(@)s Fn(\003)p Fi(;)37 473 y Fn(and)616 560 y(\000\()669 524 y Fu(\016)662 564 y Fn(\003)693 560 y(\))j(=)780 517 y Fe([)772 635 y Fh(k)q Fu(2)819 609 y Fp(\016)815 638 y Fv(\003)847 560 y Fi(k)e Fn(+)f Fi(Q)j Fn(=)g(\000\(\003\))c Fg(n)h Fn(\000\()p Fi(@)s Fn(\003\))p Fi(:)37 716 y Fn(W)l(e)20 b(build)h(a)f(cut-o\013)g(function)g Fi(\037)640 723 y Fv(\003)687 716 y Fg(2)g(D)q Fn(\()p Fj(R)830 699 y Fv(3)850 716 y Fn(\))f(satisfying)g(the)h(follo)o(wing)g(prop)q (erties:)29 b(0)20 b Fg(\024)g Fi(\037)1738 723 y Fv(\003)1785 716 y Fg(\024)g Fn(1,)37 787 y Fi(\037)65 794 y Fv(\003)105 787 y Fg(\021)13 b Fn(1)i(on)g(\000\()307 751 y Fu(\016)300 791 y Fn(\003)332 787 y(\),)f Fi(\037)405 794 y Fv(\003)445 787 y Fg(\021)f Fn(0)i(on)g(\000\(\003\))690 770 y Fh(c)706 787 y Fn(.)20 b(In)c(particular,)g(w)o(e)f(ha)o(v)o(e)558 833 y Fe(Z)583 936 y Fd(R)613 927 y Ft(3)640 895 y Fi(\037)668 876 y Fv(2)668 906 y(\003)695 895 y Fn(\()p Fi(x)p Fn(\))p Fi(\032)p Fn(\()p Fi(x;)8 b(x)p Fn(\))p Fi(dx)j Fn(=)i Fg(j)p Fn(\003)p Fg(j)c Fn(+)h Fi(o)p Fn(\()p Fg(j)p Fn(\003)p Fg(j)p Fn(\))h Fg(\024)i(j)p Fn(\003)p Fg(j)p Fi(:)403 b Fn(\(5.26\))37 1009 y(W)l(e)16 b(next)f(consider)h(the)f(op) q(erator)g Fi(K)696 1016 y Fv(\003)737 1009 y Fn(on)g Fi(L)831 993 y Fv(2)851 1009 y Fn(\()p Fj(R)908 993 y Fv(3)928 1009 y Fn(\))f(whose)h(k)o(ernel)h(is)662 1097 y Fi(\032)685 1104 y Fv(\003)712 1097 y Fn(\()p Fi(x;)8 b(y)r Fn(\))j(=)i Fi(\037)906 1104 y Fv(\003)933 1097 y Fn(\()p Fi(x)p Fn(\))p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fi(\037)1153 1104 y Fv(\003)1178 1097 y Fn(\()p Fi(y)r Fn(\))p Fi(:)37 1184 y Fn(Then,)17 b(b)q(ecause)g(of)f (\(5.26\),)e(and)i(since)h Fi(I)762 1168 y Fh(H)s(F)758 1198 y Fv(\003)839 1184 y Fn(is)f(also)g(obtained)h(b)o(y)f(minimizing) j Fi(E)1502 1168 y Fh(H)s(F)1499 1198 y Fv(\003)1578 1184 y Fn(when)d(the)g(trace)37 1241 y(constrain)o(t)f(is)h(relaxed,)g (that)e(is)i(on)f(the)g(set)g(of)g(self-adjoin)o(t)h(op)q(erators)359 1328 y Fg(K)402 1309 y Fu(0)402 1339 y Fv(\003)441 1328 y Fn(=)d Fg(f)p Fn(0)f Fg(\024)h Fi(K)i Fg(\024)31 b Fj(1)p Fi(;)22 b Fn(T)l(r)15 b Fi(K)h Fg(\024)d(j)p Fn(\003)p Fg(j)p Fi(;)21 b Fn(T)l(r)1080 1291 y Fe(\002)1099 1328 y Fn(\()p Fg(\000)p Fn(\001)10 b Fg(\000)g Fi(V)1271 1335 y Fv(\003)1298 1328 y Fn(\))p Fi(K)1358 1291 y Fe(\003)1389 1328 y Fi(<)j Fn(+)p Fg(1g)p Fi(;)37 1415 y Fn(w)o(e)i(ha)o(v)o(e)745 1490 y Fi(I)769 1473 y Fh(H)s(F)765 1503 y Fv(\003)p 745 1510 85 2 v 759 1552 a Fg(j)p Fn(\003)p Fg(j)847 1521 y(\024)917 1490 y Fn(1)p 900 1510 57 2 v 900 1552 a Fg(j)p Fn(\003)p Fg(j)962 1521 y Fi(E)999 1502 y Fh(H)s(F)996 1532 y Fv(\003)1059 1521 y Fn(\()p Fi(K)1116 1528 y Fv(\003)1142 1521 y Fn(\))p Fi(:)37 1630 y Fn(W)l(e)h(sho)o(w)e(exactly)i(as)f(in)h (the)f(pro)q(of)g(of)g(Prop)q(osition)g(4.2)g(that)381 1740 y(lim)354 1770 y Fv(\003)p Fu(\000)-5 b(!1)500 1709 y Fn(1)p 483 1729 V 483 1771 a Fg(j)p Fn(\003)p Fg(j)552 1676 y Fe(\024)576 1740 y Fn(T)l(r)623 1750 y Fh(L)647 1740 y Ft(2)665 1750 y Fv(\()p Fd(R)709 1740 y Ft(3)726 1750 y Fv(\))742 1740 y Fn([)p Fg(\000)p Fn(\001)p Fi(K)867 1747 y Fv(\003)893 1740 y Fn(])9 b Fg(\000)961 1678 y Fe(Z)986 1781 y Fd(R)1016 1772 y Ft(3)1043 1740 y Fi(V)1069 1747 y Fv(\003)1097 1740 y Fi(\032)1120 1747 y Fv(\003)1157 1740 y Fn(+)1207 1709 y(1)p 1207 1729 23 2 v 1207 1771 a(2)1235 1740 y Fi(D)q Fn(\()p Fi(\032)1315 1747 y Fv(\003)1341 1740 y Fi(;)f(\032)1385 1747 y Fv(\003)1412 1740 y Fn(\))h(+)1490 1709 y(1)p 1490 1729 V 1490 1771 a(2)1518 1740 y Fi(U)1549 1747 y Fv(\003)1575 1676 y Fe(\025)395 1862 y Fn(=)472 1800 y Fe(Z)497 1903 y Fh(Q)525 1894 y Fc(?)553 1862 y Fn(T)l(r)600 1874 y Fh(L)624 1862 y Ft(2)624 1886 y Fc(\030)641 1874 y Fv(\()p Fh(Q)p Fv(\))698 1862 y Fn([)p Fg(\000)p Fn(\001)784 1869 y Fh(\030)803 1862 y Fi(K)842 1869 y Fh(\030)860 1862 y Fn(])916 1831 y Fi(d\030)p 886 1852 106 2 v 886 1893 a Fn(\(2)p Fi(\031)r Fn(\))973 1880 y Fv(3)1006 1862 y Fg(\000)1052 1800 y Fe(Z)1077 1903 y Fh(Q)1114 1862 y Fi(G\032)i Fn(+)1234 1831 y(1)p 1234 1852 23 2 v 1234 1893 a(2)1262 1862 y Fi(D)1300 1869 y Fh(G)1329 1862 y Fn(\()p Fi(\032;)d(\032)p Fn(\))i(+)1492 1831 y Fi(M)p 1492 1852 50 2 v 1505 1893 a Fn(2)1546 1862 y Fi(:)37 1979 y Fn(According)i(to)d(the)i(de\014nition)h(\(2.5\)) c(of)i(the)g(Hartree-F)l(o)q(c)o(k)g(energy)l(,)h(to)f(reac)o(h)g(the)g (conclusion,)j(it)e(remains)37 2035 y(no)o(w)k(to)g(compute)g(the)g (exc)o(hange)h(term)f(corresp)q(onding)h(to)e Fi(K)1140 2042 y Fv(\003)1166 2035 y Fn(,)h(and)h(to)e(sho)o(w)h(that)372 2148 y(lim)345 2178 y Fv(\003)p Fu(\000)-5 b(!1)492 2117 y Fn(1)p 475 2137 57 2 v 475 2179 a Fg(j)p Fn(\003)p Fg(j)544 2086 y Fe(Z)5 b(Z)600 2189 y Fd(R)630 2180 y Ft(3)647 2189 y Fu(\002)p Fd(R)704 2180 y Ft(3)737 2117 y Fg(j)p Fi(\032)773 