Content-Type: multipart/mixed; boundary="-------------0303091646680" This is a multi-part message in MIME format. ---------------0303091646680 Content-Type: text/plain; name="03-101.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-101.keywords" Dirac operator, two-centre Coulomb potential, potential energy curves, molecular virial theorem, relativistic hydrogen molecular ion ---------------0303091646680 Content-Type: application/postscript; name="bhd1a.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bhd1a.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: bhd1a.dvi %%Pages: 39 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o bhd1a.ps bhd1a %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.04:2228 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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y(A)h(literal)g(transcription)e(of)i(the)g(ab)r(o)n(v)n(e)e(pro)r (cedure)h(to)g(relativistic)g(H)2938 2775 y Ft(+)2938 2832 y(2)3021 2810 y FA(is)h(ham-)599 2910 y(p)r(ered)c(b)n(y)f(the)g (fact)h(that)f(the)h(Coulom)n(b-Dirac)e(Hamiltonian)h(is)g(not)g(semib) r(ounded)599 3010 y(from)40 b(b)r(elo)n(w)g(\(or,)j(from)d(ab)r(o)n(v)n (e,)i(either\).)75 b(Th)n(us,)43 b(the)e(con)n(v)n(en)n(tional)d(Ra)n (yleigh-)599 3109 y(Ritz)45 b(v)-5 b(ariational)42 b(principle)i (cannot)g(b)r(e)g(applied.)87 b(Moreo)n(v)n(er,)45 b(despite)f(of)g (re-)599 3209 y(cen)n(t)31 b(progress)d(in)j(v)-5 b(ariational)29 b(c)n(haracterizations)f(of)i(Dirac)g(eigen)n(v)-5 b(alues)30 b([23)o(,)g(24)o(,)599 3308 y(25)o(],)i(unfortunately)e(these)h (rigorous)d(minimax)j(theorems)f(do)g(not)h(lead)f(to)g(upp)r(er)599 3408 y(b)r(ounds)f(along)e(the)i(same)f(straigh)n(tforw)n(ard)e(lines)i (as)g(for)g(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erators.)599 3508 y(Due)34 b(to)e(these)h(di\016culties)h(and)e(the)h (nonseparabilit)n(y)e(of)i(the)g(t)n(w)n(o-cen)n(tre)e(Dirac-)599 3607 y(Coulom)n(b)41 b(Hamiltonian,)i(apparen)n(tly)d(a)g(rigorous)f (demonstration)h(of)g(b)r(onding)599 3707 y(prop)r(erties)28 b(had)h(not)g(y)n(et)g(b)r(een)g(accomplished)f(for)g(relativistic)h (molecules.)40 b(Here,)599 3807 y(the)34 b(stratagem)e(w)n(e)h(are)f (resorting)g(to)h(is)g(to)g(consider)g(the)g(square)f(of)i(the)f (Dirac-)599 3906 y(Coulom)n(b)22 b(Hamiltonian.)35 b(This)23 b(op)r(erator)e(is)i(clearly)e(nonnegativ)n(e;)i(so,)h(w)n(e)e(can)g (fall)599 4006 y(bac)n(k)e(on)n(to)g(the)h(standard)f(minimax)h (principle)g(and)f(the)i(LCA)n(O)e(upp)r(er)h(b)r(ound.)35 b(On)599 4105 y(the)30 b(other)f(hand,)h(ev)n(en)f(for)g(the)h(v)n(ery) e(simple)h(ansatz)g(of)g(the)h(test)g(function)g(as)f(an)599 4205 y(atomic)40 b(ground)f(state)h(attac)n(hed)g(to)f(merely)h(one)g (n)n(uclear)f(cen)n(tre,)j(the)f(arising)599 4305 y(in)n(tegrals)25 b(cannot)h(b)r(e)h(computed)f(in)h(closed)e(form.)36 b(Nonetheless,)27 b(b)n(y)f(determining)599 4404 y(asymptotic)k (expressions)e(for)h(these)g(in)n(tegrals)f(or)h(b)n(y)h(estimating)f (certain)g(LCA)n(O)p eop %%Page: 4 4 4 3 bop 599 75 a FA(4)905 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(in)n(tegrals)h(w)n(e)g(are)g(able)h(to)f(sa)n(v)n(e)g(the)h (approac)n(h)e(and)h(to)h(deduce)g(c)n(hemical)f(b)r(onding)599 419 y(for)k(the)f(relativistic)g(t)n(w)n(o-cen)n(tre)f(one-electron)g (system.)682 519 y(Belo)n(w,)g(in)h(c)n(hapter)e(2,)i(w)n(e)f(start)f (our)h(exp)r(osition)g(b)n(y)g(recalling)f(the)i(precise)e(def-)599 619 y(inition)39 b(and)e(the)i(self-adjoin)n(tness)e(b)r(eha)n(viour)f (of)i(the)h(studied)f(Dirac-Coulom)n(b)599 718 y(op)r(erators.)c(Since) 25 b(also)f(symmetry)h(and)f(scaling)g(argumen)n(ts)g(en)n(ter)g(in)n (to)g(our)g(anal-)599 818 y(ysis,)34 b(they)f(are)f(explained)h(in)g (section)g(2.2.)52 b(Kno)n(wn)32 b(results)h(ab)r(out)f(the)i(limit)f (of)599 917 y(in\014nite)i(in)n(tern)n(uclear)d(separation)g(are)h (outlined)h(in)g(section)g(2.3,)g(and)g(the)g(lo)n(w)n(er)599 1017 y(b)r(ound)d(on)e(molecular)g(energies)g(in)h(terms)g(for)f(the)h ("united)g(atom")f(energy)g(is)h(es-)599 1117 y(tablished)35 b(in)f(section)g(2.4.)56 b(W)-7 b(e)34 b(sho)n(w)g(the)g(regularit)n(y) e(of)j(the)f(electronic)g(curv)n(es)599 1216 y(in)e(c)n(hapter)f(3;)j (\014rst,)e(in)g(section)g(3.1,)g(the)g(con)n(tin)n(uit)n(y)-7 b(,)32 b(then,)i(in)e(section)f(3.2,)h(dif-)599 1316 y(feren)n(tiablit)n(y;)27 b(in)f(section)g(3.3)g(w)n(e)g(pro)n(v)n(e)f (the)i(F)-7 b(eynman-Hellman)26 b(theorem)g(.)36 b(The)599 1416 y(relativistic)c(virial)f(theorem)g(\(section)h(4.1\))f(and)h(our) f(discussion)h(of)g(b)r(onding)f(are)599 1515 y(presen)n(ted)23 b(in)h(c)n(hapter)e(4.)35 b(In)24 b(app)r(endix)f(A)h(w)n(e)f(collect)g (a)g(few)h(recen)n(t)f(results)f(ab)r(out)599 1615 y(v)-5 b(ariational)18 b(metho)r(ds)i(for)f(Dirac)g(eigen)n(v)-5 b(alues,)21 b(and)e(in)h(app)r(endix)g(B)f(w)n(e)g(displa)n(y)g(the)599 1714 y(matrix)27 b(elemen)n(ts)h(that)g(are)e(needed)i(for)f(our)g (stabilit)n(y)g(pro)r(of.)37 b(Finally)-7 b(,)27 b(c)n(hapter)g(5)599 1814 y(is)h(dev)n(oted)f(to)g(concluding)h(remarks)d(and)j(to)f(an)h (outlo)r(ok.)684 2080 y Fu(2.)75 b(THE)32 b(TW)m(O-CENTRE)f(DIRA)m (C-COULOMB)h(OPERA)-8 b(TOR)682 2229 y FA(W)h(e)31 b(study)f (one-electron)f(systems)g(for)h(whic)n(h)g(the)g(electronic)g(states)f (are)g(iden-)599 2328 y(ti\014ed)g(with)f(v)n(ectors)e(in)i(the)g(Hilb) r(ert)g(space)1328 2560 y Fs(H)c FA(=)f Fr(L)1567 2525 y Ft(2)1603 2560 y FA(\()p Fq(R)1690 2525 y Ft(3)1733 2560 y Fr(;)14 b Fq(C)1824 2525 y Ft(4)1867 2560 y FA(\))1922 2537 y Fs(\030)1922 2564 y FA(=)2010 2560 y Fr(L)2067 2525 y Ft(2)2104 2560 y FA(\()p Fq(R)2190 2525 y Ft(3)2233 2560 y Fr(;)g Fq(C)h FA(\))25 b Fs(\012)18 b Fq(C)2518 2525 y Ft(4)3119 2560 y FA(\(2.1\))599 2791 y(equipp)r(ed)29 b(with)f(the)g(inner)f(pro)r(duct)1401 3097 y Fs(h)p Fr( )s(;)h(\036)p Fs(i)23 b FA(=)1733 2984 y Fp(Z)1873 2993 y Ft(4)1830 3018 y Fp(X)1836 3195 y Fo(i)p Ft(=1)p 1964 3029 82 4 v 1964 3097 a Fr( )2018 3109 y Fo(i)2045 3097 y FA(\()p Fr(x)p FA(\))p Fr(\036)2205 3109 y Fo(i)2235 3097 y FA(\()p Fr(x)p FA(\))14 b Fr(d)2403 3062 y Ft(3)2441 3097 y Fr(x)631 b FA(\(2.2\))599 3405 y(for)29 b Fr( )s(;)14 b(\036)27 b Fs(2)g(H)q FA(.)42 b(The)30 b(clamp)r(ed)f(n)n(uclei)h (\(or)e(BO\))i(appro)n(ximation)d(means)i(that)h(the)599 3505 y(t)n(w)n(o)23 b(n)n(uclear)f(cen)n(tres)g(are)g(\014xed)h(at)g(p) r(ositions)g(+)2158 3472 y Ft(1)p 2157 3486 34 4 v 2157 3533 a(2)2200 3505 y Fn(R)h FA(and)f Fs(\000)2528 3472 y Ft(1)p 2527 3486 V 2527 3533 a(2)2571 3505 y Fn(R)p FA(,)h(resp)r(ectiv)n(ely)-7 b(,)24 b(and)599 3604 y(the)33 b(electronic)e(quan)n(tities)h(are)f(regarded)f(as)h(function)i(of)f (the)g(in)n(tern)n(uclear)f(sep-)599 3704 y(aration)e Fn(R)e Fs(2)f Fq(R)1122 3674 y Ft(3)1165 3704 y FA(.)43 b(Here,)30 b(w)n(e)f(fo)r(cus)h(on)n(to)e(homon)n(uclear)g(systems,)i (viz.,)g(iden)n(tical)599 3804 y(c)n(harges)37 b(+)p Fr(Z)6 b(e)1071 3816 y Ft(0)1145 3804 y FA(for)38 b(b)r(oth)g(n)n (uclei.)69 b(In)39 b(this)f(con)n(text,)j(it)d(will)h(b)r(e)f(con)n(v)n (enien)n(t)g(to)599 3903 y(c)n(ho)r(ose)33 b(the)g(mass)g Fr(m)1300 3915 y Fo(e)1369 3903 y FA(of)g(the)h(electron)f(as)f(new)i (mass)e(unit,)k(and)d(the)h(Compton)599 4003 y(w)n(a)n(v)n(elength)d (of)i(the)g(electron)f Fr(\025)1648 4015 y Fo(e)1716 4003 y FA(=)f Fq(~)p Fr(=m)1972 4015 y Fo(e)2007 4003 y Fr(c)i FA(as)f(the)h(length)g(unit.)52 b(F)-7 b(urthermore,)599 4102 y(denoting)32 b(the)g(Sommerfeld)g(\014ne)g(structure)f(constan)n (t)g(b)n(y)h Fr(\013)2584 4114 y Fo(S)2662 4102 y FA(=)e Fr(e)2796 4072 y Ft(2)2796 4123 y(0)2832 4102 y Fr(=)p Fq(~)p Fr(c)p FA(,)j(w)n(e)e(tak)n(e)599 4202 y Fr(\020)36 b FA(:=)28 b Fr(\013)840 4214 y Fo(S)888 4202 y Fr(Z)37 b FA(as)31 b(the)g(n)n(uclear)f(c)n(harge)f(parameter.)46 b(Then,)32 b(an)f(appropriate)e(scaling)599 4302 y(of)f(the)g(length)g (v)-5 b(ariables)26 b(casts)h(the)h(free)f(Dirac)g(op)r(erator)f(in)n (to)h(the)h(form:)1656 4533 y Fr(T)1705 4545 y Fo(D)1788 4533 y FA(=)22 b Fn(\013)g Fm(\001)e Fn(p)f FA(+)f Fr(\014)t(:)886 b FA(\(2.3\))p eop %%Page: 5 5 5 4 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)477 b FA(5)599 320 y(Here,)27 b Fn(p)c FA(=)g Fs(\000)p Fr(i)p Fm(r)n FA(,)28 b(and)e(the)h(standard)f(Dirac)g (matrices)g Fr(\013)2459 332 y Ft(1)2497 320 y Fr(;)14 b(\013)2587 332 y Ft(2)2624 320 y Fr(;)g(\013)2714 332 y Ft(3)2751 320 y Fr(;)g(\014)31 b FA(are)26 b(giv)n(en)g(in)599 419 y(terms)i(of)f(the)h(P)n(auli)f(matrices)g(or)f(the)i(2)18 b Fs(\002)g FA(2)28 b(unit)g(matrix)f Fq(I)2557 431 y Ft(2)2616 419 y FA(b)n(y)1281 666 y Fr(\013)1334 678 y Fo(j)1392 666 y FA(=)1480 549 y Fp(\022)1561 615 y FA(0)70 b Fr(\033)1720 627 y Fo(j)1541 715 y Fr(\033)1588 727 y Fo(j)1693 715 y FA(0)1755 549 y Fp(\023)1830 666 y Fr(;)180 b(\014)28 b FA(=)2195 549 y Fp(\022)2256 615 y Fq(I)2295 627 y Ft(2)2422 615 y FA(0)2270 715 y(0)64 b Fs(\000)p Fq(I)2479 727 y Ft(2)2510 549 y Fp(\023)2585 666 y Fr(:)511 b FA(\(2.4\))599 912 y(The)32 b(molecular)e(Hamiltonian) h(con)n(tains)g Fr(T)2016 924 y Fo(D)2107 912 y FA(as)f(its)i(kinetic)g (energy)e(part,)i(and)f(a)599 1012 y(p)r(oten)n(tial)d Fr(V)47 b FA(that)27 b(dep)r(ends)h(parametrically)e(on)h Fn(R)i FA(and)e Fr(\020)6 b FA(:)1484 1213 y Fr(H)h FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))24 b(=)f Fr(T)1937 1225 y Fo(D)2015 1213 y FA(+)18 b Fr(V)h FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))p Fr(:)715 b FA(\(2.5\))599 1414 y(In)28 b Fr(V)19 b FA(,)28 b(w)n(e)f(distinguish)h(b)r(et)n(w)n(een)f (di\013eren)n(t)h(con)n(tributions,)1346 1615 y Fr(V)19 b FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))24 b(=)e Fr(V)1788 1627 y Ft(ne)1859 1615 y FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))19 b(+)f Fr(V)2225 1627 y Ft(nn)2303 1615 y FA(\()p Fn(R)q Fr(;)c(\020)6 b FA(\))p Fr(;)577 b FA(\(2.6\))599 1816 y(where)28 b(the)f(n)n(uclei-electron)g(in)n(teractions)1184 2017 y(\()p Fr(V)1264 2029 y Ft(ne)1335 2017 y FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))p Fr( )s FA(\)\()p Fn(x)q FA(\))23 b(=)g Fr(V)1919 2029 y Ft(ne)1990 2017 y FA(\()p Fn(x)p FA(;)k Fn(R)q Fr(;)14 b(\020)6 b FA(\)\))19 b Fs(\012)f Fq(I)2484 2029 y Ft(4)2529 2017 y Fr( )s FA(\()p Fn(x)p FA(\))414 b(\(2.7\))599 2219 y(is)28 b(determined)g(b)n(y)f(the) h(function)1087 2420 y Fr(V)1135 2432 y Ft(ne)1206 2420 y FA(\()p Fn(x)p FA(;)g Fn(R)p Fr(;)14 b(\020)6 b FA(\)\))24 b(=)f Fs(\000)p Fr(\020)6 b FA(\()p Fs(j)p Fn(x)19 b Fs(\000)2000 2387 y Ft(1)p 2000 2401 34 4 v 2000 2448 a(2)2043 2420 y Fn(R)p Fs(j)2138 2386 y Fl(\000)p Ft(1)2246 2420 y FA(+)f Fs(j)p Fn(x)g FA(+)2518 2387 y Ft(1)p 2518 2401 V 2518 2448 a(2)2561 2420 y Fn(R)q Fs(j)2657 2386 y Fl(\000)p Ft(1)2746 2420 y FA(\))p Fr(;)318 b FA(\(2.8\))599 2621 y(whereas)27 b(the)h(n)n(ucleus-n)n(ucleus)e(term)i(acts)f(simply) h(as)697 2822 y(\()p Fr(V)777 2834 y Ft(nn)856 2822 y FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))p Fr( )s FA(\)\()p Fn(x)q FA(\))23 b(=)g Fr(V)1440 2834 y Ft(nn)1518 2822 y FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))19 b Fs(\012)f Fq(I)1874 2834 y Ft(4)1919 2822 y Fr( )s FA(\()p Fn(x)p FA(\))24 b(=)e Fr(\013)2259 2787 y Fl(\000)p Ft(1)2259 2847 y Fo(S)2349 2822 y Fr(\020)2391 2788 y Ft(2)2428 2822 y Fs(j)p Fn(R)q Fs(j)2547 2788 y Fl(\000)p Ft(1)2655 2822 y Fs(\012)c Fq(I)2776 2834 y Ft(4)2821 2822 y Fr( )s FA(\()p Fn(x)p FA(\))p Fr(:)99 b FA(\(2.9\))599 3023 y(F)-7 b(or)35 b(an)n(y)g(giv)n(en)g(v)n(ector)f Fn(x)o FA(,)k(b)n(y)d(translating)f(and)i(rotating)e Fn(x)p FA(,)j(one)e(can)g(\014nd)h Fn(x)3229 3035 y Fk(R)599 3123 y FA(suc)n(h)31 b(that)g(the)h(n)n(uclei-electron)e(p)r(oten)n (tial)h(function)g(has)g(the)g(same)g(v)-5 b(alue)31 b(as)f(for)599 3223 y(the)g(n)n(uclei)f(placed)f(along)g(a)h ("standard")e(direction,)h(i.e.,)i(at)f Fs(\006)p Fn(R)2736 3235 y Fo(s)2797 3223 y FA(=)c(\(0)p Fr(;)14 b FA(0)p Fr(;)g Fs(\006)3152 3190 y Ft(1)p 3150 3204 V 3150 3251 a(2)3193 3223 y Fr(R)q FA(\))599 3322 y(on)28 b(the)g Fr(z)t FA(-axis,)1356 3477 y Fr(V)1404 3489 y Ft(ne)1475 3477 y FA(\()p Fn(x)p FA(;)g Fn(R)q Fr(;)14 b(\020)6 b FA(\)\))24 b(=)e Fr(V)1988 3489 y Ft(ne)2059 3477 y FA(\()p Fn(x)2146 3489 y Fk(R)2207 3477 y FA(;)27 b Fn(R)2330 3489 y Fo(s)2366 3477 y Fr(;)14 b(\020)6 b FA(\)\))p Fr(:)545 b FA(\(2.10\))599 3633 y(Implemen)n(ting)31 b(the)g(transformation)e Fn(x)e Fs(7!)h Fn(x)2091 3645 y Fk(R)2182 3633 y FA(b)n(y)i(the)g(unitary)g(op)r(erator)f Fr(U)3136 3645 y Fo(s)3202 3633 y FA(on)599 3732 y Fs(H)q FA(,)37 b(then)f(the)f(in)n(v)-5 b(ariance)34 b(of)h(the)g(kinetic)g (energy)f Fr(T)2328 3744 y Fo(D)2423 3732 y FA(under)h Fr(U)2723 3744 y Fo(s)2793 3732 y FA(together)f(with)599 3832 y(Eq.)27 b(\(2.10\))g(imply)h(the)g(unitary)f(equiv)-5 b(alence)1444 4033 y Fr(H)7 b FA(\()p Fn(R)q Fr(;)14 b(\020)6 b FA(\))23 b(=)g Fr(U)1904 4045 y Fo(s)1939 4033 y Fr(H)7 b FA(\()p Fn(R)2120 4045 y Fo(s)2155 4033 y Fr(;)14 b(\020)6 b FA(\))p Fr(U)2332 3999 y Fl(\000)p Ft(1)2323 4053 y Fo(s)2422 4033 y Fr(:)632 b FA(\(2.11\))599 4234 y(Therefore,)25 b(for)f(a)h(sp)r(ectral)f(analysis)g(it)h(is)g (su\016cien)n(t)g(merely)g(to)g(study)g(the)g(dep)r(en-)599 4334 y(dence)20 b(of)g Fr(H)27 b FA(on)20 b(the)h(mo)r(dulus)f Fr(R)h FA(of)f Fn(R)p FA(.)35 b(In)20 b(the)g(sequel,)i(w)n(e)d(will)i (alw)n(a)n(ys)d(assume)h(the)599 4433 y(n)n(uclei)28 b(lo)r(cated)e(at)i(the)f Fr(z)t FA(-axis)f(symmetrically)g(around)g (the)i(origin,)e(and)h(con)n(tin)n(ue)599 4533 y(to)20 b(\(ab\)use)g(the)h(same)e(sym)n(b)r(ols)h(for)f(the)i(op)r(erators,)e (that)i(is,)g Fr(V)e FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(:=)e Fr(V)d FA(\()p Fn(R)3119 4545 y Fo(s)3155 4533 y Fr(;)14 b(\020)6 b FA(\),)p eop %%Page: 6 6 6 5 bop 599 75 a FA(6)905 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(and)30 b(similarly)e(for)h Fr(H)7 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\),)31 b Fr(V)1618 332 y Ft(ne)1688 320 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))30 b(and)g Fr(V)2137 332 y Ft(nn)2215 320 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\).)43 b(The)29 b(non)n(trivial)g(part)f(of)599 419 y Fr(H)35 b FA(is)27 b(called)h(the)g("electronic)e(Hamiltonian")h (and)g(denoted)h(b)n(y)1493 615 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)e Fr(T)1908 627 y Fo(D)1986 615 y FA(+)c Fr(V)2117 627 y Ft(ne)2188 615 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))682 b(\(2.12\))682 810 y(Tw)n(o)20 b(particular)g(cases)g (deserv)n(e)f(to)i(b)r(e)g(singled)f(out,)i(namely)f(the)g("united)g (atoms)599 910 y(limit")28 b(\(UAL\))1741 1060 y(lim)1740 1114 y Fo(R)p Fl(#)p Ft(0)1871 1060 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(;)928 b FA(\(2.13\))599 1251 y(and)28 b(the)g("separated)e(atoms)h(limit")g(\(SAL\))1741 1447 y(lim)1707 1500 y Fo(R)p Fl(!1)1903 1447 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(:)896 b FA(\(2.14\))599 1670 y(F)-7 b(ormally)g(,)26 b(one)h(w)n(ould)f(exp)r(ect)h(the)g(UAL)g (b)r(eing)g(describ)r(ed)g(b)n(y)f(an)h(atomic)f(h)n(ydro-)599 1770 y(genic)i(Hamiltonian)f(for)g(t)n(wice)h(the)g(n)n(uclear)e(c)n (harge,)g(viz.,)h(b)n(y)1519 1966 y Fr(h)p FA(\(0)p Fr(;)14 b(\020)6 b FA(\))23 b(=)g Fr(T)1912 1978 y Fo(D)1990 1966 y Fs(\000)18 b FA(2)p Fr(\020)6 b Fs(j)p Fn(x)p Fs(j)2258 1931 y Fl(\000)p Ft(1)2347 1966 y Fr(;)707 b FA(\(2.15\))599 2161 y(whereas)24 b(the)h(SAL)f(w)n(ould)g(b)r(e)h (go)n(v)n(erned)e(b)n(y)h(the)h(h)n(ydrogen)d(op)r(erator)h(with)i(c)n (harge)599 2261 y Fr(\020)6 b FA(,)1519 2410 y Fr(h)p FA(\()p Fs(1)p Fr(;)14 b(\020)6 b FA(\))24 b(=)e Fr(T)1953 2422 y Fo(D)2031 2410 y Fs(\000)c Fr(\020)6 b Fs(j)p Fn(x)q Fs(j)2258 2376 y Fl(\000)p Ft(1)2347 2410 y Fr(:)707 b FA(\(2.16\))599 2557 y(Precise)27 b(results)g(for)g(b)r(oth)h(limits) g(will)g(b)r(e)g(discussed)f(b)r(elo)n(w.)1271 2723 y Fu(2.1.)94 b(Self-adjoin)m(tness)31 b(prop)s(erties)682 2839 y FA(T)-7 b(o)22 b(examine)f(domain)h(and)f(self-adjoin)n(tness)g (questions)h(for)f Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\),)23 b(w)n(e)f(ha)n(v)n(e)f(to)599 2939 y(distinguish)28 b(b)r(et)n(w)n(een)f(the)h(situations)f Fr(R)c FA(=)g(0)k(and)g Fr(R)d(>)e FA(0.)37 b(This)27 b(is)g(in)h(sharp)e(con-)599 3039 y(trast)32 b(to)f(analogous)f(one-electron)g(Sc)n(hr\177)-42 b(odinger)30 b(op)r(erators,)h(and)h(related)f(to)h(the)599 3138 y(existence)j(of)g(a)f(critical)g(c)n(harge)f(separating)g (di\013eren)n(t)i(self-adjoin)n(tness)f(regimes)599 3238 y(for)g(the)h(Dirac-Coulom)n(b)e(Hamiltonian.)57 b(As)34 b(a)g(heuristic)g(explanation)g(w)n(e)g(note)599 3337 y(that)28 b(apparen)n(tly)f(the)h(\(relativ)n(e\))f(con)n(tribution)g (of)g(the)i(kinetic)e(energy)g Fr(T)2993 3349 y Fo(D)3080 3337 y FA(is)h(less)599 3437 y(than)36 b(the)f(Sc)n(hr\177)-42 b(odinger)33 b(kinetic)i(energy)f(con)n(tribution)h Fr(T)2493 3449 y Fo(S)2576 3437 y FA(=)g Fn(p)2726 3407 y Ft(2)2763 3437 y Fr(=)p FA(2)f(\(con)n(taining)599 3537 y(the)25 b(square)f(of)g(the)h(momen)n(tum\),)h(so)e(that)h(for)f(Dirac)g (systems)g(the)h(electron)f(could)599 3636 y(collapse)33 b(within)i(a)f(\014nite)g(time)h(in)n(to)f(the)g(\(p)r(oin)n(tlik)n (e\))h(n)n(ucleus)f(if)g(the)h(attractiv)n(e)599 3736 y(p)r(oten)n(tial)28 b(gets)f(to)r(o)g(strong)g(for)g(large)f(enough)h (n)n(uclear)f(c)n(harge.)682 3836 y(T)-7 b(o)24 b(construct)e (systematically)h(the)h(molecular)e(Hamiltonian,)i(w)n(e)f(\014rst)g (consider)599 3935 y(the)k(kinetic)g(part)e Fr(T)1237 3947 y Fo(D)1297 3935 y FA(;)i(straigh)n(tforw)n(ard)c(extension)j(of)g (argumen)n(ts)f(from)h(Ref.)h([26)o(])599 4035 y(sho)n(ws)d(that)h Fr(T)1061 4047 y Fo(D)1145 4035 y FA(is)g(essen)n(tially)f(self-adjoin) n(t)g(on)g Fr(C)2222 4005 y Fl(1)2216 4055 y Ft(0)2293 4035 y FA(\()p Fq(R)2379 4005 y Ft(3)2435 4035 y Fs(n)13 b Fr(K)2561 4047 y Fo(c)2593 4035 y FA(\))26 b(where)e Fr(K)2959 4047 y Fo(c)3017 4035 y FA(denotes)599 4134 y(a)d(\014nite)h(set)g(of)f(p)r(oin)n(ts)g(in)h Fq(R)1468 4104 y Ft(3)1511 4134 y FA(,)h(e.g.,)f Fr(K)1798 4146 y Fo(c)1855 4134 y FA(=)g Fs(f)p FA(0)p Fs(g)p FA(,)g(or)e Fr(K)2279 4146 y Fo(c)2335 4134 y FA(=)j Fs(f\000)2540 4102 y Ft(1)p 2539 4116 34 4 v 2539 4163 a(2)2582 4134 y Fn(R)2655 4146 y Fo(s)2690 4134 y Fr(;)2737 4102 y Ft(1)p 2737 4116 V 2737 4163 a(2)2780 4134 y Fn(R)2853 4146 y Fo(s)2889 4134 y Fs(g)p FA(.)34 b(F)-7 b(urther-)599 4234 y(more,)22 b Fr(T)874 4246 y Fo(D)954 4234 y FA(is)e(self-adjoin)n (t)g(on)g(the)h(Sob)r(olev)f(space)f Fs(D)r FA(\()p Fr(T)2357 4246 y Fo(D)2418 4234 y FA(\))k(=)g Fr(H)2637 4204 y Ft(1)2674 4234 y FA(\()p Fq(R)2760 4204 y Ft(3)2803 4234 y Fr(;)14 b Fq(C)2894 4204 y Ft(4)2937 4234 y FA(\),)23 b(and)d(has)599 4334 y(a)28 b(purely)g(absolutely)f(con)n(tin)n(uous)g (sp)r(ectrum)i Fr(\033)s FA(\()p Fr(T)2232 4346 y Fo(D)2292 4334 y FA(\))24 b(=)g Fr(\033)2484 4346 y Ft(ac)2551 4334 y FA(\()p Fr(T)2632 4346 y Fo(D)2692 4334 y FA(\))g(=)g Fq(R)g Fs(n)19 b FA(\()p Fs(\000)p FA(1)p Fr(;)14 b FA(+1\))599 4433 y(with)26 b(the)f(gap)f(\()p Fs(\000)p FA(1)p Fr(;)14 b FA(1\).)35 b(Since)26 b Fs(j)p Fn(x)12 b Fs(\006)1781 4401 y Ft(1)p 1781 4415 V 1781 4462 a(2)1824 4433 y Fn(R)1897 4445 y Fo(s)1932 4433 y Fs(j)1955 4403 y Fl(\000)p Ft(1)2070 4433 y FA(is)24 b(relativ)n(ely)g(b)r(ounded)h(with)h(resp)r(ect)599 4533 y(to)31 b Fr(T)753 4545 y Fo(D)843 4533 y FA(with)g(relativ)n(e)f (b)r(ound)g(2,)h(one)g(can)f(in)n(v)n(ok)n(e)f(the)h(Kato-Rellic)n(h)g (theorem)g(to)p eop %%Page: 7 7 7 6 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)477 b FA(7)599 320 y(arriv)n(e)22 b(at)h(a)g(self-adjoin)n(t)g (molecular)f(Hamiltonian)h(as)f(long)h(as)f(the)i(n)n(uclear)e(c)n (harge)599 419 y(parameter)37 b(ob)r(eys)g(2)p Fs(j)p Fr(\020)6 b Fs(j)40 b Fr(<)f FA(1)p Fr(=)p FA(2)d(if)i Fr(R)j FA(=)e(0,)h(or)d Fs(j)p Fr(\020)6 b Fs(j)40 b Fr(<)f FA(1)p Fr(=)p FA(2)e(if)h Fr(R)i(>)f FA(0.)67 b(F)-7 b(or)37 b(the)599 519 y(one-cen)n(tre)27 b(\(atomic\))g(op)r (erator)1496 718 y Fr(h)1544 730 y Ft(at)1607 718 y FA(\()p Fr(\020)6 b FA(\))24 b(=)f Fr(T)1874 730 y Fo(D)1952 718 y FA(+)18 b Fr(V)2083 730 y Ft(ne)2154 718 y FA(\(0)p Fr(;)2275 685 y Ft(1)p 2275 699 34 4 v 2275 747 a(2)2318 718 y Fr(\020)6 b FA(\))685 b(\(2.17\))599 917 y(and)29 b(more)g(general)f Fr(\020)k Fs(\025)25 b FA(0,)k(the)h(self-adjoin)n (tness)e(prop)r(erties)g(of)h Fr(h)2770 929 y Ft(at)2863 917 y FA(are)f(summa-)599 1017 y(rized)g(b)n(y:)682 1183 y Fj(Theorem)j(2.1.)66 b Fi(\(i\))28 b(If)g FA(2)14 b Fr(\020)28 b(<)1685 1114 y Fs(p)p 1754 1114 42 4 v 69 x FA(3)p Fr(=)p FA(2)p Fi(,)f(then)g Fr(h)2162 1195 y Ft(at)2225 1183 y FA(\(2)p Fr(\020)6 b FA(\))28 b Fi(is)g(essential)t (ly)g(self-adjoint)599 1283 y(on)i Fr(C)783 1252 y Fl(1)777 1303 y Ft(0)854 1283 y FA(\()p Fq(R)940 1252 y Ft(3)1002 1283 y Fs(n)18 b(f)p FA(0)p Fs(g)p Fr(;)c Fq(C)1277 1252 y Ft(4)1320 1283 y FA(\))p Fi(,)31 b(and)f(self-adjoint)i(on)e Fr(H)2195 1252 y Ft(1)2232 1283 y FA(\()p Fq(R)2318 1252 y Ft(3)2361 1283 y Fr(;)14 b Fq(C)2452 1252 y Ft(4)2495 1283 y FA(\))p Fi(.)682 1382 y(\(ii\))47 b(If)950 1314 y Fs(p)p 1019 1314 V 68 x FA(3)p Fr(=)p FA(2)k Fs(\024)h FA(2)14 b Fr(\020)57 b(<)52 b FA(1)p Fi(,)e(then)45 b Fr(h)1944 1394 y Ft(at)2007 1382 y FA(\(2)p Fr(\020)6 b FA(\))47 b Fi(is)f(not)f(essential)t(ly)i(self-adjoint)599 1482 y(on)d Fr(C)797 1452 y Fl(1)791 1502 y Ft(0)867 1482 y FA(\()p Fq(R)954 1452 y Ft(3)1025 1482 y Fs(n)28 b(f)p FA(0)p Fs(g)p Fr(;)14 b Fq(C)1310 1452 y Ft(4)1353 1482 y FA(\))p Fi(.)79 b(Ther)l(e)44 b(exists,)j(however,)i(a)43 b(unique)g(self-adjoint)i(ex-)599 1581 y(tension)c(that)f(is)h(char)l (acterize)l(d)i(by)e(\014niteness)e(of)j(the)f(kinetic)g(ener)l(gy,)j (i.e.,)h(by)599 1681 y Fs(D)r FA(\()p Fr(h)745 1693 y Ft(at)809 1681 y FA(\(2)p Fr(\020)6 b FA(\)\))24 b Fs(\032)f(D)r FA(\()p Fs(j)p Fr(T)1271 1693 y Fo(D)1331 1681 y Fs(j)1354 1651 y Ft(1)p Fo(=)p Ft(2)1458 1681 y FA(\))p Fi(,)29 b(or,)g(alternatively)g(\(and)f(e)l(quivalently\),)i(by)e(\014niteness) 599 1781 y(of)j(the)f(p)l(otential)g(ener)l(gy,)h Fs(D)r FA(\()p Fr(h)1608 1793 y Ft(at)1672 1781 y FA(\(2)p Fr(\020)6 b FA(\)\))23 b Fs(\032)g(D)r FA(\()2061 1710 y Fp(p)p 2145 1710 288 4 v 2145 1781 a FA(2)14 b Fr(\020)6 b Fs(j)p Fn(x)o Fs(j)2343 1757 y Fl(\000)p Ft(1)2432 1781 y FA(\))p Fi(.)682 2030 y FA(The)40 b(pro)r(ofs)e(for)h(the)h(assertions)d(of)j (theorem)e(\(2.1\))h(are)g(distributed)h(among)599 2129 y(sev)n(eral)26 b(articles)g(in)i(the)g(literature;)e(see,)h(e.g.)37 b([27)o(,)27 b(28)o(,)h(29)o(])f(for)g(\(i\),)h(or)e([27)o(,)i(30)o(,)f (31)o(,)599 2229 y(32)o(,)c(33)o(,)g(34)o(])g(in)h(case)e(of)g(\(ii\).) 36 b(F)-7 b(or)23 b(m)n(ulticen)n(tre)f(systems)h(\(also)f(with)h(more) f(than)h(t)n(w)n(o)599 2329 y(n)n(uclei\),)k(corresp)r(onding)c (self-adjoin)n(tness)i(results)g(w)n(ere)g(demonstrated)g(b)n(y)g (Klaus)599 2428 y([19]:)682 2594 y Fj(Theorem)31 b(2.2.)69 b Fi(Supp)l(ose)30 b Fr(R)24 b(>)e FA(0)p Fi(.)682 2694 y(\(i\))i(If)f Fs(j)p Fr(\020)6 b Fs(j)23 b Fr(<)1078 2625 y Fs(p)p 1147 2625 42 4 v 69 x FA(3)p Fr(=)p FA(2)p Fi(,)g(then)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b Fi(is)g(self-adjoint)i(on)d(the)h(domain)h Fs(D)r FA(\()p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\)\))25 b(=)599 2794 y Fs(D)r FA(\()p Fr(T)746 2806 y Fo(D)807 2794 y FA(\))e(=)g Fr(H)1026 2763 y Ft(1)1063 2794 y FA(\()p Fq(R)1149 2763 y Ft(3)1192 2794 y Fr(;)14 b Fq(C)1283 2763 y Ft(4)1326 2794 y FA(\))p Fi(.)37 b(This)22 b(implies,)j(in)d(p)l (articular,)i(that)d Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))22 b Fi(is)g(essential)t(ly)599 2893 y(self-adjoint)i(on)d(any) h(c)l(or)l(e)f(for)h Fr(T)1622 2905 y Fo(D)1682 2893 y Fi(,)h(like,)i(e.g.,)f Fs(D)2141 2905 y Ft(0)2202 2893 y FA(=)e Fr(C)2354 2863 y Fl(1)2348 2914 y Ft(0)2425 2893 y FA(\()p Fq(R)2511 2863 y Ft(3)2555 2893 y Fs(nf\000)2714 2860 y Ft(1)p 2712 2874 34 4 v 2712 2922 a(2)2755 2893 y Fn(R)2828 2905 y Fo(s)2863 2893 y Fr(;)2910 2860 y Ft(1)p 2910 2874 V 2910 2922 a(2)2953 2893 y Fn(R)3026 2905 y Fo(s)3062 2893 y Fs(g)p Fr(;)14 b Fq(C)3194 2863 y Ft(4)3237 2893 y FA(\))p Fi(.)682 3005 y(\(ii\))22 b(F)-6 b(or)969 2937 y Fs(p)p 1038 2937 42 4 v 68 x FA(3)p Fr(=)p FA(2)22 b Fr(<)g Fs(j)p Fr(\020)6 b Fs(j)24 b Fr(<)f FA(1)p Fi(,)g(ther)l(e)e(exists)g(a)g(self-adjoint)j(extension)d (of)h Fr(h)2962 3017 y Ft(0)2999 3005 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(:=)599 3105 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))854 3117 y Fh(\030)p Fl(D)935 3125 y Fg(0)1012 3105 y Fi(that)41 b(is)g(uniquely)g(determine)l(d)h(by)f Fs(D)r FA(\()p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\)\))44 b Fs(\032)e(D)r FA(\()p Fs(j)p Fr(T)2897 3117 y Fo(D)2958 3105 y Fs(j)2981 3075 y Ft(1)p Fo(=)p Ft(2)3085 3105 y FA(\))p Fi(,)i(or,)599 3205 y(e)l(quivalently,)32 b(by)e Fs(D)r FA(\()p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\)\))24 b Fs(\032)f(D)r FA(\()p Fs(j)p Fr(V)1847 3217 y Ft(ne)1918 3205 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fs(j)2148 3175 y Ft(1)p Fo(=)p Ft(2)2253 3205 y FA(\))p Fi(.)682 3304 y(\(iii\))39 b(F)-6 b(or)37 b(al)t(l)g Fs(j)p Fr(\020)6 b Fs(j)37 b Fr(<)e FA(1)i Fi(and)g(the)g(self-adjoint) i(op)l(er)l(ators)f Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))37 b Fi(liste)l(d)g(in)g(\(i\))599 3404 y(or)d(\(ii\),)h(the)e (essential)h(sp)l(e)l(ctrum)e(is)i(indep)l(endent)f(of)h Fr(\020)40 b Fi(and)33 b Fr(R)q Fi(,)h Fr(\033)2789 3416 y Ft(ess)2876 3404 y FA(\()p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\)\))30 b(=)599 3504 y Fr(\033)646 3516 y Ft(ess)733 3504 y FA(\()p Fr(T)814 3516 y Fo(D)874 3504 y FA(\))24 b(=)e Fq(R)j Fs(n)18 b FA(\()p Fs(\000)p FA(1)p Fr(;)c FA(+1\))1329 3819 y Fu(2.2.)94 b(Symmetries)28 b(and)33 b(scaling)682 3935 y FA(Since)g(some)g(symmetry)f(prop)r (erties)g(of)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))33 b(will)g(b)r(e)h(of)e(relev)-5 b(ance)32 b(for)g(our)599 4035 y(in)n(v)n(estigations,)j(let)g(us)g(brie\015y)f(recall)g(that)h (the)g(symmetry)g(transformations)d(of)599 4134 y(a)38 b(homon)n(uclear)e(diatomic)i(molecule)g(form)g(the)g(group)f Fn(D)2567 4146 y Fl(1)p Fo(h)2714 4134 y FA(\(see,)k(e.g.,)f([35)o (]\).)599 4234 y(W)-7 b(e)37 b(can)f(express)g Fn(D)1291 4246 y Fl(1)p Fo(h)1438 4234 y FA(=)i Fn(C)1614 4246 y Fl(1)p Fo(h)1748 4234 y Fs(\002)24 b Fn(C)1911 4246 y Fo(i)1975 4234 y FA(as)36 b(the)g(pro)r(duct)h(of)f(its)h(subgroups)e Fn(C)3262 4246 y Fo(i)599 4334 y FA(and)j Fn(C)844 4346 y Fl(1)p Fo(h)954 4334 y FA(.)66 b(In)38 b(addition)f(to)h(the)f(group) g(iden)n(tit)n(y)g Fr(E)5 b FA(,)41 b(the)c(latter)h(con)n(tains)e(the) 599 4433 y(follo)n(wing)d(symmetry)h(op)r(erations:)49 b Fn(C)1874 4445 y Fl(1)p Fo(h)2017 4433 y FA(=)34 b Fs(f)p Fr(E)5 b(;)14 b(C)2326 4403 y Fo(\036)2320 4454 y Fl(1)2390 4433 y Fr(;)g(\033)2474 4445 y Fo(v)2548 4433 y Fs(j)34 b Fr(\036)g Fs(2)g FA(\(0)p Fr(;)14 b FA(2)p Fr(\031)s FA(\))p Fs(g)p FA(;)37 b(here,)599 