2124 y Fv(\003)800 2117 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)920 2101 y Fv(2)p 737 2137 201 2 v 772 2179 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)955 2148 y Fi(dxdy)k Fn(=)1113 2086 y Fe(Z)1138 2189 y Fh(Q)1176 2148 y Fi(dx)1234 2086 y Fe(Z)1258 2189 y Fd(R)1288 2180 y Ft(3)1320 2117 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1476 2101 y Fv(2)p 1320 2137 174 2 v 1342 2179 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1507 2148 y Fi(dy)r(:)37 2262 y Fn(It)16 b(is)f(easily)i(seen)e(that)189 2344 y(1)p 172 2364 57 2 v 172 2406 a Fg(j)p Fn(\003)p Fg(j)242 2313 y Fe(Z)5 b(Z)317 2416 y Fd(R)347 2407 y Ft(3)365 2416 y Fu(\002)p Fd(R)422 2407 y Ft(3)455 2344 y Fg(j)p Fi(\032)491 2351 y Fv(\003)517 2344 y Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)637 2328 y Fv(2)p 455 2364 201 2 v 490 2406 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)668 2375 y Fi(dxdy)209 2505 y Fn(=)308 2474 y(1)p 291 2494 57 2 v 291 2536 a Fg(j)p Fn(\003)p Fg(j)360 2443 y Fe(Z)5 b(Z)436 2546 y Fd(R)466 2537 y Ft(3)483 2546 y Fu(\002)p Fd(R)540 2537 y Ft(3)573 2474 y Fi(\037)601 2481 y Fv(\003)628 2474 y Fn(\()p Fi(x)p Fn(\))690 2458 y Fv(2)709 2474 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)865 2458 y Fv(2)883 2474 y Fi(\037)911 2481 y Fv(\003)938 2474 y Fn(\()p Fi(y)r Fn(\))998 2458 y Fv(2)p 573 2494 444 2 v 730 2536 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1030 2505 y Fi(dxdy)209 2635 y Fn(=)308 2604 y(1)p 291 2624 57 2 v 291 2666 a Fg(j)p Fn(\003)p Fg(j)360 2573 y Fe(Z)5 b(Z)436 2676 y Fv(\000\()476 2650 y Fp(\016)472 2679 y Fv(\003)496 2676 y(\))p Fu(\002)p Fv(\000\()577 2650 y Fp(\016)573 2679 y Fv(\003)597 2676 y(\))626 2604 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)782 2587 y Fv(2)p 626 2624 174 2 v 647 2666 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)812 2635 y Fi(dxdy)h Fn(+)987 2604 y(2)p 970 2624 57 2 v 970 2666 a Fg(j)p Fn(\003)p Fg(j)1039 2573 y Fe(Z)6 b(Z)1115 2676 y Fv(\000\()1155 2650 y Fp(\016)1151 2679 y Fv(\003)1175 2676 y(\))p Fu(\002)p Fv(\000\()p Fh(@)r Fv(\003\))1326 2604 y Fi(\037)1354 2611 y Fv(\003)1381 2604 y Fn(\()p Fi(x)p Fn(\))1443 2587 y Fv(2)1462 2604 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)i(y)r Fn(\))p Fg(j)1618 2587 y Fv(2)p 1326 2624 310 2 v 1415 2666 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1648 2635 y Fi(dxdy)331 2769 y Fn(+)396 2739 y(1)p 379 2759 57 2 v 379 2801 a Fg(j)p Fn(\003)p Fg(j)448 2708 y Fe(Z)c(Z)524 2811 y Fv(\000\()p Fh(@)r Fv(\003\))p Fu(\002)p Fv(\000\()p Fh(@)r Fv(\003\))755 2739 y Fi(\037)783 2746 y Fv(\003)810 2739 y Fn(\()p Fi(x)p Fn(\))872 2722 y Fv(2)891 2739 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)i(y)r Fn(\))p Fg(j)1047 2722 y Fv(2)1065 2739 y Fi(\037)1093 2746 y Fv(\003)1120 2739 y Fn(\()p Fi(y)r Fn(\))1180 2722 y Fv(2)p 755 2759 444 2 v 912 2801 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1212 2769 y Fi(dxdy)r(:)p eop %%Page: 65 65 65 64 bop 53 50 a Fm(5.)23 b Fl(The)16 b(Har)m(tree-F)o(ock)f(model) 1177 b Fn(65)37 199 y(Let)16 b(us)f(\014rst)f(concen)o(trate)h(on)g (the)g(last)g(t)o(w)o(o)f(terms)g(and)h(sho)o(w)g(that)f(they)h(con)o (v)o(erge)g(to)f(0)h(as)f(\003)h(go)q(es)g(to)37 256 y(in\014nit)o(y)l(.)27 b(Indeed,)19 b(since)f(0)d Fg(\024)h Fi(\037)606 263 y Fv(\003)649 256 y Fg(\024)g Fn(1,)g(the)h(sum)g(of)g (these)g(t)o(w)o(o)f(terms)g(is)h Ff(a)i(fortiori)j Fn(less)17 b(or)g(equal)h(to)37 312 y(the)e(follo)o(wing)g(quan)o(tit)o(y)579 439 y Fi(Q)615 446 y Fv(\003)654 439 y Fn(=)724 409 y(2)p 707 429 57 2 v 707 471 a Fg(j)p Fn(\003)p Fg(j)777 378 y Fe(Z)5 b(Z)853 481 y Fd(R)883 471 y Ft(3)900 481 y Fu(\002)p Fv(\000\()p Fh(@)r Fv(\003\))1037 409 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1193 392 y Fv(2)p 1037 429 174 2 v 1058 471 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1223 439 y Fi(dxdy)r(;)37 570 y Fn(whic)o(h)17 b(mak)o(es)d(sense)i(since)658 627 y Fe(Z)684 730 y Fh(Q)721 689 y Fi(dx)779 627 y Fe(Z)804 730 y Fd(R)834 721 y Ft(3)866 658 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1022 642 y Fv(2)p 866 678 V 888 720 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1052 689 y Fi(dy)k(<)e Fn(+)p Fg(1)p Fi(:)37 824 y Fn(Besides,)k(using)f(the)f(fact)f(that)h Fi(\032)p Fn(\()p Fi(x)10 b Fn(+)h Fi(k)q(;)d(y)j Fn(+)f Fi(k)q Fn(\))j(=)g Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\),)14 b(for)g(ev)o(ery)h Fi(k)h Fn(in)g Fj(Z)1419 808 y Fv(3)1439 824 y Fn(,)f(w)o(e)g(ha)o(v)o(e)79 966 y Fi(Q)115 973 y Fv(\003)154 966 y Fn(=)224 936 y(2)p 207 956 57 2 v 207 998 a Fg(j)p Fn(\003)p Fg(j)288 923 y Fe(X)276 1022 y Fh(k)q Fu(2)p Fh(@)r Fv(\003)372 905 y Fe(Z)397 1008 y Fh(Q)434 966 y Fi(dx)492 905 y Fe(Z)517 1008 y Fd(R)547 998 y Ft(3)579 936 y Fg(j)p Fi(\032)p Fn(\()p Fi(x)10 b Fn(+)h Fi(k)q(;)d(y)r Fn(\))p Fg(j)816 919 y Fv(2)p 579 956 255 2 v 601 998 a Fg(j)p Fi(x)i Fn(+)g Fi(k)h Fg(\000)g Fi(y)r Fg(j)846 966 y Fi(dy)448 1114 y Fn(=)518 1083 y(2)p 501 1104 57 2 v 501 1145 a Fg(j)p Fn(\003)p Fg(j)581 1071 y Fe(X)570 1170 y Fh(k)q Fu(2)p Fh(@)r Fv(\003)665 1052 y Fe(Z)691 1155 y Fh(Q)728 1114 y Fi(dx)786 1052 y Fe(Z)810 1155 y Fd(R)840 1146 y Ft(3)873 1083 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)d(y)j Fg(\000)f Fi(k)q Fn(\))p Fg(j)1108 1067 y Fv(2)p 873 1104 255 2 v 877 1145 a Fg(j)p Fi(x)f Fg(\000)i Fn(\()p Fi(y)h Fg(\000)e Fi(k)q Fn(\))p Fg(j)1140 1114 y Fi(dy)k Fn(=)f(2)1283 1083 y Fg(j)p Fi(@)s Fn(\003)p Fg(j)p 1283 1104 84 2 v 1297 1145 a(j)p Fn(\003)p Fg(j)1379 1052 y Fe(Z)1405 1155 y Fh(Q)1442 1114 y Fi(dx)1500 1052 y Fe(Z)1524 1155 y Fd(R)1554 1146 y Ft(3)1587 1083 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1743 1067 y Fv(2)p 1587 1104 174 2 v 1608 1145 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1773 1114 y Fi(dy)r(:)37 1260 y Fn(W)l(e)19 b(conclude)h(easily)l(,)g(since)f(b)o(y)f(de\014nition)i(of)e(V)l(an)h (Ho)o(v)o(e)e(sequences,)j Fg(j)p Fi(@)s Fn(\003)p Fg(j)c Fn(=)i Fi(o)p Fn(\()p Fg(j)p Fn(\003)p Fg(j)p Fn(\),)f(as)h(\003)g(go)q (es)g(to)37 1317 y(in\014nit)o(y)l(.)k(Therefore,)15 b(it)g(remains)h(to)e(c)o(hec)o(k)i(that)351 1413 y(1)p 334 1433 57 2 v 334 1475 a Fg(j)p Fn(\003)p Fg(j)404 1382 y Fe(Z)5 b(Z)459 1485 y Fv(\000\(\003\))p Fu(\002)p Fv(\000\(\003\))649 1413 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)805 1397 y Fv(2)p 649 1433 174 2 v 671 1475 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)836 1444 y Fi(dxdy)i Fn(=)993 1382 y Fe(Z)1019 1485 y Fh(Q)1056 1444 y Fi(dx)1114 1382 y Fe(Z)1139 1485 y Fd(R)1169 1476 y Ft(3)1201 1413 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)8 b(y)r Fn(\))p Fg(j)1357 1397 y Fv(2)p 1201 1433 V 1222 1475 a Fg(j)p Fi(x)i Fg(\000)h Fi(y)r Fg(j)1387 1444 y Fi(dy)h Fn(+)e Fi(o)p Fn(\(1\))p Fi(;)37 1575 y Fn(since)17 b(it)e(is)h(clear)g(that,)e(for)g(the)i (same)f(reasons)f(as)h(ab)q(o)o(v)o(e,)270 1671 y(1)p 253 1692 57 2 v 253 1733 a Fg(j)p Fn(\003)p Fg(j)323 1640 y Fe(Z)5 b(Z)378 1743 y Fv(\000\(\003\))p Fu(\002)p Fv(\000\(\003\))568 1671 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)724 1655 y Fv(2)p 568 1692 174 2 v 590 1733 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)755 1702 y Fi(dxdy)i Fn(=)934 1671 y(1)p 917 1692 57 2 v 917 1733 a Fg(j)p Fn(\003)p Fg(j)987 1640 y Fe(Z)5 b(Z)1042 1743 y Fv(\000\()1082 1717 y Fp(\016)1078 1746 y Fv(\003)1103 1743 y(\))p Fu(\002)p Fv(\000\()1184 1717 y Fp(\016)1180 1746 y Fv(\003)1204 1743 y(\))1232 1671 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1388 1655 y Fv(2)p 1232 1692 174 2 v 1254 1733 a Fg(j)p Fi(x)i Fg(\000)g Fi(y)r Fg(j)1419 1702 y Fi(dxdy)h Fn(+)g Fi(o)p Fn(\(1\))p Fi(:)37 1833 y Fn(W)l(e)16 b(rewrite)f(this)h(sum)f(in)h(the)f(follo)o(wing)h(w)o(a)o(y)187 1930 y(1)p 170 1950 57 2 v 170 1992 a Fg(j)p Fn(\003)p Fg(j)239 1898 y Fe(Z)6 b(Z)295 2002 y Fv(\000\(\003\))p Fu(\002)p Fv(\000\(\003\))485 1930 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)i(y)r Fn(\))p Fg(j)641 1913 y Fv(2)p 485 1950 174 2 v 507 1992 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)671 1960 y Fi(dxdy)43 b Fn(=)909 1930 y(1)p 892 1950 57 2 v 892 1992 a Fg(j)p Fn(\003)p Fg(j)975 1917 y Fe(X)961 2016 y Fh(k)q(;)6 b(l)p Fu(2)p Fv(\003)1063 1898 y Fe(Z)f(Z)1118 2002 y Fh(Q)p Fu(\002)p Fh(Q)1216 1930 y Fg(j)p Fi(\032)p Fn(\()p Fi(x)10 b Fn(+)g Fi(k)q(;)e(y)k Fn(+)e Fi(l)q Fn(\))p Fg(j)1522 1913 y Fv(2)p 1216 1950 325 2 v 1238 1992 a Fg(j)p Fi(x)f Fn(+)i Fi(k)g Fg(\000)f Fi(y)i Fg(\000)f Fi(l)q Fg(j)1553 1960 y Fi(dxdy)810 2112 y Fn(=)909 2081 y(1)p 892 2102 57 2 v 892 2143 a Fg(j)p Fn(\003)p Fg(j)975 2069 y Fe(X)961 2168 y Fh(k)q(;)6 b(l)p Fu(2)p Fv(\003)1063 2050 y Fe(Z)f(Z)1118 2153 y Fh(Q)p Fu(\002)p Fh(Q)1216 2081 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)j Fn(+)f Fi(l)h Fg(\000)f Fi(k)q Fn(\))p Fg(j)1521 2065 y Fv(2)p 1216 2102 325 2 v 1220 2143 a Fg(j)p Fi(x)g Fg(\000)g Fn(\()p Fi(y)i Fn(+)e Fi(l)h Fg(\000)f Fi(k)q Fn(\))p Fg(j)1553 2112 y Fi(dxdy)810 2264 y Fn(=)909 2233 y(1)p 892 2254 57 2 v 892 2295 a Fg(j)p Fn(\003)p Fg(j)975 2221 y Fe(X)961 2320 y Fh(k)q(;)c(l)p Fu(2)p Fv(\003)1063 2202 y Fe(Z)f(Z)1118 2305 y Fh(Q)p Fu(\002)p Fd(R)1203 2296 y Ft(3)1231 2264 y Fi(\037)1259 2271 y Fh(l)p Fu(\000)p Fh(k)q Fv(+)p Fh(Q)1374 2264 y Fn(\()p Fi(y)r Fn(\))1451 2233 y Fg(j)p Fi(\032)p Fn(\()p Fi(x;)j(y)r Fn(\))p Fg(j)1607 2217 y Fv(2)p 1451 2254 174 2 v 1473 2295 a Fg(j)p Fi(x)h Fg(\000)i Fi(y)r Fg(j)1638 2264 y Fi(dxdy)r(:)37 2412 y Fn(The)18 b(conclusion)h(is)f(then)g(easily)g (reac)o(hed)g(b)o(y)f(sho)o(wing)h(that)e(the)i(sequence)g(of)f (functions)h(de\014ned)h(b)o(y)56 2451 y Fv(1)p 42 2458 45 2 v 42 2485 a Fu(j)p Fv(\003)p Fu(j)99 2435 y Fe(P)147 2482 y Fh(k)q(;)6 b(l)p Fu(2)p Fv(\003)251 2469 y Fi(\037)279 2476 y Fh(l)p Fu(\000)p Fh(k)q Fv(+)p Fh(Q)394 2469 y Fn(\()p Fi(y)r Fn(\))18 b(con)o(v)o(erges)h(to)f(1)h(in)h Fi(L)869 2452 y Fu(1)906 2469 y Fn(\()p Fj(R)963 2452 y Fv(3)983 2469 y Fn(\))p Fi(?)p Fn(-w)o(eak,)f(and)g(almost)f(ev)o (erywhere)i(on)f Fj(R)1759 2452 y Fv(3)1779 2469 y Fn(.)31 b(T)l(o)37 2531 y(pro)o(v)o(e)14 b(this)g(claim,)h(w)o(e)f(could)h (apply)g(directly)g(the)f(tec)o(hnical)h(lemma)g(giv)o(en)f(in)h (Chapter)f(2)f(of)h([11)o(],)f(but,)37 2588 y(since)j(the)e(pro)q(of)g (is)h(simple,)h(w)o(e)e(repro)q(duce)h(the)g(argumen)o(t)e(here.)20 b(On)15 b(the)g(one)f(hand,)h(w)o(e)f(clearly)h(ha)o(v)o(e)429 2707 y(0)d Fg(\024)534 2676 y Fn(1)p 517 2696 57 2 v 517 2738 a Fg(j)p Fn(\003)p Fg(j)601 2663 y Fe(X)587 2762 y Fh(k)q(;)6 b(l)p Fu(2)p Fv(\003)688 2707 y Fi(\037)716 2714 y Fh(l)p Fu(\000)p Fh(k)q Fv(+)p Fh(Q)832 2707 y Fn(\()p Fi(y)r Fn(\))11 b Fg(\024)973 2676 y Fn(1)p 956 2696 V 956 2738 a Fg(j)p Fn(\003)p Fg(j)1027 2663 y Fe(X)1026 2762 y Fh(k)q Fu(2)p Fv(\003)1106 2663 y Fe(X)1101 2765 y Fh(l)p Fu(2)p Fd(Z)1161 2755 y Ft(3)1185 2707 y Fi(\037)1213 2714 y Fh(l)p Fu(\000)p Fh(k)q Fv(+)p Fh(Q)1328 2707 y Fn(\()p Fi(y)r Fn(\))h Fg(\024)h Fn(1)p Fi(:)p eop %%Page: 66 66 66 65 bop 53 50 a Fm(6.)23 b Fl(Extensions)16 b(and)h(perspectives)1109 b Fn(66)37 199 y(Therefore,)20 b(the)f(sequence)i(is)e(b)q(ounded)i(in) f Fi(L)863 183 y Fu(1)900 199 y Fn(\()p Fj(R)957 183 y Fv(3)977 199 y Fn(\).)31 b(On)19 b(the)h(other)e(hand,)i(let)g Fi(y)h Fn(b)q(e)f(\014xed)g(in)f Fj(R)1842 183 y Fv(3)1862 199 y Fn(.)37 256 y(Then,)c(there)g(exists)g Fi(m)d Fg(2)h Fj(Z)536 239 y Fv(3)570 256 y Fn(suc)o(h)i(that)f Fi(y)i Fn(lies)g(in)f Fi(m)8 b Fn(+)h Fi(Q)p Fn(,)15 b(and)f(for)g(\003)g (large)g(enough)h Fi(m)g Fn(lies)g(in)h(\003)8 b Fg(\000)h Fn(\003,)37 312 y(b)o(y)16 b(de\014nition)h(of)d(a)h(V)l(an)h(Ho)o(v)o (e)e(sequence.)21 b(Hence,)16 b(for)f(suc)o(h)g(an)g Fi(y)r Fn(,)g(w)o(e)g(obtain)373 431 y(1)42 b(=)537 400 y(1)p 520 420 57 2 v 520 462 a Fg(j)p Fn(\003)p Fg(j)590 388 y Fe(X)589 486 y Fh(k)q Fu(2)p Fv(\003)669 388 y Fe(X)664 489 y Fh(l)p Fu(2)p Fd(Z)724 479 y Ft(3)748 431 y Fi(\037)776 438 y Fh(l)p Fu(\000)p Fh(k)q Fv(+)p Fh(Q)891 431 y Fn(\()p Fi(y)r Fn(\))12 b(=)1033 400 y(1)p 1016 420 V 1016 462 a Fg(j)p Fn(\003)p Fg(j)1078 431 y Fn(#)p Fg(f)p Fi(k)q(;)i(l)f Fg(2)g Fn(\003;)i Fi(l)10 b Fg(\000)h Fi(k)i Fn(=)g Fi(m)p Fg(g)438 572 y Fn(=)537 541 y(1)p 520 562 V 520 603 a Fg(j)p Fn(\003)p Fg(j)581 572 y Fn(#)p Fg(f)p Fi(k)h Fg(2)f Fn(\003;)h Fi(k)d Fg(\000)g Fi(m)h Fg(2)h Fn(\003)p Fg(g)f Fn(=)1095 541 y(1)p 1078 562 V 1078 603 a Fg(j)p Fn(\003)p Fg(j)1140 572 y Fn(#[\003)d Fg(\\)h Fn(\(\003)g(+)g Fi(m)p Fn(\)])438 710 y Fg(\025)537 679 y Fn(1)p 520 699 V 520 741 a Fg(j)p Fn(\003)p Fg(j)581 710 y Fn(#[\003)f Fg(n)h Fn(\003)738 691 y Fu(j)p Fh(m)p Fu(j)p Fv(+)821 677 y Ft(1)p 821 683 16 2 v 821 704 a(2)843 710 y Fn(])58 b Fg(\025)h Fn(1)9 b Fg(\000)1091 679 y(j)p Fn(\003)1136 661 y Fu(j)p Fh(m)p Fu(j)p Fv(+)1219 647 y Ft(1)p 1218 653 V 1218 674 a(2)1241 679 y Fg(j)p 1091 699 163 2 v 1144 741 a(j)p Fn(\003)p Fg(j)1258 710 y Fi(;)37 850 y Fn(where)21 b(\003)206 832 y Fu(j)p Fh(m)p Fu(j)p Fv(+)289 818 y Ft(1)p 288 824 16 2 v 288 845 a(2)331 850 y Fn(=)g Fg(f)p Fi(k)g Fg(2)g Fn(\003;)15 b Fi(d)p Fn(\()p Fi(k)q Fn(;)f Fi(@)s Fn(\000\(\003\)\))19 b Fg(\024)i(j)p Fi(m)p Fg(j)12 b Fn(+)1008 832 y Fv(1)p 1008 839 18 2 v 1008 865 a(2)1030 850 y Fg(g)p Fn(,)21 b(and)f(where)g(the)g (notation)g(#)p Fi(S)i Fn(stands)e(for)37 906 y(the)d(n)o(um)o(b)q(er)g (of)g(elemen)o(ts)g(in)h(the)e(set)h Fi(S)s Fn(.)24 b(W)l(e)17 b(conclude)h(easily)g(since,)g(b)o(y)e(de\014nition)j(of)d(a)h(V)l(an)g (Ho)o(v)o(e)37 990 y(sequence,)243 921 y Fe(\014)243 949 y(\014)259 960 y Fv(\003)283 945 y Fp(j)p Fc(m)p Fp(j)p Ft(+)358 935 y(1)p 358 939 16 2 v 358 956 a(2)380 921 y Fe(\014)380 949 y(\014)p 243 980 152 2 v 297 1006 a Fu(j)p Fv(\003)p Fu(j)415 990 y Fn(go)q(es)e(to)g(0)g(as)f(\003)h(go) q(es)g(to)g(in\014nit)o(y)l(.)824 b Fg(})37 1133 y Fk(6)67 b(Extensions)23 b(and)g(p)r(ersp)r(ectiv)n(es)37 1235 y Fn(W)l(e)17 b(list)f(in)h(this)g(last)f(section)g(a)g(few)g(commen)o (ts)f(on)h(the)g(ab)q(o)o(v)o(e)g(results,)g(and)g(indicate)i(some)d(p) q(ossible)37 1291 y(extensions)h(of)f(our)g(w)o(ork.)108 1373 y(So)22 b(far,)g(w)o(e)f(ha)o(v)o(e)g(assumed)h(that)f(the)h(p)q (erio)q(dic)h(lattice)g(that)e(is)h(co)o(v)o(ered)f(in)i(the)e(limit)i (b)o(y)f(the)37 1429 y(sequence)f(\003)d(is)h Fj(Z)361 1413 y Fv(3)381 1429 y Fn(,)g(and)g(th)o(us)g(that)f(the)h(p)q(erio)q (dic)i(cell)f