4533 y Fr(C)664 4503 y Fo(\036)658 4554 y Fl(1)750 4533 y FA(denotes)21 b(the)h(rotation)e(of)h(angle)f Fr(\036)i FA(around)e(the)i(in)n(tern)n(uclear)d(axis,)j(and)f Fr(\033)3113 4545 y Fo(v)3174 4533 y FA(the)p eop %%Page: 8 8 8 7 bop 599 75 a FA(8)905 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(re\015ection)g(at)g(a)g(v)n(ertical)f(plane)h(through)g(the)g (in)n(tern)n(uclear)f(axis.)35 b(The)23 b(\(sub\)group)599 419 y Fn(C)673 431 y Fo(i)728 419 y FA(=)28 b Fs(f)p Fr(E)5 b(;)14 b(i)995 431 y Fo(O)1050 419 y Fs(g)30 b FA(has)g(as)g(only)g(non)n(trivial)f(elemen)n(t)i(the)f(in)n(v)n (ersion)f(transformation)599 519 y Fr(i)628 531 y Fo(O)712 519 y FA(at)f(the)g(origin)f Fr(O)j FA(\(=)e(midp)r(oin)n(t)g(b)r(et)n (w)n(een)g(the)g(n)n(uclear)f(cen)n(tres\).)37 b(Giv)n(en)27 b(these)599 619 y(symmetry)e(op)r(erations,)g(the)h(asso)r(ciated)e (unitary)h(transformations)f(on)h Fs(H)h FA(lead)g(to)599 718 y(unitary)e(represen)n(tations)e(of)i(the)g(\(discrete\))g(group)f Fn(C)2362 730 y Fo(i)2414 718 y FA(or)g(the)h(\(Lie\))h(group)d Fn(C)3157 730 y Fl(1)p Fo(h)3266 718 y FA(.)599 818 y(If)30 b Fr(P)737 830 y Fo(i)760 838 y Ff(O)843 818 y FA(stands)f(for)g(the)h (op)r(erator)d(\()p Fr(P)1799 830 y Fo(i)1822 838 y Ff(O)1896 818 y Fs(\012)19 b Fq(I)2018 830 y Ft(4)2050 818 y Fr( )s FA(\)\()p Fn(x)p FA(\))26 b(=)f Fr( )s FA(\()p Fr(i)2492 830 y Fo(O)2549 818 y Fn(x)o FA(\),)31 b(then)e(the)h(unitary)599 917 y(transformation)36 b(on)i Fs(H)g FA(corresp)r(onding)e(to)h(the)h (in)n(v)n(ersion)e(op)r(eration)h Fr(i)2987 929 y Fo(O)3081 917 y FA(is)g(the)599 1017 y(parit)n(y)27 b(op)r(erator)1681 1166 y Fr(P)1734 1178 y Fo(O)1814 1166 y FA(=)c Fr(P)1955 1178 y Fo(i)1978 1186 y Ff(O)2050 1166 y Fs(\012)18 b Fr(\014)t(:)870 b FA(\(2.18\))599 1305 y(In)28 b(the)g(same)f(w)n(a)n (y)-7 b(,)27 b(de\014ning)1684 1536 y(\006)1744 1548 y Fo(i)1795 1536 y FA(=)1882 1419 y Fp(\022)1944 1485 y Fr(\033)1991 1497 y Fo(i)2085 1485 y FA(0)1960 1585 y(0)66 b Fr(\033)2115 1597 y Fo(i)2144 1419 y Fp(\023)3077 1536 y FA(\(2.19\))599 1771 y(in)31 b(terms)e(of)h(the)g(P)n(auli)f (matrices)g Fr(\033)1774 1783 y Fo(i)1802 1771 y FA(,)i(and)f Fr(s)2059 1783 y Fo(v)2128 1771 y FA(b)n(y)g(\()p Fr(s)2317 1783 y Fo(v)2376 1771 y Fs(\012)20 b Fq(I)2499 1783 y Ft(4)2530 1771 y Fr( )s FA(\)\()p Fn(x)q FA(\))27 b(=)f Fr( )s FA(\()p Fr(\033)2993 1783 y Fo(v)3033 1771 y Fn(x)p FA(\),)31 b(the)599 1871 y(unitary)24 b(op)r(erator)d(asso)r(ciated)i (with)h(the)g(v)n(ertical)e(re\015ection)h Fr(\033)2629 1883 y Fo(v)2693 1871 y FA(is)g(determined)h(b)n(y)1694 2056 y Fr(S)1745 2068 y Fo(v)1807 2056 y FA(=)f Fr(s)1934 2068 y Fo(v)1992 2056 y Fs(\012)18 b FA(\006)2135 2068 y Ft(3)2172 2056 y Fr(:)882 b FA(\(2.20\))599 2242 y(T)-7 b(o)39 b(sp)r(ecify)g(the)h(op)r(erator)d(implemen)n(ting)i(rotations)f (around)g(the)h(in)n(tern)n(uclear)599 2341 y(axis,)25 b(let)h(\003)f(b)r(e)g(the)h(pro)5 b(jection)24 b(of)h(the)h(angular)d (momen)n(tum)j(op)r(erator)d Fn(L)g FA(=)f Fn(x)14 b Fs(^)g Fn(p)599 2441 y FA(on)n(to)38 b(the)i(in)n(tern)n(uclear)d(axis) h(\(viz.,)k(\003)f(=)g Fr(L)2086 2453 y Ft(3)2162 2441 y FA(when)e(w)n(e)f(assume)h(the)g(molecule)599 2540 y(orien)n(ted)24 b(along)g(the)h Fr(z)t FA(-axis\),)e(and)i(\006)1786 2552 y Ft(3)1848 2540 y FA(the)g(corresp)r(onding)d(pro)5 b(jection)24 b(of)g(the)h(spin)599 2640 y(op)r(erator.)38 b(Com)n(bining)28 b(b)r(oth)h(pro)5 b(jections,)28 b(\012)c(=)g(\003)19 b Fs(\012)f Fq(I)2413 2652 y Ft(4)2463 2640 y FA(+)2557 2618 y Fg(1)p 2557 2627 29 3 v 2557 2660 a(2)2595 2640 y Fq(I)2633 2652 y Fl(H)2707 2640 y Fs(\012)h FA(\006)2851 2652 y Ft(3)2888 2640 y FA(,)29 b(yields)f(the)599 2740 y(hermitian)34 b(generator)f(\012)h(for)f(the)i(unitary)f(rotation)f (op)r(erators)f(exp\()p Fr(i)p FA(\012)p Fr(\036)p FA(\).)57 b(The)599 2839 y(generator)36 b(\012)i(furnishes)f(the)h (generalization)e(of)h Fr(J)2280 2851 y Ft(3)2355 2839 y FA(to)h(nonradially)e(symmetric)599 2939 y(systems.)h(Ob)n(viously)-7 b(,)27 b(for)g(molecules)g(with)h(total)g(angular)e(momen)n(tum)i Fn(J)k FA(=)23 b Fn(L)18 b Fs(\012)599 3039 y Fq(I)638 3051 y Ft(2)687 3039 y FA(+)g Fq(I)809 3051 y Fl(H)882 3039 y Fs(\012)g Fu(\006)28 b FA(the)g(op)r(erator)e Fr(J)1594 3008 y Ft(2)1659 3039 y FA(no)h(longer)f(comm)n(utes)h(with)i Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))28 b(if)g Fr(R)23 b(>)g FA(0.)682 3138 y(Since)c(they)g(em)n(b)r(o)r(dy)f (symmetry)g(op)r(erations,)h(the)g(op)r(erators)d Fr(P)2716 3150 y Fo(O)2773 3138 y Fr(;)e(S)2861 3150 y Fo(v)2900 3138 y Fr(;)g FA(exp\()p Fr(i)p FA(\012)p Fr(\036)p FA(\)\))599 3238 y(comm)n(ute)28 b(with)h Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\).)39 b(F)-7 b(or)28 b(ev)n(ery)f(eigen)n(v)-5 b(alue)27 b Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))28 b(of)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\),)29 b(the)g(asso)r(ci-)599 3337 y(ated)38 b(eigen)n(v)n(ectors)c(span)j(a)g (subspace)f(in)n(v)-5 b(arian)n(t)36 b(under)h(these)h(op)r(erators,)f (th)n(us)599 3437 y(leading)42 b(to)g(represen)n(tation)f(spaces)g(for) g(\(irreducible\))i(represen)n(tations)d(of)i(the)599 3537 y(symmetry)d(group)f Fn(D)1327 3549 y Fl(1)p Fo(h)1436 3537 y FA(.)72 b(These)39 b(represen)n(tations)e(can)i(also)g(b)r(e)g (emplo)n(y)n(ed)g(to)599 3636 y(mark)34 b(the)g(eigen)n(v)-5 b(alues,)34 b(i.e.,)i(the)f(PECs)d Fr(E)5 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))35 b(=)e Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(+)f Fr(\013)2799 3601 y Fl(\000)p Ft(1)2799 3661 y Fo(S)2889 3636 y Fr(\020)2931 3606 y Ft(2)2969 3636 y Fr(R)3033 3606 y Fl(\000)p Ft(1)3156 3636 y FA(and)599 3736 y(electronic)28 b(curv)n(es)f Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\).)39 b(An)29 b(appropriate)e(lab)r(el)h(set)g(w)n(ould)g(include)h(a)f(princi-)599 3836 y(pal)h(quan)n(tum)f(n)n(um)n(b)r(er)g(\(to)g(distinguish)h(m)n (ultiple)g(o)r(ccurrence)e(of)h(the)h(same)f(irre-)599 3935 y(ducible)21 b(represen)n(tation\),)f(and)g(the)h(eigen)n(v)-5 b(alues)19 b(of)h Fr(P)2322 3947 y Fo(O)2379 3935 y Fr(;)14 b(S)2467 3947 y Fo(v)2526 3935 y FA(and)20 b(\012.)35 b(Alternativ)n(ely)-7 b(,)599 4035 y(for)26 b(one-electron)f(systems,)i (together)e(with)i(the)g(\(mo)r(dulus)g(of)g(the\))g(eigen)n(v)-5 b(alues)25 b Fr(\026)599 4134 y FA(of)33 b(\012)f(and)g(with)h(parit)n (y)-7 b(,)32 b(one)g(ma)n(y)g(sp)r(ecify)g(the)h(h)n(ydrogenic)d (states)i(that)g(emerge)599 4234 y(in)40 b(the)f(UAL)h([36)o(].)72 b(If)39 b Fs(f)p Fr(nl)r(j)5 b Fs(g)38 b FA(denotes)g(suc)n(h)h(a)g (set)g(of)g(\(relativistic\))g(h)n(ydrogen)599 4334 y(quan)n(tum)27 b(n)n(um)n(b)r(ers)f(\(e.g.,)g(1s)1564 4349 y Ft(1)p Fo(=)p Ft(2)1668 4334 y FA(,)40 b(2p)1819 4349 y Ft(1)p Fo(=)p Ft(2)1923 4334 y FA(,)h(2p)2075 4349 y Ft(3)p Fo(=)p Ft(2)2179 4334 y FA(,)f Fr(:)14 b(:)g(:)p FA(\),)27 b(then)g(the)g(molecular)e(elec-)599 4433 y(tronic)30 b(states)f(are)g(lab)r(eled)h(b)n(y)f Fs(f)p Fr(nl)r(j)5 b Fs(gj)p Fr(\026)p Fs(j)1920 4445 y Fo(g)r(;u)2046 4433 y FA(where)29 b(the)h(subscripts)g Fr(g)s FA(\(erade\))f(and)599 4533 y Fr(u)p FA(\(ngerade\))36 b(indicate)h(the)g(eigen)n(v)-5 b(alues)35 b(+1)h(and)h Fs(\000)p FA(1)f(of)g Fr(P)2556 4545 y Fo(O)2613 4533 y FA(,)j(resp)r(ectiv)n(ely)-7 b(,)39 b(and)p eop %%Page: 9 9 9 8 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)477 b FA(9)599 320 y(where)29 b(one)g(usually)g(adopts)g(the)h (traditional)e(sp)r(ectroscopic)g(notation)h Fr(\033)n(;)14 b(\031)s(;)g(\016)o(;)g(:)g(:)g(:)599 419 y FA(for)28 b Fs(j)p Fr(\026)p Fs(j)23 b FA(=)f(1)p Fr(=)p FA(2)p Fr(;)14 b FA(3)p Fr(=)p FA(2)p Fr(;)g FA(5)p Fr(=)p FA(2)p Fr(;)g(:)g(:)g(:)9 b FA(.)682 519 y(An)26 b(imp)r(ortan)n(t)g(result)f (for)g(the)h(electronic)f(curv)n(es)f(follo)n(ws)h(from)g(the)h(b)r (eha)n(viour)599 619 y(of)i Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))29 b(under)f(c)n(hanges)e(of)i(the)h(length)f(scale)f (of)h(its)g(dynamical)f(v)-5 b(ariable.)37 b(Let)599 718 y Fr(U)656 730 y Fo(a)720 718 y FA(denote)24 b(the)g(unitary)g (\(dilatation\))g(op)r(erator)e(on)i Fs(H)h FA(that)f(e\013ects)g(the)h (scaling)d(of)599 818 y(the)k Fn(x)p FA(-co)r(ordinates)d(b)n(y)i Fr(a)e Fs(2)g Fq(R)p FA(,)32 b(i.e.,)26 b(\()p Fr(U)1872 830 y Fo(a)1912 818 y Fr( )s FA(\)\()p Fn(x)p FA(\))e(=)e Fr(a)2275 788 y Fl(\000)p Ft(3)p Fo(=)p Ft(2)2432 818 y Fr( )s FA(\()p Fr(a)p Fn(x)o FA(\).)37 b(Then,)26 b(for)e Fr(R)g(>)f FA(0)599 1008 y Fr(U)656 1020 y Fo(R)711 1008 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(U)1032 972 y Fl(\000)p Ft(1)1023 1032 y Fo(R)1144 1008 y FA(=)23 b Fr(R)1296 973 y Fl(\000)p Ft(1)1384 1008 y Fn(\013)f Fm(\001)f Fn(p)c FA(+)g Fr(\014)22 b FA(+)17 b Fr(R)1881 973 y Fl(\000)p Ft(1)1969 1008 y Fr(V)2017 1020 y Ft(ne)2088 1008 y FA(\(1)p Fr(;)d(\020)6 b FA(\))24 b(=)e Fr(R)2448 973 y Fl(\000)p Ft(1)2537 940 y Fp(\000)2575 1008 y Fr(h)p FA(\(1)p Fr(;)14 b(\020)6 b FA(\))18 b(+)f(\()p Fr(R)h Fs(\000)f FA(1\))p Fr(\014)3228 940 y Fp(\001)3266 1008 y Fr(:)3077 1107 y FA(\(2.21\))599 1207 y(Since)36 b Fs(k)p Fr(\014)t Fs(k)f FA(=)h(1)e(and)h(th)n(us)h Fr(\014)j FA(is)d(a)e(b)r(ounded)i(op)r(erator,)g(for)e(all)h Fr( )k Fs(2)d(D)r FA(\()p Fr(h)p FA(\))h(and)599 1307 y Fr(\036)24 b Fs(2)f(H)q FA(,)639 1496 y Fr(R)703 1462 y Fl(\000)p Ft(1)791 1496 y Fs(h)p Fr(\036;)924 1429 y Fp(\000)962 1496 y Fr(h)p FA(\(1)p Fr(;)14 b(\020)6 b FA(\))19 b(+)f(\()p Fr(R)h Fs(\000)f FA(1\))p Fr(\014)1619 1429 y Fp(\001)1657 1496 y Fr( )s Fs(i)24 b FA(=)e Fr(R)1921 1462 y Fl(\000)p Ft(1)2010 1496 y Fs(h)p Fr(\036;)29 b(h)p FA(\(1)p Fr(;)14 b(\020)6 b FA(\))p Fr( )s Fs(i)19 b FA(+)f Fr(R)2631 1462 y Fl(\000)p Ft(1)2720 1496 y FA(\()p Fr(R)h Fs(\000)f FA(1\))p Fs(h)p Fr(\036;)c(\014)t( )s Fs(i)3077 1596 y FA(\(2.22\))599 1696 y(is)24 b(ob)n(viously)f(an)h(analytic)f (function)i(in)f Fr(R)h FA(as)e(long)h(as)f Fr(R)h Fs(6)p FA(=)e(0.)36 b(This)24 b(together)f(with)599 1795 y(Eq.)f(\(2.21\))f (and)h(the)h(indep)r(endence)g(of)f(the)g(domain)g(of)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(on)f Fr(R)h FA(implies)f(that)599 1895 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))30 b(is)g(a)f(holomorphic)f(family)i(of)f(t)n(yp)r(e)h (\(A\))g(in)g(the)g(sense)f(of)g(Kato)f([37)o(].)43 b(W)-7 b(e)599 1995 y(form)n(ulate)27 b(the)h(consequences)f(as)g(a)g(prop)r (osition:)682 2151 y Fj(Pr)n(oposition)32 b(2.1.)69 b Fi(If)31 b(for)g(some)f Fr(R)1893 2163 y Ft(0)1954 2151 y Fr(>)24 b FA(0)30 b Fi(the)g(eigenvalue)h Fr(e)p FA(\()p Fr(R)2788 2163 y Ft(0)2826 2151 y Fr(;)14 b(\020)6 b FA(\))24 b Fs(2)g FA(\()p Fs(\000)p FA(1)p Fr(;)14 b FA(1\))599 2251 y Fi(is)41 b(nonde)l(gener)l(ate)g(\(p)l(ossibly,)46 b(after)41 b(r)l(estricting)g(to)f(an)h(appr)l(opriate)i(symmetry)599 2351 y(\(sub\)se)l(ctor)36 b(of)g Fs(D)1198 2363 y Fl(1)p Fo(h)1308 2351 y Fi(\),)h(then)f(ther)l(e)f(exists)h(a)g(neighb)l (ourho)l(o)l(d)i(of)e Fr(R)2829 2363 y Ft(0)2866 2351 y Fi(,)i(such)e(that,)599 2450 y(for)c(al)t(l)h Fr(R)e Fi(in)h(this)f(neighb)l(ourho)l(o)l(d,)j Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))31 b Fi(is)h(analytic)g(in)f Fr(R)h Fi(with)g Fs(j)p Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fs(j)26 b Fr(<)f FA(1)599 2550 y Fi(and)31 b(a)f(nonde)l(gener)l(ate)g(eigenvalue)h(of)f Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fi(.)714 2699 y FA(Analogous)31 b(results)g(could)h(b)r(e)g(deriv)n(ed)f(for)g(the)h (dep)r(endence)h(of)e(the)i(energies)599 2799 y Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))33 b(on)g(the)g(c)n(harge)e (parameter)g Fr(\020)6 b FA(;)36 b(these,)e(ho)n(w)n(ev)n(er,)e(will)h (not)f(b)r(e)h(needed)g(in)599 2899 y(our)27 b(in)n(v)n(estigations)f (here.)1285 3065 y Fu(2.3.)94 b(The)32 b(sp)s(ectrum)e(in)i(the)f(SAL) 682 3181 y FA(As)j(already)e(indicated,)k(ph)n(ysical)c(and)i(n)n (umerical)f(exp)r(erience)g(motiv)-5 b(ates)33 b(the)599 3280 y(exp)r(ectation)21 b(that)h(for)e Fr(R)k Fs(!)f(1)e FA(the)h(molecular)e(energies)g(con)n(v)n(erge)e(to)j(those)g(of)g(the) 599 3380 y(constituen)n(t)35 b(atoms.)55 b(F)-7 b(or)34 b(the)h(considered)e(homon)n(uclear)f(one-electron)h(system,)599 3480 y(the)20 b(situation)f(b)r(ecomes)f(particularly)g(simple,)j(b)r (ecause)e(there)g(is)f(only)h(one)g(p)r(ossible)599 3579 y(limit,)35 b(viz.,)f(the)e(h)n(ydrogen)f(atomic)h(ion)g(with)h(c)n (harge)e Fr(\020)39 b FA(separated)31 b(from)h(a)g(bare)599 3679 y(proton)d(b)n(y)g(an)g(in\014nite)h(distance.)41 b(Rigorous)28 b(results)h(b)n(y)g(Harrell)f(and)h(Klaus)f([38)o(])599 3779 y(\(form)n(ulated)f(for)f(more)g(general)g(p)r(oten)n(tials\))h (can)f(b)r(e)i(transcrib)r(ed)e(to)h(this)g(sp)r(ecial)599 3878 y(case)g(in)h(the)g(follo)n(wing)f(w)n(a)n(y:)682 4035 y Fj(Theorem)k(2.3.)71 b Fi(F)-6 b(or)32 b(e)l(ach)g(atomic)h (ener)l(gy)f Fr(e)2200 4005 y Ft(at)2263 4035 y FA(\()p Fr(\020)6 b FA(\))27 b Fs(2)g Fr(\033)2525 4047 y Ft(pp)2603 4035 y FA(\()p Fr(h)2683 4005 y Ft(at)2746 4035 y FA(\()p Fr(\020)6 b FA(\)\))33 b Fi(ther)l(e)f(exist)599 4134 y(mole)l(cular)39 b(ener)l(gies)f Fr(e)1350 4146 y Fo(j)1385 4134 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))38 b Fi(that)g(have)h Fr(e)2046 4104 y Ft(at)2109 4134 y FA(\()p Fr(\020)6 b FA(\))39 b Fi(as)f(their)h(limit,)h(i.e.,)i FA(lim)3103 4146 y Fo(R)p Fl(!1)599 4234 y Fr(e)638 4246 y Fo(j)673 4234 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))26 b(=)e Fr(e)1034 4204 y Ft(at)1097 4234 y FA(\()p Fr(\020)6 b FA(\))p Fi(,)33 b(wher)l(e)e(the)g(numb)l(er)f(of)i(PECs)g(or)f(ele)l (ctr)l(onic)g(curves)g Fr(e)3047 4246 y Fo(j)3082 4234 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))599 4334 y Fi(c)l(onver)l(ging) 32 b(to)f Fr(e)1149 4304 y Ft(at)1212 4334 y FA(\()p Fr(\020)6 b FA(\))32 b Fi(e)l(quals)g(the)f(multiplicity)h(of)g Fr(e)2311 4304 y Ft(at)2374 4334 y FA(\()p Fr(\020)6 b FA(\))p Fi(.)44 b(Conversely,)33 b(the)f(only)599 4433 y(ac)l(cumulation)i(p)l(oints)g(of)1460 4371 y Fp(S)1529 4458 y Fo(R>R)1681 4467 y Ff(l)1723 4433 y Fr(\033)1770 4445 y Ft(pp)1848 4433 y FA(\()p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\)\))35 b Fi(for)f Fr(R)2401 4445 y Fo(l)2457 4433 y Fs(!)c(1)k Fi(ar)l(e)g(the)g(p)l(oints)f(of)599 4533 y Fr(\033)646 4545 y Ft(pp)724 4533 y FA(\()p Fr(h)804 4503 y Ft(at)868 4533 y FA(\()p Fr(\020)6 b FA(\)\))19 b Fs([)g(f)p FA(1)p Fs(g)p Fi(.)p eop %%Page: 10 10 10 9 bop 599 75 a FA(10)863 b Fz(BRIET)24 b(AND)f(HOGREVE)719 320 y FA(Besides)36 b(the)h(pro)r(of)f(giv)n(en)g(in)g(Ref.)h([38],)i (it)e(also)e(w)n(ould)h(b)r(e)h(p)r(ossible)f(to)h(de-)599 419 y(riv)n(e)d(this)g(theorem)g(b)n(y)g(the)g(more)g("geometrical")d (tec)n(hniques)j(emplo)n(y)n(ed)g(in)g(the)599 519 y(discussion)27 b(of)h(the)g(SAL)g(within)g(Sc)n(hr\177)-42 b(odinger)26 b(theory)h([39)o(,)g(40)o(,)h(41)o(].)1350 685 y Fu(2.4.)94 b(The)32 b(UAL)h(lo)m(w)m(er)e(b)s(ound)682 801 y FA(F)-7 b(or)28 b(molecular)f(Sc)n(hr\177)-42 b(odinger)27 b(op)r(erators,)g (also)g(for)h(those)g(with)h(more)e(than)i(one)599 901 y(electron)24 b(and)f(with)i(more)e(than)h(t)n(w)n(o)f(n)n(uclei,)i (the)f(electronic)g(energies)e(of)i(all)g(states)599 1000 y(and)36 b(all)e(n)n(uclear)h(geometries)e(of)j(the)f(molecule)g (can)g(b)r(e)h(sho)n(wn)e(to)h(b)r(e)h(b)r(ounded)599 1100 y(b)r(elo)n(w)k(b)n(y)f(the)h(electronic)e(ground)h(state)g (energy)g(of)g(the)h(system)f(in)h(the)g(UAL)599 1200 y(\(see,)33 b(e.g.,)f([42]\).)49 b(The)32 b(pro)r(of)f(of)g(this)h (fact)g(is)g(rather)e(easily)h(ac)n(hiev)n(ed)g(b)n(y)g(in)n(v)n(ok-) 599 1299 y(ing)i(v)-5 b(ariational)32 b(argumen)n(ts,)i(i.e.,)g(the)g (minimax)f(principle)g([43].)53 b(Since)34 b Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))599 1399 y(is)36 b(not)g(b)r(ounded)g(b)r (elo)n(w,)h(here,)g(unfortunately)-7 b(,)38 b(the)e(standard)f(\(Ra)n (yleigh-Ritz\))599 1499 y(v)-5 b(ariational)20 b(principles)h(cannot)g (b)r(e)g(applied.)35 b(On)21 b(the)g(other)g(hand,)h(rigorous)d(v)-5 b(aria-)599 1598 y(tional)23 b(c)n(haracterizations)d(for)i(eigen)n(v) -5 b(alues)22 b(of)h(Dirac)f(\(-Coulom)n(b\))h(op)r(erators)d(w)n(ere) 599 1698 y(recen)n(tly)27 b(established)g(in)h(the)g(literature)e([23)o (,)i(24)o(,)f(25)o(].)37 b(Belo)n(w,)27 b(w)n(e)g(shall)g(fall)g(bac)n (k)599 1797 y(on)n(to)33 b(the)h(corresp)r(onding)d(results)i(of)g (Dolb)r(eau,)i(Esteban,)f(and)f(S)n(\023)-39 b(er)n(\023)g(e)31 b([23)o(].)55 b(Nev-)599 1897 y(ertheless,)38 b(ev)n(en)e(with)g(these) h(adv)-5 b(anced)35 b(v)-5 b(ariational)35 b(to)r(ols)g(at)i(disp)r (osal,)g(due)g(to)599 1997 y(the)29 b(presence)e(of)h(the)h(negativ)n (e)e(energy)g(\(scattering\))h(states,)f(the)i(argumen)n(tation)599 2096 y(cannot)24 b(simply)g(b)r(e)g(adapted)g(from)g(Sc)n(hr\177)-42 b(odinger)22 b(to)i(Dirac)f(Hamiltonians;)i(in)f(case)599 2196 y(of)39 b Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\),)41 b(the)e(demonstration)e(of)h(the)h(UAL)f(lo)n(w)n(er)f(b)r (ound)i(turns)f(out)g(to)g(b)r(e)599 2296 y(distinctly)29 b(harder)d(than)i(for)f(similar)g(Sc)n(hr\177)-42 b(odinger)25 b(op)r(erators.)682 2395 y(In)33 b(the)g(sequel,)h(w)n(e)e(encoun)n (ter)g(the)h(atomic)f(system)g(with)i(the)f(n)n(uclear)e(cen)n(tre)599 2495 y(not)24 b(only)g(lo)r(cated)f(at)h(the)h(origin;)f(th)n(us,)g (with)h(\()p Fr(\034)2156 2507 y Fo(a)2197 2495 y Fr( )s FA(\)\()p Fn(x)p FA(\))e(:=)g Fr( )s FA(\()p Fn(x)11 b Fs(\000)g Fr(a)p Fn(R)2886 2507 y Fo(s)2922 2495 y Fr(=)p Fs(j)p Fn(R)3059 2507 y Fo(s)3095 2495 y Fs(j)p FA(\))24 b(the)599 2594 y(unitary)29 b(translation)g(op)r(erator)e(b)n (y)i Fr(a)d Fs(2)h Fq(R)35 b FA(along)29 b(the)g(unit)i(v)n(ector)d Fn(R)2880 2606 y Fo(s)2916 2594 y Fr(=)p Fs(j)p Fn(R)3053 2606 y Fo(s)3089 2594 y Fs(j)p FA(\),)i(let)599 2694 y Fr(h)647 2664 y Ft(at)710 2694 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))29 b(=)e Fr(\034)1074 2706 y Fo(R)1129 2694 y Fr(h)1177 2664 y Ft(at)1240 2694 y FA(\()p Fr(\020)6 b FA(\))p Fr(\034)1391 2659 y Fl(\000)p Ft(1)1382 2718 y Fo(R)1512 2694 y FA(stand)30 b(for)g(the)h(atomic)f(Hamiltonian)g (with)h(Coulom)n(b)599 2794 y(p)r(oten)n(tial)25 b(cen)n(tred)g(at)g Fr(R)15 b Fn(R)1487 2806 y Fo(s)1523 2794 y Fr(=)p Fs(j)p Fn(R)1660 2806 y Fo(s)1696 2794 y Fs(j)p FA(.)36 b(Similarly)-7 b(,)25 b(let)g Fr( )2319 2764 y Ft(at)2383 2794 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))25 b(b)r(e)h(the)f(corresp)r(ond-)599 2893 y(ing)19 b(translated)e(atomic)h(ground)f(state)i (eigenfunction\(s\))f(ob)r(eying)g Fr(h)2787 2863 y Ft(at)2850 2893 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))p Fr( )3114 2863 y Ft(at)3178 2893 y FA(\()p Fr(R)q(;)599 3002 y(\020)g FA(\))24 b(=)f Fr(e)824 2972 y Ft(at)887 3002 y FA(\()p Fr(\020)6 b FA(\))p Fr( )1050 2972 y Ft(at)1114 3002 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))28 b(with)g Fr(e)1577 2972 y Ft(at)1640 3002 y FA(\()p Fr(\020)6 b FA(\))24 b(=)1858 2929 y Fp(p)p 1941 2929 223 4 v 73 x FA(1)18 b Fs(\000)g Fr(\020)2126 2978 y Ft(2)2164 3002 y FA(.)682 3102 y(T)-7 b(o)18 b(construct)g(the)h(v)-5 b(ariational)17 b(represen)n(tation)g(of)h(the)h(relev)-5 b(an)n(t)17 b(Dirac-Coulom)n(b)599 3201 y(eigenfunctions)39 b(b)n(y)g(the)g(metho)r (d)h(of)f(Dolb)r(eau,)j(Esteban,)f(and)e(S)n(\023)-39 b(er)n(\023)g(e)36 b(\(DES\),)k(it)599 3301 y(w)n(ould)33 b(suggest)f(itself)h(to)g(follo)n(w)f(these)g(authors)g(in)h(their)g(c) n(hoice)f(of)h(the)g(decom-)599 3400 y(p)r(osition)j(of)h Fs(H)q FA(.)62 b(Namely)-7 b(,)39 b(de\014ning)d(pro)5 b(jections)35 b Fr(\022)2313 3412 y Fl(\006)2405 3400 y FA(in)h(terms)g(of)g(the)h(sp)r(ectral)599 3500 y(functions)1211 3663 y Fr(\022)1250 3675 y Ft(+)1329 3663 y FA(=)22 b Fr(\037)1468 3678 y Ft(\(0)p Fo(;)p Fl(1)p Ft(\))1643 3663 y FA(\()p Fr(T)1724 3675 y Fo(D)1784 3663 y FA(\))p Fr(;)180 b(\022)2058 3675 y Fl(\000)2138 3663 y FA(=)22 b Fr(\037)2277 3678 y Ft(\()p Fl(\0001)p Fo(;)p Ft(0\))2504 3663 y FA(\()p Fr(T)2585 3675 y Fo(D)2645 3663 y FA(\))400 b(\(2.23\))599 3826 y(for)28 b(the)h(free)g(Dirac)f(op)r(erator,)f(the) i(asso)r(ciated)e(subspaces)h Fs(H)2608 3838 y Fl(\006)2689 3826 y FA(=)c Fr(\022)2817 3838 y Fl(\006)2873 3826 y Fs(H)29 b FA(yield)g(the)599 3926 y(orthogonal)d(decomp)r(osition)1677 4130 y Fs(H)e FA(=)f Fs(H)1929 4142 y Ft(+)2002 4130 y Fs(\010)18 b(H)2155 4142 y Fl(\000)3077 4130 y FA(\(2.24\))599 4334 y(F)-7 b(or)24 b(these)g(pro)5 b(jections)23 b(and)g(subspaces)g Fs(H)1981 4346 y Fl(\006)2062 4334 y FA(it)h(is)g(not)g(di\016cult)h (to)f(obtain)f(a)h(v)-5 b(aria-)599 4433 y(tional)21 b(c)n(haracterization)d(of)j(the)g(atomic)g(energies)e([23)o(];)k (completely)e(analogous)e(ar-)599 4533 y(gumen)n(ts)k(also)f(pro)r (duce)h(v)-5 b(ariational)21 b(form)n(ulas)h(for)g(the)i(energies)d(of) i(the)h(molecular)p eop %%Page: 11 11 11 10 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(11)599 320 y(Hamiltonian.)37 b(Unfortunately)-7 b(,)28 b(ho)n(w)n(ev)n(er,)d(w)n(e)i(could)h(not)f(\014gure)g(out)h(an) n(y)e(reason-)599 419 y(ably)31 b(feasible)f(w)n(a)n(y)g(to)g(relate)g (the)h(resulting)g(v)-5 b(ariational)29 b(atomic)h(and)h(molecular)599 519 y(expressions)26 b(suc)n(h)i(that)f(a)h(UAL)g(lo)n(w)n(er)e(b)r (ound)i(can)f(b)r(e)h(deduced.)682 619 y(Therefore,)d(here)g(w)n(e)h (resort)e(to)h(an)g(alternativ)n(e)g(decomp)r(osition)g(of)g Fs(H)i FA(in)n(to)e(p)r(os-)599 718 y(itiv)n(e)31 b(and)g(negativ)n(e)e (subspaces)h Fs(H)1736 688 y Ft(at)1735 739 y Fl(\006)1828 718 y FA(=)e Fr(\022)1962 688 y Ft(at)1960 739 y Fl(\006)2025 718 y Fs(H)k FA(b)n(y)e(de\014ning)h(pro)5 b(jections)30 b Fr(\022)3034 688 y Ft(at)3032 739 y Fl(\006)3128 718 y FA(with)599 818 y(the)e(help)g(of)g(the)g(atomic)f(sp)r(ectral)g (functions)731 1002 y Fr(\022)772 968 y Ft(at)770 1022 y(+)859 1002 y FA(=)22 b Fr(\037)998 1017 y Ft(\(0)p Fo(;)p Fl(1)p Ft(\))1173 935 y Fp(\000)1211 1002 y Fr(h)1259 968 y Ft(at)1322 1002 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b FA(2)p Fr(\020)6 b FA(\))1655 935 y Fp(\001)1692 1002 y Fr(;)180 b(\022)1936 968 y Ft(at)1934 1022 y Fl(\000)2023 1002 y FA(=)23 b Fr(\037)2163 1017 y Ft(\()p Fl(\0001)p Fo(;)p Ft(0\))2389 935 y Fp(\000)2427 1002 y Fr(h)2475 968 y Ft(at)2539 1002 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)j FA(2)p Fr(\020)6 b FA(\))2884 935 y Fp(\001)2923 1002 y Fr(:)131 b FA(\(2.25\))599 1186 y(F)-7 b(urthermore,)42 b(let)e Fs(F)47 b FA(b)r(e)40 b(a)f(core)f(for)h Fr(h)1938 1156 y Ft(at)2001 1186 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)27 b FA(2)p Fr(\020)6 b FA(\);)46 b(since)39 b(b)r(elo)n(w)g(w)n(e)g(ha)n (v)n(e)g(to)599 1285 y(restrict)g(the)h Fr(\020)6 b FA(-range)39 b(to)g(\(0)p Fr(;)14 b(\020)1622 1297 y Ft(max)1749 1285 y FA(\))40 b(with)g Fr(\020)2058 1297 y Ft(max)2228 1285 y Fr(<)2336 1217 y Fs(p)p 2405 1217 42 4 v 68 x FA(3)p Fr(=)p FA(4,)i(w)n(e)d(ma)n(y)g(tak)n(e)g Fs(F)51 b FA(=)599 1385 y Fs(D)r FA(\()p Fs(j)p Fr(T)769 1397 y Fo(D)830 1385 y Fs(j)853 1355 y Ft(1)p Fo(=)p Ft(2)957 1385 y FA(\))28 b(without)g(an)n(y)f(problems.)36 b(As)28 b(for)f Fs(H)q FA(,)h(also)e Fs(F)31 b FA(=)23 b Fs(F)2648 1397 y Ft(+)2721 1385 y Fs(\010)18 b(F)2864 1397 y Fl(\000)2948 1385 y FA(is)27 b(decom-)599 1485 y(p)r(osed)c(in)n(to)g(the)h (orthogonal)c(sum)k(of)f(subspaces)f Fs(F)2238 1497 y Fl(\006)2316 1485 y FA(=)h Fr(\022)2445 1455 y Ft(at)2443 1505 y Fl(\006)2509 1485 y Fs(F)31 b(\032)22 b(H)2757 1497 y Fl(\006)2813 1485 y FA(.)36 b(T)-7 b(o)22 b(k)n(eep)h(the)599 1584 y(notation)j(succinct,)h(w)n(e)f(will)g(write)g([)p Fr(A)p FA(])1866 1596 y Fo( )1940 1584 y FA(:=)c Fs(h)p Fr( )s(;)28 b(A )s Fs(i)p Fr(=)p Fs(h)p Fr( )s(;)14 b( )s Fs(i)27 b FA(for)f(the)g(normalized)599 1684 y(exp)r(ectation)i(v)-5 b(alue)27 b(of)h(an)f(op)r(erator)f Fr(A)i FA(within)g(the)g(state)g Fr( )e Fs(2)d(D)r FA(\()p Fr(A)p FA(\))p Fr(;)29 b( )d Fs(6)p FA(=)d(0.)682 1784 y(In)33 b(case)e(of)h(the)g(atomic)g (Hamiltonian,)h(our)e(particular)g(c)n(hoice)g(of)h(pro)5 b(jections)599 1883 y(\(2.25\))29 b(renders)f(the)i(v)n(eri\014cation)d (of)j(the)f(prerequisites)f(for)h(the)h(DES)f(v)-5 b(ariational)599 1983 y(represen)n(tation)26 b(almost)h(trivial.)37 b(Clearly)-7 b(,)26 b(b)n(y)i(construction,)1508 2167 y(sup)1387 2240 y Fo(\036)1427 2248 y Fe(\000)1476 2240 y Fl(2F)1569 2248 y Fe(\000)1618 2240 y Fl(nf)p Ft(0)p Fl(g)1753 2167 y FA([)p Fr(h)1824 2133 y Ft(at)1887 2167 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)f FA(2)p Fr(\020)6 b FA(\)])2256 2179 y Fo(\036)2296 2187 y Fe(\000)2372 2167 y Fs(\024)23 b FA(0)575 b(\(2.26\))599 2419 y(and)33 b(th)n(us)g(assumption)f(\(ii\))i(of)f(theorem)f(1.1)g (of)h(DES)g([23)o(])g(\(repro)r(duced)f(as)g(the-)599 2519 y(orem)g(A.1)g(in)h(the)g(app)r(endix)f(for)g(easy)g(reference\))g (is)g(satis\014ed.)51 b(The)32 b(v)-5 b(alidit)n(y)32 b(of)599 2618 y(assumption)c(\(iii\))g(for)f(that)h(theorem)f(is)g(a)h (consequence)e(of)868 2802 y Fr(\025)916 2767 y Ft(at)980 2802 y FA(\(2)p Fr(\020)6 b FA(\))23 b(:=)76 b(inf)1253 2856 y Fo(\036)1293 2864 y Fg(+)1339 2856 y Fl(2F)1432 2864 y Fg(+)1667 2802 y FA(sup)1525 2872 y Fo( )r Ft(=)p Fo(\036)1662 2880 y Fg(+)1709 2872 y Fl(\010)p Fo(\036)1801 2880 y Fe(\000)1850 2872 y Fl(6)p Ft(=0)1620 2938 y Fo(\036)1660 2946 y Fe(\000)1709 2938 y Fl(2F)1802 2946 y Fe(\000)1967 2802 y FA([)p Fr(h)2038 2767 y Ft(at)2101 2802 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)27 b FA(2)p Fr(\020)6 b FA(\)])2470 2814 y Fo( )1174 3064 y Fs(\025)144 b FA(inf)1253 3122 y Fo(\036)1293 3130 y Fg(+)1339 3122 y Fl(2F)1432 3130 y Fg(+)1479 3122 y Fl(nf)p Ft(0)p Fl(g)1660 3064 y FA([)p Fr(h)1731 3030 y Ft(at)1794 3064 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)27 b FA(2)p Fr(\020)6 b FA(\)])2163 3076 y Fo(\036)2203 3084 y Fg(+)2291 3064 y FA(=)36 b Fr(e)2431 3030 y Ft(at)2494 3064 y FA(\(2)p Fr(\020)6 b FA(\))37 b Fr(>)g FA(0)3077 2959 y(\(2.27\))599 3296 y(where)j(the)h(second)f (line)h(follo)n(ws)e(from)h(the)h(\014rst)f(one)g(b)n(y)g("freezing")f Fr(\036)3038 3308 y Fl(\000)3139 3296 y FA(=)44 b(0)599 3395 y(to)c(b)r(ound)h(the)g(sup)f(from)g(b)r(elo)n(w.)74 b(Theorem)39 b(1.1)h(of)g(DES)g(then)h(assures)d(that)599 3495 y(the)f(expression)d(\(2.27\))h(for)g Fr(\025)1590 3465 y Ft(at)1654 3495 y FA(\(2)p Fr(\020)6 b FA(\))37 b(indeed)f(furnishes)f(the)i(correct)d(v)-5 b(ariational)599 3595 y(represen)n(tation)30 b(of)h(the)h(UAL)g(atomic)f(ground)f(state) h(energy)f(\(i.e.,)j(in)e(\(2.27\))g(the)599 3694 y(inequalit)n(y)g (can)f(b)r(e)h(replaced)f(b)n(y)g(an)h(equalit)n(y\).)45 b(The)31 b(corresp)r(onding)e(c)n(haracter-)599 3794 y(ization)h(of)h(the)g(molecular)e(energies)g(for)h Fr(R)f(>)f FA(0)i(requires)f(additional)h(argumen)n(ts)599 3893 y(and)e(is)f(form)n(ulated)g(in)h(the)g(next)g(lemma.)682 4044 y Fj(Lemma)i(2.1.)72 b Fi(De\014ne)32 b Fr(\020)1487 4056 y Ft(max)1646 4044 y Fi(as)h(the)f(r)l(e)l(al)h(r)l(o)l(ot)f(of)h (the)g(p)l(olynomial)h FA(4)p Fr(x)2977 4014 y Ft(3)3035 4044 y Fs(\000)20 b Fr(x)3167 4014 y Ft(2)3225 4044 y FA(+)599 4144 y Fr(x)f Fs(\000)f FA(1)p Fr(=)p FA(8)p Fi(;)29 b(c)l(onsider)825 4327 y Fr(\020)g Fs(2)q FA(\(0)p Fr(;)14 b(\020)1093 4339 y Ft(max)1219 4327 y FA(\))p Fr(;)783 4512 y(\020)819 4524 y Ft(max)969 4512 y FA(=)1083 4480 y Ft(1)p 1066 4494 67 4 v 1066 4541 a(12)1142 4420 y Fp(\020)1192 4512 y FA(1)k(+)1358 4457 y Fg(3)1340 4397 y Fp(q)p 1423 4397 420 4 v 115 x FA(10)g(+)g(3)1650 4440 y Fs(p)p 1718 4440 125 4 v 1718 4512 a FA(159)f Fs(\000)1967 4457 y Fg(3)1950 4397 y Fp(q)p 2033 4397 485 4 v 115 x Fs(\000)p FA(10)g(+)h(3)2324 4440 y Fs(p)p 2392 4440 125 4 v 2392 4512 a FA(159)2517 4420 y Fp(\021)2590 4512 y Fs(\031)k FA(0)p Fr(:)p FA(1333)3077 4433 y(\(2.28\))p eop %%Page: 12 12 12 11 bop 599 75 a FA(12)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y Fi(and)36 b Fs(F)826 332 y Fl(\006)918 320 y Fi(c)l(onstructe)l (d)f(fr)l(om)h(the)f(atomic)i(pr)l(oje)l(ctions)g Fr(\022)2451 290 y Ft(at)2449 340 y Fl(\006)2550 320 y Fi(in)e(\(2.25\).)58 b(Then)36 b(the)599 419 y(gr)l(ound)30 b(state)g(ener)l(gy)g(of)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))30 b Fi(c)l(an)g(b)l(e)g (expr)l(esse)l(d)g(as)1196 600 y Fr(e)1235 612 y Ft(0)1272 600 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)85 b(inf)1590 655 y Fo(\036)1630 663 y Fg(+)1677 655 y Fl(2F)1770 663 y Fg(+)2005 600 y FA(sup)1863 671 y Fo( )r Ft(=)p Fo(\036)2000 679 y Fg(+)2046 671 y Fl(\010)p Fo(\036)2138 679 y Fe(\000)2187 671 y Fl(6)p Ft(=0)1957 736 y Fo(\036)1997 744 y Fe(\000)2047 736 y Fl(2F)2140 744 y Fe(\000)2304 600 y FA([)p Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\)])2619 612 y Fo( )2670 600 y Fr(:)384 b FA(\(2.29\))682 1007 y Fi(Pr)l(o)l(of.)102 b FA(The)35 b(pro)r(of)f(of)h(lemma)f(2.2)g (consists)g(of)h(v)n(erifying)f(the)h(prerequisites)599 1107 y(that)d(allo)n(w)e(the)h(application)f(of)h(a)g("con)n(tin)n (uation")e(theorem)i(\(viz.,)h(theorem)e(3.1)599 1206 y(from)25 b(DES)g([23)o(],)h(summarized)f(in)g(theorem)f(A.2)h(in)h (the)f(app)r(endix\).)37 b(Since)25 b(in)g(this)599 1306 y(theorem)e(p)r(erturbations)g(of)g Fr(T)1573 1318 y Fo(D)1656 1306 y FA(b)n(y)g(a)f(family)i(of)f(b)r(ounded)h(op)r (erators)d(are)h(studied,)599 1406 y(w)n(e)27 b(start)g(b)n(y)g (treating)f(appropriately)g(regularized)f(v)n(ersions)g Fr(V)2668 1375 y Fo(")2649 1426 y Ft(ne)2720 1406 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\),)41 b(0)23 b Fs(\024)g Fr(\026)g Fs(\024)599 1505 y Fr(\020)48 b(<)41 b(\020)825 1517 y Ft(max)952 1505 y FA(,)h(of)c(the)h(molecular)f(p)r(oten)n(tials,)j (where)d(the)h(Coulom)n(b)f(singularities)599 1617 y Fs(j)p Fn(x)q Fs(\006)q Fn(R)q Fr(=)p FA(2)p Fs(j)924 1587 y Fl(\000)p Ft(1)1031 1617 y FA(are)18 b(replaced)g(b)n(y)g(b)r (ounded)h(functions)2266 1550 y Fp(\000)2304 1617 y Fs(j)p Fn(x)p Fs(\006)q Fn(R)q Fr(=)p FA(2)p Fs(j)q FA(+)q Fr(")2734 1550 y Fp(\001)2770 1567 y Fl(\000)p Ft(1)2859 1617 y FA(;)j(ev)n(en)n(tually)-7 b(,)599 1717 y(the)28 b(limit)h Fr(")23 b Fs(#)f FA(0)27 b(will)h(b)r(e)g(tak)n(en.)682 1816 y(Our)f(\014rst)h(goal)e(is)h(to)h(sho)n(w)1682 1998 y(sup)1561 2071 y Fo(\036)1601 2079 y Fe(\000)1651 2071 y Fl(2F)1744 2079 y Fe(\000)1792 2071 y Fl(nf)p Ft(0)p Fl(g)1927 1998 y FA([)p Fr(T)1999 2010 y Fo(D)2059 1998 y FA(])2082 2010 y Fo(\036)2122 2018 y Fe(\000)2198 1998 y Fs(\024)23 b FA(0)749 b(\(2.30\))599 2247 y(for)33 b(all)h(0)e Fr(<)h(\020)39 b(<)33 b(\020)1235 2259 y Ft(max)1362 2247 y FA(.)54 b(Due)34 b(to)g(the)g(negativit)n(y)e([)p Fr(V)2361 2217 y Fo(")2342 2267 y Ft(ne)2413 2247 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\)])2651 2259 y Fo( )2735 2247 y Fs(\024)32 b FA(0,)j(inequalit)n(y)599 2346 y(\(2.30\))27 b(then)h(immediately)g(implies)1605 2528 y(sup)1484 2601 y Fo(\036)1524 2609 y Fe(\000)1573 2601 y Fl(2F)1666 2609 y Fe(\000)1715 2601 y Fl(nf)p Ft(0)p Fl(g)1850 2528 y FA([)p Fr(h)p FA(\()p Fr(R)q(;)14 b(\026)p FA(\)])2159 2540 y Fo(\036)2199 2548 y Fe(\000)2275 2528 y Fs(\024)23 b FA(0)672 b(\(2.31\))599 2777 y(for)24 b(all)h(0)d Fs(\024)h Fr(\026)g Fs(\024)g Fr(\020)6 b FA(.)36 b(T)-7 b(o)24 b(pro)n(v)n(e)f(\(2.30\),)h(w)n(e)g(in)n(v)n(ok)n(e)f(a)h(relation)g(b) r(et)n(w)n(een)g(the)h(kinetic)599 2876 y(part)39 b(of)h(the)g(Dirac)f (op)r(erators)e(and)j(the)g(atomic)f(pro)5 b(jections)38 b(put)i(forw)n(ard)e(b)n(y)599 2976 y(Griesemer,)31 b(Lewis,)h(and)f (Sieden)n(top;)i(in)e(Ref.)h([25)o(])f(\(Corollary)e(1\))i(these)h (authors)599 3076 y(established)c(the)g(b)r(ound)1398 3257 y Fs(jj)c(j)p Fr(T)1540 3269 y Fo(D)1599 3257 y Fs(j)1622 3223 y Ft(1)p Fo(=)p Ft(2)1727 3257 y Fr(\022)1766 3269 y Ft(+)1821 3257 y Fr(\022)1862 3223 y Ft(at)1860 3277 y Fl(\000)1925 3257 y Fs(j)p Fr(T)1997 3269 y Fo(D)2057 3257 y Fs(j)2080 3223 y Fl(\000)p Ft(1)p Fo(=)p Ft(2)2260 3257 y Fs(jj)f(\024)f Fr(c)2452 3269 y Fo(\020)3077 3257 y FA(\(2.32\))599 3438 y(for)40 b(0)j Fr(<)g(\020)51 b(<)43 b FA(1)p Fr(=)p FA(4,)f(and)d(where)h Fr(c)1780 3450 y Fo(\020)1862 3438 y FA(=)j(2)p Fr(\020)6 b FA(\(1)26 b Fs(\000)h FA(4)p Fr(\020)6 b FA(\))2362 3408 y Fl(\000)p Ft(1)2451 3375 y Fs(p)p 2520 3375 227 4 v 63 x FA(1)18 b(+)g(4)p Fr(\020)6 b FA(.)74 b(F)-7 b(or)40 b(all)f Fr( )47 b Fs(2)599 3538 y(D)665 3470 y Fp(\000)704 3538 y Fs(j)p Fr(T)776 3550 y Fo(D)836 3538 y Fs(j)859 3508 y Ft(1)p Fo(=)p Ft(2)963 3470 y Fp(\001)1020 3538 y Fs(\\)18 b FA(Ran\()p Fr(\022)1315 3508 y Ft(at)1313 3558 y Fl(\000)1379 3538 y FA(\))28 b(the)g(b)r(ound)g(\(2.32\))f(en)n(tails)1153 3728 y Fs(h)p Fr(\022)1226 3694 y Ft(at)1224 3749 y Fl(\000)1290 3728 y Fr( )s(;)h(\022)1437 3740 y Ft(+)1492 3728 y Fs(j)p Fr(T)1564 3740 y Fo(D)1624 3728 y Fs(j)p Fr(\022)1686 3740 y Ft(+)1741 3728 y Fr(\022)1782 3694 y Ft(at)1780 3749 y Fl(\000)1845 3728 y Fr( )s Fs(i)c(\024)e Fr(c)2081 3694 y Ft(2)2081 3749 y Fo(\020)2119 3728 y Fs(h)p Fr( )s(;)28 b(\022)2300 3694 y Ft(at)2298 3749 y Fl(\000)2364 3728 y Fs(j)p Fr(T)2436 3740 y Fo(D)2496 3728 y Fs(j)p Fr(\022)2560 3694 y Ft(at)2558 3749 y Fl(\000)2623 3728 y Fr( )s Fs(i)p Fr(:)342 b FA(\(2.33\))599 3910 y(Th)n(us)28 b(w)n(e)f(can)g(estimate) 1097 4069 y Fr(\022)1138 4035 y Ft(at)1136 4090 y Fl(\000)1201 4069 y Fs(j)p Fr(T)1273 4081 y Fo(D)1333 4069 y Fs(j)p Fr(\022)1397 4035 y Ft(at)1395 4090 y Fl(\000)1483 4069 y FA(=)c Fr(\022)1612 4035 y Ft(at)1610 4090 y Fl(\000)1676 4069 y Fr(\022)1715 4081 y Ft(+)1770 4069 y Fs(j)p Fr(T)1842 4081 y Fo(D)1901 4069 y Fs(j)p Fr(\022)1963 4081 y Ft(+)2019 4069 y Fr(\022)2060 4035 y Ft(at)2058 4090 y Fl(\000)2155 4069 y FA(+)32 b Fr(\022)2293 4035 y Ft(at)2291 4090 y Fl(\000)2357 4069 y Fr(\022)2396 4081 y Fl(\000)2452 4069 y Fs(j)p Fr(T)2524 4081 y Fo(D)2583 4069 y Fs(j)p Fr(\022)2645 4081 y Fl(\000)2702 4069 y Fr(\022)2743 4035 y Ft(at)2741 4090 y Fl(\000)1483 4204 y Fs(\024)23 b FA(\(1)18 b Fs(\000)g Fr(c)1782 4170 y Ft(2)1782 4225 y Fo(\020)1820 4204 y FA(\))1852 4170 y Fl(\000)p Ft(1)1942 4204 y Fr(\022)1983 4170 y Ft(at)1981 4225 y Fl(\000)2046 4204 y Fr(\022)2085 4216 y Fl(\000)2141 4204 y Fs(j)p Fr(T)2213 4216 y Fo(D)2273 4204 y Fs(j)p Fr(\022)2335 4216 y Fl(\000)2391 4204 y Fr(\022)2432 4170 y Ft(at)2430 4225 y Fl(\000)3077 4138 y FA(\(2.34\))599 4384 y(and)1102 4533 y Fr(\022)1143 4499 y Ft(at)1141 4554 y Fl(\000)1207 4533 y Fr(\022)1246 4545 y Ft(+)1301 4533 y Fs(j)p Fr(T)1373 4545 y Fo(D)1433 4533 y Fs(j)p Fr(\022)1495 4545 y Ft(+)1550 4533 y Fr(\022)1591 4499 y Ft(at)1589 4554 y Fl(\000)1677 4533 y Fs(\024)23 b Fr(c)1801 4499 y Ft(2)1801 4554 y Fo(\020)1839 4533 y FA(\(1)18 b Fs(\000)g Fr(c)2050 4499 y Ft(2)2050 4554 y Fo(\020)2088 4533 y FA(\))2120 4499 y Fl(\000)p Ft(1)2210 4533 y Fr(\022)2251 4499 y Ft(at)2249 4554 y Fl(\000)2314 4533 y Fr(\022)2353 4545 y Fl(\000)2409 4533 y Fs(j)p Fr(T)2481 4545 y Fo(D)2541 4533 y Fs(j)p Fr(\022)2603 4545 y Fl(\000)2659 4533 y Fr(\022)2700 4499 y Ft(at)2698 4554 y Fl(\000)2764 4533 y Fr(:)290 b FA(\(2.35\))p eop %%Page: 13 13 13 12 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(13)599 320 y(F)-7 b(or)27 b Fr(T)797 332 y Fo(D)885 320 y FA(this)h(means)1060 478 y Fr(\022)1101 444 y Ft(at)1099 499 y Fl(\000)1165 478 y Fr(T)1214 490 y Fo(D)1273 478 y Fr(\022)1314 444 y Ft(at)1312 499 y Fl(\000)1401 478 y FA(=)23 b Fr(\022)1530 444 y Ft(at)1528 499 y Fl(\000)1593 478 y Fr(\022)1632 490 y Ft(+)1687 478 y Fr(T)1736 490 y Fo(D)1796 478 y Fr(\022)1835 490 y Ft(+)1890 478 y Fr(\022)1931 444 y Ft(at)1929 499 y Fl(\000)2027 478 y FA(+)32 b Fr(\022)2165 444 y Ft(at)2163 499 y Fl(\000)2228 478 y Fr(\022)2267 490 y Fl(\000)2323 478 y Fr(T)2372 490 y Fo(D)2432 478 y Fr(\022)2471 490 y Fl(\000)2527 478 y Fr(\022)2568 444 y Ft(at)2566 499 y Fl(\000)1401 613 y Fs(\024)23 b FA(\(1)18 b Fs(\000)g Fr(c)1700 579 y Ft(2)1700 634 y Fo(\020)1738 613 y FA(\))1770 579 y Fl(\000)p Ft(1)1859 613 y FA(\(1)h Fs(\000)f FA(2)p Fr(c)2113 579 y Ft(2)2113 634 y Fo(\020)2150 613 y FA(\))p Fr(\022)2223 579 y Ft(at)2221 634 y Fl(\000)2287 613 y Fr(\022)2326 625 y Fl(\000)2382 613 y Fr(T)2431 625 y Fo(D)2491 613 y Fr(\022)2530 625 y Fl(\000)2586 613 y Fr(\022)2627 579 y Ft(at)2625 634 y Fl(\000)2713 613 y Fs(\024)23 b FA(0)3077 548 y(\(2.36\))599 789 y(as)32 b(long)g(as)g(1)22 b Fs(\000)f FA(2)p Fr(c)1226 759 y Ft(2)1226 813 y Fo(\020)1295 789 y Fr(>)31 b FA(0,)j(i.e.,)g(as)e(long) g(as)g Fr(\020)39 b FA(ob)r(eys)32 b(the)h(condition)f(sp)r(eci\014ed)h (in)599 889 y(\(2.28\).)44 b(Hence)30 b(estimates)g(\(2.30\))f(and)h (\(2.31\))f(hold,)i(and)f(w)n(e)f(can)h(set)g Fr(a)2974 901 y Fl(\000)3057 889 y FA(=)d(0)i(in)599 988 y(theorem)f(A.2.)682 1088 y(T)-7 b(o)38 b(v)n(erify)f(the)h(next)g(assumption)f(for)g (theorem)g(A.2,)k(viz.,)f(the)e(existence)f(of)599 1188 y(a)44 b(constan)n(t)g Fr(a)1081 1200 y Ft(+)1187 1188 y Fr(>)50 b(a)1346 1200 y Fl(\000)1446 1188 y FA(suc)n(h)44 b(that)h(all)f(eigen)n(v)-5 b(alues)43 b Fr(e)2464 1157 y Fo(")2464 1211 y(k)2505 1188 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))44 b(of)g Fr(h)2923 1157 y Fo(")2959 1188 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))51 b(=)599 1287 y Fr(T)648 1299 y Fo(D)734 1287 y FA(+)26 b Fr(V)892 1257 y Fo(")873 1308 y Ft(ne)944 1287 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))39 b(remain)g(ab)r(o)n(v)n(e)f(this)h(constan)n(t,)i Fr(e)2318 1257 y Fo(")2318 1311 y(k)2359 1287 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))42 b Fs(\025)g Fr(a)2767 1299 y Ft(+)2822 1287 y FA(,)g(w)n(e)d(observ)n(e)599 1387 y(that)27 b(the)f Fr(e)958 1357 y Fo(")958 1410 y(k)999 1387 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))26 b(are)f(b)r(ounded)i(b)r (elo)n(w)f(b)n(y)f(the)i(eigen)n(v)-5 b(alues)25 b(of)h(the)g(corresp)r (ond-)599 1486 y(ing)37 b(Bro)n(wn-Ra)n(v)n(enhall)c(op)r(erators)h (\(theorem)i(A.3)h(in)f(the)h(app)r(endix\).)64 b(In)36 b(fact,)599 1586 y(due)f(to)f(the)h(monotonicit)n(y)e(of)i(the)g(p)r (oten)n(tials)f(in)g Fr(")p FA(,)i(w)n(e)e(ma)n(y)g(tak)n(e)f(the)i (Bro)n(wn-)599 1686 y(Ra)n(v)n(enhall)28 b(op)r(erators)e(with)j Fr(")24 b FA(=)g(0.)39 b(Moreo)n(v)n(er,)26 b(since)j(for)f(the)g(Bro)n (wn-Ra)n(v)n(enhall)599 1785 y(Hamiltonian)36 b(it)g(is)f(easy)f(\(lik) n(e)i(for)f(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erators\))h(to)h (demonstrate)g(a)599 1885 y(UAL)29 b(lo)n(w)n(er)d(b)r(ound,)i(w)n(e)f (infer)871 2053 y Fr(e)910 2018 y Fo(")910 2073 y(k)950 2053 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))23 b Fs(\025)g Fr(e)1315 2065 y Ft(BR)1414 2053 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))23 b Fs(\025)f Fr(e)1778 2018 y Ft(at)1778 2073 y(BR)1877 2053 y FA(\(2)p Fr(\026)p FA(\))h Fs(\025)g Fr(e)2183 2018 y Ft(at)2183 2073 y(BR)2281 2053 y FA(\(2)p Fr(\020)2391 2065 y Ft(max)2518 2053 y FA(\))g(=:)g Fr(a)2728 2065 y Ft(+)2783 2053 y Fr(:)271 b FA(\(2.37\))599 2220 y(for)28 b(all)g Fr(")23 b Fs(\025)h FA(0)p Fr(;)j(R)d Fs(\025)g FA(0)p Fr(;)j FA(0)d Fs(\024)f Fr(\026)h Fs(\024)f Fr(\020)31 b Fs(\024)23 b Fr(\020)1861 2232 y Ft(max)1988 2220 y FA(.)38 b(Then,)29 b Fr(a)2334 2232 y Ft(+)2413 2220 y Fr(>)23 b FA(0,)28 b(b)r(ecause)f(the)i(atomic)599 2320 y(Bro)n(wn-Ra)n(v)n(enhall)g(ground)i(state)g(energy)g Fr(e)2056 2290 y Ft(at)2056 2343 y(BR)2154 2320 y FA(\(2)p Fr(\020)2264 2332 y Ft(max)2391 2320 y FA(\))h(is)g(kno)n(wn)f(to)h(b)r (e)g(strictly)599 2420 y(p)r(ositiv)n(e)c([66)o(])f(if)i(2)p Fr(\020)1218 2432 y Ft(max)1367 2420 y Fs(\024)23 b FA(2)p Fr(=)p FA(\()1581 2387 y Fo(\031)p 1580 2401 41 4 v 1584 2448 a Ft(2)1649 2420 y FA(+)1746 2387 y Ft(2)p 1742 2401 V 1742 2448 a Fo(\031)1793 2420 y FA(\).)682 2519 y(F)-7 b(or)27 b(the)h(last)g(assumption)f(in)h(theorem)f(A.2)g(w)n(e)g (ha)n(v)n(e)g(to)g(sho)n(w)1407 2687 y(inf)1345 2741 y Fo(\036)1385 2749 y Fg(+)1431 2741 y Fl(2F)1524 2749 y Fg(+)1759 2687 y FA(sup)1617 2757 y Fo( )r Ft(=)p Fo(\036)1754 2765 y Fg(+)1801 2757 y Fl(\010)p Fo(\036)1893 2765 y Fe(\000)1942 2757 y Fl(6)p Ft(=0)1712 2823 y Fo(\036)1752 2831 y Fe(\000)1801 2823 y Fl(2F)1894 2831 y Fe(\000)2059 2687 y FA([)p Fr(T)2131 2699 y Fo(D)2191 2687 y FA(])2214 2699 y Fo( )2310 2687 y Fr(>)46 b(a)2465 2699 y Fl(\000)2521 2687 y Fr(:)533 b FA(\(2.38\))599 2985 y(Ho)n(w)n(ev)n(er,)22 b(since)h(adding)f(the)i(\(atomic\))e(p)r(oten)n(tial)h(giv)n(es)f(a)g (negativ)n(e)g(con)n(tribution,)599 3084 y(inequalit)n(y)28 b(\(2.38\))e(follo)n(ws)h(directly)g(from)695 3251 y(inf)633 3305 y Fo(\036)673 3313 y Fg(+)720 3305 y Fl(2F)813 3313 y Fg(+)1047 3251 y FA(sup)906 3322 y Fo( )r Ft(=)p Fo(\036)1043 3330 y Fg(+)1089 3322 y Fl(\010)p Fo(\036)1181 3330 y Fe(\000)1230 3322 y Fl(6)p Ft(=0)1000 3387 y Fo(\036)1040 3395 y Fe(\000)1089 3387 y Fl(2F)1182 3395 y Fe(\000)1347 3251 y FA([)p Fr(T)1419 3263 y Fo(D)1479 3251 y FA(])1502 3263 y Fo( )1599 3251 y Fs(\025)166 b FA(inf)1700 3309 y Fo(\036)1740 3317 y Fg(+)1787 3309 y Fl(2F)1880 3317 y Fg(+)1926 3309 y Fl(nf)p Ft(0)p Fl(g)2107 3251 y FA([)p Fr(T)2179 3263 y Fo(D)2239 3251 y FA(])2262 3263 y Fo(\036)2302 3271 y Fg(+)1599 3514 y Fs(\025)g FA(inf)1700 3571 y Fo(\036)1740 3579 y Fg(+)1787 3571 y Fl(2F)1880 3579 y Fg(+)1926 3571 y Fl(nf)p Ft(0)p Fl(g)2107 3514 y FA([)p Fr(h)2178 3480 y Ft(at)2241 3514 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)27 b FA(2)p Fr(\020)6 b FA(\)])2610 3526 y Fo(\036)2650 3534 y Fg(+)2738 3514 y FA(=)36 b Fr(e)2878 3480 y Ft(at)2941 3514 y FA(\(2)p Fr(\020)6 b FA(\))38 b Fr(>)e FA(0)3077 3665 y(\(2.39\))599 3765 y(as)27 b(in)h(\(2.27\).) 682 3865 y(No)n(w,)h(with)g(all)f(prerequisites)f(for)h(theorem)g(A.2)h (b)r(eing)f(met,)i(w)n(e)e(can)g(con)n(tin)n(ue)599 3964 y(from)k Fr(\026)e FA(=)g(0)i(to)g Fr(\026)e FA(=)g Fr(\020)36 b(<)30 b(\020)1533 3976 y Ft(max)1660 3964 y FA(,)j(and,)g(b)n(y)f (emplo)n(ying)f(the)h(same)g(limiting)g(argu-)599 4064 y(men)n(ts)22 b(as)f(in)i([23)o(])f(when)g Fr(")g Fs(#)h FA(0,)g(w)n(e)e(can)h(deduce)g(the)g(correctness)e(of)i(the)g (expression)599 4184 y(\(2.29\))27 b(for)g(the)h(molecular)f(ground)f (state)i(energy)-7 b(.)p 2310 4201 25 75 v 682 4334 a(With)24 b(the)g(v)-5 b(ariational)21 b(represen)n(tation)g(\(2.29\))h(at)h(our) f(disp)r(osal,)i(the)f(deriv)-5 b(ation)599 4433 y(of)31 b(the)f(lo)n(w)n(er)f(b)r(ound)i(prop)r(ert)n(y)e(of)h(the)h(UAL)f (energies)f(b)r(ecomes)h(fairly)g(straigh)n(t-)599 4533 y(forw)n(ard.)37 b(Recalling)27 b(that)h(all)g(eigenstates)f(of)h Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))28 b(can)g(b)r(e)g(c)n (hosen)g(to)f(enjo)n(y)h(a)p eop %%Page: 14 14 14 13 bop 599 75 a FA(14)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(de\014nite)30 b(parit)n(y)-7 b(,)28 b(in)i(\(2.29\))e(w)n(e)g (can)h(restrict)f(v)-5 b(ariations)27 b(to)i(subspaces)f(of)h(ev)n(en)f (or)599 419 y(o)r(dd)33 b(parit)n(y)-7 b(.)51 b(Th)n(us,)33 b(using)f Fr(P)1567 431 y Fo(O)1624 419 y Fr(h)1672 389 y Ft(at)1735 419 y FA(\()p Fs(\000)p Fr(R)q(=)p FA(2)p Fr(;)26 b FA(2)p Fr(\020)6 b FA(\))p Fr(P)2198 431 y Fo(O)2287 419 y FA(=)30 b Fr(h)2430 389 y Ft(at)2493 419 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)d FA(2)p Fr(\020)6 b FA(\),)34 b(in)f(the)g(\014rst)599 519 y(case)27 b(\(and)h(similarly) f(for)g(o)r(dd)g(parit)n(y\))615 738 y Fr(e)654 750 y Ft(0)691 738 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)76 b(inf)1000 792 y Fo(\036)1040 800 y Fg(+)1087 792 y Fl(2F)1180 800 y Fg(+)1462 738 y FA(sup)1320 808 y Fo( )r Ft(=)p Fo(\036)1457 816 y Fg(+)1504 808 y Fl(\010)p Fo(\036)1596 816 y Fe(\000)1645 808 y Fl(6)p Ft(=0)1273 874 y Fo(P)1315 882 y Ff(O)1364 874 y Fo( )r Ft(=)p Fo( )r(;)19 b(\036)1586 882 y Fe(\000)1635 874 y Fl(2F)1728 882 y Fe(\000)1809 738 y FA([)p Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\)])2124 750 y Fo( )922 1026 y FA(=)76 b(inf)1000 1080 y Fo(\036)1040 1088 y Fg(+)1087 1080 y Fl(2F)1180 1088 y Fg(+)1462 1026 y FA(sup)1320 1096 y Fo( )r Ft(=)p Fo(\036)1457 1104 y Fg(+)1504 1096 y Fl(\010)p Fo(\036)1596 1104 y Fe(\000)1645 1096 y Fl(6)p Ft(=0)1273 1162 y Fo(P)1315 1170 y Ff(O)1364 1162 y Fo( )r Ft(=)p Fo( )r(;)19 b(\036)1586 1170 y Fe(\000)1635 1162 y Fl(2F)1728 1170 y Fe(\000)1809 934 y Fp(\020)1869 993 y Ft(1)p 1869 1007 34 4 v 1869 1054 a(2)1912 1026 y FA([)p Fr(h)1983 992 y Ft(at)2046 1026 y FA(\()p Fs(\000)p Fr(R)q(=)p FA(2)p Fr(;)26 b FA(2)p Fr(\020)6 b FA(\)])2479 1038 y Fo( )2548 1026 y FA(+)2641 993 y Ft(1)p 2641 1007 V 2641 1054 a(2)2684 1026 y FA([)p Fr(h)2755 992 y Ft(at)2819 1026 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)26 b FA(2)p Fr(\020)6 b FA(\)])3187 1038 y Fo( )3238 934 y Fp(\021)922 1288 y FA(=)76 b(inf)1000 1343 y Fo(\036)1040 1351 y Fg(+)1087 1343 y Fl(2F)1180 1351 y Fg(+)1462 1288 y FA(sup)1320 1359 y Fo( )r Ft(=)p Fo(\036)1457 1367 y Fg(+)1504 1359 y Fl(\010)p Fo(\036)1596 1367 y Fe(\000)1645 1359 y Fl(6)p Ft(=0)1273 1424 y Fo(P)1315 1432 y Ff(O)1364 1424 y Fo( )r Ft(=)p Fo( )r(;)19 b(\036)1586 1432 y Fe(\000)1635 1424 y Fl(2F)1728 1432 y Fe(\000)1809 1288 y FA([)p Fr(h)1880 1254 y Ft(at)1944 1288 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)26 b FA(2)p Fr(\020)6 b FA(\)])2312 1300 y Fo( )922 1551 y Fs(\025)153 b FA(inf)1010 1608 y Fo(\036)1050 1616 y Fg(+)1097 1608 y Fl(2F)1190 1616 y Fg(+)1236 1608 y Fl(nf)p Ft(0)p Fl(g)1033 1674 y Fo(P)1075 1682 y Ff(O)1124 1674 y Fo(\036)1164 1682 y Fg(+)1211 1674 y Ft(=)p Fo(\036)1302 1682 y Fg(+)1404 1551 y FA([)p Fr(h)1475 1517 y Ft(at)1538 1551 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)27 b FA(2)p Fr(\020)6 b FA(\)])1907 1563 y Fo(\036)1947 1571 y Fg(+)922 1801 y Fs(\025)143 b FA(inf)1000 1858 y Fo(\036)1040 1866 y Fg(+)1087 1858 y Fl(2F)1180 1866 y Fg(+)1226 1858 y Fl(nf)p Ft(0)p Fl(g)1407 1801 y FA([)p Fr(h)1478 1767 y Ft(at)1542 1801 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)26 b FA(2)p Fr(\020)6 b FA(\)])1910 1813 y Fo(\036)1950 1821 y Fg(+)2048 1801 y FA(=)45 b Fr(e)2197 1767 y Ft(at)2260 1801 y FA(\(2)p Fr(\020)6 b FA(\))3077 1952 y(\(2.40\))599 2052 y(where)26 b(the)f(last)h(inequalit)n(y)f(results)g(from)g (enlarging)f(the)i(set)g(of)f Fr(\036)2741 2064 y Ft(+)2797 2052 y FA(.)36 b(By)25 b(estimate)599 2152 y(\(2.40\))i(w)n(e)h(ha)n(v) n(e)e(pro)n(v)n(ed:)682 2358 y Fj(Theorem)31 b(2.4.)72 b Fi(If)34 b FA(0)28 b Fr(<)g(\020)35 b(<)28 b(\020)1731 2370 y Ft(max)1891 2358 y Fi(as)33 b(sp)l(e)l(ci\014e)l(d)h(in)f (\(2.28\))h(and)f Fr(R)d Fs(\025)e FA(0)p Fi(,)33 b(then)599 2458 y(the)k(mole)l(cular)h(gr)l(ound)e(state)g(ener)l(gy)h(c)l(annot)f (b)l(e)h(lower)h(than)e(the)h(gr)l(ound)g(state)599 2558 y(ener)l(gy)30 b(of)h(the)f(UAL,)1604 2768 y Fr(e)1643 2780 y Ft(0)1680 2768 y FA(\()p Fr(R)q(;)d(\020)6 b FA(\))24 b Fs(\025)f Fr(e)2051 2734 y Ft(at)2114 2768 y FA(\(2)p Fr(\020)6 b FA(\))p Fr(:)792 b FA(\(2.41\))682 3130 y(While)27 b(our)e(metho)r(ds)h(are)f(not)h(strong)f(enough)g(to)h(demonstrate)f (\(2.41\))g(for)g Fr(\020)30 b Fs(\025)599 3230 y Fr(\020)635 3242 y Ft(max)762 3230 y FA(,)37 b(w)n(e)e(conjecture)f(that)h(the)h (UAL)f(lo)n(w)n(er)e(b)r(ound)j(con)n(tin)n(ues)e(to)h(hold)f(for)h (all)599 3330 y Fr(\020)47 b Fs(2)41 b FA(\(0)p Fr(;)14 b FA(1)p Fr(=)p FA(2\),)39 b(indep)r(enden)n(tly)f(of)g(whether)g(the)g (v)-5 b(ariational)37 b(expression)f(\(2.29\))599 3429 y(and)28 b(lemma)f(2.2)g(can)g(b)r(e)h(extended)g(b)r(ey)n(ond)f Fr(\020)2104 3441 y Ft(max)2231 3429 y FA(.)815 3695 y Fu(3.)32 b(REGULARITY)h(OF)e(THE)h(ELECTR)m(ONIC)g(CUR)-11 b(VES)1600 3819 y(3.1.)94 b(Con)m(tin)m(uit)m(y)682 3935 y FA(The)23 b(scaling)f(pro)r(cedure)f(leading)h(to)h(the)g(analyticit) n(y)f(prop)r(erties)f(form)n(ulated)h(in)599 4035 y(prop)r(osition)29 b(2.1)g(are)g(not)h(applicable)f(if)h Fr(R)d FA(=)f(0.)43 b(In)30 b(view)g(of)g(the)g(drastic)f(c)n(hange)599 4134 y(of)34 b(the)f(self-adjoin)n(tness)f(b)r(eha)n(viour)g(of)i Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))33 b(\(cf.)55 b(theorems)32 b(2.1)g(and)i(2.2\))e(in)599 4234 y(the)d(UAL,)g(a)f (priori)f(ev)n(en)g(the)i(con)n(tin)n(uit)n(y)f(of)g(the)g(electronic)g (curv)n(es)f(ma)n(y)g(app)r(ear)599 4334 y(unpredictable)g(for)g Fr(R)c Fs(#)g FA(0.)36 b(T)-7 b(o)27 b(reco)n(v)n(er)d(this)k(con)n (tin)n(uit)n(y)e(at)h(least)f(for)h Fr(\020)i Fs(2)24 b FA(\(0)p Fr(;)14 b FA(1)p Fr(=)p FA(2\))599 4433 y(will)21 b(b)r(e)h(our)d(next)i(goal;)h(here,)g(if)1637 4365 y Fs(p)p 1706 4365 42 4 v 68 x FA(3)p Fr(=)p FA(2)g Fr(<)g FA(2)p Fr(\020)29 b(<)23 b FA(1,)f(w)n(e)e(shall)g(adopt)h(the)g (self-adjoin)n(t)599 4533 y(extension)28 b(describ)r(ed)f(in)h(part)f (\(ii\))h(of)g(theorem)f(2.1)g(for)g Fr(h)2476 4503 y Ft(at)2539 4533 y FA(\(2)p Fr(\020)6 b FA(\).)p eop %%Page: 15 15 15 14 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(15)682 320 y Fj(Theorem)31 b(3.1.)65 b Fi(If)27 b FA(0)22 b Fr(<)h(\020)30 b(<)22 b FA(1)p Fr(=)p FA(2)p Fi(,)k(the)h(mole)l(cular)g(op)l(er)l(ator)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))27 b Fi(c)l(onver)l(ges)599 429 y(to)f(the)g(UAL)f(op)l(er)l(ator)i Fr(h)1401 399 y Ft(at)1464 429 y FA(\(2)p Fr(\020)6 b FA(\))26 b Fi(in)f(norm)h(r)l (esolvent)g(sense,)h Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))2810 382 y Ft(n)p Fo(:)p Ft(r)p Fo(:)2794 429 y Fs(\000)-14 b(!)p Fr(h)2976 399 y Ft(at)3039 429 y FA(\(2)p Fr(\020)6 b FA(\))26 b Fi(as)599 529 y Fr(R)e Fs(#)f FA(0)p Fi(.)682 790 y(Pr)l(o)l(of.)88 b FA(The)30 b(ab)r(o)n(v)n(e)f(assertion)g(extends)h(results)f(b)n(y)h(Klaus)f([19) o(])i(in)f(whic)n(h)g(the)599 890 y(norm)h(resolv)n(en)n(t)f(con)n(v)n (ergence)e(w)n(as)i(sho)n(wn)h(if)h(0)c Fr(<)h(\020)35 b(<)29 b FA(1)p Fr(=\031)s FA(;)j(for)f(the)g(remaining)599 989 y(in)n(terv)-5 b(al)37 b(1)p Fr(=\031)42 b Fs(\024)c Fr(\020)46 b(<)39 b FA(1)p Fr(=)p FA(2,)f(a)f(tec)n(hnical)g(problem)g (forced)f(Klaus)h(to)g(exclude)g(a)599 1089 y(coun)n(table)e(set)h Fs(f)p Fr(\020)1199 1101 y Fo(i)1226 1089 y Fs(g)f FA(of)h("critical")e (coupling)h(parameters)f Fr(\020)2588 1101 y Fo(i)2651 1089 y FA(so)h(that)h(the)g(norm)599 1189 y(resolv)n(en)n(t)i(con)n(v)n (ergence)g(could)h(only)g(b)r(e)h(demonstrated)f(if)h Fr(\020)50 b Fs(6)p FA(=)42 b Fr(\020)2835 1201 y Fo(i)2863 1189 y FA(.)73 b(Ho)n(w)n(ev)n(er,)599 1288 y(it)41 b(w)n(as)e (conjectured)g(that)h(actually)f(this)h(set)g Fs(f)p Fr(\020)2238 1300 y Fo(i)2266 1288 y Fs(g)f FA(is)h(empt)n(y)g([19)o (].)74 b(Here,)42 b(our)599 1388 y(pro)r(of)31 b(follo)n(ws)g(the)h (general)e(strategy)g(of)h(Klaus')g(reasoning;)g(at)g(a)g(certain)g(p)r (oin)n(t,)599 1488 y(though,)26 b(some)f(alternativ)n(e)f(argumen)n(ts) g(allo)n(w)g(simpli\014cations)h(and)g(an)g(extension)599 1587 y(of)38 b(the)h Fr(\020)6 b FA(-range.)66 b(While)39 b(w)n(e)e(refer)g(for)h(details)f(to)h(the)h(original)d(article)h([19)o (],)k(a)599 1687 y(comprehensible)33 b(presen)n(tation)g(requires)g (the)h(rep)r(etition)g(of)f(the)i(essen)n(tial)e(steps)599 1786 y(of)28 b(Klaus')f(argumen)n(tation.)682 1886 y(Let)1010 2036 y Fr(K)1081 2048 y Fo(\020)1119 2036 y FA(\()p Fr(R)q(;)14 b(E)5 b FA(\))23 b(=)f Fr(\020)1502 1961 y Fp(p)p 1586 1961 391 4 v 1586 2036 a Fs(\000)p Fr(V)1699 2048 y Ft(ne)1769 2036 y FA(\()p Fr(R)q(;)14 b FA(1\)\()p Fr(T)2057 2048 y Fo(D)2135 2036 y Fs(\000)k Fr(E)5 b FA(\))2316 2001 y Fl(\000)p Ft(1)2406 1961 y Fp(p)p 2489 1961 V 75 x Fs(\000)p Fr(V)2602 2048 y Ft(ne)2672 2036 y FA(\()p Fr(R)q(;)14 b FA(1\))240 b(\(3.1\))599 2182 y(stand)20 b(for)g(the)h(Birman-Sc)n(h)n(winger)c(op)r(erator)i(asso)r(ciated)f (with)j Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\).)35 b(With)21 b(the)599 2282 y(help)30 b(of)e Fr(K)6 b FA(,)29 b(for)f(all)h Fr(E)34 b FA(suc)n(h)28 b(that)h(the)g(sp)r(ectrum)g(of)g Fr(K)2389 2294 y Fo(\020)2427 2282 y FA(\()p Fr(R)q(;)14 b(E)5 b FA(\))29 b(do)r(es)f(not)h(con)n(tain)599 2382 y Fs(\000)p FA(1,)e(the)h(resolv)n(en)n(t)e(of)i Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))28 b(can)f(b)r(e)h(expressed)e(as) 717 2527 y Fp(\000)755 2594 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)1178 2527 y Fp(\001)1216 2544 y Fl(\000)p Ft(1)1328 2594 y FA(=)23 b(\()p Fr(T)1497 2606 y Fo(D)1557 2594 y Fs(\000)o Fr(E)5 b FA(\))1719 2560 y Fl(\000)p Ft(1)1827 2594 y FA(+)18 b(\()p Fr(T)1991 2606 y Fo(D)2070 2594 y Fs(\000)g Fr(E)5 b FA(\))2251 2560 y Fl(\000)p Ft(1)2340 2519 y Fp(p)p 2423 2519 V 75 x Fs(\000)p Fr(V)2536 2606 y Ft(ne)2606 2594 y FA(\()p Fr(R)q(;)14 b FA(1\))p Fs(\002)1621 2677 y Fp(\000)1659 2745 y FA(1)k Fs(\000)g Fr(K)1873 2757 y Fo(\020)1911 2745 y FA(\()p Fr(R)q(;)c(E)5 b FA(\))2142 2677 y Fp(\001)2180 2695 y Fl(\000)p Ft(1)2269 2670 y Fp(p)p 2352 2670 V 75 x Fs(\000)p Fr(V)2465 2757 y Ft(ne)2536 2745 y FA(\()p Fr(R)q(;)14 b FA(1\))o(\()p Fr(T)2823 2757 y Fo(D)2902 2745 y Fs(\000)k Fr(E)5 b FA(\))3083 2710 y Fl(\000)p Ft(1)3119 2853 y FA(\(3.2\))599 3049 y(Clearly)-7 b(,)44 b Fr(E)50 b FA(=)45 b(0)c(b)r(elongs)f(to)h(the)g(resolv)n(en)n(t)e (set)i(of)g Fr(T)2469 3061 y Fo(D)2529 3049 y FA(,)j(0)h Fs(2)h Fr(\032)p FA(\()p Fr(T)2908 3061 y Fo(D)2968 3049 y FA(\).)77 b(Since)599 3149 y Fr(V)647 3161 y Ft(ne)718 3149 y FA(\()p Fr(R)q(;)14 b FA(1\))28 b(con)n(v)n(erges)d(strongly)h (to)h Fr(V)1795 3161 y Ft(ne)1866 3149 y FA(\(0)p Fr(;)14 b FA(1\))27 b(as)g Fr(R)d Fs(#)e FA(0,)28 b(also)1576 3344 y Fr(K)1647 3356 y Fo(\020)1684 3344 y FA(\()p Fr(R)q(;)14 b FA(0\))23 b Fs(!)g Fr(K)2091 3356 y Fo(\020)2129 3344 y FA(\(0)p Fr(;)14 b FA(0\))805 b(\(3.3\))599 3540 y(in)28 b(the)g(strong)f(sense.)36 b(F)-7 b(urthermore,)27 b(lemma)g(5.2)g(of)g (Ref.)i([19)o(])e(states)h(that)1276 3671 y Fp(p)p 1359 3671 V 75 x Fs(\000)p Fr(V)1472 3758 y Ft(ne)1542 3746 y FA(\()p Fr(R)q(;)14 b FA(1\))p Fr(T)1810 3711 y Fl(\000)p Ft(1)1798 3771 y Fo(D)1921 3746 y Fs(!)2027 3671 y Fp(p)p 2110 3671 368 4 v 75 x Fs(\000)p Fr(V)2223 3758 y Ft(ne)2293 3746 y FA(\(0)p Fr(;)g FA(1\))p Fr(T)2539 3711 y Fl(\000)p Ft(1)2527 3771 y Fo(D)1276 3891 y Fr(T)1337 3856 y Fl(\000)p Ft(1)1325 3916 y Fo(D)1425 3816 y Fp(p)p 1508 3816 391 4 v 75 x Fs(\000)p Fr(V)1621 3903 y Ft(ne)1691 3891 y FA(\()p Fr(R)q(;)g FA(1\))23 b Fs(!)g Fr(T)2088 