Fi(Q)f Fn(is)h(a)e(cub)q(e)i(of)e(unit)i(size.)32 b(The)19 b(\014rst)f(basic)37 1485 y(observ)m(ation)e(to)e(mak)o(e)h (is)h(that)e(our)h(whole)h(w)o(ork)e(go)q(es)h(through)f Ff(mutatis)j(mutandis)e Fn(if)h(w)o(e)f(replace)h(the)37 1542 y(cub)q(e)h(of)e(unit)i(size)f(b)o(y)g(a)f(cub)q(e)i(of)e(size)h Fi(R)p Fn(.)22 b(Sligh)o(t)16 b(mo)q(di\014cations)h(m)o(ust)e(b)q(e)h (made)g(in)g(the)g(de\014nition)h(of)37 1598 y(the)f(p)q(oten)o(tial)g Fi(G)f Fn(in)h(particular,)f(and)g(w)o(e)g(refer)g(the)h(reader)f(to)f ([11])g(for)h(suc)o(h)g(mo)q(di\014cations.)108 1655 y(Replacing)i(the)e(cub)q(e)i(b)o(y)e(another)f(shap)q(e)i(of)f(unit)g (cell)i(is)f(another)e(story)l(.)20 b(As)15 b(w)o(e)f(ha)o(v)o(e)h(men) o(tioned)37 1711 y(ab)q(o)o(v)o(e,)d(Theorems)g(2.1)f(and)h(2.3)f (still)j(hold.)19 b(On)13 b(the)f(con)o(trary)l(,)f(our)h(strategy)f (of)g(pro)q(of)h(for)f(Theorem)h(2.2)37 1768 y(dep)q(ends)19 b(up)q(on)e(the)g(shap)q(e)h(of)e(the)h(cell.)26 b(It)17 b(is)h(an)e(op)q(en)i(\(but)f(rather)f(tec)o(hnical\))i(question)f(to)f (extend)37 1824 y(this)g(result)g(to)e(other)h(shap)q(es)h(of)f(cells.) 108 1881 y(Lik)o(ewise,)i(w)o(e)e(ha)o(v)o(e)g(men)o(tioned)h(ab)q(o)o (v)o(e)f(that)f(the)i(assumption)f(\(2.21\))f(is)i(a)f(tec)o(hnical)i (assumption)37 1937 y(required)g(only)f(for)e(the)i(pro)q(of)f(of)g (Theorem)g(2.2.)k(W)l(e)c(recall)i(w)o(e)e(b)q(eliev)o(e)i(it)f(can)f (b)q(e)h(left)g(apart,)e(but)h(w)o(e)37 1994 y(do)h(not)e(kno)o(w)h(ho) o(w.)108 2050 y(Concerning)j(the)f(p)q(erio)q(dic)i(problems)f Ff(p)n(er)g(se)p Fn(,)f(it)g(w)o(ould)g(b)q(e)h(an)f(in)o(teresting)h (question)f(to)g(address)37 2106 y(to)i(see)g(whether)g(one)g(can)g(sa) o(y)f(something)h(on)g(the)f(minimizers.)33 b(In)20 b(the)e(HF)h (setting,)g(for)g(instance,)37 2163 y(w)o(e)c(are)g(not)g(able)h(to)f (c)o(hec)o(k,)g(for)g(the)g(time)h(b)q(eing,)g(whether)f(or)g(not)g (the)g(minimizing)j(p)q(erio)q(dic)f(densit)o(y)37 2219 y(matrix)f Fi(K)j Fn(is)e(a)f(pro)s(jector)f(\(whic)o(h)h(is)h(equiv)m (alen)o(t)h(to)d(the)h(fact)g(that)f Fi(K)1303 2226 y Fh(\030)1338 2219 y Fn(is)h(a)g(\(\014nite)h(rank\))f(pro)s(jector,)37 2276 y(for)f(almost)g(all)h Fi(\030)h Fn(in)f Fi(Q)443 2259 y Fh(?)463 2276 y Fn(\),)e(as)h(it)g(is)h(the)f(case)h(for)e(the)i (Hartree-F)l(o)q(c)o(k)e(mo)q(del)i(for)f(molecules.)108 2332 y(Apart)g(from)g(these)g(somewhat)g(secondary)h(questions,)f(the)h (main)g(issue)g(to)f(tac)o(kle)h(is)g(the)f Ff(pr)n(o)n(of)25 b Fn(of)37 2389 y(the)15 b(thermo)q(dynamic)g(limit)h(for)e(the)h (energy)f(p)q(er)i(unit)f(v)o(olume)g(in)g(the)g(H)f(and)h(HF)f (settings.)20 b(As)14 b(far)g(as)37 2445 y(this)i(question)g(is)g (concerned,)g(m)o(uc)o(h)f(remains)g(to)g(b)q(e)h(done.)108 2502 y(Ev)o(en)i(in)h(some)f(simpli\014ed)j(framew)o(ork,)c(trying)h (to)f(understand)i(Hartree-F)l(o)q(c)o(k)e(t)o(yp)q(e)i(mo)q(dels)g (for)37 2558 y(quasicrystals)j(w)o(ould)g(also)f(b)q(e)h(of)f(in)o (terest.)39 b(Our)22 b(study)f([11)o(])g(and)h(references)g([1)o(,)f (3,)g(8,)g(47)o(])g(could)37 2615 y(constitute)16 b(a)f(starting)f(p)q (oin)o(t.)108 2671 y(Finally)l(,)20 b(let)f(us)g(men)o(tion)f(that)g (the)g(p)q(erio)q(dic)i(problems)f(w)o(e)f(ha)o(v)o(e)g(de\014ned)i(in) f(this)g(w)o(ork)e(can)h(b)q(e)37 2727 y(treated)e(n)o(umerically)l(,)h (and)f(w)o(e)f(indeed)j(in)o(tend)e(to)f(treat)g(them)h(n)o(umerically) l(.)23 b(Numerical)17 b(exp)q(erimen)o(t)p eop %%Page: 67 67 67 66 bop 53 50 a Fl(References)1536 b Fn(67)37 199 y(migh)o(t)15 b(in)h(particular)g(giv)o(e)f(some)f(insigh)o(t)i(in)o(to)f(the)g (mathematical)g(nature)g(of)g(these)g(mo)q(dels)h(and)f(help)37 256 y(oneself)h(to)f(mak)o(e)g(up)g(his)h(mind)h(on)e(some)g(of)f(the)i (questions)f(men)o(tioned)h(ab)q(o)o(v)o(e.)37 395 y Fj(Ac)o(kno)o(wledgmen)o(ts:)25 b Fn(The)17 b(authors)g(w)o(ould)h(lik) o(e)g(to)f(thank)g(V.)g(Bac)o(h)h(for)e(sev)o(eral)i(stim)o(ulating)g (dis-)37 452 y(cussions,)d(and)e(J.)h(P)l(.)f(Solo)o(v)o(ej)h(for)e(v)m (aluable)k(commen)o(ts)d(on)g(a)g(preliminary)j(v)o(ersion)d(of)g(the)h (man)o(uscript.)37 651 y Fk(References)60 753 y Fn([1])22 b(L.)11 b(Amerio)g(&)g(G.)f(Prouse,)24 b Fm(Almost)13 b(Periodic)f(Functions)g(and)h(Functional)f(Equa)m(tions)p Fn(,)131 809 y(V)l(an)j(Nostrand)g(Reinhold)j(Compan)o(y)l(,)c(1971.)60 903 y([2])22 b(N.)14 b(W.)g(Ashcroft)g(&)h(N.)f(D.)g(Mermin,)32 b Fm(Solid-st)m(a)m(te)16 b(Physics)p Fn(,)d(Saunders)i(College)h (Publishing,)131 960 y(1976.)60 1053 y([3])22 b(F.)14 b(Axel)i(&)g(D.)e(Gratias)g(ed.,)32 b Fm(Beyond)17 b(Quasicr)m(yst)m (als)p Fn(,)d(Cen)o(tre)h(de)h(Ph)o(ysique)f(Les)h(Houc)o(hes,)131 1110 y(Les)g(Editions)g(de)f(Ph)o(ysique,)h(Springer,)g(1995.)60 1204 y([4])22 b(V.)16 b(Bac)o(h,)g(E.)g(H.)g(Lieb)i(&)f(J.)f(P)l(.)g (Solo)o(v)o(ej,)g Ff(Gener)n(alize)n(d)g(Hartr)n(e)n(e-F)m(o)n(ck)g (The)n(ory)i(and)f(the)h(Hubb)n(ar)n(d)131 1260 y(Mo)n(del)p Fn(,)c(J.)h(Stat.)f(Ph)o(ys.)h(76)p 563 1267 46 2 v -1 w(,)g(1994,)f(pp.)h(3{90.)60 1354 y([5])22 b(V.)12 b(Bac)o(h,)h Ff(Err)n(or)h(b)n(ound)g(for)g(the)g(Hartr)n(e)n(e-F)m(o)n(ck)f(ener)n (gy)g(of)h(atoms)g(and)g(mole)n(cules)p Fn(,)e(Comm.)f(Math.)131 1410 y(Ph)o(ys.,)j(147)p 268 1417 69 2 v -1 w(,)h(1992,)f(pp.)h (527{548.)60 1504 y([6])22 b(R.)c(Balian,)42 b Fm(Fr)o(om)21 b(micr)o(ophysics)e(to)i(ma)o(cr)o(ophysics)e(;)k(methods)e(and)f (applica)m(tions)131 1561 y(of)d(St)m(a)m(tistical)f(Physics)p Fn(,)e(I)h(&)h(I)q(I,)f(Springer)i(V)l(erlag,)e(1991.)60 1654 y([7])22 b(F.)10 b(Blo)q(c)o(h,)330 1643 y Ff(\177)321 1654 y(Ub)n(er)i(die)h(Quantenme)n(chanik)e(der)i(Ele)n(ctr)n(onen)d (in)i(Kristal)r(lgittern)p Fn(,)f(Z.)f(Ph)o(ys.,)h(52)p 1702 1661 46 2 v -1 w(,)g(1928,)131 1711 y(pp.)k(555{560.)60 1805 y([8])22 b(H.)15 b(Bohr,)32 b Fm(Almost)17 b(Periodic)g(Functions) p Fn(,)d(Chelsea,)h(1947.)60 1898 y([9])22 b(J.)15 b(Calla)o(w)o(a)o(y) l(,)32 b Fm(Quantum)16 b(Theor)m(y)i(of)f(the)g(Solid)h(St)m(a)m(te)p Fn(,)c(Academic)i(Press,)f(1974.)37 1992 y([10])22 b(I.)c(Catto,)f(C.)g (Le)i(Bris)g(&)f(P)l(.-L.)g(Lions,)h Ff(Limite)h(thermo)n(dynamique)g (p)n(our)f(des)g(mo)n(d)o(\022)-22 b(eles)18 b(de)h(typ)n(e)131 2049 y(Thomas-F)m(ermi)e([Thermo)n(dynamic)h(limit)f(for)g(Thomas-F)m (ermi)g(typ)n(e)g(mo)n(dels])p Fn(,)e(C.)g(R.)h(Acad.)f(Sci.)131 2105 y(P)o(aris)g(S)o(\023)-21 b(er.)14 b(I)h(Math.,)f(t.)h(322,)f(pp.) h(357{364,)e(1996.)37 2199 y([11])22 b(I.)e(Catto,)g(C.)g(Le)h(Bris)g (&)f(P)l(.-L.)h(Lions,)46 b Fm(Ma)m(thema)m(tical)22 b(Theor)m(y)g(of)h(thermod)o(ynamic)131 2255 y(limits)17 b(:)23 b(Thomas-Fermi)17 b(type)g(models)p Fn(,)f(Oxford)f(Univ)o (ersit)o(y)h(Press,)e(1998.)37 2349 y([12])22 b(I.)16 b(Catto,)f(C.)h(Le)g(Bris)h(&)g(P)l(.-L.)f(Lions,)h Ff(Sur)g(la)g (limite)g(thermo)n(dynamique)h(p)n(our)g(des)f(mo)n(d)o(\022)-22 b(eles)17 b(de)131 2406 y(typ)n(e)f(Hartr)n(e)n(e)f(et)g(Hartr)n(e)n (e-F)m(o)n(ck)g([On)g(the)h(thermo)n(dynamic)g(limit)f(for)h(Hartr)n(e) n(e)g(and)f(Hartr)n(e)n(e-F)m(o)n(ck)131 2462 y(typ)n(e)h(mo)n(dels])p Fn(,)e(C.)h(R.)g(Acad.)g(Sci.)h(P)o(aris)f(S)o(\023)-21 b(er.)14 b(I)i(Math.,)d(t.)i(327,)f(1998,)f(pp.)j(259{266.)37 2556 y([13])22 b(I.)c(Catto,)e(C.)i(Le)g(Bris)g(&)g(P)l(.-L.)g(Lions,)h Ff(On)f(some)h(p)n(erio)n(dic)f(Hartr)n(e)n(e-typ)n(e)h(mo)n(dels)f (for)h(crystals)p Fn(,)131 2612 y(submitted)d(to)e(Comm.)g(Math.)h(Ph)o (ys.)f(Also)i(a)o(v)m(ailable)h(as)d(:)131 2669 y Ff (http://www.math.utexas.e)n(du/mp)p 772 2669 14 2 v 20 w(ar)n(c/c/99/99-392.ps.gz)p eop %%Page: 68 68 68 67 bop 53 50 a Fl(References)1536 b Fn(68)37 199 y([14])22 b(L.)g(Chen,)i(R.V.)f(Mo)q(o)q(dy)f(&)g(J.)h(P)o(atera,)f Ff(Non-crystal)r(lo)n(gr)n(aphic)h(r)n(o)n(ot)g(systems)p Fn(,)f(Quasicrystals)131 256 y(and)c(discrete)h(geometry)e(\(T)l(oron)o (to,)f(ON,)i(1995\),)f(pp.)h(135{178,)e(Fields)j(Institute)g(Monogr.)d (10,)131 312 y(American)g(Mathematical)f(So)q(ciet)o(y)l(,)h(Pro)o (vidence,)f(RI,)h(1998.)37 406 y([15])22 b(C.)14 b(Conca)g(&)i(M.)d(V)l (anninathan,)j Ff(Homo)n(genization)f(of)h(p)n(erio)n(dic)g(structur)n (es)f(via)h(Blo)n(ch)g(de)n(c)n(omp)n(o-)131 462 y(sition)p Fn(,)e(SIAM)i(J.)f(Appl.)h(Math.,)e(V)l(ol.)h(57,)f(No.)h(6,)f(1997,)g (pp.)h(1639{1659.)37 556 y([16])22 b(C.)g(Conca,)h(J.)g(Planc)o(hard)g (&)g(M.)e(V)l(anninathan,)52 b Fm(Fluids)25 b(and)g(Periodic)g(Str)o (uctures)p Fn(,)131 613 y(Collection)17 b(RAM,)e(38)p 483 620 46 2 v -1 w(,)g(J.)g(Wiley)h(&)g(Sons/Masson,)e(P)o(aris,)g (1995.)37 707 y([17])22 b(M.S.P)l(.)15 b(Eastham,)33 b Fm(The)18 b(Spectral)h(Theor)m(y)f(of)h(Periodic)f(Differential)h (Equa)m(tions)p Fn(,)131 763 y(Scottish)c(Acad.)g(Press,)g(Edin)o (burgh-London,)i(1973.)37 857 y([18])22 b(A.)11 b(Figotin)h(&)g(P)l(.)f (Kuc)o(hmen)o(t,)i Ff(Band-gap)g(structur)n(e)g(of)h(sp)n(e)n(ctr)n(a)e (of)h(p)n(erio)n(dic)g(diele)n(ctric)f(and)h(ac)n(ous-)131 913 y(tic)j(me)n(dia.)h(I.)f(Sc)n(alar)g(mo)n(del)p Fn(,)f(SIAM)g(J.)g (Appl.)h(Math.,)e(56)p 1127 920 V -1 w(,)h(No.)g(1,)f(1996,)g(pp.)h (68{88.)37 1007 y([19])22 b(A.)11 b(Figotin)h(&)g(P)l(.)f(Kuc)o(hmen)o (t,)i Ff(Band-gap)g(structur)n(e)g(of)h(sp)n(e)n(ctr)n(a)e(of)h(p)n (erio)n(dic)g(diele)n(ctric)f(and)h(ac)n(ous-)131 1064 y(tic)i(me)n(dia.)g(II.)g(Two-dimensional)g(photonic)g(crystals)p Fn(,)e(SIAM)h(J.)g(Appl.)g(Math.,)e(56)p 1567 1071 V(,)h(No.)g(6,)g (1996,)131 1120 y(pp.)i(1561{1620.)37 1214 y([20])22 b(G.)f(Flo)q(quet,)i Ff(Sur)g(les)d(\023)-22 b(equations)23 b(di\013)o(\023)-22 b(er)n(entiel)r(les)20 b(lin)o(\023)-22 b(eair)n(es)21 b(\022)-23 b(a)23 b(c)n(o)n(e\016cients)e(p)o(\023)-22 b(erio)n(diques)p Fn(,)23 b(Ann.)131 1270 y(Ecole)16 b(Norm.)e(S)o(\023)-21 