3856 y Fl(\000)p Ft(1)2076 3916 y Fo(D)2176 3816 y Fp(p)p 2259 3816 368 4 v 75 x Fs(\000)p Fr(V)2372 3903 y Ft(ne)2443 3891 y FA(\(0)p Fr(;)14 b FA(1\))3119 3814 y(\(3.4\))599 4113 y(in)22 b(norm)e(as)g Fr(R)k Fs(#)e FA(0.)34 b(As)21 b(long)g(as)f(the)h(resolv)n(en)n(t)2108 4046 y Fp(\000)2146 4113 y FA(1)5 b Fs(\000)g Fr(K)2334 4125 y Fo(\020)2371 4113 y FA(\()p Fr(R)q(;)14 b FA(0\))2578 4046 y Fp(\001)2615 4063 y Fl(\000)p Ft(1)2725 4113 y FA(is)21 b(de\014ned)g(for)g(all)599 4221 y Fr(R)k Fs(\025)e FA(0)28 b(in)g(a)g(neigh)n(b)r(ourho)r(o)r(d)f Fs(U)1632 4233 y Ft(0)1693 4221 y Fs(6)p FA(=)c Fs(;)28 b FA(of)g(zero,)f(i.e.,)h(as)g(long)f(as)g(1)d Fs(2)g Fr(\032)2875 4154 y Fp(\000)2913 4221 y Fr(K)2984 4233 y Fo(\020)3022 4221 y FA(\()p Fr(R)q(;)14 b FA(0\))3229 4154 y Fp(\001)3266 4221 y FA(,)599 4320 y(relation)27 b(\(3.3\))g(implies)h(the)g(strong)f(con)n(v)n(ergence)1267 4466 y Fp(\000)1305 4533 y FA(1)18 b Fs(\000)g Fr(K)1519 4545 y Fo(\020)1557 4533 y FA(\()p Fr(R)q(;)c FA(0\))1764 4466 y Fp(\001)1802 4483 y Fl(\000)p Ft(1)1914 4533 y Fs(!)2020 4466 y Fp(\000)2058 4533 y FA(1)k Fs(\000)g Fr(K)2272 4545 y Fo(\020)2310 4533 y FA(\(0)p Fr(;)c FA(0\))2495 4466 y Fp(\001)2532 4483 y Fl(\000)p Ft(1)3119 4533 y FA(\(3.5\))p eop %%Page: 16 16 16 15 bop 599 75 a FA(16)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 337 y FA(for)18 b Fr(R)24 b Fs(#)f FA(0.)33 b(Hence,)20 b(using)e(that)1608 266 y Fp(p)p 1691 266 391 4 v 71 x Fs(\000)p Fr(V)1804 349 y Ft(ne)1874 337 y FA(\()p Fr(R)q(;)c FA(1\))p Fr(T)2142 301 y Fl(\000)p Ft(1)2130 361 y Fo(D)2249 337 y FA(and)k(th)n(us)g(also)2732 269 y Fp(\000)2770 337 y FA(1)p Fs(\000)p Fr(K)2948 349 y Fo(\020)2984 337 y FA(\()p Fr(R)q(;)c FA(0\))3191 269 y Fp(\001)3229 287 y Fl(\000)p Ft(1)3318 337 y Fs(\002)599 381 y Fp(p)p 682 381 V 71 x Fs(\000)p Fr(V)795 464 y Ft(ne)866 452 y FA(\()p Fr(R)q(;)g FA(1\))o Fr(T)1133 416 y Fl(\000)p Ft(1)1121 476 y Fo(D)1250 452 y FA(are)28 b(compact)g(op)r(erators,)f(and,)h(b)n(y)g(\(3.4\),)h(con)n(v)n(erge)d (in)j(norm,)599 551 y(from)d(the)g(second)e(relation)h(in)h(\(3.4\))f (w)n(e)g(conclude)h(that)f(when)h Fr(R)e Fs(#)e FA(0)k(the)f(pro)r (duct)796 765 y Fr(T)857 729 y Fl(\000)p Ft(1)845 789 y Fo(D)945 690 y Fp(p)p 1028 690 V 75 x Fs(\000)p Fr(V)1141 777 y Ft(ne)1212 765 y FA(\()p Fr(R)q(;)14 b FA(1\))1418 697 y Fp(\000)1456 765 y FA(1)k Fs(\000)g Fr(K)1670 777 y Fo(\020)1708 765 y FA(\()p Fr(R)q(;)c FA(0\))1915 697 y Fp(\001)1953 715 y Fl(\000)p Ft(1)2042 690 y Fp(p)p 2125 690 V 75 x Fs(\000)p Fr(V)2238 777 y Ft(ne)2308 765 y FA(\()p Fr(R)q(;)g FA(1\))p Fr(T)2576 729 y Fl(\000)p Ft(1)2564 789 y Fo(D)991 915 y Fs(!)47 b Fr(T)1182 880 y Fl(\000)p Ft(1)1170 940 y Fo(D)1270 840 y Fp(p)p 1353 840 368 4 v 75 x Fs(\000)p Fr(V)1466 927 y Ft(ne)1536 915 y FA(\(0)p Fr(;)14 b FA(1\))1721 848 y Fp(\000)1759 915 y FA(1)k Fs(\000)g Fr(K)1973 927 y Fo(\020)2010 915 y FA(\(0)p Fr(;)c FA(0\))2195 848 y Fp(\001)2233 865 y Fl(\000)p Ft(1)2322 840 y Fp(p)p 2405 840 V 75 x Fs(\000)p Fr(V)2518 927 y Ft(ne)2588 915 y FA(\(0)p Fr(;)g FA(1\))p Fr(T)2834 880 y Fl(\000)p Ft(1)2822 940 y Fo(D)3119 832 y FA(\(3.6\))599 1112 y(also)28 b(con)n(v)n(erges)f(in)i(norm.)40 b(Therefore,)28 b(the)h(norm)f(con)n(v)n(ergence)f(of)i(\(3.1\))f(and)h (the)599 1212 y(assertion)36 b(of)i(theorem)e(3.1)h(can)g(b)r(e)h (deduced)f(from)g(\(3.6\))h(as)e(so)r(on)h(as)g(w)n(e)g(ha)n(v)n(e)599 1312 y(con\014rmed)27 b(that)h(for)f(all)h Fr(R)23 b Fs(\025)g FA(0)p Fr(;)k(R)d Fs(2)g(U)1889 1324 y Ft(0)1926 1312 y FA(,)1644 1508 y(1)f Fs(2)g Fr(\032)1830 1441 y Fp(\000)1868 1508 y Fr(K)1939 1520 y Fo(\020)1977 1508 y FA(\()p Fr(R)q(;)14 b FA(0\))2184 1441 y Fp(\001)2222 1508 y Fr(:)874 b FA(\(3.7\))599 1705 y(As)33 b(observ)n(ed)d(b)n(y)i (Klaus,)g(if)g(0)e Fr(<)g(\020)37 b(<)30 b FA(1)p Fr(=\031)k FA(the)f(relation)e(\(3.7\))g(follo)n(ws)g(from)h(the)599 1804 y(inequalit)n(y)c(\(lemma)f(5.1)g(in)h(Ref.)g([19)o(]\))1263 2012 y Fs(k)1305 1937 y Fp(p)p 1388 1937 391 4 v 75 x Fs(\000)p Fr(V)1501 2024 y Ft(ne)1571 2012 y FA(\()p Fr(R)q(;)14 b FA(1\))p Fr(T)1839 1976 y Fl(\000)p Ft(1)1827 2036 y Fo(D)1927 1937 y Fp(p)p 2010 1937 V 75 x Fs(\000)p Fr(V)2123 2024 y Ft(ne)2193 2012 y FA(\()p Fr(R)q(;)g FA(1\))p Fs(k)23 b(\024)f Fr(\031)s(:)494 b FA(\(3.8\))599 2229 y(Here,)28 b(w)n(e)f(infer)h(\(3.7\))f(for)g(all)g(0)c Fr(<)g(\020)29 b(<)23 b FA(1)p Fr(=)p FA(2)j(from)h(the)h(subsequen)n (t)f(lemma.)p 3141 2246 25 75 v 682 2392 a Fj(Lemma)j(3.1.)64 b Fi(L)l(et)25 b Fr(P)1370 2404 y Ft(\012)1421 2392 y FA(\()p Fr(K)6 b FA(\))25 b Fi(denote)h(the)f(pr)l(oje)l(ction-value)l (d)i(me)l(asur)l(e)e(asso)l(ciate)l(d)599 2492 y(with)35 b(the)g(self-adjoint)h(op)l(er)l(ator)g Fr(K)6 b Fi(.)52 b(If)35 b FA(0)c Fr(<)g(\020)38 b(<)31 b FA(1)p Fr(=)p FA(2)i Fi(and)i Fs(\000)p FA(1)c Fr(<)g(E)37 b Fs(\024)31 b FA(0)p Fi(,)k(then)599 2592 y(ther)l(e)j(exists)f(a)g(nonvoid)i (neighb)l(ourho)l(o)l(d)g Fs(U)2047 2604 y Ft(0)2121 2592 y Fs(6)p FA(=)d Fs(;)h Fi(of)h Fr(R)g FA(=)e(0)h Fi(such)g(that)g(for)h(al)t(l)599 2691 y Fr(R)24 b Fs(2)g(U)817 2703 y Ft(0)883 2691 y Fi(with)31 b FA(0)22 b Fs(\024)h Fr(R)h(<)e Fs(1)1458 2888 y FA(dim)14 b Fr(P)1663 2903 y Ft(\()p Fo(\020)t(;)p Fl(1)p Ft(\))1839 2821 y Fp(\000)1878 2888 y Fr(K)1949 2900 y Fo(\020)1986 2888 y FA(\()p Fr(R)q(;)g(E)5 b FA(\))2217 2821 y Fp(\001)2279 2888 y FA(=)22 b(0)p Fr(:)688 b FA(\(3.9\))682 3246 y Fi(Pr)l(o)l(of.)87 b FA(This)30 b(result)f(is)h(the)g(analogue)e(of)h(prop)r(osition)g (E-\(iii\))h(in)g(section)f(2)g(of)599 3346 y(Ref.)34 b([19)o(];)i(so,)d(again,)g(w)n(e)g(can)f(mostly)h(mimic)g(the)g (reasoning)e(b)n(y)i(Klaus.)51 b(First,)599 3445 y(since)25 b(for)g Fs(\000)p FA(1)d Fr(<)g(E)28 b Fs(\024)23 b FA(0)i(the)g(b)r (ound)g Fr(K)1849 3457 y Fo(\020)1887 3445 y FA(\()p Fr(R)q(;)14 b(E)5 b FA(\))23 b Fs(\024)g Fr(K)2300 3457 y Fo(\020)2337 3445 y FA(\()p Fr(R)q(;)14 b FA(0\))25 b(holds,)g(it)h(is)e(su\016cien)n(t)599 3545 y(to)k(restrict)f(atten)n (tion)g(to)h Fr(K)1518 3557 y Fo(\020)1555 3545 y FA(\()p Fr(R)q(;)14 b FA(0\).)37 b(Assuming)1470 3742 y(dim)15 b Fr(P)1676 3757 y Ft(\()p Fo(\020)t(;)p Fl(1)p Ft(\))1852 3674 y Fp(\000)1890 3742 y Fr(K)1961 3754 y Fo(\020)1998 3742 y FA(\()p Fr(R)q(;)f FA(0\))2205 3674 y Fp(\001)2266 3742 y Fr(>)23 b FA(0)p Fr(;)658 b FA(\(3.10\))599 3938 y(for)35 b(some)f(0)h Fr(<)g(\020)42 b(<)35 b FA(1)p Fr(=)p FA(2)f(and)h(some)f Fr(R)j Fs(\025)e FA(0,)h(w)n(e)f(w)n(an)n(t) f(to)h(deriv)n(e)f(a)h(con)n(tradic-)599 4038 y(tion.)57 b(Inequalit)n(y)34 b(\(3.10\))f(implies)h(the)h(existence)f(of)g(an)g (eigen)n(v)-5 b(alue)33 b Fr(\025)h Fs(2)g FA(\()p Fr(\020)6 b(;)14 b Fs(1)p FA(\))599 4137 y(of)37 b Fr(K)774 4149 y Fo(\020)811 4137 y FA(\()p Fr(R)q(;)14 b FA(0\),)39 b(and,)f(after)e(scaling,)h(the)g(existence)f(of)g(the)h(eigen)n(v)-5 b(alue)35 b(1)h(for)g(the)599 4237 y(op)r(erator)28 b Fr(K)1007 4249 y Fo(\026)1051 4237 y FA(\()p Fr(R)q(;)14 b FA(0\))30 b(with)g Fr(\026)c FA(=)g Fr(\020)6 b(=\025)27 b Fs(\024)f Fr(\020)6 b FA(.)44 b(By)29 b(the)h(Birman-Sc)n(h)n(winger) d(principle)599 4337 y([44,)g(45)o(],)h(this)g(is)f(equiv)-5 b(alen)n(t)28 b(to)1648 4533 y(0)22 b Fs(2)i Fr(\033)1838 4545 y Ft(d)1879 4466 y Fp(\000)1917 4533 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\026)p FA(\))2180 4466 y Fp(\001)2218 4533 y Fr(;)836 b FA(\(3.11\))p eop %%Page: 17 17 17 16 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(17)599 320 y(i.e.,)29 b(to)e(the)i(existence)e(of)h(an) g(eigen)n(v)-5 b(alue)27 b Fr(e)p FA(\()p Fr(R)q(;)14 b(\026)p FA(\))23 b(=)g(0)28 b(for)f(the)h(electronic)g(Hamil-)599 419 y(tonian)37 b Fr(h)p FA(\()p Fr(R)q(;)14 b(\026)p FA(\).)63 b(Let)37 b Fr( )1428 431 y Ft(0)1502 419 y FA(b)r(e)g(the)f(asso)r(ciated)g(zero-energy)d(eigenstate,)38 b(c)n(hosen)599 519 y(to)c(ha)n(v)n(e)e(a)i(de\014nite)g(parit)n(y)-7 b(.)54 b(F)-7 b(urthermore,)34 b(w)n(e)g(denote)f(b)n(y)h Fr(V)2672 489 y Fl(\006)2653 539 y Ft(ne)2728 519 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))34 b(the)g(m)n(ul-)599 619 y(tiplication)41 b(op)r(erators)1390 551 y Fp(\000)1428 619 y Fr(V)1495 588 y Fl(\000)1476 639 y Ft(ne)1551 619 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))p Fr( )1823 551 y Fp(\001)1861 619 y FA(\()p Fn(x)p FA(\))45 b(=)g Fs(\000)p Fr(\026)p Fs(j)p Fn(x)26 b Fs(\006)2456 586 y Ft(1)p 2456 600 34 4 v 2456 647 a(2)2499 619 y Fn(R)2572 631 y Fo(s)2607 619 y Fs(j)2630 588 y Fl(\000)p Ft(1)2720 619 y Fr( )s FA(\()p Fn(x)p FA(\))41 b(suc)n(h)f(that)599 718 y Fr(V)647 730 y Ft(ne)741 718 y FA(=)23 b Fr(V)896 688 y Ft(+)877 739 y(ne)969 718 y FA(+)18 b Fr(V)1119 688 y Fl(\000)1100 739 y Ft(ne)1175 718 y FA(.)37 b(F)-7 b(or)27 b(an)n(y)g(giv)n(en)g Fr(")c(>)f FA(0,)27 b(the)h(follo)n(wing) f(w)n(eigh)n(t)g(function)1410 985 y Fr(w)r FA(\()p Fn(x)q FA(\))c(=)1702 843 y Fp(\()1769 928 y Fr(")p Fs(j)p Fn(x)o Fs(j)83 b FA(if)90 b Fs(j)p Fn(x)p Fs(j)23 b(\024)g Fr(")2380 898 y Fl(\000)p Ft(1)1769 1048 y FA(1)180 b(if)90 b Fs(j)p Fn(x)p Fs(j)23 b(\025)g Fr(")2380 1018 y Fl(\000)p Ft(1)3077 985 y FA(\(3.12\))599 1247 y(w)n(as)g(in)n(tro)r(duced)h(b)n(y)f(W)r (\177)-44 b(ust)24 b(and)g(led)g(to)f(a)h(relation)e(b)r(et)n(w)n(een)i (the)g(w)n(eigh)n(ted)f(kinetic)599 1347 y(and)f(Coulom)n(b)e(parts)h (of)h(the)f(atomic)g(Dirac)g(Hamiltonian)g(\(lemma)h(3)f(in)h(Ref.)g ([32)o(]\).)599 1446 y(T)-7 b(o)24 b(apply)g(a)g(sp)r(ecialized)f(v)n (ersion)g(of)h(W)r(\177)-44 b(ust's)24 b(inequalit)n(y)g(\(Eq.)g (\(20\))f(in)i([32)o(]\))f(to)g(the)599 1546 y(molecular)j(situation,)g (viz.,)1155 1738 y Fs(jj)14 b Fr(w)1276 1704 y Ft(1)p Fo(=)p Ft(2)1381 1738 y Fr(T)1430 1750 y Fo(D)1490 1738 y Fr( )j Fs(jj)1607 1704 y Ft(2)1681 1738 y Fs(\025)37 b FA(\(1)18 b Fs(\000)g Fr(")1997 1704 y Ft(2)2034 1738 y Fr(=)p FA(4\))c Fs(jj)g Fr(w)2285 1704 y Ft(1)p Fo(=)p Ft(2)2389 1738 y Fs(j)p Fn(x)p Fs(j)2490 1704 y Fl(\000)p Ft(1)2579 1738 y Fr( )j Fs(jj)2696 1704 y Ft(2)3077 1738 y FA(\(3.13\))599 1930 y(if)34 b(0)f Fr(<)f(")g(<)h FA(2)g(and)g Fr( )j Fs(2)d Fr(H)1518 1900 y Ft(1)1555 1930 y FA(\()p Fq(R)1641 1900 y Ft(3)1684 1930 y Fr(;)14 b Fq(C)1775 1900 y Ft(4)1818 1930 y FA(\),)35 b(w)n(e)e(translate)g Fr(w)j FA(along)c(the)i(in)n(tern)n(uclear)599 2030 y(axis.)43 b(With)30 b Fr(w)1082 2042 y Fl(\006)1165 2030 y FA(:=)c Fr(\034)1315 2045 y Fl(\006)p Fo(R=)p Ft(2)1489 2030 y Fr(w)r FA(\()p Fr(\034)1618 2045 y Fl(\006)p Fo(R=)p Ft(2)1793 2030 y FA(\))1825 2000 y Fl(\000)p Ft(1)1944 2030 y FA(regarded)i(as)h(m)n(ultiplication)g(op)r(erators,)599 2130 y(inequalit)n(y)f(\(3.13\))e(can)i(b)r(e)g(recast)e(in)n(to)1029 2338 y Fs(jj)14 b Fr(w)1150 2295 y Ft(1)p Fo(=)p Ft(2)1148 2358 y(+)1255 2338 y Fr(T)1304 2350 y Fo(D)1363 2338 y Fr( )j Fs(jj)37 b(\025)1619 2263 y Fp(p)p 1702 2263 302 4 v 75 x FA(1)18 b Fs(\000)g Fr(")1884 2314 y Ft(2)1921 2338 y Fr(=)p FA(4)13 b Fr(\026)2068 2304 y Fl(\000)p Ft(1)2157 2338 y Fs(jj)h Fr(w)2278 2295 y Ft(1)p Fo(=)p Ft(2)2276 2358 y(+)2383 2338 y Fr(V)2450 2304 y Ft(+)2431 2358 y(ne)2505 2338 y FA(\()p Fr(R)q(;)g(\026)p FA(\))p Fr( )j Fs(jj)p Fr(:)217 b FA(\(3.14\))599 2530 y(Emplo)n(ying)27 b(\(3.14\),)g(w)n(e)g(estimate)g(for)g(0)c Fr(<)g(R)g(<)g Fs(1)661 2738 y FA(0)46 b(=)g Fs(jj)p Fr(w)967 2695 y Ft(1)p Fo(=)p Ft(2)965 2759 y(+)1072 2738 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\026)p FA(\))p Fr( )1389 2750 y Ft(0)1426 2738 y Fs(jj)906 2888 y(\025)22 b(jj)p Fr(w)1100 2845 y Ft(1)p Fo(=)p Ft(2)1098 2909 y(+)1206 2888 y Fr(T)1255 2900 y Fo(D)1314 2888 y Fr( )1368 2900 y Ft(0)1406 2888 y Fs(jj)c(\000)g(jj)p Fr(w)1660 2845 y Ft(1)p Fo(=)p Ft(2)1658 2909 y(+)1765 2888 y Fr(V)1832 2854 y Ft(+)1813 2909 y(ne)1887 2888 y FA(\()p Fr(R)q(;)c(\026)p FA(\))p Fr( )2156 2900 y Ft(0)2194 2888 y Fs(jj)k(\000)g(jj)p Fr(w)2448 2845 y Ft(1)p Fo(=)p Ft(2)2446 2909 y(+)2554 2888 y Fr(V)2620 2854 y Fl(\000)2602 2909 y Ft(ne)2677 2888 y FA(\()p Fr(R)q(;)c(\026)p FA(\))p Fr( )2946 2900 y Ft(0)2983 2888 y Fs(jj)906 3038 y(\025)22 b FA(2)1035 2971 y Fp(\000)1073 2963 y(p)p 1156 2963 V 75 x FA(1)c Fs(\000)g Fr(")1338 3014 y Ft(2)1375 3038 y Fr(=)p FA(4)o Fr(=)p FA(\(2)p Fr(\026)p FA(\))g Fs(\000)g FA(1)1799 2971 y Fp(\001)1850 3038 y Fs(jj)p Fr(w)1957 2995 y Ft(1)p Fo(=)p Ft(2)1955 3059 y(+)2063 3038 y Fr(V)2129 3004 y Ft(+)2111 3058 y(ne)2185 3038 y FA(\()p Fr(R)q(;)c(\026)p FA(\))p Fr( )2454 3050 y Ft(0)2491 3038 y Fs(jj)23 b FA(+)1381 3188 y Fs(jj)p Fr(w)1488 3145 y Ft(1)p Fo(=)p Ft(2)1486 3208 y(+)1593 3188 y Fr(V)1660 3153 y Ft(+)1641 3208 y(ne)1715 3188 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))p Fr( )1984 3200 y Ft(0)2021 3188 y Fs(jj)19 b(\000)f(jj)p Fr(w)2276 3145 y Ft(1)p Fo(=)p Ft(2)2274 3208 y(+)2381 3188 y Fr(V)2448 3153 y Fl(\000)2429 3208 y Ft(ne)2504 3188 y FA(\()p Fr(R)q(;)c(\026)p FA(\))p Fr( )2773 3200 y Ft(0)2811 3188 y Fs(jj)3077 2955 y FA(\(3.15\))599 3376 y(where)38 b(in)h(the)g(\014rst)f(line)g(at)g(the)h(RHS)g(all)f (terms)g(are)f(w)n(ell)h(de\014ned)h(since)f(0)j Fr(<)599 3476 y(\026)26 b(<)f FA(1)p Fr(=)p FA(2)j(\(and)h(therefore)f Fr( )1518 3488 y Ft(0)1581 3476 y Fs(2)e(D)r FA(\()p Fr(V)1827 3446 y Fl(\006)1808 3496 y Ft(ne)1883 3476 y FA(\)\).)42 b(No)n(w,)29 b Fr( )2280 3488 y Ft(0)2347 3476 y FA(b)r(eing)g(an)g(eigenstate)f(of)h(the)599 3591 y(parit)n(y)f(op)r(erator)f Fr(P)1234 3603 y Fo(O)1290 3591 y FA(,)i(the)g(equalit)n(y)f Fs(jj)p Fr(w)1910 3548 y Ft(1)p Fo(=)p Ft(2)1908 3612 y(+)2015 3591 y Fr(V)2082 3561 y Ft(+)2063 3612 y(ne)2137 3591 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))p Fr( )2406 3603 y Ft(0)2443 3591 y Fs(jj)25 b FA(=)f Fs(jj)p Fr(w)2710 3548 y Ft(1)p Fo(=)p Ft(2)2708 3612 y Fl(\000)2815 3591 y Fr(V)2882 3561 y Ft(+)2863 3612 y(ne)2937 3591 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))p Fr( )3206 3603 y Ft(0)3243 3591 y Fs(jj)599 3691 y FA(and)28 b(the)g(b)r(ound)697 3808 y Fp(\014)697 3858 y(\014)738 3879 y Fs(jj)p Fr(w)845 3836 y Ft(1)p Fo(=)p Ft(2)843 3899 y(+)951 3879 y Fr(V)1017 3844 y Ft(+)999 3899 y(ne)1073 3879 y FA(\()p Fr(R)q(;)14 b(\026)p FA(\))p Fr( )1342 3891 y Ft(0)1379 3879 y Fs(jj)19 b(\000)f(jj)p Fr(w)1634 3836 y Ft(1)p Fo(=)p Ft(2)1632 3899 y Fl(\000)1739 3879 y Fr(V)1806 3844 y Ft(+)1787 3899 y(ne)1861 3879 y FA(\()p Fr(R)q(;)c(\026)p FA(\))p Fr( )2130 3891 y Ft(0)2167 3879 y Fs(jj)2227 3808 y Fp(\014)2227 3858 y(\014)2301 3879 y Fs(\024)46 b(jj)p Fr(w)2517 3891 y Ft(+)2591 3879 y Fs(\000)18 b Fr(w)2733 3891 y Fl(\000)2789 3879 y Fs(jj)2835 3891 y Fl(1)2929 3879 y Fs(\002)1871 4009 y(jj)p Fr(V)1984 3979 y Ft(+)1965 4030 y(ne)2039 4009 y FA(\()p Fr(R)q(;)c(\026)p FA(\))p Fr( )2308 4021 y Ft(0)2345 4009 y Fs(jj)2391 3979 y Ft(2)p 1412 4046 1476 4 v 1412 4143 a Fs(jj)p Fr(w)1519 4100 y Ft(1)p Fo(=)p Ft(2)1517 4164 y(+)1624 4143 y Fr(V)1691 4108 y Ft(+)1672 4153 y(ne)1746 4143 y FA(\()p Fr(R)q(;)g(\026)p FA(\))p Fr( )2015 4155 y Ft(0)2053 4143 y Fs(jj)k FA(+)g Fs(jj)p Fr(w)2307 4100 y Ft(1)p Fo(=)p Ft(2)2305 4164 y Fl(\000)2413 4143 y Fr(V)2479 4108 y Ft(+)2461 4153 y(ne)2535 4143 y FA(\()p Fr(R)q(;)c(\026)p FA(\))p Fr( )2804 4155 y Ft(0)2841 4143 y Fs(jj)3077 4002 y FA(\(3.16\))599 4334 y(together)24 b(with)h Fs(jj)p Fr(w)1215 4346 y Ft(+)1282 4334 y Fs(\000)12 b Fr(w)1418 4346 y Fl(\000)1474 4334 y Fs(jj)1520 4346 y Fl(1)1614 4334 y FA(=)23 b Fr("R)h FA(ensue)h(a)f(con)n(tradiction)f (in)h(Eq.)g(\(3.15\))g(if)g Fr(")h FA(is)599 4433 y(c)n(hosen)h(suc)n (h)h(that)1230 4362 y Fp(p)p 1313 4362 302 4 v 71 x FA(1)18 b Fs(\000)g Fr(")1495 4409 y Ft(2)1532 4433 y Fr(=)p FA(4)k Fr(>)h FA(2)p Fr(\026)j FA(and)h Fr(R)h FA(is)f(con\014ned)f(to) h(the)g(in)n(terv)-5 b(al)27 b(\(0)p Fr(;)14 b(R)3222 4445 y Fo(")3257 4433 y FA(\))599 4533 y(with)36 b Fr(R)859 4545 y Fo(")929 4533 y FA(su\016cien)n(tly)f(small.)59 b(If)35 b Fr(R)h FA(=)f(0,)i(the)e(energies)f(are)g(explicit,)j Fr(e)p FA(\(0)p Fr(;)14 b(\026)p FA(\))35 b(=)p eop %%Page: 18 18 18 17 bop 599 75 a FA(18)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 256 y Fp(p)p 682 256 337 4 v 71 x FA(1)18 b Fs(\000)g FA(\(2)p Fr(\026)p FA(\))981 303 y Ft(2)1019 327 y FA(.)44 b(Hence,)31 b(if)f Fr(\026)g FA(ob)r(eys)g(0)c Fr(<)h(\026)g(<)g FA(1)p Fr(=)p FA(2,)i(there)h(is)g(a)f(neigh)n(b)r(ourho)r(o)r(d)g Fs(U)3252 339 y Ft(0)599 426 y FA(of)21 b Fr(R)j FA(=)f(0)d(\(that)i (in)f(our)g(construction)f(also)g(dep)r(ends)h(on)g Fr(\026)p FA(\))h(suc)n(h)e(that)i(the)f(molecu-)599 526 y(lar)f(Hamiltonians)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\026)p FA(\))21 b(only)f(ha)n(v)n(e)g(non) n(v)-5 b(anishing)20 b(eigen)n(v)-5 b(alues,)21 b(whic)n(h)f(con)n (tra-)599 646 y(dicts)28 b(\(3.11\))f(for)g(these)h Fr(R)q FA(.)p 1548 663 25 75 v 682 801 a(It)h(ma)n(y)f(b)r(e)h(w)n(orth)e(to)i (p)r(oin)n(t)f(out)h(that)f(the)h(argumen)n(ts)e(ab)r(o)n(v)n(e)g(for)h (the)h(absence)599 901 y(of)f(zero-energy)d(eigenstates)i(can)h(also)f (b)r(e)h(utilized)g(to)g(sho)n(w)f(the)h(nonexistence)f(of)599 1001 y(negativ)n(e)33 b(eigen)n(v)-5 b(alues)32 b(for)h(one-cen)n(tre)f (p)r(oten)n(tials)h(and)h(w)n(ould)f(considerably)f(fa-)599 1100 y(cilitate)h(and)g(shorten)f(the)i(relev)-5 b(an)n(t)32 b(analysis)f(in)j(Ref.)f([34)o(])g(\(for)g(details,)h(cf.)53 b(the)599 1200 y(app)r(endix\).)41 b(Strictly)29 b(p)r(ositiv)n(e)g(lo) n(w)n(er)e(b)r(ounds)i(on)f Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))30 b(pro)n(vide)d(alternativ)n(e)h({)599 1299 y(though)d(rather)f(direct)h({)f(pro)r(cedures)g(to)h(rule)f(out)h (molecular)f(zero-energy)e(b)r(ound)599 1399 y(states)32 b(in)h(lemma)f(3.1.)51 b(If)33 b(w)n(e)f(in)n(v)n(ok)n(e)e(the)j(UAL)g (lo)n(w)n(er)e(b)r(ound)i(\(2.41\),)f(this,)i(un-)599 1499 y(fortunately)-7 b(,)36 b(w)n(ould)e(only)f(co)n(v)n(er)g(c)n (harge)f(parameters)h(up)h(to)g(the)h Fr(\020)2836 1511 y Ft(max)2997 1499 y FA(giv)n(en)e(in)599 1598 y(\(2.28\).)67 b(Larger)36 b Fr(\020)44 b FA(up)38 b(to)g Fr(\020)46 b Fs(\024)40 b FA(\()1728 1566 y Fo(\031)p 1728 1580 41 4 v 1732 1627 a Ft(2)1804 1598 y FA(+)1908 1566 y Ft(2)p 1904 1580 V 1904 1627 a Fo(\031)1955 1598 y FA(\))1987 1568 y Fl(\000)p Ft(1)2117 1598 y Fs(\031)f FA(0)p Fr(:)p FA(4530)d(can)h(b)r(e)h(reac)n(hed)f(if)h(the)599 1698 y(UAL)26 b(ground)e(state)h(energy)f(of)h(the)g(\(atomic\))h(Bro)n (wn-Ra)n(v)n(enhall)c(Hamiltonian)j(is)599 1798 y(tak)n(en)c(as)g(lo)n (w)n(er)e(b)r(ound)j(\(see)f(theorem)g(A.3)g(and)g(the)h(discussion)f (in)g(the)h(app)r(endix\).)599 1897 y(Compared)28 b(with)i(the)f (maximal)g Fr(\020)i FA(=)25 b(1)p Fr(=\031)j Fs(\031)d FA(0)p Fr(:)p FA(3183)i(allo)n(w)n(ed)g(b)n(y)i(Klaus')f(original)599 1997 y(metho)r(d,)i(this)f(is)g(already)e(quite)i(an)g(impro)n(v)n (emen)n(t.)39 b(Of)29 b(course,)f(the)h(conjectured)599 2096 y(extension)h(of)h(the)g(UAL)f(lo)n(w)n(er)f(b)r(ound)i(to)f(all)g (0)e Fr(<)f(\020)34 b(<)27 b FA(1)p Fr(=)p FA(2)i(w)n(ould)h (immediately)599 2196 y(dispro)n(v)n(e)19 b(the)h(existence)f(of)h (molecular)f(zero-energy)e(eigenstates)i(for)g(these)h Fr(\020)6 b FA(;)23 b(here,)599 2296 y(from)28 b(the)h(con)n(tin)n(uit) n(y)e(of)h(the)g(electronic)g(curv)n(es)e(and)i(the)h(p)r(ositivit)n(y) f(of)g(the)g(UAL)599 2395 y(energies)g(w)n(e)g(only)g(can)h(infer)f (the)i(existence)e(of)h(a)f Fr(R)2295 2407 y Fo(p)2358 2395 y Fr(>)d FA(0)j(suc)n(h)g(that)h Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))25 b Fr(>)g FA(0)599 2495 y(for)j(all)f Fr(\020)i Fs(2)24 b FA(\(0)p Fr(;)14 b FA(1)p Fr(=)p FA(2\))26 b(and)h(0)c Fs(\024)g Fr(R)g(<)g(R)1832 2507 y Fo(p)1870 2495 y FA(.)1492 2661 y Fu(3.2.)94 b(Di\013eren)m (tiabilit)m(y)682 2777 y FA(The)37 b(norm)f(resolv)n(en)n(t)f(con)n(v)n (ergence)g(of)h Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))38 b(established)e(in)h(theorem)f(3.1)599 2877 y(en)n(tails)e(con)n(tin)n(uit)n(y)f(of)h(the)h(eigen)n(v)-5 b(alues)33 b Fr(e)p FA(\()p Fr(r)n(;)14 b(\020)6 b FA(\))35 b(and)f(their)g(total)f(pro)5 b(jections)33 b(as)599 2976 y Fr(R)24 b Fs(#)f FA(0)i(\(see)h(theorem)f(VI)r(I)r(I.23)h(in)f (Ref.)i([46)o(]\).)36 b(Th)n(us)26 b(w)n(e)f(kno)n(w)g(that)h(the)g (electronic)599 3076 y(curv)n(es)j Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))31 b(are)e(con)n(tin)n(uous)g(functions)i(of)f Fr(R)e Fs(\025)f FA(0,)j(and)g(analytic)g(a)n(w)n(a)n(y)e(from)599 3176 y Fr(R)41 b FA(=)e(0.)67 b(Our)37 b(next)h(aim)f(is)h(to)f (demonstrate)g(that)h Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))38 b(are)f(actually)g(once)599 3275 y(di\013eren)n(tiable)28 b(also)e(at)i Fr(R)c FA(=)e(0.)682 3551 y Fj(Theorem)31 b(3.2.)63 b Fi(L)l(et)25 b FA(0)d Fr(<)h(\020)29 b(<)23 b FA(1)p Fr(=)p FA(2)p Fi(.)35 b(Then)26 b(for)f(al)t(l)h Fr(R)e Fs(\025)e FA(0)i Fi(and)h Fr(E)j Fs(2)c Fr(\032)2958 3484 y Fp(\000)2996 3551 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))3251 3484 y Fp(\001)599 3676 y Fi(the)30 b(r)l(esolvent)1084 3609 y Fp(\000)1122 3676 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)1545 3609 y Fp(\001)1583 3626 y Fl(\000)p Ft(1)1702 3676 y Fi(is)30 b(str)l(ongly)g(c)l(ontinuously)g(di\013er)l(entiable)h(in)f Fr(R)q Fi(.)682 3944 y(Pr)l(o)l(of.)78 b FA(T)-7 b(o)20 b(deduce)g(the)g(assertion,)g(w)n(e)g(ha)n(v)n(e)f(to)g(v)n(erify)h (the)g(strong)f(con)n(v)n(ergence)599 4043 y(of)667 4278 y(\(\001)p Fr(R)r FA(\))865 4244 y Fl(\000)p Ft(1)954 4211 y Fp(\000)992 4278 y FA(\()p Fr(h)p FA(\()p Fr(R)h FA(+)e(\001)p Fr(R)q(;)27 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))1727 4244 y Fl(\000)p Ft(1)1835 4278 y Fs(\000)18 b FA(\()p Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))2419 4244 y Fl(\000)p Ft(1)2509 4211 y Fp(\001)2593 4278 y FA(=)46 b(\(\001)p Fr(R)q FA(\))2901 4244 y Fl(\000)p Ft(1)2990 4278 y Fs(\002)833 4361 y Fp(\000)871 4428 y Fr(h)p FA(\()p Fr(R)19 b FA(+)f(\001)p Fr(R)q(;)27 b(\020)6 b FA(\))20 b Fs(\000)e Fr(E)1542 4361 y Fp(\001)1580 4378 y Fl(\000)p Ft(1)1669 4361 y Fp(\000)1707 4428 y Fr(h)p FA(\()p Fr(R)h FA(+)f(\001)p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))2581 4361 y Fp(\001\000)2657 4428 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)3094 4361 y Fp(\001)3132 4378 y Fl(\000)p Ft(1)3077 4536 y FA(\(3.17\))p eop %%Page: 19 19 19 18 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(19)599 320 y(for)28 b(\001)p Fr(R)d Fs(!)e FA(0,)28 b(or,)g(if)g Fr(R)d FA(=)e(0,)28 b(for)f(\001)p Fr(R)e Fs(#)e FA(0.)38 b(With)29 b(the)f(help)h(of)f(the)g(unitary)f (trans-)599 419 y(lation)i(op)r(erator)e Fr(\034)1208 431 y Ft(\001)p Fo(R)1346 419 y FA(w)n(e)i(can)f(rewrite)g(the)h (di\013erence)g(on)g(the)g(RHS)g(of)g(equation)599 519 y(\(3.17\))e(as)723 745 y Fr(h)p FA(\()p Fr(R)20 b FA(+)e(\001)p Fr(R)q(;)27 b(\020)6 b FA(\))19 b Fs(\000)f Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))24 b(=)e Fr(V)1756 757 y Ft(ne)1827 745 y FA(\()p Fr(R)d FA(+)g(\001)p Fr(R)q(;)27 b(\020)6 b FA(\))19 b Fs(\000)f Fr(V)2432 757 y Ft(ne)2503 745 y FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))890 881 y(=)23 b Fr(\034)1014 893 y Ft(\001)p Fo(R)1123 881 y Fr(V)1190 847 y Ft(+)1171 902 y(ne)1245 881 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(\034)1497 845 y Fl(\000)p Ft(1)1488 905 y(\001R)1614 881 y Fs(\000)18 b Fr(V)1764 847 y Ft(+)1745 902 y(ne)1819 881 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))19 b(+)f Fr(\034)2164 893 y Ft(\001)p Fo(R)2274 881 y Fr(V)2341 847 y Fl(\000)2322 902 y Ft(ne)2397 881 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))p Fr(\034)2649 845 y Fl(\000)p Ft(1)2640 905 y(\001R)2766 881 y Fs(\000)18 b Fr(V)2916 847 y Fl(\000)2897 902 y Ft(ne)2972 881 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))890 1017 y(=)23 b Fr(\034)1014 1029 y Ft(\001)p Fo(R)1123 1017 y Fr(V)1190 983 y Ft(+)1171 1038 y(ne)1245 1017 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\)\()p Fr(\034)1529 982 y Fl(\000)p Ft(1)1520 1041 y(\001)p Fo(R)1649 1017 y Fs(\000)18 b FA(1\))g(+)g(\()p Fr(\034)1975 1029 y Ft(\001)p Fo(R)2103 1017 y Fs(\000)g FA(1\))p Fr(V)2327 983 y Ft(+)2308 1038 y(ne)2382 1017 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))1218 1153 y(+)18 b Fr(\034)1346 1118 y Fl(\000)p Ft(1)1337 1178 y(\001)p Fo(R)1446 1153 y Fr(V)1513 1119 y Fl(\000)1494 1174 y Ft(ne)1569 1153 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\)\()p Fr(\034)1844 1165 y Ft(\001)p Fo(R)1972 1153 y Fs(\000)18 b FA(1\))h(+)f(\()p Fr(\034)2308 1118 y Fl(\000)p Ft(1)2299 1178 y(\001)p Fo(R)2427 1153 y Fs(\000)g FA(1\))p Fr(V)2650 1119 y Fl(\000)2632 1174 y Ft(ne)2707 1153 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))3077 1254 y(\(3.18\))599 1353 y(where)32 b Fr(V)911 1323 y Fl(\006)892 1374 y Ft(ne)967 1353 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))32 b(are)f(the)h(m)n(ultiplication)g(op)r(erators)e(de\014ned)i(in)g(the)g (preceding)599 1453 y(section)c(3.1.)36 b(On)27 b(the)h(domain)f Fs(D)r FA(\()p Fn(p)q FA(\))h(of)f(the)h(momen)n(tum)g(op)r(erator)e(w) n(e)h(ha)n(v)n(e)641 1680 y(\(\001)p Fr(R)q FA(\))838 1645 y Fl(\000)p Ft(1)928 1680 y FA(\()p Fr(\034)996 1692 y Ft(\001)p Fo(R)1117 1680 y Fs(\000)12 b FA(1\))22 b(=)g(\(\001)p Fr(R)q FA(\))1574 1645 y Fl(\000)p Ft(1)1664 1612 y Fp(\000)1702 1680 y FA(exp\()p Fs(\000)p Fr(i)p FA(\001)p Fr(R)14 b(p)2143 1692 y Fo(s)2178 1680 y FA(\))e Fs(\000)g FA(1)2341 1612 y Fp(\001)2401 1680 y FA(=)23 b Fs(\000)p Fr(ip)2625 1692 y Fo(s)2671 1680 y FA(+)12 b Fs(O)r FA(\(\001)p Fr(R)q FA(\))p Fr(;)41 b FA(\(3.19\))599 1907 y(with)d Fr(p)840 1919 y Fo(s)915 1907 y FA(=)h Fs(j)p Fn(R)1115 1919 y Fo(s)1150 1907 y Fs(j)1173 1876 y Fl(\000)p Ft(1)1262 1907 y Fn(R)1335 1919 y Fo(s)1371 1907 y Fm(\001)p Fn(p)d FA(b)r(eing)i(the)g(pro)5 b(jection)36 b(of)h(the)h(momen)n(tum)f(on)n(to)g(the)599 2006 y(in)n(tern)n(uclear) k(axis.)81 b(Th)n(us,)46 b(\(\001)p Fr(R)q FA(\))1766 1976 y Fl(\000)p Ft(1)1856 2006 y FA(\()p Fr(\034)1924 2018 y Ft(\001)p Fo(R)2062 2006 y Fs(\000)28 b FA(1\))48 b Fs(!)g(\000)p Fr(ip)2544 2018 y Fo(s)2578 2006 y FA(,)f(and,)f (analogously)-7 b(,)599 2106 y(\(\001)p Fr(R)q FA(\))796 2076 y Fl(\000)p Ft(1)886 2106 y FA(\()p Fr(\034)963 2070 y Fl(\000)p Ft(1)954 2130 y(\001)p Fo(R)1079 2106 y Fs(\000)16 b FA(1\))23 b Fs(!)g Fr(ip)1434 2118 y Fo(s)1495 2106 y FA(con)n(v)n(erge)h(strongly)h(as)g(\001)p Fr(R)f Fs(!)f FA(0.)36 b(F)-7 b(o)r(cusing)26 b(\014rst)g(on)n(to)599 2205 y(the)34 b(terms)e(con)n(taining)g Fr(V)1460 2175 y Ft(+)1441 2226 y(ne)1515 2205 y FA(,)i(and)f(using)f(that)h Fr(\034)2182 2217 y Ft(\001)p Fo(R)2325 2205 y FA(and)f(\()p Fr(T)2572 2217 y Fo(D)2654 2205 y Fs(\000)22 b FA(1\))2815 2175 y Fl(\000)p Ft(1)2937 2205 y FA(comm)n(ute,)599 2305 y(the)28 b(expression)699 2462 y Fp(\000)737 2530 y Fr(h)p FA(\()p Fr(R)q FA(+\001)p Fr(R)q(;)f(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))1403 2495 y Fl(\000)p Ft(1)1493 2530 y Fr(\034)1529 2542 y Ft(\001)p Fo(R)1638 2530 y Fr(V)1705 2495 y Ft(+)1686 2550 y(ne)1760 2530 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\)\(\001)p Fr(R)q FA(\))2164 2495 y Fl(\000)p Ft(1)2254 2530 y FA(\()p Fr(\034)2331 2494 y Fl(\000)p Ft(1)2322 2554 y(\001)p Fo(R)2450 2530 y Fs(\000)18 b FA(1\))2607 2462 y Fp(\000)2645 2530 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))3114 2495 y Fl(\000)p Ft(1)969 2665 y FA(=)1056 2597 y Fp(\000)1094 2665 y Fr(h)p FA(\()p Fr(R)20 b FA(+)e(\001)p Fr(R)q(;)27 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))1797 2630 y Fl(\000)p Ft(1)1887 2665 y FA(\()p Fr(T)1968 2677 y Fo(D)2046 2665 y Fs(\000)18 b Fr(E)5 b FA(\))p Fr(\034)2263 2677 y Ft(\001)p Fo(R)2373 2665 y FA(\()p Fr(T)2454 2677 y Fo(D)2533 2665 y Fs(\000)18 b Fr(E)5 b FA(\))2714 2630 y Fl(\000)p Ft(1)2803 2665 y Fr(V)2870 2630 y Ft(+)2851 2685 y(ne)2925 2665 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fs(\002)1278 2801 y FA(\(\001)p Fr(R)q FA(\))1475 2766 y Fl(\000)p Ft(1)1564 2801 y FA(\()p Fr(\034)1641 2765 y Fl(\000)p Ft(1)1632 2825 y(\001)p Fo(R)1760 2801 y Fs(\000)18 b FA(1\))1917 2733 y Fp(\000)1955 2801 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))2424 2766 y Fl(\000)p Ft(1)3077 2901 y FA(\(3.20\))599 3001 y(is)44 b(recognized)e(to)i(b)r(e)g(a)f (pro)r(duct)g(of)h(b)r(ounded)g(op)r(erators;)50 b(here,)d(due)d(to)f (the)599 3100 y(norm)29 b(resolv)n(en)n(t)e(con)n(tin)n(uit)n(y)h(of)h Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\),)29 b(the)g(\014rst)g (factor)f(con)n(v)n(erges)e(strongly)i(to)599 3150 y Fp(\000)637 3217 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))20 b Fs(\000)e Fr(E)1061 3150 y Fp(\001)1100 3167 y Fl(\000)p Ft(1)1189 3217 y FA(\()p Fr(T)1270 3229 y Fo(D)1349 3217 y Fs(\000)g Fr(E)5 b FA(\),)29 b(while)g(as)e(a)h (consequence)g(of)g(\(3.19\))g(the)g(last)g(factor)599 3342 y(has)f Fs(\000)p Fr(ip)883 3354 y Fo(s)917 3274 y Fp(\000)955 3342 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))18 b Fs(\000)e Fr(E)1375 3274 y Fp(\001)1414 3292 y Fl(\000)p Ft(1)1530 3342 y FA(as)26 b(its)h(strong)f(limit;)i (moreo)n(v)n(er,)c Fr(\034)2637 3354 y Ft(\001)p Fo(R)2769 3342 y Fs(!)f Fq(I)2914 3354 y Fl(H)2996 3342 y FA(strongly)599 3441 y(for)28 b(\001)p Fr(R)23 b Fs(!)h FA(0.)36 b(In)28 b(the)g(same)f(w)n(a)n(y)-7 b(,)26 b(in)620 3598 y Fp(\000)658 3665 y Fr(h)p FA(\()p Fr(R)q FA(+)o(\001)p Fr(R)q(;)i(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))1324 3630 y Fl(\000)p Ft(1)1414 3665 y FA(\(\001)p Fr(R)q FA(\))1611 3630 y Fl(\000)p Ft(1)1700 3665 y FA(\()p Fr(\034)1768 3677 y Ft(\001)p Fo(R)1896 3665 y Fs(\000)18 b FA(1\))p Fr(V)2120 3630 y Ft(+)2101 3685 y(ne)2175 3665 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))2382 3598 y Fp(\000)2421 3665 y Fr(h)p FA(\()p Fr(R)q(;)27 b(\020)6 b FA(\))20 b Fs(\000)e Fr(E)5 b FA(\))2890 3630 y Fl(\000)p Ft(1)890 3800 y FA(=)977 3732 y Fp(\000)1015 3800 y Fr(h)p FA(\()p Fr(R)19 b FA(+)f(\001)p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))1718 3765 y Fl(\000)p Ft(1)1808 3800 y FA(\()p Fr(T)1889 3812 y Fo(D)1967 3800 y Fs(\000)18 b Fr(E)5 b FA(\)\(\001)p Fr(R)q FA(\))2345 3765 y Fl(\000)p Ft(1)2435 3800 y FA(\()p Fr(\034)2503 3812 y Ft(\001)p Fo(R)2631 3800 y Fs(\000)18 b FA(1\)\()p Fr(T)2869 3812 y Fo(D)2948 3800 y Fs(\000)g Fr(E)5 b FA(\))3129 3765 y Fl(\000)p Ft(1)3218 3800 y Fs(\002)1365 3934 y Fr(V)1431 3900 y Ft(+)1413 3955 y(ne)1487 3934 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))1694 3867 y Fp(\000)1732 3934 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)5 b FA(\))2201 3900 y Fl(\000)p Ft(1)3077 4035 y FA(\(3.21\))599 4134 y(the)33 b(factor)f(\(\001)p Fr(R)q FA(\))1187 4104 y Fl(\000)p Ft(1)1277 4134 y FA(\()p Fr(\034)1345 4146 y Ft(\001)p Fo(R)1476 4134 y Fs(\000)22 b FA(1\)\()p Fr(T)1718 4146 y Fo(D)1799 4134 y Fs(\000)f Fr(E)5 b FA(\))1983 4104 y Fl(\000)p Ft(1)2106 4134 y FA(is)32 b(a)g(b)r(ounded)h(op)r(erator)e(that)i(con-)599 4234 y(v)n(erges)d(strongly)g(to)h Fr(ip)1354 4246 y Fo(s)1388 4234 y FA(\()p Fr(T)1469 4246 y Fo(D)1550 4234 y Fs(\000)21 b Fr(E)5 b FA(\))1734 4204 y Fl(\000)p Ft(1)1823 4234 y FA(.)48 b(Completely)31 b(analogous)e(argumen)n(ts)h(hold)599 4334 y(for)e(terms)f(con)n(taining)g Fr(V)1428 4304 y Fl(\000)1410 4354 y Ft(ne)1484 4334 y FA(.)38 b(Therefore,)26 b(equations)h(\(3.20\))g(and)g(\(3.21\))g(and)g(their)599 4433 y(coun)n(terparts)i(for)g Fr(V)1281 4403 y Fl(\000)1262 4454 y Ft(ne)1368 4433 y FA(together)g(with)i(\(3.18\))e(pro)n(v)n(e)f (the)j(strong)e(con)n(v)n(ergence)e(of)599 4533 y(the)d(expression)e (\(3.17\).)35 b(F)-7 b(urthermore,)23 b(com)n(bining)g(the)h(form)n (ulas,)f(the)h(deriv)-5 b(ativ)n(e)p eop %%Page: 20 20 20 19 bop 599 75 a FA(20)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(of)28 b(the)g(resolv)n(en)n(t)e(b)r(ecomes)734 458 y Fo(@)p 709 472 90 4 v 709 519 a(@)t(R)808 423 y Fp(\000)846 491 y Fr(h)p FA(\()p Fr(R)q(;)i(\020)6 b FA(\))19 b Fs(\000)f Fr(E)1283 423 y Fp(\001)1321 441 y Fl(\000)p Ft(1)1433 491 y FA(=)1521 423 y Fp(\000)1559 491 y Fr(h)p FA(\()p Fr(R)q(;)c(\020)6 b FA(\))19 b Fs(\000)f Fr(E)1982 423 y Fp(\001)2020 441 y Fl(\000)p Ft(1)2109 423 y Fp(\000)2147 491 y Fs(\000)p Fr(i)p FA(\()p Fr(p)2315 503 y Fo(s)2350 491 y Fr(V)2417 456 y Ft(+)2398 511 y(ne)2472 491 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\)\))1744 641 y(+)18 b Fr(i)p FA(\()p Fr(p)1930 653 y Fo(s)1966 641 y Fr(V)2032 607 y Fl(\000)2014 662 y Ft(ne)2088 641 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\)\))2327 574 y Fp(\001)q(\000)2404 641 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)2841 574 y Fp(\001)2879 591 y Fl(\000)p Ft(1)2968 641 y FA(;)3077 558 y(\(3.22\))599 821 y(since)32 b Fr(p)849 833 y Fo(s)884 821 y Fr(V)951 791 y Ft(+)932 841 y(ne)1006 821 y FA(\(0)p Fr(;)14 b(\020)6 b FA(\))31 b(=)e Fr(p)1358 833 y Fo(s)1394 821 y Fr(V)1461 791 y Fl(\000)1442 841 y Ft(ne)1517 821 y FA(\(0)p Fr(;)14 b(\020)6 b FA(\),)33 b(equation)e(\(3.22\))g(rev)n (eals)f(that)j(at)e Fr(R)g FA(=)f(0)h(the)599 920 y(deriv)-5 b(ativ)n(e)25 b(v)-5 b(anishes.)35 b(F)-7 b(rom)25 b(the)g(RHS)h(of)f (\(3.22\))f(w)n(e)h(also)e(can)i(deduce)g(the)h(strong)599 1037 y(con)n(tin)n(uit)n(y)42 b(when)g Fr(R)48 b Fs(#)e FA(0.)80 b(Namely)-7 b(,)46 b(b)r(oth)c(factors)2415 970 y Fp(\000)2453 1037 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))28 b Fs(\000)g Fr(E)2909 970 y Fp(\001)2947 987 y Fl(\000)p Ft(1)3036 1037 y Fr(p)3078 1049 y Fo(s)3156 1037 y FA(and)599 1162 y Fr(V)666 1132 y Fl(\006)647 1182 y Ft(ne)722 1162 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))929 1094 y Fp(\000)968 1162 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))21 b Fs(\000)f Fr(E)1409 1094 y Fp(\001)1448 1112 y Fl(\000)p Ft(1)1568 1162 y FA(con)n(v)n(erge)29 b(strongly)h(as)g(b)r(ounded)i(op)r(erators)d(in)i (this)599 1261 y(limit,)f(where,)f(in)g(turn,)g(the)h(con)n(v)n (ergence)c(of)j(the)g(latter)f(factor)g(is)h(easily)f(seen)h(b)n(y)599 1373 y(writing)i(it)h(as)e(a)h(pro)r(duct)g(of)g Fr(V)1634 1343 y Fl(\006)1615 1394 y Ft(ne)1690 1373 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))1897 1306 y Fp(\000)1935 1373 y Fr(T)1984 1385 y Fo(D)2065 1373 y Fs(\000)20 b Fr(E)2216 1306 y Fp(\001)2254 1323 y Fl(\000)p Ft(1)2375 1373 y FA(and)2540 1306 y Fp(\000)2578 1373 y Fr(T)2627 1385 y Fo(D)2707 1373 y Fs(\000)h Fr(E)2859 1306 y Fp(\001\000)2935 1373 y Fr(h)p FA(\()p Fr(R)q(;)27 b(\020)6 b FA(\))22 b Fs(\000)599 1494 y Fr(E)665 1426 y Fp(\001)704 1444 y Fl(\000)p Ft(1)793 1494 y FA(.)p 899 1510 25 75 v 682 1643 a(W)-7 b(e)26 b(summarize)f(the)g(consequences)f(of)h(the)h (preceding)e(results)h(for)g(the)h(regular-)599 1743 y(it)n(y)i(of)g(electronic)e(curv)n(es)h(and)g(eigenfunctions:)682 1884 y Fj(Cor)n(ollar)-5 b(y)33 b(3.1.)69 b Fi(Assume)29 b FA(0)23 b Fr(<)f(\020)30 b(<)22 b FA(1)p Fr(=)p FA(2)p Fi(.)599 1984 y(\(i\))40 b(If)g(for)g Fr(R)1034 1996 y Ft(0)1112 1984 y Fs(\025)g FA(0)f Fi(the)h(ener)l(gy)g Fr(e)p FA(\()p Fr(R)1852 1996 y Ft(0)1889 1984 y Fr(;)14 b(\020)6 b FA(\))41 b Fs(2)g Fr(\033)2184 1996 y Ft(d)2225 1917 y Fp(\000)2263 1984 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))2518 1917 y Fp(\001)2596 1984 y Fi(is)40 b(a)g(nonde)l(gener)l(ate)599 2084 y(eigenvalue)35 b(of)e Fr(h)p FA(\()p Fr(R)1247 2096 y Ft(0)1285 2084 y Fr(;)14 b(\020)6 b FA(\))p Fi(,)35 b(then)e Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))33 b Fi(is)g(at)g(le)l(ast)g(onc)l(e)g(c)l (ontinuously)g(di\013er)l(en-)599 2183 y(tiable)h(in)e Fr(R)h Fi(on)f(a)h(neighb)l(ourho)l(o)l(d)h(of)f Fr(R)1924 2195 y Ft(0)1961 2183 y Fi(;)h(if)f Fr(R)2166 2195 y Ft(0)2231 2183 y Fr(>)27 b FA(0)p Fi(,)33 b(then)f Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))32 b Fi(is)h(analytic.)599 2283 y(Mor)l(e)l(over,)g(the)d(ele)l(ctr)l(onic)h(curves)f(arrive)i(at) e Fr(R)25 b FA(=)e(0)30 b Fi(with)h(a)g(horizontal)h(tangent,)599 2382 y(that)e(is,)h FA(lim)999 2394 y Fo(R)p Fl(#)p Ft(0)1154 2350 y Fo(@)t(e)p 1144 2364 90 4 v 1144 2411 a(@)t(R)1244 2382 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g(0)p Fi(.)599 2482 y(\(ii\))38 b(If)f Fr(P)12 b FA(\()p Fr(R)1009 2494 y Ft(0)1046 2482 y Fr(;)i(\020)6 b FA(\))38 b Fi(stands)e(for)i (the)f(total)g(pr)l(oje)l(ction)h(asso)l(ciate)l(d)g(with)f Fr(e)p FA(\()p Fr(R)3049 2494 y Ft(0)3087 2482 y Fr(;)14 b(\020)6 b FA(\))36 b Fs(2)599 2582 y Fr(\033)646 2594 y Ft(d)688 2514 y Fp(\000)726 2582 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))981 2514 y Fp(\001)1019 2582 y Fi(,)33 b(then)e Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))32 b Fi(is)g(at)g(le)l(ast)f(onc)l(e)h(c)l(ontinuously)g(di\013er)l (entiable)h(in)f Fr(R)599 2681 y Fi(on)e(a)g(neighb)l(ourho)l(o)l(d)i (of)f Fr(R)1489 2693 y Ft(0)1526 2681 y Fi(.)682 2938 y(Pr)l(o)l(of.)80 b FA(Giv)n(en)26 b(the)g(results)g(in)g(c)n(hapter)f (2)h(and)g(3,)g(the)g(assertions)f(of)h(the)g(corol-)599 3038 y(lary)38 b(ensue)h(from)f(application)g(of)h(standard)e(theorems) h(\(see)h(section)f(VI)r(I)r(I.7)h(in)599 3138 y(Ref.)33 b([46)o(],)g(and)e(VI)r(I)i Fs(x)p FA(3.1)e(in)h([37)o(]\).)50 b(Concerning)30 b(the)i(b)r(eha)n(viour)f(around)g Fr(R)f FA(=)g(0)599 3237 y(w)n(e)22 b(recall)f(that)45 b({after)21 b(reducing)h(the)g(atomic)g(Hamiltonian)g(b)n(y)g(its)g(full)h (symmetry)599 3337 y(group)c({)h(the)g(atomic)g(energies)e(are)h (nondegenerate)g(on)h(the)g(irreducible)f(subspaces;)599 3436 y(whence,)30 b(for)f(su\016cien)n(tly)g(small)g Fr(R)d(>)f FA(0,)k(the)h(molecular)e(energies,)g(to)r(o,)i(are)e(non-) 599 3536 y(degenerate)c(on)g(their)h(symmetry)f(subspaces.)35 b(F)-7 b(urthermore,)24 b(for)h(\(ii\),)h(w)n(e)e(observ)n(e)599 3636 y(that)35 b(there)g(exists)f(a)g(circle)g(\000\()p Fr(R)1689 3648 y Ft(0)1727 3636 y FA(\))h(around)f Fr(R)2146 3648 y Ft(0)2218 3636 y FA(suc)n(h)g(that)h(for)f(all)g Fr(R)i Fs(2)f(N)12 b FA(\()p Fr(R)3219 3648 y Ft(0)3257 3636 y FA(\))599 3735 y(from)43 b(a)f(neigh)n(b)r(ourho)r(o)r(d)f Fs(N)12 b FA(\()p Fr(R)1652 3747 y Ft(0)1690 3735 y FA(\))43 b(of)g Fr(R)1938 3747 y Ft(0)2018 3735 y FA(w)n(e)f(ha)n(v)n(e)f (\000\()p Fr(R)2508 3747 y Ft(0)2546 3735 y FA(\))48 b Fs(\032)f Fr(\032)2781 3668 y Fp(\000)2819 3735 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))3074 3668 y Fp(\001)3156 3735 y FA(and)599 3860 y(\000\()p Fr(R)746 3872 y Ft(0)784 3860 y FA(\))23 b Fs(\\)p 918 3779 762 4 v 918 3798 a Fp(S)987 3885 y Fo(R)p Fl(2N)9 b Ft(\()p Fo(R)1221 3893 y Fg(0)1254 3885 y Ft(\))1298 3860 y Fr(\033)1348 3793 y Fp(\000)1386 3860 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))1641 3793 y Fp(\001)1715 3860 y FA(=)35 b Fs(;)p FA(.)58 b(With)36 b(the)f(circle)f(\000\()p Fr(R)2681 3872 y Ft(0)2718 3860 y FA(\))i(the)f(total)g(pro-)599 3959 y(jection)28 b(can)f(b)r(e)h(expressed)f(as)g(con)n(tour)f(in)n (tegral,)1161 4172 y Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))24 b(=)e Fs(\000)1651 4139 y Ft(1)p 1619 4153 98 4 v 1619 4201 a(2)p Fo(\031)r(i)1740 4059 y Fp(Z)1786 4248 y Ft(\000\()p Fo(R)1903 4256 y Fg(0)1936 4248 y Ft(\))1979 4172 y Fr(dE)2103 4105 y Fp(\000)2141 4172 y Fr(h)p FA(\()p Fr(R)q(;)27 b(\020)6 b FA(\))19 b Fs(\000)f Fr(E)2577 4105 y Fp(\001)2616 4122 y Fl(\000)p Ft(1)2705 4172 y Fr(;)349 b FA(\(3.23\))599 4408 y(so)30 b(that)h(the)g (di\013eren)n(tiabilit)n(y)f(of)h Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))31 b(follo)n(ws)e(from)h(the)h(di\013eren)n(tiabilit)n(y)f(of) 599 4533 y(the)e(resolv)n(en)n(t.)p 1170 4550 25 75 v eop %%Page: 21 21 21 20 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(21)682 320 y(F)-7 b(or)25 b(Sc)n(hr\177)-42 b(odinger)24 b(op)r(erators)g(with)i(the)g(kinetic)f(energy)g(con)n (taining)f(the)i(second)599 419 y(deriv)-5 b(ativ)n(es)25 b(with)i(resp)r(ect)e(to)h(the)h(electronic)e(co)r(ordinates,)g(it)h (is)g(kno)n(wn)f([40)o(])h(that)599 519 y(the)37 b(resolv)n(en)n(t)d (at)i Fr(R)i FA(=)f(0)f(is)g(not)g(solely)f(once,)j(but)e(t)n(wice)g (di\013eren)n(tiable)g(in)h Fr(R)q FA(.)599 619 y(Indeed,)21 b(the)e(o)r(ccurrence)e(of)i(second)f(deriv)-5 b(ativ)n(es)17 b(in)i(the)g(Sc)n(hr\177)-42 b(odinger)17 b(Hamiltonian)599 718 y(pro)n(vides)23 b(the)h(crucial)f(to)r(ol)h(to)g(con)n(trol)f(the) h(terms)g(analogous)d(to)j(\(3.12\))f(that)i(arise)599 818 y(when)38 b(the)g(resolv)n(en)n(t)e(is)i(deriv)n(ed)f(t)n(wice)h (with)g(resp)r(ect)f(to)h Fr(R)q FA(;)k(a)c(kinetic)g(energy)599 917 y(linear)24 b(in)h(the)g(momen)n(tum)g({)f(lik)n(e)g Fr(T)1758 929 y Fo(D)1817 917 y FA(,)i(the)f(Dirac)f(one)g({)g(is)g (not)h(su\016cien)n(t)f(for)g(that)599 1017 y(purp)r(ose.)57 b(Hence,)36 b(one)e(cannot)g(exp)r(ect)h(to)f(extend)g(theorem)g(3.2)g (and)g(corollary)599 1117 y(3.1)27 b(to)h(second)f(deriv)-5 b(ativ)n(es)26 b(in)i Fr(R)q FA(.)1254 1283 y Fu(3.3.)94 b(F)-8 b(eynman-Hellman)28 b(theorem)682 1399 y FA(A)i(rather)d (immediate)j(consequence)d(of)i(the)g(regularit)n(y)e(of)i(the)g(pro)5 b(jections)28 b(for)599 1499 y(the)j(electronic)f(curv)n(es)f(is)h(the) h(F)-7 b(eynman-Hellman)30 b(theorem)g(that)h(simpli\014es)f(the)599 1598 y(deriv)-5 b(ativ)n(e)29 b(of)h(energy)e(exp)r(ectation)h(v)-5 b(alues)30 b(within)g(eigenstates)e(of)i(the)g(Hamilto-)599 1698 y(nian.)37 b(The)28 b(form)n(ulation)f(b)r(elo)n(w)g(also)f(co)n (v)n(ers)g(the)i(degenerate)e(case.)682 1843 y Fj(Theorem)31 b(3.3.)69 b Fi(F)-6 b(or)30 b FA(0)22 b Fr(<)h(\020)30 b(<)22 b FA(1)p Fr(=)p FA(2)28 b Fi(and)j FA(0)22 b Fs(\024)h Fr(R)2265 1855 y Ft(0)2332 1843 y Fi(let)30 b Fr(e)p FA(\()p Fr(R)2583 1855 y Ft(0)2620 1843 y Fr(;)14 b(\020)6 b FA(\))24 b Fs(2)f Fr(\033)2880 1855 y Ft(d)2921 1776 y Fp(\000)2959 1843 y Fr(h)p FA(\()p Fr(R)3102 1855 y Ft(0)3140 1843 y Fr(;)14 b(\020)6 b FA(\))3251 1776 y Fp(\001)599 1943 y Fi(b)l(e)31 b(a)f(discr)l(ete)h(eigenvalue)g(of)g Fr(h)p FA(\()p Fr(R)1721 1955 y Ft(0)1758 1943 y Fr(;)14 b(\020)6 b FA(\))p Fi(,)31 b(and)g Fr(P)12 b FA(\()p Fr(R)2247 1955 y Ft(0)2284 1943 y Fr(;)i(\020)6 b FA(\))31 b Fi(the)f(asso)l(ciate)l(d)i(total)e(pr)l(o-)599 2043 y(je)l(ction.)37 b(We)21 b(denote)h(by)g Fr(e)1438 2055 y Fo(j)1472 2043 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fi(,)24 b Fr(j)k FA(=)23 b(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(d)21 b Fi(the)h(eigenvalues)g(of)g Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))599 2142 y Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))46 b Fi(for)g Fr(R)g Fi(in)g(a)f(neighb)l(ourho)l(o)l(d)j(of)e Fr(R)2109 2154 y Ft(0)2146 2142 y Fi(.)86 b(Then)46 b(the)f (eigenvalues)i Fr(\025)3141 2154 y Fo(j)3221 2142 y Fi(of)599 2242 y Fr(P)12 b FA(\()p Fr(R)759 2254 y Ft(0)797 2242 y Fr(;)i(\020)6 b FA(\)\()p Fr(@)f(h=@)g(R)q FA(\)\()p Fr(R)1319 2254 y Ft(0)1356 2242 y Fr(;)14 b(\020)6 b FA(\))p Fr(P)12 b FA(\()p Fr(R)1627 2254 y Ft(0)1665 2242 y Fr(;)i(\020)6 b FA(\))32 b Fi(ob)l(ey)h FA(\()p Fr(@)5 b(e)2115 2254 y Fo(j)2149 2242 y Fr(=@)g(R)q FA(\)\()p Fr(R)2431 2254 y Ft(0)2468 2242 y Fr(;)14 b(\020)6 b FA(\))27 b(=)f Fr(\025)2745 2254 y Fo(j)2781 2242 y Fi(,)32 b Fr(j)g FA(=)26 b(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(d)p Fi(.)599 2341 y(A)n(n)29 b(analo)l(gous)i(r)l(esult)e(holds)i(for)g (the)f Fr(\020)6 b Fi(-derivative.)682 2603 y(Pr)l(o)l(of.)105 b FA(Since)37 b(the)f(reasoning)e(is)i(the)g(same)f(for)h Fr(@)5 b(=@)g(\020)41 b FA(and)36 b Fr(@)5 b(=@)g(R)q FA(,)36 b(w)n(e)g(only)599 2703 y(treat)c(the)h(latter)e(case.)50 b(The)32 b(pro)r(of)f(pro)r(ceeds)h(via)f(the)i(standard)e(metho)r(d)h (of)g(re-)599 2802 y(ducing)j(the)f(problem)g(to)g (\014nite-dimensional)g(matrices)f(acting)h(on)g(a)f(\014xed)i(sub-)599 2902 y(space.)62 b(As)36 b(explained)f(in)h(Ref.)h([37)o(],)h(it)f(is)e (p)r(ossible)h(to)g(construct)f(unitary)h(and)599 3001 y(once-di\013eren)n(tiable)k(op)r(erators)e Fr(U)9 b FA(\()p Fr(R)q FA(\))41 b(with)g Fr(U)9 b FA(\()p Fr(R)2278 3013 y Ft(0)2315 3001 y FA(\))45 b(=)f Fq(I)2539 3013 y Fl(H)2635 3001 y FA(and)c(suc)n(h)g(that)h(on)599 3101 y(a)g(neigh)n(b)r(ourho)r(o)r(d)g(of)g Fr(R)1435 3113 y Ft(0)1514 3101 y FA(the)h(pro)5 b(jections)40 b(can)h(b)r(e)g (expressed)g(as)f Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))47 b(=)599 3201 y Fr(U)9 b FA(\()p Fr(R)q FA(\))p Fr(P)j FA(\()p Fr(R)953 3213 y Ft(0)991 3201 y Fr(;)i(\020)6 b FA(\))p Fr(U)j FA(\()p Fr(R)q FA(\))1296 3171 y Fl(\000)p Ft(1)1386 3201 y FA(.)34 b(Whence,)23 b Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(P)12 b FA(\()p Fr(R)q(;)i(\020)6 b FA(\))23 b(is)e(isosp)r(ectral)f(to)h(the)599 3300 y(Hermitian)e(matrix)f Fr(P)12 b FA(\()p Fr(R)1406 3312 y Ft(0)1443 3300 y Fr(;)i(\020)6 b FA(\))p Fr(U)j FA(\()p Fr(R)q FA(\))1748 3270 y Fl(\000)p Ft(1)1838 3300 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(U)j FA(\()p Fr(R)q FA(\))p Fr(P)j FA(\()p Fr(R)2447 3312 y Ft(0)2485 3300 y Fr(;)i(\020)6 b FA(\))19 b(acting)f(on)g Fr(P)12 b FA(\()p Fr(R)3121 3312 y Ft(0)3158 3300 y Fr(;)i(\020)6 b FA(\))p Fs(H)q FA(.)599 3400 y(This)33 b(matrix)f(is)g(di\013eren)n (tiable)g(with)h(resp)r(ect)f(to)g Fr(R)q FA(,)i(and)e(the)h(deriv)-5 b(ativ)n(es)31 b(of)h(its)599 3500 y(eigen)n(v)-5 b(alues)23 b(are)f(giv)n(en)h(b)n(y)g(the)h(eigen)n(v)-5 b(alues)22 b(of)i(the)g(deriv)-5 b(ativ)n(e)22 b(of)i(the)g(matrix)f(\(cf.,)599 3599 y(Kato)c([37]\).)34 b(The)20 b(form)n(ula)g(for)f Fr(@)5 b(e=@)g(R)20 b FA(no)n(w)f(follo)n(ws)g(from)h(the)g(observ)-5 b(ation)19 b(that)h(in)599 3632 y Fp(\000)637 3699 y Fr(@)f(P)12 b FA(\()p Fr(R)860 3711 y Ft(0)897 3699 y Fr(;)i(\020)6 b FA(\))p Fr(U)j FA(\()p Fr(R)q FA(\))1202 3669 y Fl(\000)p Ft(1)1292 3699 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(U)j FA(\()p Fr(R)q FA(\))p Fr(P)j FA(\()p Fr(R)1901 3711 y Ft(0)1939 3699 y Fr(;)i(\020)6 b FA(\))p Fr(=@)f(R)2205 3632 y Fp(\001)2243 3732 y Fh(\030)p Fo(R)2322 3740 y Fg(0)2381 3699 y FA(=)22 b Fr(P)12 b FA(\()p Fr(R)2628 3711 y Ft(0)2666 3699 y Fr(;)i(\020)6 b FA(\)\()p Fr(@)f(h=@)g(R)q FA(\)\()p Fr(R)3188 3711 y Ft(0)3225 3699 y Fr(;)14 b(\020)6 b FA(\))599 3819 y Fr(P)12 b FA(\()p Fr(R)759 3831 y Ft(0)797 3819 y Fr(;)i(\020)6 b FA(\))28 b(the)g(deriv)-5 b(ativ)n(es)27 b(of)g Fr(U)1653 3789 y Fl(\000)p Ft(1)1770 3819 y FA(and)g Fr(U)37 b FA(cancel)27 b(at)g Fr(R)2438 3831 y Ft(0)2475 3819 y FA(.)682 3919 y(Whereas)d(for)g Fr(\020)29 b(<)1296 3850 y Fs(p)p 1365 3850 42 4 v 69 x FA(3)p Fr(=)p FA(4)23 b(there)h(are)f(no)i(problems)e(with)i(the)g(op)r(erator)d Fr(P)12 b FA(\()p Fr(R)3140 3931 y Ft(0)3178 3919 y Fr(;)i(\020)6 b FA(\))599 4018 y(\()p Fr(@)f(h=@)g(R)q FA(\)\()p Fr(R)1010 4030 y Ft(0)1047 4018 y Fr(;)14 b(\020)6 b FA(\))p Fr(P)12 b FA(\()p Fr(R)1318 4030 y Ft(0)1356 4018 y Fr(;)i(\020)6 b FA(\))29 b(in)g(the)f(united)h(atom)f(situation)g Fr(R)2619 4030 y Ft(0)2681 4018 y FA(=)c(0,)k(if)h Fr(\020)i Fs(\025)3096 3950 y(p)p 3165 3950 V 68 x FA(3)o Fr(=)p FA(4)599 4118 y(and)26 b Fr(R)822 4130 y Ft(0)882 4118 y FA(=)d(0)i(the)h(domain)f Fs(D)1537 4051 y Fp(\000)1575 4118 y Fr(@)5 b(h=@)g(R)13 b FA(\(0)p Fr(;)h(\020)6 b FA(\))2024 4051 y Fp(\001)2088 4118 y FA(of)26 b(the)g(op)r(erator)e Fr(@)5 b(h=@)g(R)13 b FA(\(0)p Fr(;)h(\020)6 b FA(\))26 b(do)r(es)599 4226 y(not)h(necessarily)e(con)n(tain)g(Ran)1599 4159 y Fp(\000)1637 4226 y Fr(P)12 b FA(\(0)p Fr(;)i(\020)6 b FA(\))1887 4159 y Fp(\001)1952 4226 y FA(as)26 b(subset.)36 b(Nev)n(ertheless,)26 b(ev)n(en)g(for)g Fr( )g Fs(2)599 4334 y FA(Ran)748 4267 y Fp(\000)786 4334 y Fr(P)12 b FA(\(0)p Fr(;)i(\020)6 b FA(\))1036 4267 y Fp(\001)1099 4334 y FA(with)25 b Fr( )35 b(=)-51 b Fs(2)23 b(D)1509 4267 y Fp(\000)1548 4334 y Fr(@)5 b(h=@)g(R)13 b FA(\(0)p Fr(;)h(\020)6 b FA(\))1997 4267 y Fp(\001)2060 4334 y FA(the)25 b(expression)d Fs(h)p Fr( )s(;)29 b(@)5 b(h=@)g(R)13 b FA(\(0)p Fr(;)h(\020)6 b FA(\))14 b Fr( )s Fs(i)599 4433 y FA(can)28 b(b)r(e)h(de\014ned,)f (e.g.,)g(b)n(y)g(form)n(ulating)f(it)i(as)e(a)h(principal)g(v)-5 b(alue)28 b(in)n(tegral)e(around)599 4533 y(the)19 b(singularit)n(y)e (at)h Fn(x)23 b FA(=)g(0;)e(then)e(the)g(principal)e(v)-5 b(alue)19 b(in)n(tegral)e Fs(h)p Fr( )s(;)28 b(@)5 b(h=@)g(R)13 b FA(\(0)p Fr(;)h(\020)6 b FA(\))14 b Fr( )s Fs(i)p eop %%Page: 22 22 22 21 bop 599 75 a FA(22)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(can)32 b(b)r(e)h(sho)n(wn)e(to)h(v)-5 b(anish.)51 b(On)32 b(the)h(other)e(hand,)j(from)e(\(i\))h(of)f(corollary)d(3.1)j (w)n(e)599 419 y(kno)n(w)40 b(the)g(v)-5 b(anishing)40 b(of)1471 352 y Fp(\000)1509 419 y Fr(@)5 b(e=@)g(R)1752 352 y Fp(\001)1789 419 y FA(\(0)p Fr(;)14 b(\020)6 b FA(\).)75 b(Hence,)43 b(with)e(suc)n(h)f(a)g(de\014nition)g(of)599 540 y Fs(h)p Fr( )s(;)29 b(@)5 b(h=@)g(R)13 b FA(\(0)p Fr(;)h(\020)6 b FA(\))14 b Fr( )s Fs(i)p FA(,)23 b(theorem)d(3.3)g (also)g(holds)g(for)h(all)f(0)j Fr(<)g(\020)29 b(<)23 b FA(1)p Fr(=)p FA(2)c(at)i Fr(R)2989 552 y Ft(0)3049 540 y FA(=)i(0.)p 3264 556 25 75 v 1252 855 a Fu(4.)32 b(BONDING)g(AND)g(ST)-8 b(ABILITY)974 979 y(4.1.)94 b(The)32 b(relativistic)g(molecular)e(virial)h(theorem)682 1095 y FA(F)-7 b(or)20 b(molecular)e(Sc)n(hr\177)-42 b(odinger)18 b(op)r(erators)g(con)n(taining)g(m)n(ulticen)n(tre)i(Coulom)n(b)f(p)r (o-)599 1195 y(ten)n(tials,)28 b(the)g(virial)f(theorem)g(can)g(b)r(e)h (expressed)e(as)h(\(see,)h(e.g.,)f([42)o(]\))1292 1378 y Fr(@)5 b(e)1380 1390 y Fo(S)p 1292 1416 136 4 v 1304 1492 a Fr(@)g(R)1438 1435 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g Fs(\000)p Fr(R)1885 1400 y Fl(\000)p Ft(1)1973 1367 y Fp(\000)2011 1435 y Fr(e)2050 1447 y Fo(S)2098 1435 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))19 b(+)f Fs(h)p Fr(T)2488 1447 y Fo(S)2536 1435 y Fs(i)2568 1367 y Fp(\001)3119 1435 y FA(\(4.1\))599 1659 y(where)i Fr(e)871 1671 y Fo(S)919 1659 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))21 b(stands)f(for)f(an)h(eigen)n(v)-5 b(alue)19 b(of)i(the)f(Sc)n(hr\177)-42 b(odinger)19 b(Hamiltonian)h(and)599 1758 y Fs(h)p Fr(T)680 1770 y Fo(S)728 1758 y Fs(i)f FA(denotes)g(the)g(exp)r(ectation)f(v)-5 b(alue)19 b(of)f(the)i (kinetic)e(energy)g(op)r(erator)f Fr(T)2961 1770 y Fo(S)3032 1758 y FA(=)22 b Fn(p)3169 1728 y Ft(2)3206 1758 y Fr(=)p FA(2)599 1858 y(within)28 b(the)f(asso)r(ciated)e(eigenfunction.)36 b(T)-7 b(o)26 b(deriv)n(e)g(an)g(analogue)f(for)h(the)h(studied)599 1958 y(Dirac-Coulom)n(b)e(Hamiltonian,)h(let)g Fr(e)1832 1970 y Fo(j)1867 1958 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))27 b(b)r(e)f(a)g(nondegenerate)e(eigen)n(v)-5 b(alue)25 b(of)599 2057 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))38 b(with)f(eigenfunction)g Fr( )1663 2069 y Fo(j)1698 2057 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\);)43 b(in)37 b(case)f(of)h(degeneracy)-7 b(,)37 b(w)n(e)g(assume)f(a)599 2157 y(lab)r(eling)d(c)n(hosen)f(suc)n(h)h (that)g Fr(e)1604 2169 y Fo(j)1639 2157 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))34 b(is)f(di\013eren)n(tiable)g(in)g(a)f(neigh)n(b)r (ourho)r(o)r(d)g(of)599 2257 y(the)25 b(considered)d Fr(R)q FA(.)36 b(F)-7 b(urthermore,)23 b(let)h Fs(h)p Fr(\014)t Fs(i)1996 2269 y Fo(j)2055 2257 y FA(=)f Fs(h)p Fr( )2229 2269 y Fo(j)2264 2257 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(;)28 b(\014)t( )2627 2269 y Fo(j)2663 2257 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fs(i)p FA(.)36 b(Then)24 b(the)599 2356 y(virial)j(theorem)g(for)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))28 b(reads:)682 2519 y Fj(Theorem)j(4.1.)77 b Fi(F)-6 b(or)38 b FA(0)f Fr(<)g(\020)44 b(<)36 b FA(1)p Fr(=)p FA(2)h Fi(and)h Fr(R)g(>)e FA(0)p Fi(,)k(the)e Fr(R)q Fi(-derivative)h(of)f(the)599 2619 y(eigenvalue)31 b Fr(e)1039 2631 y Fo(j)1074 2619 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))30 b Fi(ob)l(eys)1299 2805 y Fr(@)5 b(e)1387 2817 y Fo(j)p 1299 2842 123 4 v 1304 2918 a Fr(@)g(R)1431 2862 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)f Fs(\000)p Fr(R)1879 2827 y Fl(\000)p Ft(1)1967 2794 y Fp(\000)2005 2862 y Fr(e)2044 2874 y Fo(j)2079 2862 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))19 b Fs(\000)f(h)p Fr(\014)t Fs(i)2503 2874 y Fo(j)2539 2794 y Fp(\001)2577 2862 y Fr(:)519 b FA(\(4.2\))599 3086 y Fi(A)n(lternatively,)31 b(we)f(c)l(an)g(e)l(qual)t(ly)h(expr)l (ess)f(Eq.)g(\(4.2\))h(in)f(inte)l(gr)l(ate)l(d)g(form)1473 3344 y Fr(e)1512 3356 y Fo(j)1547 3344 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g Fr(R)1929 3309 y Fl(\000)p Ft(1)2032 3231 y Fp(Z)2115 3251 y Fo(R)2078 3419 y Ft(0)2169 3344 y Fs(h)p Fr(\014)t Fs(i)2284 3356 y Fo(j)2334 3344 y Fr(ds)703 b FA(\(4.3\))706 3636 y(Since)25 b(the)f(pro)r(of)f(of)h (Eq.)g(\(4.2\))f(consists)g(of)h(an)g(easy)f(application)g(of)h(the)g (scaling)599 3736 y(prop)r(erties)19 b(discussed)h(in)g(section)f(2.2)g (and)g(the)i(second)e(form)n(ula)g(\(4.3\))g(follo)n(ws)g(from)599 3836 y(\(4.2\))32 b(b)n(y)g(straigh)n(tforw)n(ard)c(in)n(tegration,)k (w)n(e)f(will)h(skip)g(the)g(details)g(here.)49 b(In)32 b(the)599 3935 y(limit)g Fr(R)e Fs(#)f FA(0,)j(Eq.)e(\(4.3\))h(reduces) g(to)g(the)g(atomic)g(v)n(ersion)f(of)h(the)h(virial)e(theorem)599 4035 y(for)i(Dirac-Coulom)n(b)d(Hamiltonians,)k(viz.,)f Fr(e)2072 4047 y Fo(j)2107 4035 y FA(\(0)p Fr(;)14 b(\020)6 b FA(\))30 b(=)f Fs(h)p Fr(\014)t Fs(i)2531 4047 y Fo(j)2599 4035 y FA(\(cf.)50 b(Ref.)f([26)o(]\).)g(As)599 4134 y(for)28 b(atomic)g(systems,)g(also)f(for)g(molecules)h(w)n(e)f(can)h (in)n(v)n(ok)n(e)f(the)h(virial)f(theorem)h(as)599 4234 y(a)34 b(con)n(v)n(enien)n(t)e(to)r(ol)i(to)g(dispro)n(v)n(e)e(the)i (existence)f(of)h(eigen)n(v)-5 b(alues)32 b(em)n(b)r(edded)j(in)n(to) 599 4334 y(the)f(con)n(tin)n(uous)f(part)g(\()p Fs(\0001)p Fr(;)14 b Fs(\000)p FA(1])21 b Fs([)i FA([1)p Fr(;)14 b Fs(1)p FA(\))34 b(of)f(the)h(sp)r(ectrum.)55 b(The)34 b(subsequen)n(t)599 4433 y(corollary)24 b(is)i(an)f(immediate)i (consequence)d(of)i(Eq.)g(\(4.3\))f(and)h(of)g(the)g(b)r(ound)h Fs(\000)p FA(1)22 b Fs(\024)599 4533 y(h)p Fr( )s(;)29 b(\014)t( )s Fs(i)23 b(\024)g FA(1)k(that)h(holds)f(for)g(all)h(normed) f Fr( )f Fs(2)d(H)q FA(.)p eop %%Page: 23 23 23 22 