b(er.)14 b(2,)h(12)p 535 1277 V -1 w(,)g(1883,)f(pp.)h(47{89.)37 1364 y([21])22 b(G.)g(F)l(riesec)o (k)o(e,)i Ff(Pair)f(Corr)n(elations)g(and)g(Exchange)f(Phenomena)h(in)f (the)h(F)m(r)n(e)n(e)f(Ele)n(ctr)n(on)f(Gas)p Fn(,)131 1421 y(Comm.)14 b(Math.)g(Ph)o(ys.)h(184)p 550 1428 69 2 v -1 w(,)g(1997,)e(pp.)j(143{171.)37 1514 y([22])22 b(Y.)15 b(E.)g(Karp)q(eshina,)34 b Fm(Per)m(turba)m(tion)17 b(theor)m(y)h(f)o(or)g(the)f(Schr)1340 1510 y(\177)1338 1514 y(odinger)h(opera)m(tor)h(with)131 1571 y(a)e(periodic)h (potential)p Fn(,)d(Lecture)h(Notes)e(in)i(Mathematics)f(1663)p 1254 1578 91 2 v -1 w(,)g(Springer-V)l(erlag,)h(1997.)37 1665 y([23])22 b(C.)11 b(Kittel,)i Fm(Intr)o(oduction)g(to)g(Solid)h (St)m(a)m(te)f(Physics)p Fn(,)d(6th)h(Edition,)i(Wiley)g(and)e(Sons,)h (1986.)37 1758 y([24])22 b(P)l(.)14 b(Kuc)o(hmen)o(t,)32 b Fm(Floquet)18 b(Theor)m(y)f(f)o(or)g(P)l(ar)m(tial)g(Differential)g (Equa)m(tions)p Fn(,)c(Op)q(erator)131 1815 y(Theory)i(Adv)m(ances)h (and)g(Applications)h(v)o(ol.)e(60,)f(Birkh\177)-23 b(auser,)15 b(Basel,)h(1993.)37 1909 y([25])22 b(J.)14 b(L.)g(Leb)q(o)o(witz)h(&)g (E.)e(H.)h(Lieb,)h Ff(Existenc)n(e)f(of)i(thermo)n(dynamics)f(for)h(r)n (e)n(al)e(matter)i(with)g(Coulomb)131 1965 y(for)n(c)n(es)p Fn(,)e(Ph)o(ys.)g(Rev.)i(Lett.,)e(22)p 625 1972 46 2 v(,)g(13,)h(1969,)e(pp.)j(631{634.)37 2059 y([26])22 b(E.)17 b(H.)g(Lieb)h(&)g(J.L.)f(Leb)q(o)o(witz,)i Ff(The)f(c)n (onstitution)g(of)g(matter)h(:)25 b(existenc)n(e)16 b(of)j(thermo)n (dynamics)131 2115 y(for)e(systems)e(c)n(omp)n(ose)n(d)h(of)g(ele)n (ctr)n(ons)f(and)h(nuclei)p Fn(,)e(Adv.)h(in)h(Maths.,)e(9)p 1375 2122 23 2 v(,)g(1972,)g(pp.)h(316{398.)37 2209 y([27])22 b(E.)15 b(H.)g(Lieb)h(&)g(J.L.)f(Leb)q(o)o(witz,)h Ff(L)n(e)n(ctur)n (es)g(on)g(the)g(thermo)n(dynamic)h(limit)f(for)h(Coulomb)g(systems)p Fn(,)131 2266 y(Springer)f(Lecture)g(Notes)f(in)h(Ph)o(ysics,)f(20)p 834 2273 46 2 v(,)f(1973,)g(pp.)h(136{161.)37 2359 y([28])22 b(E.)17 b(H.)g(Lieb,)j Ff(The)f(stability)f(of)g(matter)i(:)25 b(fr)n(om)18 b(atoms)h(to)g(stars)p Fn(,)f(Bull.)g(Amer.)g(Math.)e(So)q (c.,)i(22)p 1817 2366 V -1 w(,)131 2416 y(No.)d(1,)f(1990,)g(pp.)h (1{49.)37 2510 y([29])22 b(E.)d(H.)g(Lieb,)i Ff(Thomas-F)m(ermi)g(and)f (r)n(elate)n(d)f(the)n(ories)h(of)g(atoms)g(and)g(mole)n(cules)p Fn(,)f(Rev.)h(Mo)q(dern)131 2566 y(Ph)o(ys.,)14 b(53)p 268 2573 V(,)g(No.)h(4,)g(1981,)e(pp.)i(603{641.)e Ff(Err)n(ata)19 b Fn(ibid)e(54)p 1098 2573 V(,)e(1982,)e(pp.)j(311.)37 2660 y([30])22 b(E.)12 b(H.)g(Lieb,)i Ff(A)f(V)m(ariational)g (principle)h(for)g(many-fermion)f(systems,)f Fn(Ph)o(ys.)g(Rev.)h (Lett.)f(46)p 1701 2667 V -1 w(,)g(1981,)131 2716 y(pp.)j(457{459.)e Ff(Err)n(ata)j Fn(ibid)h(47)p 631 2723 V -1 w(,)e(1981,)e(pp.)j(69.)p eop %%Page: 69 69 69 68 bop 53 50 a Fl(References)1536 b Fn(69)37 199 y([31])22 b(E.)12 b(H.)h(Lieb)h(&)f(S.)g(Oxford,)f Ff(A)o(n)i(Impr)n(ove)n(d)g(L) n(ower)h(Bound)f(on)g(the)h(Indir)n(e)n(ct)e(Coulomb)i(Ener)n(gy)p Fn(,)d(In)o(t.)131 256 y(J.)j(Quan)o(tum)g(Chem.,)g(19)p 542 263 46 2 v -1 w(,)g(1981,)f(pp.)h(427{439.)37 350 y([32])22 b(E.)14 b(H.)h(Lieb)h(&)f(B.)g(Simon,)g Ff(The)i(Thomas-F)m (ermi)g(the)n(ory)f(of)g(atoms,)g(mole)n(cules)f(and)h(solids)p Fn(,)e(Adv.)131 406 y(Math.,)g(23)p 279 413 V -1 w(,)h(1977,)f(pp.)h (22{116.)37 500 y([33])22 b(E.)c(H.)g(Lieb)h(&)g(B.)f(Simon,)h Ff(The)h(Hartr)n(e)n(e-F)m(o)n(ck)e(the)n(ory)i(for)f(Coulomb)h (systems)p Fn(,)e(Comm.)f(Math.)131 556 y(Ph)o(ys.,)d(53)p 268 563 V(,)g(1977,)g(pp.)h(185{194.)37 650 y([34])22 b(E.)14 b(H.)h(Lieb,)g(J.)g(P)l(.)f(Solo)o(v)o(ej)h(&)g(J.)g(Yngv)m (ason,)f Ff(Asymptotics)i(of)g(He)n(avy)f(A)o(toms)h(in)f(High)h (Magnetic)131 707 y(Fields:)28 b(I.)21 b(L)n(owest)g(L)n(andau)g(Band)f (R)n(e)n(gions)p Fn(,)f(Comm.)g(Pure.)g(Appl.)i(Math.,)e(47)p 1547 714 V(,)h(No.)f(4,)h(1994,)131 763 y(pp.)15 b(513{591.)37 857 y([35])22 b(E.)j(H.)g(Lieb)i(&)f(W.)e(Thirring,)29 b Ff(Ine)n(qualities)c(for)h(the)g(moments)f(of)h(the)g(eigenvalues)f (of)h(the)131 913 y(Schr\177)-23 b(odinger)15 b(Hamiltonian)g(and)h (their)f(r)n(elation)g(to)h(Sob)n(olev)f(ine)n(qualities.)e Fn(In:)20 b(Studies)15 b(in)h(Math-)131 970 y(ematical)k(Ph)o(ysics)h (\()e(E.)h(H.)f(Lieb,)j(B.)e(Simon)g(&)h(A.)e(Wigh)o(tman)h(eds.\),)g (Princeton)g(Univ)o(ersit)o(y)131 1026 y(Press,)15 b(1976,)e(pp.)i (269{303.)37 1120 y([36])22 b(E.)12 b(H.)g(Lieb)i(&)f(W.)f(Thirring,)h Ff(Bounds)h(for)g(the)g(kinetic)f(ener)n(gy)g(of)h(fermions)g(which)g (pr)n(ove)g(the)g(sta-)131 1176 y(bility)h(of)g(matter)p Fn(,)f(Ph)o(ys.)g(Rev.)g(Lett.,)f(35)p 804 1183 V(,)g(1975,)g(pp.)h (687{689.)e Ff(Err)n(ata)17 b Fn(ibid.)f(35)p 1505 1183 V -1 w(,)e(1975,)e(pp.)i(1116.)37 1270 y([37])22 b(P)l(.