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(23)682 320 y Fj(Cor)n(ollar)-5 b(y)33 b(4.1.)70 b Fi(If)30 b FA(0)24 b Fr(<)g(\020)30 b(<)24 b FA(1)p Fr(=)p FA(2)29 b Fi(and)i Fr(R)25 b Fs(\025)e FA(0)p Fi(,)31 b(the)f(eigenvalues)i Fr(e)2949 332 y Fo(j)2984 320 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))30 b Fi(of)599 419 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))31 b Fi(ar)l(e)f(b)l(ounde)l(d)g(by)1619 569 y Fs(\000)p FA(1)22 b Fs(\024)h Fr(e)1875 581 y Fo(j)1910 569 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b Fs(\024)g FA(1)849 b(\(4.4\))710 762 y(Similar)27 b(as)h(in)f(Sc)n(hr\177)-42 b(odinger)26 b(theory)h([47)o(,)h(48)o(,)g(49)o(],)g(a)f(lo)n(w)n(er)f (b)r(ound)i(on)g(the)g(size)599 862 y(of)e(the)g(molecule)g(ensues)f (as)h(another)e(o\013-spring)h(from)g(the)i(virial)e(theorem.)35 b(More)599 961 y(precisely)-7 b(,)28 b(w)n(e)g(can)f(estimate)h Fr(R)1627 973 y Fo(e)1691 961 y FA(from)g(b)r(elo)n(w,)g(where)f Fr(R)2450 973 y Fo(e)2514 961 y FA(marks)g(the)i(p)r(osition)e(of)599 1061 y(the)32 b(\(leftmost\))h(minim)n(um)f(for)e(a)h(giv)n(en)g(PEC,)f (i.e.,)j(the)f(equilibrium)f(separation)599 1161 y(for)d(static)f(n)n (uclei)h(within)g(the)g(giv)n(en)f(electronic)f(state.)682 1319 y Fj(Cor)n(ollar)-5 b(y)33 b(4.2.)82 b Fi(L)l(et)41 b FA(0)j Fr(<)f(\020)51 b(<)44 b(\020)1941 1331 y Ft(max)2110 1319 y Fi(with)e Fr(\020)2338 1331 y Ft(max)2506 1319 y Fi(fr)l(om)g(\(2.38\).)76 b(F)-6 b(or)41 b(al)t(l)599 1418 y(ele)l(ctr)l(onic)f(b)l(ound)f(states)g(the)g(nucle)l(ar)g(e)l (quilibrium)h(p)l(ositions)g Fr(R)2778 1430 y Fo(e)2853 1418 y Fi(ar)l(e)f(b)l(ounde)l(d)599 1518 y(b)l(elow)31 b(by)1357 1680 y Fr(R)1420 1692 y Fo(e)1479 1680 y Fs(\025)22 b Fr(\013)1619 1644 y Fl(\000)p Ft(1)1619 1704 y Fo(S)1709 1680 y Fr(\020)1751 1645 y Ft(2)1789 1613 y Fp(\000)1827 1680 y FA(1)c Fs(\000)1970 1605 y Fp(p)p 2053 1605 329 4 v 75 x FA(1)g Fs(\000)g FA(\(2)p Fr(\020)6 b FA(\))2344 1656 y Ft(2)2381 1613 y Fp(\001)2420 1630 y Fl(\000)p Ft(1)2509 1680 y Fr(:)587 b FA(\(4.5\))682 1985 y Fi(Pr)l(o)l(of.)92 b FA(T)-7 b(o)31 b(deriv)n(e)f(the)i(b)r(ound)g(\(4.5\),)g(w)n(e)f (observ)n(e)e(that)j(the)g(repulsiv)n(e)e(parts)599 2085 y(of)e(the)g(PECs)e(are)h(c)n(haracterized)e(b)n(y)1454 2259 y Fr(@)5 b(e)1542 2271 y Fo(j)p 1454 2296 123 4 v 1459 2372 a Fr(@)g(R)1586 2315 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))19 b Fs(\000)f Fr(\013)1948 2279 y Fl(\000)p Ft(1)1948 2339 y Fo(S)2037 2315 y Fr(\020)2079 2281 y Ft(2)2117 2315 y Fr(R)2181 2281 y Fl(\000)p Ft(2)2293 2315 y Fs(\024)23 b FA(0)p Fr(;)673 b FA(\(4.6\))599 2530 y(where)19 b(for)f(su\016cien)n(tly)g(small)h Fr(R)g FA(the)g(second)f(term)h(pro)n(vides)e(the)i(leading)f(con)n(tribu-)599 2629 y(tion.)35 b(Using)19 b(Eq.)f(\(4.2\))h(and,)i(again,)f Fs(h)p Fr(\014)t Fs(i)1888 2641 y Fo(j)1947 2629 y Fs(\024)j FA(1)c(to)g(estimate)g Fr(@)5 b(e)2601 2641 y Fo(j)2635 2629 y Fr(=@)g(R)20 b FA(together)e(with)599 2750 y(the)28 b(UAL)h(lo)n(w)n(er)d(energy)g(b)r(ound)i(\(2.41\))f(leads)g(to)g (\(4.5\).)p 2511 2766 25 75 v 682 2899 a(The)h(RHS)g(of)g(the)g (inequalit)n(y)f(\(4.5\))g(is)h(monotonically)e(deca)n(ying)g(in)i Fr(\020)34 b FA(from)1421 3129 y Fr(R)1484 3141 y Fo(e)1543 3129 y Fs(\025)23 b Fr(\013)1684 3094 y Fl(\000)p Ft(1)1684 3154 y Fo(S)1773 3062 y Fp(\000)1821 3073 y FA(1)p 1821 3110 42 4 v 1821 3186 a(2)1891 3129 y Fs(\000)1984 3073 y FA(1)p 1984 3110 V 1984 3186 a(4)2036 3129 y Fr(\020)2078 3095 y Ft(2)2134 3129 y FA(+)18 b Fs(O)r FA(\()p Fr(\020)2359 3095 y Ft(4)2397 3129 y FA(\))2429 3062 y Fp(\001)3119 3129 y FA(\(4.7\))599 3354 y(for)30 b(small)g Fr(\020)6 b FA(,)32 b(to)e(\(4)p Fr(\013)p FA(\))1309 3324 y Fl(\000)p Ft(1)1429 3354 y FA(for)g Fr(\020)k FA(=)27 b(1)p Fr(=)p FA(2)i(\(where)h(w)n(e)f(assume)h(that)h(the)f(UAL)h(lo)n(w)n(er)599 3454 y(b)r(ound)g(can)e(b)r(e)h(extended)g(to)g(all)g Fr(\020)j Fs(2)27 b FA(\(0)p Fr(;)14 b FA(1)p Fr(=)p FA(2\)\).)42 b(Comparison)28 b(of)i(\(4.7\))g(with)g(the)599 3553 y(analogous)25 b(b)r(ound)j(for)f(molecular)f(Sc)n(hr\177)-42 b(odinger)26 b(op)r(erators)f([48)o(])j(rev)n(eals)d(that)j(the)599 3653 y(latter)23 b(is)f(sharp)r(er)f(b)n(y)h(a)g(factor)g(of)g(t)n(w)n (o;)i(this)f(is)f(essen)n(tially)f(a)h(consequence)g(of)g(more)599 3753 y(e\016cien)n(t)28 b(con)n(trol)e(of)i(the)g(term)g Fs(h)p Fr(T)1703 3765 y Fo(S)1750 3753 y Fs(i)g FA(within)h(Sc)n (hr\177)-42 b(odinger)25 b(theory)-7 b(.)1417 3919 y Fu(4.2.)94 b(Stabilit)m(y)32 b(prop)s(erties)682 4035 y FA(Within)i(the)f(Born-Opp)r(enheimer)e(appro)n(ximation,)g(a)h (molecular)g(system)g(can)599 4134 y(b)r(e)h(regarded)e(as)h(stable)g (against)g(disso)r(ciation)f(if)i(its)g(ground)e(state)i(PEC)e(enjo)n (ys)599 4234 y(a)36 b(global)f(minim)n(um)h(at)g(some)f(\014nite)i(0)f Fr(<)h(R)2099 4246 y Fo(e)2171 4234 y Fr(<)g Fs(1)f FA(\(see,)i(e.g.,)f (Refs.)62 b([47)o(,)36 b(50)o(,)599 4334 y(51)o(,)30 b(52)o(,)f(53)o(,)h(54)o(])f(for)g(a)f(discussion)h(of)g(Sc)n(hr\177) -42 b(odinger)27 b(systems\).)42 b(This)29 b(de\014nition)g(of)599 4433 y(stabilit)n(y)21 b(is)g(motiv)-5 b(ated)21 b(b)n(y)f(the)h (observ)-5 b(ation)20 b(that)h(a)f(global)g(minim)n(um)h(of)g(the)g (PEC)599 4533 y(ma)n(y)g(accommo)r(date)g(b)r(ound)h(vibrational)e(lev) n(els,)i(i.e.,)h(assuming)e(su\016cien)n(tly)h(large)p eop %%Page: 24 24 24 23 bop 599 75 a FA(24)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(\(reduced\))d(n)n(uclear)e(mass)g Fr(M)28 b FA(and)19 b(nonrelativistic)g(kinematics)f(for)h(the)h(vibrational)599 419 y(motion)27 b(of)f(the)h(n)n(uclei,)g(the)g(op)r(erator)e Fr(H)1920 431 y Ft(vib)2038 419 y FA(=)d Fs(\000)p FA(\(2)p Fr(M)9 b FA(\))2386 389 y Fl(\000)p Ft(1)2475 419 y Fr(d)2518 389 y Ft(2)2555 419 y Fr(=dR)2704 389 y Ft(2)2758 419 y FA(+)16 b Fr(E)5 b FA(\()p Fr(R)q FA(\))27 b(on)f(the)599 519 y(half-line)36 b Fq(R)982 489 y Ft(+)1067 519 y Fs([)24 b(f)p FA(0)p Fs(g)34 b FA(with)i(Diric)n(hlet)f(b)r(oundary)g (conditions)g(at)h Fr(R)g FA(=)g(0)f(has)g(\(at)599 619 y(least\))f(one)f(b)r(ound)h(state.)55 b(T)-7 b(o)33 b(compare)f(the)i(molecular)f(ground)f(state)i(energies)599 718 y Fr(E)5 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))43 b(=)f Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))26 b(+)f Fr(\013)1437 683 y Fl(\000)p Ft(1)1437 743 y Fo(S)1527 718 y Fr(\020)1569 688 y Ft(2)1607 718 y Fr(R)1671 688 y Fl(\000)p Ft(1)1798 718 y FA(with)40 b(the)f(the)g(corresp)r(onding)e (SAL)j(energy)599 818 y(lim)715 830 y Fo(R)p Fl(!1)915 818 y Fr(E)5 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)f(lim)1415 830 y Fo(R)p Fl(!1)1616 818 y Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g Fr(e)2012 788 y Ft(at)2075 818 y FA(\()p Fr(\020)6 b FA(\),)26 b(w)n(e)d(ma)n(y)h (assume)f(the)i(p)r(ositivit)n(y)599 917 y(of)e Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(and)f(v)n(erify)f(the)h (existence)g(of)g(a)f(global)g(minim)n(um)i(b)n(y)f(lo)r(oking)f(for)g (some)599 1017 y(0)h Fr(<)g(R)g(<)g Fs(1)28 b FA(with)1336 1150 y Fp(\000)1374 1217 y Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))19 b(+)f Fr(\013)1775 1181 y Fl(\000)p Ft(1)1775 1241 y Fo(S)1865 1217 y Fr(\020)1907 1183 y Ft(2)1944 1217 y Fr(R)2008 1183 y Fl(\000)p Ft(1)2097 1150 y Fp(\001)2135 1167 y Ft(2)2196 1217 y Fr(<)k(e)2322 1183 y Ft(at)2385 1217 y FA(\()p Fr(\020)6 b FA(\))2491 1183 y Ft(2)2529 1217 y Fr(:)567 b FA(\(4.8\))599 1417 y(As)30 b(a)e(consequence)g(of)h(the)g(sp)r(ectral)g(theorem,)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))2421 1387 y Ft(2)2484 1417 y Fs(\025)25 b Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))2820 1387 y Ft(2)2857 1417 y Fq(I)2896 1429 y Fl(H)2951 1417 y FA(;)29 b(whence,)599 1517 y(b)n(y)f(standard)e (Ra)n(yleigh-Ritz)h(v)-5 b(ariational)26 b(argumen)n(ts,)g(the)i(upp)r (er)g(b)r(ound)1479 1717 y Fs(h)p Fr( )s(;)g(h)p FA(\()p Fr(R)q(;)g(\020)6 b FA(\))1888 1682 y Ft(2)1926 1717 y Fr( )s Fs(i)23 b(\025)g Fr(e)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))2372 1682 y Ft(2)3119 1717 y FA(\(4.9\))599 1917 y(for)28 b(all)f Fr( )f Fs(2)d(D)r FA(\()p Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))1353 1886 y Ft(2)1392 1917 y FA(\))28 b(can)f(b)r(e)h(inferred.)36 b(Therefore,)27 b(setting)1032 2133 y Fr(D)r FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g Fs(h)p Fr( )s(;)1561 2066 y Fp(\000)1599 2133 y Fr(h)p FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))19 b(+)f Fr(\013)2023 2098 y Fl(\000)p Ft(1)2023 2158 y Fo(S)2112 2133 y Fr(\020)2154 2099 y Ft(2)2192 2133 y Fr(R)2256 2099 y Fl(\000)p Ft(1)2345 2066 y Fp(\001)2383 2083 y Ft(2)2420 2133 y Fr( )s Fs(i)h(\000)f Fr(e)2650 2099 y Ft(at)2713 2133 y FA(\()p Fr(\020)6 b FA(\))2819 2099 y Ft(2)3077 2133 y FA(\(4.10\))599 2333 y(the)30 b(inequalit)n(y)e Fr(D)r FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))26 b Fr(<)f FA(0)k(for)f(some)h(0)c Fr(<)g(R)h(<)f Fs(1)k FA(will)g(guaran)n(tee)e(the)i(o)r(ccur-)599 2433 y(rence)20 b(of)g(a)g(global)f(minim)n(um)h(for)g(the)g(ground)f(state) h(PEC.)f(With)i(the)g(notation)e Fr(V)3233 2403 y Fl(\006)3214 2454 y Ft(ne)599 2533 y FA(from)i(section)f(3.1)g(w)n(e)g(can)h(write)f Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)e Fr(h)2066 2503 y Ft(at)2129 2533 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))f(+)g Fr(V)2561 2503 y Ft(+)2543 2553 y(ne)2616 2533 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\).)34 b(Since)21 b(the)599 2632 y(application)34 b(of)h Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))1392 2602 y Ft(2)1464 2632 y FA(on)n(to)34 b(the)g(atomic)g(ground)g(state)g Fr( )2642 2602 y Ft(at)2705 2632 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))35 b(is)f(w)n(ell-)599 2732 y(de\014ned)i(for)e(the)h(considered)f(0)h Fr(<)g(\020)42 b(<)34 b FA(1)p Fr(=)p FA(2,)i(and)f(using)f Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr( )2835 2702 y Ft(at)2899 2732 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))35 b(=)599 2832 y Fr(e)638 2801 y Ft(at)701 2832 y FA(\()p Fr(\020)6 b FA(\))p Fr( )864 2801 y Ft(at)929 2832 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))j(+)g Fr(V)1368 2801 y Ft(+)1351 2852 y(ne)1423 2832 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fr( )1771 2801 y Ft(at)1834 2832 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\),)23 b(b)n(y)g(c)n(ho)r(osing)e Fr( )26 b FA(=)c Fr( )2838 2801 y Ft(at)2902 2832 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))22 b(as)599 2931 y(v)-5 b(ariational)25 b(trial)i(function)g(w)n(e)f(arriv)n(e)f(at)h(the)h (follo)n(wing)f(expression)f(for)h Fr(D)r FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\):)692 3134 y Fr(D)r FA(\()p Fr(R)q(;)28 b(\020)6 b FA(\))23 b(=)g(2)1137 3066 y Fp(\000)1175 3134 y Fs(h)p Fr( )1264 3099 y Ft(at)1327 3134 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))14 b Fr(V)1698 3099 y Ft(+)1680 3154 y(ne)1754 3134 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)g(\020)6 b FA(\))p Fr( )2102 3099 y Ft(at)2164 3134 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fs(i)19 b FA(+)f Fr(\013)2642 3098 y Fl(\000)p Ft(1)2642 3158 y Fo(S)2731 3134 y Fr(\020)2773 3099 y Ft(2)2811 3134 y Fr(R)2875 3099 y Fl(\000)p Ft(1)2964 3066 y Fp(\001)3002 3134 y Fr(e)3041 3099 y Ft(at)3104 3134 y FA(\()p Fr(\020)6 b FA(\))1110 3270 y(+)p Fs(h)p Fr( )1264 3235 y Ft(at)1327 3270 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fr(;)27 b(V)1735 3235 y Ft(+)1716 3290 y(ne)1790 3270 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))2081 3235 y Ft(2)2118 3270 y Fr( )2175 3235 y Ft(at)2239 3270 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fs(i)18 b FA(+)g(2)p Fr(\013)2758 3234 y Fl(\000)p Ft(1)2758 3294 y Fo(S)2847 3270 y Fr(\020)2889 3235 y Ft(2)2927 3270 y Fr(R)2991 3235 y Fl(\000)p Ft(1)1193 3406 y Fs(\002)g(h)p Fr( )1365 3371 y Ft(at)1429 3406 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)c(\020)6 b FA(\))p Fr(;)27 b(V)1837 3371 y Ft(+)1818 3426 y(ne)1892 3406 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fr( )2240 3371 y Ft(at)2303 3406 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fs(i)18 b FA(+)g Fr(\013)2780 3370 y Fl(\000)p Ft(2)2780 3430 y Fo(S)2870 3406 y Fr(\020)2912 3371 y Ft(4)2949 3406 y Fr(R)3013 3371 y Fl(\000)p Ft(2)3077 3506 y FA(\(4.11\))599 3606 y(T)-7 b(o)28 b(decide)g(on)g(the)h(sign)f(of)g Fr(D)r FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\),)29 b(w)n(e)f(need)g (information)f(ab)r(out)h(the)h(in)n(tegrals)599 3705 y(o)r(ccurring)e(in)g(Eq.)g(\(4.11\).)682 3882 y Fj(Lemma)j(4.1.)76 b Fi(Consider)37 b FA(0)c Fr(<)h(\020)40 b(<)33 b FA(1)p Fr(=)p FA(2)p Fi(.)55 b(Then,)39 b(with)d Fr(\015)i FA(=)2719 3809 y Fp(p)p 2802 3809 223 4 v 73 x FA(1)18 b Fs(\000)g Fr(\020)2987 3858 y Ft(2)3025 3882 y Fi(,)37 b(for)g Fr(R)599 3981 y Fi(lar)l(ge)643 4181 y Fs(h)p Fr( )732 4147 y Ft(at)795 4181 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fr(;)28 b(V)1204 4147 y Ft(+)1185 4202 y(ne)1259 4181 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fr( )1607 4147 y Ft(at)1670 4181 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fs(i)22 b FA(=)h Fs(\000)p Fr(\020)6 b(R)2274 4147 y Fl(\000)p Ft(1)2381 4181 y FA(+)18 b Fs(O)r FA(\()p Fr(R)2628 4147 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2789 4181 y Fr(e)2828 4147 y Fl(\000)p Ft(2)p Fo(\020)t(R)3002 4181 y FA(\))43 b(\(4.12\))599 4381 y Fi(and)1053 4533 y Fs(h)p Fr( )1142 4499 y Ft(at)1205 4533 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fr(;)28 b(V)1613 4499 y Ft(+)1595 4554 y(ne)1669 4533 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))1960 4499 y Ft(2)1996 4533 y Fr( )2053 4499 y Ft(at)2117 4533 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))p Fs(i)23 b FA(=)f Fs(O)r FA(\()p Fr(R)2714 4499 y Fl(\000)p Ft(2)2804 4533 y FA(\))241 b(\(4.13\))p eop %%Page: 25 25 25 24 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(25)682 432 y Fi(Pr)l(o)l(of.)87 b FA(The)29 b(con)n(v)n(en)n(tional)f(quan)n(tum)i(n)n(um)n(b)r(ers)f(c)n (haracterizing)e(the)j(\(doubly)599 531 y(degenerate\))d(atomic)g (ground)g(state)g(are)g Fr(n)c FA(=)g(1)p Fr(;)k(j)h FA(=)23 b(1)p Fr(=)p FA(2)p Fr(;)j(m)2594 543 y Fo(j)2652 531 y FA(=)d Fs(\006)p FA(1)p Fr(=)p FA(2;)j(c)n(ho)r(osing)599 631 y(the)c(plus)f(sign)g(for)f(the)i(latter,)g(the)f(explicit)h(form)f (for)f Fr( )2361 601 y Ft(at)2424 631 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))23 b(=)g Fr(\034)2862 646 y Fo(R=)p Ft(2)2984 631 y Fr( )3041 601 y Ft(at)3104 631 y FA(\(0)p Fr(;)14 b(\020)6 b FA(\))599 731 y(is)28 b(determined)g(b)n(y)727 1069 y Fr( )784 1035 y Ft(at)848 1069 y FA(\(0)p Fr(;)f(\020)6 b FA(\)\()p Fn(r)t FA(\))23 b(=)1269 920 y Fp(s)p 1352 920 419 4 v 1394 1013 a Fr(\020)1436 989 y Ft(3)1473 1013 y FA(\(1)c(+)f Fr(\015)5 b FA(\))p 1362 1050 399 4 v 1362 1126 a Fr(\031)s FA(\000\(1)18 b(+)g(2)p Fr(\015)5 b FA(\))1770 1069 y(\(2)p Fr(\020)h(r)r FA(\))1957 1035 y Fo(\015)t Fl(\000)p Ft(1)2086 1069 y Fr(e)2125 1035 y Fl(\000)p Fo(\020)t(r)2261 853 y Fp(0)2261 999 y(B)2261 1049 y(B)2261 1102 y(@)2691 919 y FA(1)2691 1019 y(0)2384 1118 y Fr(i\020)2455 1088 y Fl(\000)p Ft(1)2545 1118 y FA(\(1)18 b Fs(\000)g Fr(\015)5 b FA(\))14 b(cos)o(\()p Fr(#)p FA(\))2334 1218 y Fr(i\020)2405 1188 y Fl(\000)p Ft(1)2494 1218 y FA(\(1)19 b Fs(\000)f Fr(\015)5 b FA(\))14 b(sin)o(\()p Fr(#)p FA(\))p Fr(e)3017 1188 y Fo(i')3089 853 y Fp(1)3089 999 y(C)3089 1049 y(C)3089 1102 y(A)3077 1319 y FA(\(4.14\))599 1418 y(with)29 b(resp)r(ect)f(to)g(usual)g (radial)f(co)r(ordinates)f Fr(r)n(;)14 b(#;)g(')29 b FA(\(see,)f(e.g.,)g([55)o(]\).)39 b(Emplo)n(ying)599 1518 y(the)i(abbreviations)d Fr(r)1319 1530 y Fl(\006)1418 1518 y FA(=)43 b Fs(j)p Fn(r)29 b Fs(\006)1723 1485 y Ft(1)p 1723 1499 34 4 v 1723 1546 a(2)1767 1518 y Fn(R)1839 1530 y Fo(s)1875 1518 y Fs(j)p FA(,)43 b(the)d(exp)r(ectation)f(v)-5 b(alue)40 b(in)g(Eq.)f(\(4.12\))599 1617 y(b)r(ecomes)887 1844 y Fs(\000)p Fr(\020)1008 1731 y Fp(Z)1105 1844 y Fr(d)1148 1810 y Ft(3)1186 1844 y Fr(r)16 b(r)1278 1808 y Fl(\000)p Ft(1)1276 1865 y(+)1368 1844 y Fs(j)p Fr( )1448 1810 y Ft(at)1511 1844 y FA(\(0)p Fr(;)e(\020)6 b FA(\)\()p Fn(r)t FA(\))p Fs(j)1831 1810 y Ft(2)1891 1844 y FA(=)1989 1788 y Fs(\000)p Fr(\020)24 b Fs(\002)18 b FA(\(2)p Fr(\020)6 b FA(\))2345 1758 y Ft(2)p Fo(\015)t Ft(+1)p 1989 1825 517 4 v 2027 1901 a FA(4)p Fr(\031)s FA(\000\(1)18 b(+)g(2)p Fr(\015)5 b FA(\))2516 1844 y Fr(I)i FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(:)288 b FA(\(4.15\))599 2082 y(T)-7 b(o)28 b(handle)f(the)h(in)n(tegral)1351 2314 y Fr(I)7 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)1713 2201 y Fp(Z)1809 2314 y Fr(d)1852 2280 y Ft(3)1890 2314 y Fr(r)16 b(r)1982 2278 y Fl(\000)p Ft(1)1980 2335 y(+)2072 2314 y Fr(r)2111 2274 y Ft(2)p Fo(\015)t Fl(\000)p Ft(2)2109 2335 y Fl(\000)2273 2314 y Fr(e)2312 2280 y Fl(\000)p Ft(2)p Fo(\020)t(r)2462 2288 y Fe(\000)2515 2314 y Fr(;)539 b FA(\(4.16\))599 2550 y(w)n(e)39 b(in)n(tro)r(duce)g(translated)f (prolate)g(spheroidal)g(co)r(ordinates)f Fr(\030)47 b Fs(2)42 b FA([0)p Fr(;)14 b Fs(1)p FA(\),)56 b Fr(\021)45 b Fs(2)599 2650 y FA([)p Fs(\000)p FA(2)p Fr(;)14 b FA(0\),)43 b Fr(')25 b Fs(2)h FA([0)p Fr(;)14 b FA(2)p Fr(\031)s FA(\),)29 b(suc)n(h)g(that)g Fr(r)1751 2662 y Ft(+)1832 2650 y FA(=)c Fr(R)q FA(\()p Fr(\030)f FA(+)19 b Fr(\021)k FA(+)18 b(2\))p Fr(=)p FA(2)28 b(and)h Fr(r)2695 2662 y Fl(\000)2777 2650 y FA(=)c Fr(R)q FA(\()p Fr(\030)f Fs(\000)19 b Fr(\021)s FA(\))p Fr(=)p FA(2.)599 2750 y(Then)28 b Fr(I)35 b FA(is)27 b(cast)h(in)n(to)830 2982 y Fr(I)7 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)e(2)1233 2948 y Fl(\000)p Ft(2)p Fo(\015)t Ft(+1)1444 2982 y Fr(\031)s(R)1558 2948 y Ft(2)p Fo(\015)1648 2869 y Fp(Z)1731 2890 y Fl(1)1694 3058 y Ft(0)1815 2982 y Fr(d\030)1926 2869 y Fp(Z)2009 2890 y Ft(0)1972 3058 y Fl(\000)p Ft(2)2061 2982 y FA(\()p Fr(\030)h Fs(\000)18 b Fr(\021)s FA(\))2311 2948 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2472 2982 y Fr(e)2511 2948 y Fl(\000)p Fo(\020)t(R)p Ft(\()p Fo(\030)r Fl(\000)p Fo(\021)r Ft(\))2824 2982 y Fr(:)230 b FA(\(4.17\))599 3225 y(Although)29 b(less)f(ob)n(vious)f(at)h(\014rst)g(sigh)n(t,)g(for)g(the)g(analysis)f (of)h(the)h(remaining)f(t)n(w)n(o-)599 3325 y(dimensional)23 b(in)n(tegration)f(it)h(will)g(b)r(e)h(of)f(adv)-5 b(an)n(tage)21 b(to)i(set)g Fr(\030)28 b FA(=)22 b Fr(\045)14 b FA(cos)f Fr(t)p FA(,)38 b Fr(\021)26 b FA(=)d Fr(\045)14 b FA(sin)f Fr(t)599 3425 y FA(with)29 b Fr(\045)23 b FA(=)942 3352 y Fp(p)p 1025 3352 261 4 v 73 x Fr(\030)1065 3401 y Ft(2)1121 3425 y FA(+)18 b Fr(\021)1248 3401 y Ft(2)1285 3425 y FA(,)28 b(and)g(to)f(transform)869 3681 y Fr(I)7 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g(2)1272 3646 y Fl(\000)p Ft(2)p Fo(\015)t Ft(+1)1483 3681 y Fr(\031)s(R)1597 3646 y Ft(2)p Fo(\015)1687 3568 y Fp(Z)1770 3588 y Ft(0)1733 3756 y Fl(\000)p Fo(\031)r(=)p Ft(2)1911 3681 y Fr(dt)14 b FA(\(cos)f Fr(t)19 b Fs(\000)f FA(sin)13 b Fr(t)p FA(\))2464 3646 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2625 3681 y Fr(F)f FA(\()p Fr(t)p FA(\);)270 b(\(4.18\))599 3931 y(here,)1419 4142 y Fr(F)12 b FA(\()p Fr(t)p FA(\))23 b(=)1689 4029 y Fp(Z)1772 4049 y Fo(\045)1806 4058 y Ff(b)1837 4049 y Ft(\()p Fo(t)p Ft(\))1735 4217 y(0)1932 4142 y Fr(d\045)14 b(\045)2075 4107 y Ft(2)p Fo(\015)2150 4142 y Fr(e)2189 4107 y Fl(\000)p Fo(c)p Ft(\()p Fo(t)p Ft(\))p Fo(\020)t(R\045)3077 4142 y FA(\(4.19\))599 4334 y(with)23 b Fr(c)p FA(\()p Fr(t)p FA(\))h(=)e(cos)13 b Fr(t)7 b Fs(\000)g FA(sin)14 b Fr(t)p FA(,)24 b(and)d(with)i Fr(\032)1833 4346 y Fo(b)1866 4334 y FA(\()p Fr(t)p FA(\))h(=)e Fs(\000)p FA(2)p Fr(=)14 b FA(sin)f Fr(t)22 b FA(marking)f(the)h Fr(t)p FA(-dep)r(enden)n(t)599 4433 y(upp)r(er)k(in)n(tegration)f(limit,)i(viz.,)f(the)h(line)f Fr(\021)g FA(=)d Fs(\000)p FA(2.)35 b(After)26 b(these)g (transformations,)599 4533 y(no)n(w)f(it)g(is)f(straigh)n(tforw)n(ard)e (to)j(apply)f(metho)r(ds)h(from)f(Olv)n(er)g([56)o(])h(\(c)n(hapter)f (3.7,)g(in)p eop %%Page: 26 26 26 25 bop 599 75 a FA(26)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(particular\))h(to)g(iden)n(tify)g(the)h(asymptotic)e(b)r(eha)n (viour)g(of)h Fr(F)37 b FA(and)24 b Fr(I)7 b FA(.)35 b(Since)25 b(1)d Fs(\024)h Fr(c)p FA(\()p Fr(t)p FA(\))599 419 y(and)28 b Fr(\032)804 431 y Fo(b)837 419 y FA(\()p Fr(t)p FA(\))c Fs(\025)e FA(2)28 b(for)f Fr(t)c Fs(2)g FA([)p Fs(\000)p Fr(\031)s(=)p FA(2)p Fr(;)14 b FA(0],)26 b(the)i(remainder)f(in)843 656 y Fr(F)12 b FA(\()p Fr(t)p FA(\))24 b(=)1113 589 y Fp(\000)1151 656 y Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\020)6 b(R)1387 589 y Fp(\001)1426 606 y Fl(\000)p Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1639 656 y FA(\000\(2)p Fr(\015)22 b FA(+)c(1\))h Fs(\000)2089 543 y Fp(Z)2172 563 y Fl(1)2135 732 y Fo(\045)2169 741 y Ff(b)2200 732 y Ft(\()p Fo(t)p Ft(\))2295 656 y Fr(d\045)14 b(\045)2438 622 y Ft(2)p Fo(\015)2513 656 y Fr(e)2552 622 y Fl(\000)p Fo(c)p Ft(\()p Fo(t)p Ft(\))p Fo(\020)t(R\045)3077 656 y FA(\(4.20\))599 902 y(is)28 b(estimated)g(b)n(y)661 1021 y Fp(Z)744 1042 y Fl(1)708 1210 y Fo(\045)742 1219 y Ff(b)772 1210 y Ft(\()p Fo(t)p Ft(\))867 1134 y Fr(d\045)14 b(\045)1010 1100 y Ft(2)p Fo(\015)1086 1134 y Fr(e)1125 1100 y Fl(\000)p Fo(c)p Ft(\()p Fo(t)p Ft(\))p Fo(\020)t(R\045)1429 1134 y Fs(\024)1507 1021 y Fp(Z)1591 1042 y Fl(1)1554 1210 y Fo(\045)1588 1219 y Ff(b)1618 1210 y Ft(\()p Fo(t)p Ft(\))1713 1134 y Fr(d\045)g(\045)1856 1100 y Ft(2)p Fo(\015)1932 1134 y Fr(e)1971 1100 y Fl(\000)p Fo(\020)t(R\045)2169 1134 y FA(=)1517 1329 y(=)22 b(\()p Fr(\020)6 b(R)q FA(\))1774 1295 y Fl(\000)p Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1988 1329 y FA(\000\(2)p Fr(\015)22 b FA(+)c(1)p Fr(;)28 b(\020)6 b(\045)2440 1341 y Fo(b)2473 1329 y FA(\()p Fr(t)p FA(\))p Fr(R)q FA(\))1517 1470 y(=)22 b Fr(\045)1647 1482 y Fo(b)1681 1470 y FA(\()p Fr(t)p FA(\))1775 1436 y Ft(2)p Fo(\015)1851 1470 y FA(\()p Fr(\020)6 b(R)q FA(\))2021 1436 y Fl(\000)p Ft(1)2111 1470 y Fr(e)2150 1436 y Fl(\000)p Ft(2)p Fo(\020)t(\045)2303 1445 y Ff(b)2333 1436 y Ft(\()p Fo(t)p Ft(\))p Fo(R)2465 1403 y Fp(\000)2503 1470 y FA(1)18 b(+)g Fs(O)r FA(\(1)p Fr(=)p FA(\()p Fr(\020)6 b(\045)2947 1482 y Fo(b)2980 1470 y FA(\()p Fr(t)p FA(\))p Fr(R)q FA(\)\))3202 1403 y Fp(\001)1517 1608 y FA(=)22 b Fs(O)r FA(\()p Fr(R)1768 1574 y Fl(\000)p Ft(1)1858 1608 y Fr(e)1897 1574 y Fl(\000)p Ft(2)p Fo(\020)t(R)2070 1608 y FA(\))3077 1708 y(\(4.21\))599 1808 y(uniformly)28 b(in)g Fr(t)23 b Fs(2)g FA([)p Fs(\000)p Fr(\031)s(=)p FA(2)p Fr(;)14 b FA(0].)36 b(Hence)682 2057 y Fr(I)7 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)e(2)1085 2023 y Fl(\000)p Ft(2)p Fo(\015)t Ft(+1)1297 2057 y Fr(\031)1347 1990 y Fp(\000)1385 2057 y Fr(\020)1427 2023 y Fl(\000)p Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1640 2057 y Fr(R)1704 2023 y Fl(\000)p Ft(1)1793 2057 y FA(\000\(2)p Fr(\015)h FA(+)18 b(1\))2156 1944 y Fp(Z)2238 1965 y Ft(0)2201 2133 y Fl(\000)p Fo(\031)r(=)p Ft(2)2365 2057 y FA(\(cos)c Fr(t)k Fs(\000)g FA(sin)c Fr(t)p FA(\))2832 2023 y Fl(\000)p Ft(2)2935 2057 y Fr(dt)1532 2255 y FA(+)k Fs(O)r FA(\()p Fr(R)1779 2221 y Fl(\000)p Ft(1+2)p Fo(\015)1991 2255 y Fr(e)2030 2221 y Fl(\000)p Ft(2)p Fo(\020)t(R)2203 2255 y FA(\))2235 2188 y Fp(\001)956 2393 y FA(=)k(4)p Fr(\031)s FA(\(2)p Fr(\020)6 b FA(\))1283 2358 y Fl(\000)p Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1496 2393 y FA(\000\(2)p Fr(\015)23 b FA(+)18 b(1\))p Fr(R)1909 2358 y Fl(\000)p Ft(1)2016 2393 y FA(+)g Fs(O)r FA(\()p Fr(R)2263 2358 y Fl(\000)p Ft(1+2)p Fo(\015)2475 2393 y Fr(e)2514 2358 y Fl(\000)p Ft(2)p Fo(\020)t(R)2687 2393 y FA(\))p Fr(;)3077 2196 y FA(\(4.22\))599 2583 y(whic)n(h)28 b(yields)f(Eq.)g(\(4.12\).) 682 2683 y(Apart)19 b(from)f(unimp)r(ortan)n(t)h(factors,)g(the)g(exp)r (ectation)g(v)-5 b(alue)18 b(in)h(\(4.13\))f(in)n(v)n(olving)599 2782 y Fr(V)666 2752 y Ft(+)647 2803 y(ne)721 2782 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))928 2752 y Ft(2)987 2782 y FA(is)21 b(determined)h(b)n(y)f(an)f(in)n(tegral)g(that)i(in)f (prolate)f(spheroidals)g(tak)n(es)g(the)599 2882 y(form)614 3110 y Fr(I)650 3125 y Ft([)p Fl(\000)p Ft(1)p Fo(;)p Ft(0])830 3110 y FA(\()p Fr(R)q(;\020)6 b FA(\))19 b(+)f Fr(I)1161 3125 y Ft([0)p Fo(;)p Ft(1])1289 3110 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))24 b(=)e Fr(R)1671 3076 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1845 2997 y Fp(Z)1928 3018 y Fl(1)1892 3186 y Ft(0)2013 3110 y Fr(d\030)2110 3043 y Fp(\000)2148 2997 y(Z)2231 3018 y Ft(0)2194 3186 y Fl(\000)p Ft(1)2297 3110 y Fr(d\021)17 b FA(\()p Fr(\030)23 b FA(+)18 b Fr(\021)s FA(\))2648 3076 y Fl(\000)p Ft(1)967 3356 y Fs(\002)g FA(\()p Fr(\030)23 b Fs(\000)18 b Fr(\021)s FA(\))1300 3322 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1461 3356 y Fr(e)1500 3322 y Fl(\000)p Fo(\020)t(R)p Ft(\()p Fo(\030)r Fl(\000)p Fo(\021)r Ft(\))1831 3356 y FA(+)1914 3243 y Fp(Z)1997 3264 y Ft(1)1960 3432 y(0)2048 3356 y Fr(d\021)f FA(\()p Fr(\030)23 b FA(+)18 b Fr(\021)s FA(\))2399 3322 y Fl(\000)p Ft(1)2489 3356 y FA(\()p Fr(\030)23 b Fs(\000)18 b Fr(\021)s FA(\))2739 3322 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2900 3356 y Fr(e)2939 3322 y Fl(\000)p Fo(\020)t(R)p Ft(\()p Fo(\030)r Fl(\000)p Fo(\021)r Ft(\))3251 3289 y Fp(\001)3077 3502 y FA(\(4.23\))599 3602 y(where)33 b(w)n(e)g(ha)n(v)n(e)f(split)h(the)h Fr(\021)s FA(-in)n(tegration)e(in)n(to)h(the)g(in)n(terv)-5 b(als)32 b([)p Fs(\000)p FA(1)p Fr(;)14 b FA(0])32 b(and)h([0)p Fr(;)14 b FA(1].)599 3711 y(F)-7 b(or)22 b Fr(I)779 3726 y Ft([0)p Fo(;)p Ft(1])907 3711 y FA(,)i(after)d(translating)g Fr(\030)28 b Fs(7!)23 b Fr(\030)11 b Fs(\000)d FA(1,)36 b Fr(\021)27 b Fs(7!)c Fr(\021)11 b Fs(\000)d FA(1,)21 b(w)n(e)h(again)f(set)h Fr(\045)h FA(=)2946 3638 y Fp(p)p 3029 3638 261 4 v 73 x Fr(\030)3069 3687 y Ft(2)3125 3711 y FA(+)18 b Fr(\021)3252 3687 y Ft(2)599 3810 y FA(and)28 b Fr(\030)f FA(=)c Fr(\045)14 b FA(cos)e Fr(t)p FA(,)42 b Fr(\021)26 b FA(=)d Fr(\045)14 b FA(sin)f Fr(t)p FA(.)37 b(This)28 b(leads)f(to)726 4061 y Fr(I)762 4076 y Ft([0)p Fo(;)p Ft(1])890 4061 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)e Fr(R)1272 4026 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1446 3948 y Fp(Z)1529 3968 y Ft(0)1493 4136 y Fl(\000)p Fo(\031)r(=)p Ft(2)1670 4061 y Fr(dt)14 b(c)p FA(\()p Fr(t)p FA(\))1887 4026 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2062 3948 y Fp(Z)2146 3968 y Fo(\045)2180 3977 y Ff(b)2210 3968 y Ft(\()p Fo(t)p Ft(\))2109 4136 y(0)2305 4061 y Fr(Q)2371 4073 y Fo(t)2400 4061 y FA(\()p Fr(\045)p FA(\))p Fr(e)2546 4026 y Fl(\000)p Fo(c)p Ft(\()p Fo(t)p Ft(\))p Fo(\020)t(R\045)2842 4061 y Fr(d\045:)126 b FA(\(4.24\))599 4317 y(Here,)42 b Fr(c)p FA(\()p Fr(t)p FA(\))e(has)e(the)i(same)e(meaning)h(as)f(b)r(efore,)k(whereas)c Fr(\045)2649 4329 y Fo(b)2682 4317 y FA(\()p Fr(t)p FA(\))43 b(=)e Fs(\000)p FA(\(sin)14 b Fr(t)p FA(\))3200 4287 y Fl(\000)p Ft(1)599 4433 y FA(describ)r(es)22 b(the)h(line)f Fr(\021)k FA(=)d Fs(\000)p FA(1,)f(and)g Fr(Q)1766 4445 y Fo(t)1795 4433 y FA(\()p Fr(\045)p FA(\))i(=)2013 4366 y Fp(\000)2051 4433 y FA(2)8 b(+)g Fr(\045)p FA(\(cos)k Fr(t)c FA(+)g(sin)k Fr(t)p FA(\))2660 4366 y Fp(\001)2698 4383 y Fl(\000)p Ft(1)2788 4433 y Fr(\045)2831 4403 y Ft(2)p Fo(\015)2906 4433 y FA(.)35 b(Since)23 b(2)8 b(+)599 4533 y Fr(\045)p FA(\(cos)14 b Fr(t)e FA(+)g(sin)h Fr(t)p FA(\))24 b Fs(\025)e FA(1)i(if)h Fr(\045)e Fs(2)h FA([0)p Fr(;)14 b(\045)1636 4545 y Fo(b)1669 4533 y FA(\()p Fr(t)p FA(\)])25 b(and)f Fr(t)f Fs(2)h FA([)p Fs(\000)p Fr(\031)s(=)p FA(2)p Fr(;)14 b FA(0],)23 b(for)h Fr(\045)f Fs(#)g FA(0)h(the)h(b)r (eha)n(viour)p eop %%Page: 27 27 27 26 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(27)599 320 y Fr(Q)665 332 y Fo(t)694 320 y FA(\()p Fr(\045)p FA(\))30 b Fs(/)e Fr(\045)967 290 y Ft(2)p Fo(\015)1043 320 y Fr(=)p FA(2)i(holds)h(uniformly)g(for)f (all)h Fr(t)g FA(in)h(the)f(considered)f(in)n(terv)-5 b(al.)47 b(Hence)599 419 y(w)n(e)38 b(ma)n(y)f(in)n(v)n(ok)n(e)f (theorem)i(7.1)f(from)h(Ref.)g([56)o(])g(to)g(conclude)g(that)g(the)g (leading)599 536 y(con)n(tribution)31 b(to)g(the)h Fr(\045)p FA(-in)n(tegral)e(in)h(\(4.24\))g(is)g(giv)n(en)f(b)n(y)2485 503 y Ft(1)p 2485 517 34 4 v 2485 564 a(2)2528 469 y Fp(\000)2567 536 y Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\020)2739 469 y Fp(\001)2777 486 y Fl(\000)p Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2990 536 y Fr(R)3054 506 y Fl(\000)p Ft(2)p Fo(\015)t Fl(\000)p Ft(1)3266 536 y FA(,)599 635 y(and)e(therefore)1188 807 y Fr(I)1224 822 y Ft([0)p Fo(;)p Ft(1])1352 807 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)f Fr(R)1735 773 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1895 807 y Fs(O)r FA(\()p Fr(R)2059 773 y Fl(\000)p Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2272 807 y FA(\))g(=)g Fs(O)r FA(\()p Fr(R)2579 773 y Fl(\000)p Ft(2)2668 807 y FA(\))377 b(\(4.25\))599 979 y(In)29 b(case)e(of)h Fr(I)1011 994 y Ft([)p Fl(\000)p Ft(1)p Fo(;)p Ft(0])1191 979 y FA(,)g(to)r(o,)g(w)n(e)g(adhere)f(to)h (the)g(same)g(sequence)f(of)h(transformations.)599 1078 y(First,)e(w)n(e)f(translate)g Fr(\030)i Fs(7!)c Fr(\030)c Fs(\000)14 b FA(1,)39 b Fr(\021)26 b Fs(7!)d Fr(\021)18 b FA(+)c(1,)25 b(and)h(then)g(c)n(hange)e(to)h Fr(\032;)14 b(t)26 b FA(v)-5 b(ariables.)599 1178 y(With)29 b(the)f(upp)r(er)g(b)r (oundary)e Fr(\045)1611 1190 y Fo(b)1645 1178 y FA(\()p Fr(t)p FA(\))d(=)g(\(sin)14 b Fr(t)p FA(\))2060 1148 y Fl(\000)p Ft(1)2149 1178 y FA(,)28 b(w)n(e)f(obtain)661 1415 y Fr(I)697 1430 y Ft([)p Fl(\000)p Ft(1)p Fo(;)p Ft(0])877 1415 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)f Fr(R)1260 1381 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1420 1415 y Fr(e)1459 1381 y Fl(\000)p Fo(\020)t(R)1613 1302 y Fp(Z)1696 1322 y Fo(\031)r(=)p Ft(2)1659 1490 y(0)1808 1415 y Fr(dt)14 b FA(\(cos)f Fr(t)19 b FA(+)f(sin)c Fr(t)p FA(\))2362 1381 y Fl(\000)p Ft(1)2465 1302 y Fp(Z)2548 1322 y Fo(\045)2582 1331 y Ff(b)2613 1322 y Ft(\()p Fo(t)p Ft(\))2511 1490 y(0)2694 1348 y Fp(\000)2732 1415 y FA(2)k(+)g Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\045)p FA(\))3080 1381 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1827 1602 y Fs(\002)g Fr(e)1949 1567 y Fl(\000)p Fo(\020)t(R)p Ft(\(1+)p Fo(c)p Ft(\()p Fo(t)p Ft(\))p Fo(\045)p Ft(\))2379 1602 y Fr(d\045)3077 1702 y FA(\(4.26\))599 1802 y(where)j(1)t(+)t Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\045)i Fs(\025)g FA(0)d(on)g(the)h(sp)r (eci\014ed)g(domain)f(of)h(in)n(tegration.)33 b(W)-7 b(e)21 b(no)n(w)f(split)h(the)599 1901 y Fr(t)p FA(-in)n(tegral)29 b(in)n(to)i(t)n(w)n(o)f(parts)g(in)h(accordance)d(with)j(the)g(sign)g (of)f Fr(c)p FA(\()p Fr(t)p FA(\).)47 b(T)-7 b(o)30 b(this)h(end,)599 2001 y(w)n(e)g(c)n(ho)r(ose)f Fr(\024)f(>)g FA(0)i(suc)n(h)g(that)g Fr(\032)1653 2013 y Fo(b)1686 2001 y FA(\()1728 1968 y Fo(\031)p 1729 1982 41 4 v 1733 2030 a Ft(4)1800 2001 y Fs(\000)21 b Fr(\024)p FA(\))29 b Fr(<)g FA(2)i(and)g(decomp)r(ose)f (the)i(in)n(tegration)599 2101 y(in)n(terv)-5 b(al)30 b([0)p Fr(;)1013 2068 y Fo(\031)p 1013 2082 V 1017 2129 a Ft(2)1064 2101 y FA(])d(=)f([0)p Fr(;)1317 2068 y Fo(\031)p 1317 2082 V 1321 2129 a Ft(4)1387 2101 y Fs(\000)20 b Fr(\024)p FA(])g Fs([)g FA([)1671 2068 y Fo(\031)p 1671 2082 V 1675 2129 a Ft(4)1742 2101 y Fs(\000)f Fr(\024;)1921 2068 y Fo(\031)p 1921 2082 V 1925 2129 a Ft(2)1972 2101 y FA(].)44 b(Then)30 b Fr(\045)2324 2113 y Fo(b)2357 2101 y FA(\()p Fr(t)p FA(\))d Fr(<)g FA(2)i(for)g Fr(t)e Fs(2)g FA([)2942 2068 y Fo(\031)p 2942 2082 V 2946 2129 a Ft(4)3013 2101 y Fs(\000)20 b Fr(\024;)3192 2068 y Fo(\031)p 3192 2082 V 3196 2129 a Ft(2)3243 2101 y FA(],)599 2200 y(whence)34 b(for)f(these)g Fr(t)h FA(v)-5 b(alues)33 b(the)h Fr(\032)p FA(-in)n(tegration)d(runs)i(o)n(v)n(er)f(a)h(compact) g(domain,)599 2300 y(and)28 b(is)f(easily)g(estimated)h(b)n(y)f(a)g (\014nite)h(constan)n(t;)f(th)n(us)643 2532 y Fr(R)707 2498 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)867 2532 y Fr(e)906 2498 y Fl(\000)p Fo(\020)t(R)1060 2419 y Fp(Z)1143 2440 y Fo(\031)r(=)p Ft(2)1116 2586 y Ff(\031)p 1116 2595 36 3 v 1120 2628 a Fg(4)1162 2608 y Fl(\000)p Fo(\024)1271 2532 y Fr(dt)14 b FA(\(cos)f Fr(t)18 b FA(+)g(sin)c Fr(t)p FA(\))1824 2498 y Fl(\000)p Ft(1)1927 2419 y Fp(Z)2010 2440 y Fo(\045)2044 2449 y Ff(b)2075 2440 y Ft(\()p Fo(t)p Ft(\))1974 2608 y(0)2156 2465 y Fp(\000)2195 2532 y FA(2)k(+)g Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\045)p FA(\))2543 2498 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2704 2532 y Fr(e)2743 2498 y Fl(\000)p Fo(\020)t(R)p Ft(\(1+)p Fo(c)p Ft(\()p Fo(t)p Ft(\))p Fo(\045)p Ft(\))3174 2532 y Fr(d\045)803 2736 y Fs(\024)k Fr(const)d Fs(\002)f Fr(R)1251 2701 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1411 2736 y Fr(e)1450 2701 y Fl(\000)p Fo(\020)t(R)1590 2736 y Fr(:)3077 2836 y FA(\(4.27\))599 2936 y(F)-7 b(or)22 b Fr(t)h Fs(2)g FA([0)p Fr(;)986 2903 y Fo(\031)p 986 2917 41 4 v 990 2964 a Ft(4)1044 2936 y Fs(\000)7 b Fr(\024)p FA(],)23 b(the)f(function)h Fr(c)p FA(\()p Fr(t)p FA(\))h Fr(>)e FA(0)g(is)g(nonnegativ)n(e,)g(whic)n(h)g(p)r(ermits)g(appli-)599 3060 y(cation)h(of)h(Olv)n(er's)e(theorem)h(7.1)g([56)o(])g(to)h (deduce)f(2)2226 3030 y Fo(\015)t Fl(\000)p Ft(1)2354 3060 y FA(\000\(1\))2512 2993 y Fp(\000)2550 3060 y Fr(\020)6 b(R)q(c)p FA(\()p Fr(t)p FA(\))2786 2993 y Fp(\001)2824 3010 y Fl(\000)p Ft(1)2927 3060 y FA(exp\()p Fs(\000)p Fr(\020)g(R)q FA(\))599 3188 y(as)38 b(the)h(leading)f(asymptotics)g (for)g(the)h(in)n(tegral)2239 3121 y Fp(R)2294 3141 y Fo(\045)2328 3150 y Ff(b)2359 3141 y Ft(\()p Fo(t)p Ft(\))2278 3217 y(0)2454 3188 y Fr(Q)2520 3200 y Fo(t)2549 3188 y FA(\()p Fr(\045)p FA(\))14 b(exp\()p Fs(\000)p Fr(R)q(P)3011 3200 y Fo(t)3040 3188 y FA(\()p Fr(\045)p FA(\)\))p Fr(d\045)p FA(,)599 3313 y(where)27 b Fr(Q)905 3325 y Fo(t)934 3313 y FA(\()p Fr(\045)p FA(\))d(=)1152 3246 y Fp(\000)1190 3313 y FA(2)18 b(+)f Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\045)1505 3246 y Fp(\001)1544 3263 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)1732 3313 y FA(and)27 b Fr(P)1946 3325 y Fo(t)1976 3313 y FA(\()p Fr(\045)p FA(\))c(=)g Fr(\020)6 b FA(\(1)18 b(+)g Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\045)p FA(\).)37 b(Since)28 b(in)g(addition)599 3373 y Fp(R)655 3394 y Fo(\031)r(=)p Ft(4)p Fl(\000)p Fo(\024)639 3470 y Ft(0)858 3373 y Fp(\000)896 3440 y FA(\(cos)13 b Fr(t)p FA(\))1115 3410 y Ft(2)1171 3440 y Fs(\000)18 b FA(\(sin)c Fr(t)p FA(\))1464 3410 y Ft(2)1502 3373 y Fp(\001)1540 3390 y Fl(\000)p Ft(1)1629 3440 y Fr(dt)23 b(<)g Fs(1)k FA(is)h(w)n (ell-de\014ned)f(for)g Fr(\024)c(>)g FA(0,)k(w)n(e)g(get)637 3686 y Fr(R)701 3652 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)862 3686 y Fr(e)901 3652 y Fl(\000)p Fo(\020)t(R)1054 3573 y Fp(Z)1147 3571 y Ff(\031)p 1147 3580 36 3 v 1151 3613 a Fg(4)1193 3593 y Fl(\000)p Fo(\024)1101 3762 y Ft(0)1302 3686 y Fr(dt)14 b FA(\(cos)f Fr(t)19 b FA(+)f(sin)c Fr(t)p FA(\))1856 3652 y Fl(\000)p Ft(1)1959 3573 y Fp(Z)2042 3593 y Fo(\045)2076 3602 y Ff(b)2107 3593 y Ft(\()p Fo(t)p Ft(\))2005 3762 y(0)2188 3619 y Fp(\000)2226 3686 y FA(2)k(+)g Fr(c)p FA(\()p Fr(t)p FA(\))p Fr(\045)p FA(\))2574 3652 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2735 3686 y Fr(e)2774 3652 y Fl(\000)p Fo(\020)t(R)p Ft(\(1+)p Fo(c)p Ft(\()p Fo(t)p Ft(\))p Fo(\045)3179 3686 y Fr(d\045)797 3869 y FA(=)23 b Fs(O)953 3802 y Fp(\000)991 3869 y Fr(R)1055 3835 y Ft(2)p Fo(\015)t Fl(\000)p Ft(2)1215 3869 y Fr(e)1254 3835 y Fl(\000)p Ft(2)p Fo(\020)t(R)1428 3802 y Fp(\001)1466 3869 y Fr(;)3077 3970 y FA(\(4.28\))599 4070 y(whic)n(h,)28 b(together)f(with)h(\(4.27\))f(implies)1392 4241 y Fr(I)1428 4256 y Ft([)p Fl(\000)p Ft(1)p Fo(;)p Ft(0])1608 4241 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g Fs(O)1994 4174 y Fp(\000)2033 4241 y Fr(R)2097 4207 y Ft(2)p Fo(\015)t Fl(\000)p Ft(1)2257 4241 y Fr(e)2296 4207 y Fl(\000)p Fo(\020)t(R)2436 4174 y Fp(\001)2474 4241 y Fr(;)580 b FA(\(4.29\))599 4413 y(so)25 b(that,)h(ev)n(en)n(tually)-7 b(,)25 b(the)h(b)r(eha)n(viour)e(of)h Fr(I)1963 4428 y Ft([0)p Fo(;)p Ft(1])2116 4413 y FA(in)h(Eq.)e(\(4.23\))h(dominates,) g(and)g(the)599 4533 y(asymptotics)i(as)g(stated)h(in)g(\(4.13\))e (results.)p 2090 4550 25 75 v eop %%Page: 28 28 28 27 bop 599 75 a FA(28)863 b Fz(BRIET)24 b(AND)f(HOGREVE)682 320 y FA(Inserting)28 b(the)h(form)n(ulas)e(from)g(lemma)i(4.1)e(in)n (to)h(Eq.)f(\(4.11\))h(for)g(the)g(b)r(ound)h(on)599 419 y(the)f(energy)f(di\013erence)g(yields)1093 611 y Fr(D)r FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)f(2)p Fr(\020)6 b(e)1606 576 y Ft(at)1668 611 y FA(\()p Fr(\020)g FA(\)\()p Fs(\000)p FA(1)19 b(+)f Fr(\013)2068 575 y Fl(\000)p Ft(1)2068 635 y Fo(S)2158 611 y Fr(\020)6 b FA(\))p Fr(R)2296 576 y Fl(\000)p Ft(1)2404 611 y FA(+)18 b Fs(O)r FA(\()p Fr(R)2651 576 y Fl(\000)p Ft(2)2740 611 y FA(\))p Fr(:)282 b FA(\(4.30\))599 802 y(Since)30 b Fr(e)857 772 y Ft(at)919 802 y FA(\()p Fr(\020)6 b FA(\))27 b Fr(>)e FA(0,)k(then,)g(as)g(long)f(as)g Fr(Z)j FA(=)25 b Fr(\013)2068 767 y Fl(\000)p Ft(1)2068 827 y Fo(S)2158 802 y Fr(\020)31 b(<)25 b FA(1,)k(w)n(e)g(infer)g Fr(D)r FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))26 b Fr(<)e FA(0)29 b(for)599 902 y Fr(R)g FA(su\016cien)n(tly)e(large.)36 b(This)27 b(pro)n(v)n(es)f(the)i(\014rst)f(part)h(of)f(the)h(follo)n (wing)f(theorem.)682 1060 y Fj(Theorem)k(4.2.)60 b Fi(If)23 b FA(0)f Fr(<)h(Z)29 b FA(=)22 b Fr(\013)1723 1025 y Fl(\000)p Ft(1)1723 1084 y Fo(S)1813 1060 y Fr(\020)29 b(<)23 b FA(1)p Fi(,)g(then)f(the)g(gr)l(ound)g(state)g(PEC)h Fr(E)5 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))599 1160 y Fi(for)34 b(the)g(mole)l(cular)g(system)f(mo)l(dele)l(d)h(by)f Fr(H)7 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))30 b(=)f Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))21 b(+)f Fr(V)2776 1172 y Ft(nn)2854 1160 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))34 b Fi(has)g(a)599 1259 y(glob)l(al)j(minimum)e(at)g (some)g FA(0)e Fr(<)f(R)1772 1271 y Fo(e)1840 1259 y Fr(<)h Fs(1)p Fi(.)55 b(If)35 b Fr(Z)j FA(=)33 b(1)p Fi(,)j(the)f(gr)l(ound)h(state)e(PEC)599 1359 y(for)i(the)e(hydr)l(o)l (gen)i(mole)l(cular)g(ion)f(H)1824 1323 y Ft(+)1824 1381 y(2)1913 1359 y Fi(has)g(a)g(glob)l(al)h(minimum)e(for)i(a)f(\014nite)f Fr(R)3254 1371 y Fo(e)599 1459 y Fi(at)41 b(le)l(ast)g(if)h Fr(\013)1055 1471 y Fo(S)1146 1459 y Fs(2)i FA(\(0)p Fr(;)14 b FA(23)p Fr(=)p FA(99])38 b Fi(which)k(includes)g(the)f (physic)l(al)i(value)e(of)h(the)f(\014ne)599 1558 y(structur)l(e)29 b(c)l(onstant)f Fr(\013)1333 1570 y Fo(S)1405 1558 y Fs(\031)22 b FA(1)p Fr(=)p FA(137)p Fi(.)682 1815 y(Pr)l(o)l(of.)106 b FA(Ph)n(ysically)-7 b(,)36 b(our)g(strategy)e(ab)r(o)n(v)n(e)h(for)g (0)i Fr(<)g(Z)42 b(<)37 b FA(1)e(corresp)r(onds)g(to)599 1915 y(attac)n(hing)24 b(the)g(electron)f(to)h(one)g(of)f(the)i(n)n (uclear)e(cen)n(tres)g(and)h(to)g(observ)n(e)e(that)i(the)599 2014 y(other)32 b(n)n(ucleus)h(exp)r(eriences)e(a)i(Coulom)n(b)e (attraction)h(from)g(the)h(negativ)n(e)e(excess)599 2114 y(c)n(harge)18 b Fr(Z)9 b Fs(\000)s FA(1.)33 b(If)20 b Fr(Z)29 b FA(=)23 b(1,)e(the)f(excess)f(c)n(harge)f(v)-5 b(anishes,)21 b(and)e(terms)h(of)g(higher)f(order)599 2214 y(than)34 b Fr(r)838 2183 y Fl(\000)p Ft(1)961 2214 y FA(need)g(to)f(b)r(e)h(studied.)55 b(Rather)33 b(than)h(doing)f(so,)h (it)g(app)r(ears)e(easier)h(to)599 2313 y(start)24 b(from)g(a)g(mo)r (di\014ed)g(v)-5 b(ariational)23 b(trial)h(function)g(for)g Fr(Z)29 b FA(=)22 b(1.)36 b(Since)24 b(relativistic)599 2413 y(corrections)f(can)h(b)r(e)h(exp)r(ected)g(to)f(pla)n(y)g(a)g (minor)g(role)g(for)g(the)h(b)r(onding)g(of)f(H)3091 2377 y Ft(+)3091 2435 y(2)3146 2413 y FA(,)i(w)n(e)599 2512 y(replace)c(our)h(previous)f(c)n(hoice)g Fr( )1645 2482 y Ft(at)1708 2512 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)14 b(\020)6 b FA(\))23 b(b)n(y)f(the)i("nonrelativistic")d(trial)h(spinor) 1616 2754 y Fr( )s FA(\()p Fn(r)s FA(\))h(=)1895 2637 y Fp(\022)1956 2703 y Fr(\036)p FA(\()p Fn(r)s FA(\))p Fr(\037)2168 2715 y Fo(u)2063 2803 y FA(0)2212 2637 y Fp(\023)3077 2754 y FA(\(4.31\))599 2995 y(where)33 b(the)g(lo)n(w)n (er)f(\("small"\))g(comp)r(onen)n(ts)g(are)g(assumed)h(to)g(v)-5 b(anish)33 b(iden)n(tically)-7 b(,)599 3095 y(and)28 b(where)f(the)h(t)n(w)n(o-spinor)e Fr(\037)1603 3107 y Fo(u)1674 3095 y FA(is)h(giv)n(en)g(b)n(y)1445 3331 y Fr(\037)1497 3343 y Fo(u)1564 3331 y FA(=)22 b Fr(Y)1699 3343 y Fg(00)1774 3214 y Fp(\022)1835 3281 y FA(1)1835 3380 y(0)1876 3214 y Fp(\023)1961 3331 y FA(=)2118 3275 y(1)p 2058 3312 161 4 v 2058 3329 a Fs(p)p 2127 3329 92 4 v 68 x FA(4)p Fr(\031)2243 3214 y Fp(\022)2304 3281 y FA(1)2304 3380 y(0)2346 3214 y Fp(\023)2421 3331 y Fr(:)633 b FA(\(4.32\))599 3568 y(The)30 b(function)g Fr(\036)g FA(is)f(of)g(LCA)n(O)g(t)n(yp)r(e,)h(i.e.,)g(w)n(e)f (construct)g Fr(\036)h FA(as)f(the)h(linear)e(com)n(bi-)599 3668 y(nation)g(of)f(h)n(ydrogenic)f(ground)h(state)g(w)n(a)n(v)n(e)f (functions)1668 3859 y Fr(\036)1717 3871 y Ft(h)n(yd)1828 3859 y FA(\()p Fn(r)s FA(\))d(=)g Fr(e)2089 3825 y Fl(\000)p Fo(\013r)3077 3859 y FA(\(4.33\))599 4050 y(translated)k(to)h(the)g(t)n (w)n(o)e(n)n(uclear)h(cen)n(tres)1458 4242 y Fr(\036)d FA(=)e Fr(\034)1654 4257 y Fo(R=)p Ft(2)1776 4242 y Fr(\036)1825 4254 y Ft(h)n(yd)1955 4242 y FA(+)c Fr(\034)2074 4257 y Fl(\000)p Fo(R=)p Ft(2)2247 4242 y Fr(\036)2296 4254 y Ft(h)n(yd)2408 4242 y Fr(:)646 b FA(\(4.34\))599 4433 y(While)28 b Fr(\013)c(>)e FA(0)27 b(in)g(\(4.33\))g(could)f(b)r(e)i(a) f(more)f(general)g(v)-5 b(ariational)25 b(parameter,)h(here)599 4533 y(w)n(e)k(demand)f(it)h(to)g(b)r(e)g(equal)f(to)g Fr(\013)1720 4545 y Fo(S)1769 4533 y FA(,)h(and,)g(in)g(order)e(to)h (indicate)h(that)g(w)n(e)f(do)g(not)p eop %%Page: 29 29 29 28 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(29)599 320 y(consider)29 b Fr(\013)978 332 y Fo(S)1055 320 y FA(at)h(a)f(certain)f(\014xed)i(v)-5 b(alue,)30 b(b)r(elo)n(w)f(w)n(e)g(shall)g(suppress)f(its)i(subscript) 599 419 y Fr(S)5 b FA(.)37 b(The)27 b(matrix)f(elemen)n(ts)h Fs(h)p Fr( )s(;)14 b(H)1695 389 y Ft(2)1732 419 y Fr( )s Fs(i)p Fr(=)p Fs(h)p Fr( )s(;)g( )s Fs(i)28 b FA(computed)f(with)h(the) f(ansatz)f(\(4.31\))599 519 y(are)h(listed)h(in)g(app)r(endix)g(B.)g (In)g(particular,)e(using)i(Eqs.)f(\(B.2\),)h(\(B.4\),)g(\(B.5\),)g (and)599 619 y Fr(e)638 588 y Ft(at)701 619 y FA(\()p Fr(\020)6 b FA(\))807 588 y Ft(2)869 619 y FA(=)22 b(1)c Fs(\000)g Fr(\020)1141 588 y Ft(2)1179 619 y FA(,)28 b(the)g(disso)r(ciation)e(energy)h(for)g Fr( )k FA(b)r(ecomes)674 831 y Fr(D)r FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))24 b(=)f Fr(\020)1106 797 y Ft(2)1162 831 y FA(+)1255 791 y Fl(h)p Fo( )r(;)11 b Fl(\000)p Ft(\001)p Fo( )r Fl(i)p 1255 812 284 4 v 1314 860 a(h)p Fo( )r(; )r Fl(i)1567 831 y FA(+)18 b(2)1692 764 y Fp(\000)1729 831 y FA(1)g(+)g Fr(V)1920 843 y Ft(nn)1998 831 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))2205 764 y Fp(\001)2254 791 y Fl(h)p Fo( )r(;)11 b(V)2397 799 y Fg(ne)2457 791 y Ft(\()p Fo(R;\020)t Ft(\))p Fo( )r Fl(i)p 2254 812 433 4 v 2387 860 a(h)p Fo( )r(; )r Fl(i)1567 999 y FA(+)1660 959 y Fl(h)p Fo( )r(;)g(V)1803 967 y Fg(ne)1863 959 y Ft(\()p Fo(R;\020)t Ft(\))2019 934 y Fg(2)2052 959 y Fo( )r Fl(i)p 1660 980 465 4 v 1809 1028 a(h)p Fo( )r(; )r Fl(i)2153 999 y FA(+)2236 932 y Fp(\000)2274 999 y FA(2)18 b(+)g Fr(V)2465 1011 y Ft(nn)2543 999 y FA(\()p Fr(R)q(;)c(\020)6 b FA(\))2750 932 y Fp(\001)2789 999 y Fr(V)2837 1011 y Ft(nn)2914 999 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(:)3077 1113 y FA(\(4.35\))599 1213 y(W)-7 b(e)28 b(no)n(w)e(sp)r(ecify)i(to)f Fr(\020)i FA(=)23 b Fr(\013)p FA(,)k(and,)g(motiv)-5 b(ated)28 b(b)n(y)e(the)i(equilibrium)f(geometry)f(en-)599 1312 y(coun)n(tered)21 b(in)g(nonrelativistic)f(H)1662 1277 y Ft(+)1662 1334 y(2)1738 1312 y FA([53)o(],)j(w)n(e)d(tak)n(e)g Fr(R)k FA(=)f(2)p Fr(=\013)d FA(as)h(a)f(promising)g(candi-)599 1412 y(date)i(for)e Fr(R)961 1424 y Fo(e)997 1412 y FA(.)35 b(F)-7 b(urthermore,)21 b(to)g(emplo)n(y)g(the)h(b)r(ound)f(\(B.14\))g (on)g Fs(h)p Fr( )s(;)14 b(V)2868 1382 y Ft(2)2849 1433 y(ne)2920 1412 y Fr( )s Fs(i)p Fr(=)p Fs(h)p Fr( )s(;)g( )s Fs(i)p FA(,)599 1512 y(w)n(e)45 b(restrict)e Fr(\013)i FA(to)g(the)f(in)n(terv)-5 b(al)44 b(0)51 b Fr(<)g(\013)g Fs(\024)g FA(1)p Fr(=)p FA(4.)86 b(T)-7 b(ogether,)48 b(inserting)c(also)599 1611 y(Eqs.)27 b(\(B.1\),)h(\(B.3\))g(and)f (\(B.4\))h(in)n(to)f(\(4.35\),)g(w)n(e)g(obtain)1478 1864 y Fr(D)r FA(\(2)p Fr(=\013;)g(\013)p FA(\))47 b Fr(<)2063 1807 y(\013)2116 1777 y Ft(2)2153 1807 y Fr(P)12 b FA(\()p Fr(\013)p FA(\))p 2021 1844 357 4 v 2021 1925 a(1)18 b(+)2174 1892 y Ft(13)p 2174 1906 67 4 v 2190 1953 a(3)2250 1925 y Fr(e)2289 1901 y Fl(\000)p Ft(2)2388 1864 y Fr(;)666 b FA(\(4.36\))599 2129 y(where)28 b(ha)n(v)n(e)e (estimated)i(the)g(co)r(e\016cien)n(t)f(of)h Fr(\013)2088 2098 y Ft(2)2153 2129 y FA(as)1408 2307 y Fs(\000)1483 2274 y Ft(5)p 1483 2288 34 4 v 1483 2335 a(3)1526 2307 y Fr(e)1565 2276 y Fl(\000)p Ft(2)1672 2307 y FA(+)18 b(2)p Fr(e)1836 2276 y Fl(\000)p Ft(4)p 1408 2348 517 4 v 1488 2429 a FA(1)g(+)1641 2396 y Ft(13)p 1641 2410 67 4 v 1657 2457 a(3)1717 2429 y Fr(e)1756 2405 y Fl(\000)p Ft(2)1980 2368 y Fr(<)2204 2307 y Fs(\000)2279 2274 y Ft(15)p 2279 2288 V 2279 2335 a(88)p 2101 2348 357 4 v 2101 2429 a FA(1)g(+)2254 2396 y Ft(13)p 2254 2410 67 4 v 2270 2457 a(3)2330 2429 y Fr(e)2369 2405 y Fl(\000)p Ft(2)2468 2368 y Fr(;)586 b FA(\(4.37\))599 2633 y(and)28 b(the)g(co)r(e\016cien)n(t)f(of)h Fr(\013)1439 2602 y Ft(4)1504 2633 y FA(as)1318 2778 y Ft(839)p 1318 2792 100 4 v 1318 2839 a(276)1445 2811 y Fs(\000)1538 2778 y Ft(59)p 1538 2792 67 4 v 1538 2839 a(12)1614 2811 y Fr(e)1653 2780 y Fl(\000)p Ft(2)1761 2811 y FA(+)1854 2778 y Ft(3)p 1854 2792 34 4 v 1854 2839 a(2)1897 2811 y Fr(e)1936 2780 y Fl(\000)p Ft(4)p 1308 2852 717 4 v 1488 2933 a FA(1)18 b(+)1641 2900 y Ft(13)p 1641 2914 67 4 v 1657 2961 a(3)1717 2933 y Fr(e)1756 2909 y Fl(\000)p Ft(2)2081 2872 y Fr(<)2330 2778 y Ft(197)p 2330 2792 100 4 v 2347 2839 a(82)p 2201 2852 357 4 v 2201 2933 a FA(1)g(+)2354 2900 y Ft(13)p 2354 2914 67 4 v 2371 2961 a(3)2430 2933 y Fr(e)2469 2909 y Fl(\000)p Ft(2)2568 2872 y Fr(;)486 b FA(\(4.38\))599 3127 y(and)28 b(where)f(w)n(e)g(ha)n (v)n(e)g(set)1256 3326 y Fr(P)12 b FA(\()p Fr(\013)p FA(\))47 b(=)f Fs(\000)1671 3293 y Ft(15)p 1670 3307 V 1670 3354 a(88)1765 3326 y FA(+)1858 3293 y Ft(10)p 1858 3307 V 1858 3354 a(59)1934 3326 y Fr(\013)19 b FA(+)2099 3293 y Ft(197)p 2099 3307 100 4 v 2115 3354 a(82)2208 3326 y Fr(\013)2261 3292 y Ft(2)2317 3326 y FA(+)2427 3293 y Ft(11)p 2410 3307 V 2410 3354 a(112)2519 3326 y Fr(\013)2572 3292 y Ft(3)2610 3326 y Fr(:)444 b FA(\(4.39\))599 3525 y(Clearly)-7 b(,)20 b(the)f(p)r(olynomial)e Fr(P)12 b FA(\()p Fr(\013)p FA(\))20 b(is)e(\(strictly\))h(monotonically)e (increasing)g(in)i Fr(\013)k(>)g FA(0,)599 3625 y(hence)28 b(negativit)n(y)f(of)g(\(4.36\))g(for)g(0)c Fr(<)f(\013)i Fs(\024)f FA(23)p Fr(=)p FA(99)d(=)j(0)p Fr(:)p 2381 3561 84 4 v FA(23)j(follo)n(ws)h(from)1545 3824 y Fr(P)12 b FA(\()1652 3791 y Ft(23)p 1653 3805 67 4 v 1653 3852 a(99)1729 3824 y FA(\))23 b(=)g Fs(\000)2012 3791 y Ft(4223509)p 1946 3805 365 4 v 1946 3852 a(23898287952)2320 3824 y Fr(:)734 b FA(\(4.40\))p 599 4002 25 75 v 682 4134 a(T)-7 b(o)30 b(k)n(eep)g(the)h(analysis)e(simple,)j(w)n(e)e(did)g(not)h (striv)n(e)e(for)h(optimal)g(results)g(when)599 4234 y(demonstrating)f(stabilit)n(y)g(for)g Fr(Z)j FA(=)26 b(1.)42 b(Ev)n(en)29 b(with)h(the)g("primitiv)n(e")e(form)h(\(4.31\)) 599 4334 y(for)e Fr( )s FA(,)g(a)f(b)r(etter)h(adjustmen)n(t)g(of)f (the)h(v)-5 b(ariational)25 b(parameter)h(and)g(of)h(the)g(equilib-)599 4433 y(rium)g(distance)g(and)f(sharp)r(er)g(b)r(ounds)h(w)n(ould)f(p)r (ossibly)h(yield)g(a)f(somewhat)g(larger)599 4533 y Fr(\013)652 4545 y Fo(S)701 4533 y FA(-in)n(terv)-5 b(al)30 b(in)i(theorem)f(4.2.) 49 b(W)-7 b(e)32 b(surmise)f(that)h(b)r(onding)f(in)h(H)2762 4497 y Ft(+)2762 4555 y(2)2849 4533 y FA(holds)g(for)f(all)p eop %%Page: 30 30 30 29 bop 599 75 a FA(30)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(0)h Fr(<)g(\013)807 332 y Fo(S)879 320 y Fr(<)g FA(1.)38 b(Nonetheless,)28 b(similar)g(as)f(for)h(the)h(h)n(ydrogen)d (atom)i(where)g(with)h(an)599 419 y(analogous)22 b(trial)h(function)h (the)g(ground)f(state)g(energy)g(can)g(b)r(e)h(separated)f(from)g(the) 599 519 y(upp)r(er)32 b(part)f(of)h(the)g(con)n(tin)n(uous)e(sp)r (ectrum)i(only)f(if)h Fr(\013)2376 531 y Fo(S)2454 519 y Fr(<)d FA(1)p Fr(=)2632 450 y Fs(p)p 2701 450 42 4 v 69 x FA(2)o(,)k(for)e(increasing)599 619 y Fr(\013)652 631 y Fo(S)734 619 y FA(more)i(sophisticated)g(trial)g(functions)h (with)g(relativistic)f(ingredien)n(ts)f(will)i(b)r(e)599 718 y(required)i(to)g(obtain)g(su\016cien)n(tly)g(tigh)n(t)g(v)-5 b(ariational)35 b(b)r(ounds)h(on)g(the)h(molecular)599 818 y(energies.)44 b(Since)30 b(man)n(y)g(of)g(the)g(arising)f (molecular)g(in)n(tegrals)f(will)j(b)r(e)f(di\016cult)h(or)599 917 y(imp)r(ossible)h(to)f(express)g(in)h(closed)f(form)g(and)g(the)h (n)n(um)n(b)r(er)f(of)h(terms)f(will)h(prolif-)599 1017 y(erate)i(rapidly)-7 b(,)35 b(suc)n(h)e(calculations)g(probably)g(w)n (ould)g(not)h(b)r(e)h(tractable)e(without)599 1117 y(relying)27 b(hea)n(vily)g(on)n(to)f(appropriate)g(computer)h(algebra)f(metho)r(ds) i(and)f(accepting)599 1216 y(computer)h(assisted)e(pro)r(ofs.)682 1316 y(In)21 b(the)h(nonrelativistic)d(situation,)j(n)n(umerical)e(in)n (v)n(estigations)f([53])h(suggest)g(that)599 1416 y(the)32 b(critical)f(c)n(harge)e Fr(Z)1348 1428 y Fo(c)1413 1416 y FA(un)n(til)j(whic)n(h)g(a)e(global)h(minim)n(um)h(for)f(the)g (ground)g(state)599 1515 y(PEC)24 b(exists)g(is)g(giv)n(en)g(b)n(y)g Fr(Z)1486 1527 y Fo(c)1543 1515 y Fs(\031)e FA(1)p Fr(:)p FA(2367;)i(for)f(n)n(uclear)h(c)n(harges)e Fr(Z)2666 1527 y Fo(c)2723 1515 y Fr(<)g(Z)29 b(<)23 b(Z)3041 1527 y Fo(m)3128 1515 y FA(with)599 1615 y Fr(Z)656 1627 y Fo(m)757 1615 y Fs(\031)38 b FA(1)p Fr(:)p FA(4391)d(the)i(PEC)e (retains)h(a)h(lo)r(cal)f(minim)n(um)h(ab)r(o)n(v)n(e)e(the)i(SAL)h (energy;)599 1714 y(this)c(corresp)r(onds)d(to)i(a)g(metastable)g (regime)g(for)f(the)i(system.)54 b(If)33 b Fr(Z)38 b(>)32 b(Z)3053 1726 y Fo(m)3116 1714 y FA(,)j(the)599 1814 y(PEC)25 b(has)g(a)h(purely)f(repulsiv)n(e)g(shap)r(e)g(and)h(the)g(n)n (uclear)e(motion)i(is)f(un)n(b)r(ound.)37 b(F)-7 b(or)599 1914 y(the)24 b(Dirac-Coulom)n(b)e(Hamiltonian)h(one)g(w)n(ould)f(exp)r (ect)i(sligh)n(tly)f(larger)e(v)-5 b(alues)23 b(for)599 2013 y(analogously)j(de\014ned)i(critical)f(c)n(harges)e([57)o(].)1289 2279 y Fu(5.)33 b(CONCLUDING)f(REMARKS)682 2428 y FA(In)27 b(our)f(in)n(v)n(estigations)f(ab)r(o)n(v)n(e,)g(the)i(motion)f(of)h (the)g(n)n(uclei)f(has)g(not)h(b)r(een)g(tak)n(en)599 2528 y(in)n(to)g(accoun)n(t.)36 b(T)-7 b(o)26 b(incorp)r(orate)f (dynamical)h(n)n(uclei)h(one)f(can)h(consider)e(t)n(w)n(o)h(p)r(ossi-) 599 2627 y(bilities.)40 b(First,)29 b(a)f(fully)h(relativistic)f(mo)r (del)g(of)h(the)f(molecule)h(encompassing)e(also)599 2727 y(a)34 b(relativistic)g(treatmen)n(t)g(of)h(the)g(n)n(uclei,)h (i.e.,)g(using)e(the)h(Dirac)f(kinetic)h(energy)599 2826 y(\(2)p Fr(M)9 b FA(\))795 2796 y Fl(\000)p Ft(1)884 2826 y Fn(\013)22 b Fm(\001)e Fn(P)48 b FA(with)35 b Fn(P)48 b FA(=)34 b Fs(\000)p Fr(i)p Fm(r)1701 2838 y Fk(R)1794 2826 y FA(for)g(the)h(n)n(uclear)e(motion.)57 b(Whereas)34 b(suc)n(h)g(a)599 2926 y(mo)r(del)27 b(w)n(ould)g(o\013er) f(attractiv)n(e)f(consistency)-7 b(,)26 b(apparen)n(tly)-7 b(,)26 b(ho)n(w)n(ev)n(er,)f(rigorous)f(re-)599 3026 y(sults)32 b(on)f(a)f(relativistic)h(Born-Opp)r(enheimer)f(appro)n (ximation)f(or)h(on)h(relativistic)599 3125 y(molecular)i(vibrations)f (are)g(still)i(lac)n(king)e(in)i(the)f(literature)g(\(see,)i(e.g.,)f (Ref.)g([58)o(])599 3225 y(for)c(a)g(rigorous)e(nonrelativistic)h (discussion\).)44 b(Alternativ)n(ely)-7 b(,)31 b(one)f(ma)n(y)f(en)n (visage)599 3325 y(a)j(mo)r(del)g(with)h(mixed)f(kinematics)g(for)f (the)i(molecule,)f(where)g(the)g(electrons)f(are)599 3424 y(describ)r(ed)h(relativistically)-7 b(,)32 b(but)g(the)g(n)n (uclei)g(in)g(a)g(nonrelativistic)f(w)n(a)n(y)-7 b(,)31 b(as)h(moti-)599 3524 y(v)-5 b(ated)25 b(b)n(y)f(the)h(10)1156 3494 y Ft(3)1193 3524 y FA(-10)1305 3494 y Ft(4)1365 3524 y FA(times)g(larger)e(n)n(uclear)g(masses.)35 b(Then,)26 b(the)f(electronic)f(and)599 3623 y(n)n(uclear)30 b(motion)h(can)g(b)r (e)h(separated)e(via)g(the)i(con)n(v)n(en)n(tional)d(Born-Opp)r (enheimer)599 3723 y(appro)n(ximation)d(suc)n(h)i(that)f(the)i (vibrational-)c(or)i(scattering-t)n(yp)r(e)g(tra)5 b(jectories)25 b(of)599 3823 y(the)36 b(n)n(uclei)f(are)f(go)n(v)n(erned)f(b)n(y)i(a)g (kinetic)g(energy)f(quadratic)g(in)i(the)f(momen)n(tum)599 3922 y(plus)28 b(an)f(e\013ectiv)n(e)h(p)r(oten)n(tial)g(determined)f (b)n(y)h(the)g(in)n(v)n(olv)n(ed)e(PECs.)682 4022 y(It)45 b(w)n(as)f(men)n(tioned)g(in)h(section)f(2.1)g(that)h(in)f(the)h(UAL)g (the)g(essen)n(tial)f(self-)599 4122 y(adjoin)n(tness)20 b(of)g(the)h(Dirac-Coulom)n(b)d(Hamiltonian)i(is)g(lost)g(if)2579 4053 y Fs(p)p 2649 4053 V 2649 4122 a FA(3)o Fr(=)p FA(2)i Fs(\024)h FA(2)p Fr(\020)29 b(<)23 b FA(1,)e(but)599 4221 y(a)34 b(ph)n(ysically)g(distinguished)g(self-adjoin)n(t)g (extension)g(could)g(b)r(e)h(retained.)56 b(That)599 4321 y(this)20 b(particular)f({)g(b)n(y)g(atomic)h(quan)n(tities)f (iden)n(ti\014ed)h({)f(extension)h(actually)f(em)n(b)r(o)r(d-)599 4420 y(ies)34 b(the)f("authen)n(tic")g(one)g(is)g(also)g(supp)r(orted)g (through)g(the)g(molecular)g(con)n(text,)599 4520 y(viz.,)40 b(b)n(y)d(the)g(con)n(tin)n(uit)n(y)g(of)g(the)g(electronic)f(curv)n (es)g(for)h Fr(R)i Fs(#)g FA(0)d(pro)n(v)n(ed)g(in)h(sec-)p eop %%Page: 31 31 31 30 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(31)599 320 y(tion)28 b(3.1;)g(an)n(y)f(other)g(c)n (hoice)g(of)h(self-adjoin)n(t)g(extension)f(for)g(the)i(atomic)e(op)r (erator)599 419 y(should)i(lead)f(to)h(a)f(discon)n(tin)n(uit)n(y)g(of) g(the)h(electronic)f(curv)n(es)f(at)i Fr(R)c FA(=)g(0.)39 b(If)29 b(for)f(the)599 519 y(molecular)j(system)g(1)p Fr(=)p FA(2)e Fs(\024)g Fr(\020)37 b(<)29 b FA(1,)j(then)h(the)f Fr(R)e FA(=)g(0)h(op)r(erator)f Fr(h)p FA(\(0)p Fr(;)14 b(\020)6 b FA(\))32 b(carries)e(a)599 619 y(c)n(harge)21 b(2)p Fr(\020)29 b Fs(\025)23 b FA(1)e(excluding)h(the)h(ph)n(ysically) e(distinguished)h(self-adjoin)n(t)g(extensions)599 718 y(with)h(\014nite)g(kinetic)f(or)f(\014nite)h(p)r(oten)n(tial)g(energy) f(\(the)i(construction)e(of)h(self-adjoin)n(t)599 818 y(op)r(erators)i(in)h(this)h(regime)e(is)h(studied)h(in)f(Ref.)h([59)o (]\).)37 b(Th)n(us,)25 b(the)h(pathological)d(b)r(e-)599 917 y(ha)n(viour)k(of)h(electronic)f(curv)n(es)f(with)j Fr(\020)h Fs(\025)23 b FA(1)p Fr(=)p FA(2)j(as)i Fr(R)c Fs(#)f FA(0)28 b(demonstrated)f(b)n(y)g(Klaus)599 1017 y([19])g(should)h(come)f(as)g(no)g(surprise.)682 1117 y(While)i(our)f(atten)n(tion)g(here)g(has)g(b)r(een)h(restricted)f(to)g (homon)n(uclear)f(molecules,)599 1216 y(one)35 b(of)h(the)f(next)h (steps)f(w)n(ould)g(b)r(e)h(to)f(examine)g(heteron)n(uclear)e(systems)i (b)n(y)g(al-)599 1316 y(lo)n(wing)40 b(di\013eren)n(t)g(c)n(harges)f Fr(\020)1556 1328 y Fo(i)1624 1316 y FA(attac)n(hed)h(to)h(the)f(n)n (uclear)g(cen)n(tres.)74 b(W)-7 b(e)41 b(exp)r(ect)599 1416 y(appropriate)e(generalizations)g(of)h(all)g(our)g(results)g(to)h (hold)f(true)h(also)e(in)i(those)599 1515 y(asymmetrical)f(situations.) 