-L.)f(Lions,)i Ff(Solutions)e(of)h(Hartr)n(e)n(e-F)m(o)n(ck)e(e)n(quations)h(for)h (Coulomb)h(systems)p Fn(,)e(Comm.)f(Math.)131 1327 y(Ph)o(ys.,)14 b(109)p 268 1334 69 2 v -1 w(,)h(1987,)f(pp.)h(33{97.)37 1421 y([38])22 b(P)l(.-L.)13 b(Lions,)h Ff(Hartr)n(e)n(e-F)m(o)n(ck)f (and)h(r)n(elate)n(d)g(e)n(quations.)f Fn(In)g(:)19 b(Nonlinear)14 b(P)o(artial)f(Di\013eren)o(tial)i(Equa-)131 1477 y(tions)i(and)f (their)h(Applications,)i(Lect.)d(Coll)o(\022)-21 b(ege)16 b(de)h(F)l(rance)g(Seminar,)g(V)l(ol.)g(IX,)f(P)o(aris,)g(1985{86,)131 1533 y(Pitman)f(Res.)h(Notes)e(Math.)g(Ser.)h(181,)f(1988,)g(pp.)h (304{333.)37 1627 y([39])22 b(P)l(.-L.)16 b(Lions)h(&)f(T.)g(P)o(aul,)g Ff(Sur)h(les)f(mesur)n(es)h(de)g(Wigner,)f Fn(Rev.)g(Mat.)f(Ib)q (eroamericana)i(9)p 1703 1634 23 2 v -1 w(,)f(No.)g(3,)131 1684 y(1993,)e(pp.)h(553{618.)37 1778 y([40])22 b(O.)j(Madelung,)59 b Fm(Intr)o(oduction)28 b(to)f(Solid)h(St)m(a)m(te)g(theor)m(y)p Fn(,)g(Solid)f(State)d(Sciences)k(2)p 1840 1785 V -1 w(,)131 1834 y(Springer-V)l(erlag,)16 b(Berlin,)h(1981.)37 1928 y([41])22 b(R.G.)c(P)o(arr)f(&)h(W.)g(Y)l(ang,)40 b Fm(Density-functional)19 b(theor)m(y)i(of)f(atoms)g(and)h(molecules)p Fn(,)131 1984 y(Oxford)15 b(Univ)o(ersit)o(y)h(Press,)e(Oxford)i(&)f (New)g(Y)l(ork,)g(1989.)37 2078 y([42])22 b(C.)13 b(Pisani)i(ed.,)29 b Fm(Quantum)16 b(Mechanical)f(Ab)h(Initio)f(Calcula)m(tion)g(of)h(the) g(pr)o(oper)m(ties)131 2134 y(of)h(cr)m(yst)m(alline)h(ma)m(terials)p Fn(,)d(Lecture)h(Notes)e(in)i(Chemistry)g(67)p 1303 2141 46 2 v -1 w(,)f(Springer)h(V)l(erlag,)f(1996.)37 2228 y([43])22 b(Ch.)14 b(M.)f(Quinn,)31 b Fm(An)16 b(intr)o(oduction)f(to)h (the)g(Quantum)g(Theor)m(y)f(of)i(Solids)p Fn(,)c(Clarendon)131 2285 y(Press,)i(Oxford,)f(1973.)37 2379 y([44])22 b(M.)14 b(Reed)h(&)g(B.)g(Simon,)31 b Fm(Methods)17 b(of)g(Modern)g(Ma)m(thema) m(tical)f(Physics)p Fn(,)d(I:)h Ff(F)m(unctional)131 2435 y(A)o(nalysis,)f Fn(Academic)k(Press,)d(New-Y)l(ork-London,)i (1972.)37 2529 y([45])22 b(M.)15 b(Reed)h(&)g(B.)f(Simon,)33 b Fm(Methods)17 b(of)h(Modern)g(Ma)m(thema)m(tical)f(Physics)p Fn(,)c(IV:)j Ff(A)o(nalysis)131 2585 y(of)g(Op)n(er)n(ators)p Fn(,)f(Academic)h(Press,)f(New-Y)l(ork,)g(1978.)37 2679 y([46])22 b(D.)11 b(Ruelle,)30 b Fm(St)m(a)m(tistical)12 b(Mechanics)h(:)22 b(Rigor)o(ous)14 b(Resul)m(ts)p Fn(,)e(Benjamin,)i (New-Y)l(ork,)e(1969)131 2736 y(and)p 131 2743 74 2 v 15 w(Adv)m(anced)17 b(Bo)q(oks)e(Classics,)g(Addison-W)l(esley)l(,)j (1989.)p eop %%Page: 70 70 70 69 bop 53 50 a Fl(References)1536 b Fn(70)37 199 y([47])22 b(M.)14 b(Senec)o(hal,)34 b Fm(Quasicr)m(yst)m(als)16 b(and)i(Geometr)m(y)p Fn(,)d(Cam)o(bridge)g(Univ)o(ersit)o(y)h(Press,)e (1995.)37 293 y([48])22 b(J.)15 b(C.)g(Slater,)32 b Fm(Quantum)17 b(Theor)m(y)g(of)g(Molecules)h(and)g(Solids)p Fn(,)c(Mac)h(Gra)o(w)f (Hill,)j(1963.)37 387 y([49])22 b(J.)15 b(C.)g(Slater,)32 b Fm(Symmetr)m(y)17 b(and)g(Ener)o(gy)h(Bands)f(in)g(Cr)m(yst)m(als)p Fn(,)d(Do)o(v)o(er,)g(1972.)37 481 y([50])22 b(J.)f(P)l(.)g(Solo)o(v)o (ej,)h Ff(Universality)f(in)g(the)h(Thomas-F)m(ermi-von)g(Weizs\177)-23 b(acker)21 b(mo)n(del)h(of)g(atoms)g(and)131 537 y(mole)n(cules)p Fn(,)14 b(Comm.)g(Math.)g(Ph)o(ys.,129)p 763 544 69 2 v -2 w(,)h(1990,)f(pp.)h(561{598.)37 631 y([51])22 b(J.)13 b(P)l(.)g(Solo)o(v)o(ej,)g Ff(A)o(n)h(impr)n(ovement)g(on)g(stability)g (of)h(matter)g(in)f(me)n(an)g(\014eld)f(the)n(ory.)h Fn(In.)f(Di\013eren)o(tial)131 687 y(Equations)e(and)h(Mathematical)f (Ph)o(ysics,)h(Pro)q(ceedings)h(of)e(the)g(In)o(ternational)h (Conference,)g(Univ.)131 744 y(of)j(Alabama,)g(Birmingham,)g(Marc)o(h)g (1994,)e(In)o(ternational)j(Press)f(1995.)37 838 y([52])22 b(J.)14 b(P)l(.)g(Solo)o(v)o(ej,)g Ff(Pr)n(o)n(of)i(of)f(the)h (ionization)f(c)n(onje)n(ctur)n(e)f(in)h(a)h(r)n(e)n(duc)n(e)n(d)f (Hartr)n(e)n(e-F)m(o)n(ck)f(mo)n(del)p Fn(,)g(In)o(v)o(en)o(t.)131 894 y(Math.)g(104)p 266 901 V -1 w(,)h(1991,)f(pp.)h(291{311.)37 988 y([53])22 b(E.)11 b(Stein,)28 b Fm(Singular)13 b(Integrals)i(Opera) m(tors)g(and)e(Differentiability)h(of)g(Functions)p Fn(,)131 1044 y(Princeton)i(Univ)o(ersit)o(y)g(Press,)e(Princeton,)i(1970.)37 1138 y([54])22 b(R.)30 b(T)l(emam,)70 b Fm(Infinite-Dimensional)32 b(D)o(ynamical)f(Systems)i(in)f(Mechanics)g(and)131 1195 y(Physics)p Fn(,)13 b(Springer-V)l(erlag,)k(1988.)37 1289 y([55])22 b(R.)12 b(C.)g(T)l(olman,)28 b Fm(The)14 b(principles)h(of)f(St)m(a)m(tistical)f(Mechanics)p Fn(,)e(Oxford)i (Univ)o(ersit)o(y)g(Press,)131 1345 y(1962.)37 1439 y([56])22 b(C.)15 b(Wilco)o(x,)g Ff(The)n(ory)i(of)f(Blo)n(ch)g(waves)p Fn(,)f(J.)g(Anal.)h(Math.,)d(33,)i(1978,)e(pp.)j(146{167.)37 1533 y([57])22 b(J.)13 b(Ziman,)28 b Fm(Principles)15 b(of)g(the)f(Theor)m(y)h(of)g(Solids)p Fn(,)e(2nd)g(Edition,)h(Cam)o (bridge)e(Univ)o(ersit)o(y)131 1589 y(Press,)j(1972.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0002290830390--