76 b(On)40 b(the)i(other)e(hand,)k(since)c(in)i(sections)e(2.4)g(and) 599 1615 y(3.1)34 b(v)-5 b(arious)32 b(pro)r(ofs)h(rely)h(on)f(the)i (homon)n(uclear)d(symmetry)-7 b(,)35 b(there)f(is)g(not)g(m)n(uc)n(h) 599 1714 y(hop)r(e)40 b(to)g(extend)g(their)g(strategy)e(to)i(heteron)n (uclear)e(systems;)45 b(so,)d(no)n(v)n(el)d(ideas)599 1814 y(are)34 b(called)f(for.)56 b(In)34 b(case)g(of)g(the)g(UAL)h(b)r (ound)f(for)g(the)g(molecular)f(energies,)i(w)n(e)599 1914 y(conjecture)29 b(a)h(particular)e(molecular)g(prop)r(ert)n(y)g (to)i(b)r(e)f(v)-5 b(alid)30 b(that)g(will)f(imply)h(the)599 2013 y(lo)n(w)n(er)39 b(b)r(ound)j(estimate)e(\(2.41\))g(in)h(theorem)f (2.4,)j(namely)e(the)g(monotonic)e(in-)599 2113 y(crease)23 b(of)i(the)f(molecular)f(ground)h(state)g(electronic)f(curv)n(e,)h Fr(e)2577 2125 y Ft(0)2614 2113 y FA(\()p Fr(R)2709 2125 y Ft(1)2747 2113 y Fr(;)14 b(\020)6 b FA(\))24 b Fs(\024)e Fr(e)3008 2125 y Ft(0)3045 2113 y FA(\()p Fr(R)3140 2125 y Ft(2)3178 2113 y Fr(;)14 b(\020)6 b FA(\))599 2213 y(if)37 b(0)g Fs(\024)g Fr(R)928 2225 y Ft(1)1002 2213 y Fs(\024)g Fr(R)1167 2225 y Ft(2)1240 2213 y FA(\(with)g Fr(\020)43 b FA(replaced)35 b(b)n(y)h Fr(\020)2045 2225 y Ft(1)2082 2213 y Fr(;)14 b(\020)2155 2225 y Ft(2)2229 2213 y FA(for)35 b(heteron)n(uclear)g(molecules\).)599 2312 y(Suc)n(h)27 b(a)e(monotonicit)n(y)h(has)f(b)r(een)i(indeed)f (rigorously)e(con\014rmed)i(for)f(Sc)n(hr\177)-42 b(odinger)599 2412 y(Hamiltonians)24 b([60)o(,)g(61)o(,)g(62)o(,)g(63)o(],)h(and)e (for)h(the)g(\(relativistic\))f(Herbst)h(op)r(erator)e([64)o(])599 2511 y(\(where)28 b(\()p Fs(\000)p FA(\001)18 b(+)g(1\))1213 2481 y Ft(1)p Fo(=)p Ft(2)1344 2511 y FA(serv)n(es)26 b(as)h(the)h(kinetic)f(energy)g(part\),)g(and)g(can)g(b)r(e)h(v)n (eri\014ed)599 2611 y(n)n(umerically)f(for)g(the)h(Dirac-Coulom)n(b)e (Hamiltonian)h([57].)682 2711 y(The)41 b(least)f(p)r(ositiv)n(ely)g(c)n (harged)f(heteron)n(uclear)g(one-electron)g(molecular)h(ion)599 2810 y(w)n(ould)33 b(b)r(e)g(HeH)1126 2780 y Ft(2+)1215 2810 y FA(;)j(ho)n(w)n(ev)n(er,)c(this)h(sp)r(ecies)g(is)f(far)h(from)f (b)r(eing)h(stable)g(\(cf.,)i(e.g.,)599 2910 y(Ref.)40 b([53]\).)73 b(Hence,)42 b(ev)n(en)d(an)g(enhanced)h(b)r(onding)f(in)h (a)f(relativistic)g(mo)r(del)h(is)599 3010 y(certainly)c(to)r(o)g(w)n (eak)f(to)h(yield)g(a)g(b)r(ound)g(dication)g(HeH)2471 2979 y Ft(2+)2560 3010 y FA(.)63 b(Our)35 b(discussion)g(of)599 3109 y(b)r(onding)c(in)f(section)g(4.2)g(can)f(b)r(e)i(trivially)f (extended)g(to)g(demonstrate)g(that)g(het-)599 3209 y(eron)n(uclear)j (systems)h(with)h(c)n(harges)e Fr(\020)1854 3221 y Ft(1)1891 3209 y Fr(;)14 b(\020)1964 3221 y Ft(2)2036 3209 y Fr(>)35 b FA(0)f(are)f(stable)i(if)g(max)o(\()p Fr(\020)2910 3221 y Ft(1)2948 3209 y Fr(;)14 b(\020)3021 3221 y Ft(2)3059 3209 y FA(\))35 b Fs(\024)f FA(1.)599 3308 y(More)25 b(realistic)g(or)f(sharp)r(er)h(estimates)g(for)g(stabilit)n(y)g(or)g (instabilit)n(y)h(of)f(relativistic)599 3408 y(molecules)j(still)f (need)h(to)g(b)r(e)g(found.)682 3508 y(F)-7 b(or)37 b(m)n(ulticen)n (tre)h(systems)f(with)h(more)e(than)i(t)n(w)n(o)f(n)n(uclei,)j(the)e(a) n(v)-5 b(ailable)36 b(n)n(u-)599 3607 y(clear)k(geometries)f(increase)g (rapidly)h(and)g(render)g(the)h(binding)f(problem)g(more)599 3707 y(complicated.)83 b(Nonetheless,)46 b(again)41 b(the)j(lac)n(k)d (of)i(stabilit)n(y)g(of)f(nonrelativistic)599 3818 y(H)661 3775 y Ft(\()p Fo(n)p Fl(\000)p Ft(1\)+)661 3828 y Fo(n)921 3818 y FA(for)25 b Fr(n)e(>)g FA(2)i(suggests)g(instabilit)n(y)g(of)h (the)g(relativistically)f(describ)r(ed)h(m)n(ul-)599 3918 y(tic)n(harged)h(cations,)f(to)r(o.)37 b(In)27 b(addition)h(w)n(e) f(exp)r(ect)h(m)n(ultidimensional)f(v)n(ersions)e(of)599 4017 y(our)i(smo)r(othness)g(results)g(for)g(the)h(electronic)f(h)n(yp) r(ersurfaces)f(to)i(b)r(e)g(v)-5 b(alid.)p eop %%Page: 32 32 32 31 bop 599 75 a FA(32)863 b Fz(BRIET)24 b(AND)f(HOGREVE)1632 320 y Fu(APPENDIX)32 b(A)1031 428 y(V)-8 b(ariational)32 b(represen)m(tation)g(of)g(Dirac)g(energies)682 544 y FA(Here)f(w)n(e)g(giv)n(e)g(an)g(outline)h(of)f(the)h(v)-5 b(ariational)29 b(principle)j(for)f(Dirac)f(op)r(erators)599 643 y(put)k(forw)n(ard)d(b)n(y)i(Dolb)r(eault,)i(Esteban,)e(and)g(S)n (\023)-39 b(er)n(\023)g(e)30 b([23)o(,)j(65].)53 b(Besides)32 b(pro)n(viding)599 743 y(the)d(reader)f(with)h(bac)n(kground)d (material,)j(the)g(purp)r(ose)f(of)g(this)h(app)r(endix)g(is)g(also)599 843 y(to)37 b(p)r(oin)n(t)h(out)f(a)f(lo)n(w)n(er)g(b)r(ound)h(on)g (the)h(Dirac-Coulom)n(b)d(energies)h(that)h(can)g(b)r(e)599 942 y(deduced)24 b(easily)e(from)h(their)g(v)-5 b(ariational)21 b(represen)n(tation.)34 b(The)23 b(generalized)e(mini-)599 1042 y(max)27 b(theorem)f(as)g(pro)n(v)n(ed)f(b)n(y)h(Dolb)r(eault,)h (Esteban,)f(and)h(S)n(\023)-39 b(er)n(\023)g(e)24 b(in)j(Ref.)g([23)o (])g(reads)599 1142 y(as)g(follo)n(ws:)682 1296 y Fj(Theorem)k(A.1.)63 b Fi(L)l(et)24 b Fr(H)31 b Fi(b)l(e)24 b(a)h(self-adjoint)h(op)l(er)l (ator)f(with)g(domain)g Fs(D)r FA(\()p Fr(H)7 b FA(\))24 b Fs(\032)e(H)599 1395 y Fi(with)31 b(form)g(domain)h Fs(Q)p FA(\()p Fr(H)7 b FA(\))p Fi(.)40 b(L)l(et)30 b FA(\000)1748 1407 y Fl(\006)1834 1395 y Fi(b)l(e)g(ortho)l(gonal)i(pr)l (oje)l(ctions)g(onto)e(subsp)l(ac)l(es)599 1495 y Fs(H)669 1507 y Fl(\006)757 1495 y Fs(\032)i(H)j Fi(such)g(that)g Fs(H)e FA(=)e Fs(H)1597 1507 y Ft(+)1674 1495 y Fs(\010)22 b(H)1831 1507 y Fl(\000)1922 1495 y Fi(Assume)33 b(that)i(ther)l(e)g (is)g(a)g(c)l(or)l(e)g Fs(F)42 b Fi(for)36 b Fr(H)599 1595 y Fi(with)599 1694 y(\(i\))30 b Fs(F)782 1706 y Fl(\006)868 1694 y Fi(ar)l(e)g(subsp)l(ac)l(es)g(of)h Fs(Q)p FA(\()p Fr(H)7 b FA(\))p Fi(,)30 b(wher)l(e)g Fs(F)2031 1706 y Fl(\006)2110 1694 y FA(=)23 b(\000)2250 1706 y Fl(\006)2306 1694 y Fs(F)599 1794 y Fi(\(ii\))31 b(the)f(action)g(of)h Fr(H)36 b Fi(on)30 b Fs(F)1520 1806 y Fl(\000)1606 1794 y Fi(is)g(b)l(ounde)l(d)g(ab)l(ove)h(in)f(the) g(sense)f(that)1440 2029 y Fr(a)23 b FA(=)128 b(sup)1595 2100 y Fo(\036)p Fl(2F)1728 2108 y Fe(\000)1776 2100 y Fo(;)11 b(\036)p Fl(6)p Ft(=0)1955 1973 y Fs(h)p Fr(\036;)28 b(H)7 b(\036)p Fs(i)p 1955 2010 290 4 v 2000 2086 a(h)p Fr(\036;)14 b(\036)p Fs(i)2278 2029 y Fr(<)23 b Fs(1)649 b FA(\(A.1\))599 2282 y Fi(\(iii\))31 b Fr(\025)821 2294 y Ft(1)882 2282 y Fr(>)23 b(a)p Fi(,)30 b(wher)l(e)g(for)h Fr(k)26 b FA(=)c(1)p Fr(;)14 b FA(2)p Fr(;)g(:)g(:)g(:)1354 2517 y(\025)1402 2529 y Fo(k)1467 2517 y FA(=)109 b(inf)1595 2570 y Fl(V)5 b(\032F)1742 2578 y Fg(+)1564 2636 y Ft(dim)10 b Fl(V)5 b Ft(=)p Fo(k)1957 2517 y FA(sup)1877 2587 y Fo( )r Fl(2V)g(\010F)2115 2595 y Fe(\000)1960 2653 y Fo( )r Fl(6)p Ft(=0)2219 2461 y Fs(h)p Fr( )s(;)29 b(H)7 b( )s Fs(i)p 2219 2498 306 4 v 2264 2574 a(h)p Fr( )s(;)14 b( )s Fs(i)3098 2517 y FA(\(A.2\))599 2826 y Fi(and)31 b(wher)l(e)f(the)g(in\014mum)f(runs)g(over)h(al)t(l)h(subsp)l(ac)l(es)f (of)h Fs(F)2479 2838 y Ft(+)2534 2826 y Fi(.)599 2926 y(Then)41 b Fr(\025)874 2938 y Fo(k)957 2926 y FA(=)g Fr(E)1124 2938 y Fo(k)1205 2926 y Fi(for)g(al)t(l)g(eigenvalues)h Fr(E)1984 2938 y Fo(k)2066 2926 y Fs(2)g FA(\()p Fr(a;)14 b(b)p FA(\))p Fi(,)43 b(wher)l(e)e Fr(b)g FA(=)h(inf)2942 2859 y Fp(\000)2980 2926 y FA(\()p Fr(a;)14 b Fs(1)p FA(\))26 b Fs(\\)599 3034 y Fr(\033)646 3046 y Ft(ess)733 3034 y FA(\()p Fr(H)7 b FA(\))873 2967 y Fp(\001)912 3034 y Fi(,)30 b(or)g Fr(\025)1122 3046 y Fo(k)1186 3034 y FA(=)23 b Fr(b)29 b Fi(if)i(ther)l(e)f(is)g(no)g Fr(E)1896 3046 y Fo(k)1937 3034 y Fi(.)705 3183 y FA(T)-7 b(o)22 b(facilitate)g(the)g(application)g(of)g(the)h(preceding)e(theorem)h(to) g(Dirac)f(op)r(erators)599 3283 y(with)34 b(p)r(oten)n(tials)f Fr(V)18 b FA(,)35 b(Dolb)r(eault,)g(Esteban,)e(and)g(S)n(\023)-39 b(er)n(\023)g(e)31 b(devised)h(a)h("con)n(tin)n(uation)599 3382 y(principle";)23 b(its)d(form)n(ulation)f(b)r(elo)n(w)h(is)g (adapted)g(to)n(w)n(ards)f(our)g(studied)i(atomic)f(and)599 3482 y(molecular)i(Dirac-Coulom)n(b)g(Hamiltonians)g(and)h(th)n(us)g (more)f(sp)r(ecialized)h(that)g(the)599 3582 y(original)j(v)n(ersion)h (\(theorem)g(3.1)g(in)h(Ref.)g([23)o(]\).)682 3736 y Fj(Theorem)j(A.2.)75 b Fi(L)l(et)34 b Fr(T)46 b Fi(b)l(e)35 b(a)g(self-adjoint)h(op)l(er)l(ator)g(on)f Fs(D)r FA(\()p Fr(T)12 b FA(\))32 b Fs(\032)g(H)q Fi(.)53 b(F)-6 b(or)35 b(a)599 3836 y(given)k(b)l(ounde)l(d)g(symmetric)f(op)l(er)l(ator)h Fr(V)19 b Fi(,)41 b(c)l(onsider)e(the)f(one-p)l(ar)l(ameter)h(family) 599 3935 y Fr(H)7 b FA(\()p Fr(t)p FA(\))29 b(=)f Fr(T)k FA(+)20 b Fr(tV)51 b Fi(with)34 b FA(0)28 b Fs(\024)g Fr(t)g Fs(\024)g Fr(t)1714 3947 y Ft(max)1841 3935 y Fi(,)34 b(and)f(such)g(that)f(e)l(ach)i Fr(H)7 b FA(\()p Fr(t)p FA(\))33 b Fi(ob)l(eys)g(the)g(as-)599 4035 y(sumptions)26 b(\(i\))g(and)g(\(ii\))g(of)h(the)l(or)l(em)f(A.1)g(with)g(c)l(orr)l (esp)l(onding)h Fr(a)p FA(\()p Fr(t)p FA(\))p Fi(.)38 b(In)25 b(addition,)599 4134 y(we)36 b(r)l(e)l(quir)l(e)g(that)g(ther)l (e)f(exist)h Fr(a)1638 4146 y Fl(\006)1729 4134 y Fi(such)g(that)f Fr(a)p FA(\()p Fr(t)p FA(\))g Fs(\024)e Fr(a)2414 4146 y Fl(\000)2504 4134 y Fr(<)g(a)2646 4146 y Ft(+)2735 4134 y Fs(\024)g Fr(E)2894 4146 y Ft(1)2932 4134 y FA(\()p Fr(t)p FA(\))j Fi(for)g(al)t(l)599 4234 y Fr(t)31 b Fs(2)f FA([0)p Fr(;)14 b(t)877 4246 y Ft(max)1004 4234 y FA(])p Fi(,)35 b(and)f(that,)h(for)g Fr(t)30 b FA(=)g(0)p Fi(,)35 b(the)f(analo)l(gue)g(of)h(assumption)f(\(iii\))h(in)f(A.1)599 4334 y(holds)k(for)g Fr(H)7 b FA(\(0\))35 b(=)g Fr(T)1324 4346 y Fo(D)1384 4334 y Fi(,)j(i.e.,)j Fr(\025)1674 4346 y Ft(1)1711 4334 y FA(\(0\))36 b Fr(>)f(a)1997 4346 y Fl(\000)2052 4334 y Fi(.)59 b(Then)38 b(the)l(or)l(em)e(A.1)i(is)e (applic)l(able)599 4433 y(to)c(al)t(l)h Fr(H)7 b FA(\()p Fr(t)p FA(\))32 b Fi(and)g(their)g(eigenvalues)h Fr(E)1885 4445 y Fo(k)1926 4433 y FA(\()p Fr(t)p FA(\))p Fr(;)28 b(k)i FA(=)c(1)p Fr(;)14 b FA(2)p Fr(;)g(:)g(:)g(:)30 b Fi(c)l(an)i(b)l(e)f(r)l(epr)l(esente)l(d)h(as)599 4533 y Fr(E)660 4545 y Fo(k)702 4533 y FA(\()p Fr(t)p FA(\))23 b(=)g Fr(\025)955 4545 y Fo(k)996 4533 y FA(\()p Fr(t)p FA(\))h Fs(\025)e Fr(a)1245 4545 y Ft(+)1330 4533 y Fi(for)31 b Fr(t)23 b Fs(2)g FA([0)p Fr(;)14 b(t)1726 4545 y Ft(max)1852 4533 y FA(])p Fi(.)p eop %%Page: 33 33 33 32 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(33)723 320 y(Since)41 b(the)g(Coulom)n(b)f(terms)h Fs(\000)p Fr(\020)6 b Fs(j)p Fn(x)27 b Fs(\006)g Fn(R)p Fr(=)p FA(2)p Fs(j)2203 290 y Fl(\000)p Ft(1)2332 320 y FA(are)40 b(un)n(b)r(ounded)h(op)r(erators,)599 436 y(they)g(need)g(to)f(b)r(e)h(regularized,)g(e.g.,)j(b)n(y)c(in)n(tro)r (ducing)g Fs(\000)p Fr(\020)2592 369 y Fp(\000)2630 436 y Fs(j)p Fn(x)27 b Fs(\006)f Fn(R)q Fr(=)p FA(2)p Fs(j)g FA(+)g Fr(")3162 369 y Fp(\001)3200 386 y Fl(\000)p Ft(1)599 536 y FA(with)j Fr(")23 b(>)f FA(0,)28 b(b)r(efore)f(in)n(v)n(oking)f (theorem)h(A.2;)h(then,)g(the)g(results)f(for)h(the)g(singular)599 635 y(p)r(oten)n(tial)k(ma)n(y)e(b)r(e)i(obtained)f(via)g(limiting)g (argumen)n(ts)f(for)h Fr(")e Fs(#)g FA(0)h(\(cf.)49 b(section)31 b(4)599 735 y(of)d(Ref.)g([23)o(]\).)682 835 y(The)39 b(so-called)f(Bro)n(wn-Ra)n(v)n(enhall)e(or)i("no-pair")e(Hamiltonian)j Fr(h)2905 847 y Ft(BR)3042 835 y FA(is)g(con-)599 934 y(structed)29 b(to)f(o)n(v)n(ercome)e(instabilities)j(arising)e(from)h (the)g(negativ)n(e)g(energy)f(con)n(tin-)599 1034 y(uum)32 b(and)f(asso)r(ciated)f(un)n(b)r(oundedness)h(of)g(the)h(Dirac)f(op)r (erator.)46 b(By)30 b(pro)5 b(jecting)599 1134 y(on)n(to)27 b(the)h(p)r(ositiv)n(e)f(sp)r(ectral)f(subspace)h(of)g(the)h(free)f (Dirac)g(Hamiltonian)g Fr(T)3041 1146 y Fo(D)3128 1134 y FA(with)599 1233 y(the)h(pro)5 b(jections)27 b Fr(\022)1209 1245 y Fl(\006)1292 1233 y FA(from)h(\(2.23\),)f(one)g(arriv)n(es)e(at) j(an)f(op)r(erator)1490 1438 y Fr(h)1538 1450 y Ft(BR)1637 1438 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b(=)g Fr(\022)1994 1450 y Ft(+)2049 1438 y Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(\022)2343 1450 y Ft(+)3098 1438 y FA(\(A.3\))599 1642 y(that,)34 b(at)e(least)f(for)h(not)g(to)r(o)g (large)e Fr(\020)6 b FA(,)34 b(is)e(semib)r(ounded)g(from)g(b)r(elo)n (w.)50 b(More)31 b(pre-)599 1742 y(cisely)-7 b(,)20 b(strict)f(p)r (ositivit)n(y)f(of)h(the)g(atomic)f(Bro)n(wn-Ra)n(v)n(enhall)d (energies)i(of)i Fr(h)2964 1754 y Ft(BR)3062 1742 y FA(\(0)p Fr(;)14 b FA(2)p Fr(\020)6 b FA(\))599 1841 y(can)37 b(b)r(e)h(demonstrated)f([66)o(])h(if)g(and)f(only)g(if)h(0)h Fs(\024)g Fr(\020)46 b Fs(\024)39 b FA(\()2531 1809 y Fo(\031)p 2531 1823 41 4 v 2535 1870 a Ft(2)2607 1841 y FA(+)2711 1809 y Ft(2)p 2707 1823 V 2707 1870 a Fo(\031)2757 1841 y FA(\))2789 1811 y Fl(\000)p Ft(1)2879 1841 y FA(;)j(for)37 b(larger)599 1941 y Fr(\020)6 b FA(,)46 b(the)d(op)r(erator)d Fr(h)1265 1953 y Ft(BR)1363 1941 y FA(\(0)p Fr(;)14 b FA(2)p Fr(\020)6 b FA(\))42 b(is)g(no)f(longer)g(b)r(ounded)h(from)g(b) r(elo)n(w)f([67)o(].)80 b(F)-7 b(or)599 2041 y Fr(R)45 b(>)f FA(0,)e(the)f(molecular)e(energies)g(are)g(sho)n(wn)g([68)o(,)h (69)o(])h(to)f(b)r(e)g(nonnegativ)n(e)f(if)599 2140 y(0)49 b Fs(\024)g Fr(\020)56 b Fs(\024)49 b FA(2)p Fr(=)p FA(\()1136 2107 y Fo(\031)p 1135 2121 V 1139 2169 a Ft(2)1215 2140 y FA(+)1323 2107 y Ft(2)p 1319 2121 V 1319 2169 a Fo(\031)1369 2140 y FA(\))h Fs(\031)f FA(0)p Fr(:)p FA(9060.)82 b(In)44 b(the)f(atomic)g(case,)k(it)d(is)f(w)n(ell-kno)n(wn)599 2240 y([67,)35 b(66)o(])h(that)g(the)g(Bro)n(wn-Ra)n(v)n(enhall)c (energies)j(pro)n(vide)f(lo)n(w)n(er)g(b)r(ounds)i(on)f(the)599 2339 y(corresp)r(onding)i(Dirac-Coulom)n(b)f(energies.)68 b(Here,)40 b(emplo)n(ying)d(the)i(v)-5 b(ariational)599 2439 y(principle)28 b(ab)r(o)n(v)n(e,)e(w)n(e)i(extend)f(these)h(b)r (ounds)g(to)f(the)h(molecular)f(situation.)682 2615 y Fj(Theorem)k(A.3.)76 b Fi(Assume)35 b FA(0)e Fr(<)h(\020)40 b(<)34 b FA(\()2008 2583 y Fo(\031)p 2008 2597 V 2012 2644 a Ft(2)2082 2615 y FA(+)2183 2583 y Ft(2)p 2179 2597 V 2179 2644 a Fo(\031)2230 2615 y FA(\))2262 2585 y Fl(\000)p Ft(1)2351 2615 y Fi(,)k(or,)g(if)e Fr(R)f(>)f FA(0)p Fi(,)71 b FA(0)33 b Fr(<)g(\020)41 b(<)599 2715 y FA(2)p Fr(=)p FA(\()725 2682 y Fo(\031)p 725 2696 V 729 2744 a Ft(2)782 2715 y FA(+)868 2682 y Ft(2)p 864 2696 V 864 2744 a Fo(\031)915 2715 y FA(\))p Fi(.)c(Then)25 b(the)f(eigenvalues)i FA(\()p Fr(e)1853 2727 y Ft(BR)1951 2715 y FA(\))1983 2727 y Fo(k)2025 2715 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))25 b Fi(of)g Fr(h)2397 2727 y Ft(BR)2495 2715 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))25 b Fi(ar)l(e)g(never)g (lar)l(ger)599 2815 y(than)30 b(those)h(of)f Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fi(,)1509 3019 y FA(\()p Fr(e)1580 3031 y Ft(BR)1678 3019 y FA(\))1710 3031 y Fo(k)1752 3019 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))23 b Fs(\024)g Fr(e)2109 3031 y Fo(k)2149 3019 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))p Fr(:)719 b FA(\(A.4\))682 3386 y Fi(Pr)l(o)l(of.)111 b FA(Since)37 b(the)h(prerequisites)f(of)g (theorem)h(A.1)f(are)g(satis\014ed)g(\(with)h Fr(\022)3233 3398 y Fl(\006)599 3486 y FA(b)r(eing)c(the)f(pro)5 b(jections)32 b(on)n(to)h(the)g(p)r(ositiv)n(e)g(or)g(negativ)n(e,)g(resp)r(ectiv)n (ely)-7 b(,)34 b(sp)r(ectral)599 3586 y(spaces)22 b(of)h Fr(T)988 3598 y Fo(D)1047 3586 y FA(,)h(and,)g(e.g.,)f Fs(F)31 b FA(=)23 b Fr(C)1690 3555 y Fl(1)1684 3606 y Ft(0)1760 3586 y FA(\()p Fq(R)1846 3555 y Ft(3)1890 3586 y Fr(;)14 b Fq(C)1981 3555 y Ft(4)2024 3586 y FA(\)\),)24 b(w)n(e)e(can)h(use)f(the)h(form)n(ula)f(\(A.2\))h(for)599 3685 y(the)32 b(energies)d Fr(e)1102 3697 y Fo(k)1143 3685 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))31 b(of)g Fr(h)p FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))31 b(as)f(w)n(ell)h(as)f (for)g(the)i(energies)d(\()p Fr(e)2812 3697 y Ft(BR)2911 3685 y FA(\))2943 3697 y Fo(k)2984 3685 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))31 b(of)599 3785 y Fr(h)647 3797 y Ft(BR)746 3785 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\).)39 b(In)29 b(the)f(decomp)r(osition)g Fr( )f FA(=)d Fr( )2035 3797 y Ft(+)2109 3785 y Fs(\010)18 b Fr( )2246 3797 y Fl(\000)2331 3785 y FA(w)n(e)27 b(c)n(ho)r(ose)g Fr( )2771 3797 y Fl(\000)2851 3785 y FA(=)d(0)g Fs(2)g(F)3145 3797 y Fl(\000)3225 3785 y FA(=)599 3884 y Fr(\022)638 3896 y Fl(\000)694 3884 y Fs(F)8 b FA(.)78 b(Then)41 b Fs(h)p Fr( )s(;)28 b(h)14 b( )s Fs(i)46 b FA(=)f Fs(h)p Fr( )1626 3896 y Ft(+)1681 3884 y Fr(;)28 b(h)14 b( )1848 3896 y Ft(+)1903 3884 y Fs(i)46 b FA(=)f Fs(h)p Fr(\022)2162 3896 y Ft(+)2217 3884 y Fr( )s(;)28 b(h\022)2412 3896 y Ft(+)2467 3884 y Fr( )s Fs(i)46 b FA(=)f Fs(h)p Fr( )s(;)28 b(h)2900 3896 y Ft(BR)2999 3884 y Fr( )s Fs(i)p FA(,)45 b(and)599 3984 y(therefore)27 b(the)h(b)r(ound)g(\(A.4\))g(turns)g(out)f(to)h(b)r (e)g(a)f(consequence)g(of)g(the)h(inequalit)n(y)1275 4236 y(sup)1194 4307 y Fo( )r Fl(2V)5 b(\010F)1432 4315 y Fe(\000)1278 4372 y Fo( )r Fl(6)p Ft(=0)1537 4180 y Fs(h)p Fr( )s(;)28 b(H)7 b( )s Fs(i)p 1537 4217 306 4 v 1582 4293 a(h)p Fr( )s(;)14 b( )s Fs(i)1898 4236 y(\025)139 b FA(sup)2019 4310 y Fo( )r Fl(2V)5 b(\010f)p Ft(0)p Fl(g)2105 4375 y Fo( )r Fl(6)p Ft(=0)2366 4180 y Fs(h)p Fr( )s(;)28 b(H)7 b( )s Fs(i)p 2366 4217 V 2411 4293 a(h)p Fr( )s(;)14 b( )s Fs(i)2681 4236 y Fr(:)394 b FA(\(A.5\))p 599 4550 25 75 v eop %%Page: 34 34 34 33 bop 599 75 a FA(34)863 b Fz(BRIET)24 b(AND)f(HOGREVE)682 320 y FA(Due)38 b(to)f(the)g(semib)r(oundedness)f(of)h(the)h(Bro)n (wn-Ra)n(v)n(enhall)33 b(Hamiltonian)k(for)599 419 y(0)46 b Fr(<)g(\020)53 b Fs(\024)46 b FA(\()1040 387 y Fo(\031)p 1040 401 41 4 v 1044 448 a Ft(2)1119 419 y FA(+)1225 387 y Ft(2)p 1221 401 V 1221 448 a Fo(\031)1272 419 y FA(\))1304 389 y Fl(\000)p Ft(1)1393 419 y FA(,)f(w)n(e)d(can)f(apply)g (standard)g(Ra)n(yleigh-Ritz)f(v)-5 b(ariational)599 519 y(tec)n(hniques)35 b(to)f Fr(h)1169 531 y Ft(BR)1268 519 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))35 b(=)1619 486 y Ft(1)p 1619 500 34 4 v 1619 548 a(2)1662 519 y Fr(\022)1701 531 y Ft(+)1756 519 y Fr(h)1804 489 y Ft(at)1867 519 y FA(\()p Fs(\000)p Fr(R)q(=)p FA(2)p Fr(;)27 b FA(2)p Fr(\020)6 b FA(\))p Fr(\022)2317 531 y Ft(+)2395 519 y FA(+)2493 486 y Ft(1)p 2493 500 V 2493 548 a(2)2536 519 y Fr(\022)2575 531 y Ft(+)2630 519 y Fr(h)2678 489 y Ft(at)2741 519 y FA(\()p Fr(R)q(=)p FA(2)p Fr(;)27 b FA(2)p Fr(\020)6 b FA(\))p Fr(\022)3126 531 y Ft(+)3216 519 y FA(to)599 619 y(conclude)29 b Fr(e)981 631 y Ft(BR)1079 619 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))26 b Fs(\025)e Fr(e)1440 588 y Ft(at)1440 641 y(BR)1538 619 y FA(\(2)p Fr(\020)6 b FA(\))30 b(with)f Fr(e)1945 588 y Ft(at)1945 641 y(BR)2043 619 y FA(\(2)p Fr(\020)6 b FA(\))30 b(standing)e(for)g (the)h(ground)f(state)599 718 y(energy)35 b(of)h(the)g(atomic)f(Bro)n (wn-Ra)n(v)n(enhall)e(Hamiltonian.)61 b(Hence,)38 b(as)d(a)g(conse-)599 818 y(quence)d(of)f(theorem)h(A.3,)g(for)f Fr(\020)39 b FA(in)31 b(this)h(in)n(terv)-5 b(al)31 b(the)h(atomic)g(ground)e (state)i(en-)599 917 y(ergies)27 b Fr(e)870 887 y Ft(at)870 940 y(BR)968 917 y FA(\(2)p Fr(\020)6 b FA(\))29 b(ma)n(y)e(serv)n(e)f (as)h(lo)n(w)n(er)f(b)r(ounds)i(for)f(the)h(discrete)f(sp)r(ectrum)h (of)g(the)599 1017 y(atomic)g(and)f(molecular)f(Dirac-Coulom)n(b)g (energies.)682 1117 y(F)-7 b(rom)39 b(the)g(pro)r(of)g(giv)n(en)f(ab)r (o)n(v)n(e)g(it)h(should)g(b)r(e)g(clear)f(that)h(theorem)g(A.3)g(can) 599 1216 y(b)r(e)f(extended)f(to)g(all)g(p)r(oten)n(tials)f Fr(V)56 b FA(for)37 b(whic)n(h)f(the)i(v)-5 b(ariational)35 b(represen)n(tation)599 1316 y(\(A.2\))26 b(for)e(Dirac)h(energies)e (of)i Fr(T)1615 1328 y Fo(D)1688 1316 y FA(+)13 b Fr(V)43 b FA(holds,)25 b(and)g(for)f(whic)n(h)h(sim)n(ultaneously)f(the)599 1416 y(eigen)n(v)-5 b(alues)33 b(of)h(the)h(corresp)r(onding)d(Bro)n (wn-Ra)n(v)n(enhall)f(Hamiltonian)j Fr(\022)3006 1428 y Ft(+)3061 1416 y FA(\()p Fr(T)3142 1428 y Fo(D)3225 1416 y FA(+)599 1515 y Fr(V)19 b FA(\))p Fr(\022)737 1527 y Ft(+)833 1515 y FA(exist.)76 b(In)41 b(particular,)h(if)f(for)f (0)k Fs(\024)g Fr(t)h(<)f FA(1)c(one)h(regards)d(the)j(family)f(of)599 1615 y(op)r(erators)24 b Fr(T)1014 1627 y Fo(D)1089 1615 y FA(+)15 b Fr(tV)44 b FA(with)27 b(p)r(oten)n(tials)e(that)i(satisfy)e (sup)2421 1635 y Fk(x)p Fd(R)2511 1618 y Fg(3)2542 1548 y Fp(\000)2581 1615 y Fs(j)p Fn(x)o Fs(j)p Fr(V)19 b FA(\()p Fn(x)p FA(\))2867 1548 y Fp(\001)2929 1615 y Fs(\024)j FA(1,)k(up)r(on)599 1714 y(sho)n(wing)32 b(that)i(for)e (su\016cien)n(tly)i(small)e Fr(t)h(>)e FA(0)i(the)h(asso)r(ciated)d (Bro)n(wn-Ra)n(v)n(enhall)599 1814 y(ground)i(state)h(energies)f(of)h Fr(\022)1558 1826 y Ft(+)1613 1814 y FA(\()p Fr(T)1694 1826 y Fo(D)1777 1814 y FA(+)22 b Fr(tV)d FA(\))p Fr(\022)2032 1826 y Ft(+)2121 1814 y FA(are)33 b(nonnegativ)n(e,)i(our)e(argumen)n (ts)599 1914 y(from)h(lemma)f(3.1)g(b)r(ecome)g(applicable)g(to)g(rule) h(out)f(zero-energy)e(eigenstates)h(of)599 2013 y Fr(T)648 2025 y Fo(D)730 2013 y FA(+)21 b Fr(tV)52 b FA(for)32 b(all)h(0)e Fr(<)g(t)h(<)f FA(1.)52 b(The)32 b(con)n(tin)n(uit)n(y)g (of)h(the)g(Dirac)f(eigen)n(v)-5 b(alues)32 b(then)599 2113 y(prev)n(en)n(ts)27 b(a)h(c)n(hange)e(of)i(their)g(sign)g(when)g Fr(t)g FA(gro)n(ws)e(from)h(smaller)g(to)h(larger)e(v)-5 b(alues)599 2213 y(within)27 b(the)g(in)n(terv)-5 b(al)25 b(\(0)p Fr(;)14 b FA(1\).)36 b(Hence)26 b(the)h(Hamiltonian)e Fr(T)2451 2225 y Fo(D)2526 2213 y FA(+)15 b Fr(tV)45 b FA(has)26 b(no)g(negativ)n(e)599 2312 y(discrete)f(sp)r(ectrum)f(if)i Fr(t)d Fs(2)g FA(\(0)p Fr(;)14 b FA(1\).)36 b(In)24 b(this)h(w)n(a)n(y) -7 b(,)25 b(an)f(alternativ)n(e)f(and)i(rather)f(direct)599 2412 y(pro)r(of)j(of)h(theorem)f(I)r(I.9)h(in)f(Ref.)i([34)o(])e(is)h (ac)n(hiev)n(ed.)1634 2677 y Fu(APPENDIX)k(B)1152 2785 y(Matrix)g(elemen)m(ts)d(and)k(b)s(ounds)e(for)h Fm(h)p Fn(H)2659 2755 y Fc(2)2700 2785 y Fm(i)682 2902 y FA(The)g(simple)h (ansatz)e(\(4.31\))g(for)g(the)i(spinor)e Fr( )k FA(allo)n(ws)30 b(explicit)i(calculation)f(of)599 3001 y(most)21 b(matrix)f(elemen)n (ts)h(in)g(the)h(v)-5 b(ariational)19 b(expression)g(for)i Fs(h)p Fr( )s(;)28 b(H)7 b FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))2962 2971 y Ft(2)2999 3001 y Fr( )s Fs(i)p FA(.)35 b(The)599 3101 y(remaining)g(in)n(tegrals)g(in)h Fs(h)p Fr( )s(;)28 b(V)1632 3113 y Ft(ne)1703 3101 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))1910 3071 y Ft(2)1947 3101 y Fr( )s Fs(i)37 b FA(will)f(b)r(e)g(estimated)g(b)n(y)f(appropriate) 599 3200 y(upp)r(er)26 b(b)r(ounds.)36 b(Belo)n(w,)25 b(w)n(e)f(collect)h(the)h(matrix)e(elemen)n(ts)h(required)f(in)i(the)f (pro)r(of)599 3300 y(of)j(theorem)f(4.2,)g(i.e.,)h(for)f(the)h(disso)r (ciation)e(stabilit)n(y)i(of)f(H)2531 3265 y Ft(+)2531 3322 y(2)2587 3300 y FA(.)682 3400 y(As)35 b(for)g(the)g (nonrelativistic)f(t)n(w)n(o-cen)n(tre)f(system,)j(man)n(y)e(of)h(the)g (arising)f(in)n(te-)599 3499 y(grals)d(can)h(b)r(e)h(computed)f(rather) f(straigh)n(tforw)n(ardly)e(after)j(c)n(hanging)f(to)h(prolate)599 3599 y(spheroidal)19 b(co)r(ordinates.)33 b(In)20 b(particular,)g(for)f (the)h(norm)f Fs(jj)p Fr( )s Fs(jj)2568 3569 y Ft(2)2626 3599 y FA(of)g(\(4.31\))g(w)n(e)h(obtain)1136 3925 y Fs(h)p Fr( )s(;)28 b( )s Fs(i)c FA(=)1532 3869 y(1)p 1486 3906 133 4 v 1486 3982 a(2)p Fr(\013)1581 3958 y Ft(3)1628 3833 y Fp(\020)1678 3925 y FA(1)18 b(+)g(\(1)g(+)g Fr(\013R)i FA(+)2225 3893 y Ft(1)p 2225 3907 34 4 v 2225 3954 a(3)2268 3925 y Fr(\013)2321 3891 y Ft(2)2359 3925 y Fr(R)2423 3891 y Ft(2)2459 3925 y FA(\))p Fr(e)2530 3891 y Fl(\000)p Fo(\013R)2680 3833 y Fp(\021)2729 3925 y Fr(:)349 b FA(\(B.1\))599 4246 y(T)-7 b(o)28 b(determine)f Fs(h)p Fr( )s(;)h(T)1310 4216 y Ft(2)1298 4269 y Fo(D)1358 4246 y Fr( )s Fs(i)p FA(,)g(w)n(e)f(fall)h(bac)n(k)f(on)n(to)g(the)h (relation)1701 4533 y Fr(T)1762 4499 y Ft(2)1750 4554 y Fo(D)1833 4533 y FA(=)22 b Fs(\000)p FA(\001)d(+)f Fq(I)900 b FA(\(B.2\))p eop %%Page: 35 35 35 34 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(35)599 320 y(and)28 b(compute)1128 495 y Fs(h)p Fr( )s(;)g Fs(\000)p FA(\001)p Fr( )s Fs(i)p 1128 533 364 4 v 1202 609 a(h)p Fr( )s(;)14 b( )s Fs(i)1524 552 y FA(=)23 b Fr(\013)1665 517 y Ft(2)1736 491 y FA(1)18 b(+)g(\(1)g(+)g Fr(\013R)h Fs(\000)2282 458 y Ft(1)p 2282 472 34 4 v 2282 519 a(3)2326 491 y Fr(\013)2379 460 y Ft(2)2416 491 y Fr(R)2480 460 y Ft(2)2517 491 y FA(\))p Fr(e)2588 460 y Fl(\000)p Fo(\013R)p 1736 533 1002 4 v 1736 613 a FA(1)f(+)g(\(1)g(+)g Fr(\013R)h FA(+)2282 580 y Ft(1)p 2282 594 34 4 v 2282 641 a(3)2326 613 y Fr(\013)2379 589 y Ft(2)2416 613 y Fr(R)2480 589 y Ft(2)2517 613 y FA(\))p Fr(e)2588 589 y Fl(\000)p Fo(\013R)2748 552 y Fr(:)330 b FA(\(B.3\))599 784 y(Since)26 b(in)g(our)f(ansatz)f (for)h Fr( )k FA(the)d(lo)n(w)n(er)e(comp)r(onen)n(ts)h(v)-5 b(anish,)26 b(while,)g(on)f(the)h(other)599 883 y(hand,)39 b(the)e(non)n(trivial)f(results)g(of)h(the)g(application)f(of)h Fr(T)2501 895 y Fo(D)2597 883 y FA(reside)f(in)h(the)g(lo)n(w)n(er)599 983 y(comp)r(onen)n(ts)28 b(of)f Fr(T)1200 995 y Fo(D)1260 983 y Fr( )s FA(,)h(the)g(exp)r(ectation)f(v)-5 b(alue)27 b(of)h Fr(T)2316 995 y Fo(D)2403 983 y FA(just)h(b)r(ecomes)1601 1159 y Fs(h)p Fr( )s(;)f(T)1790 1171 y Fo(D)1849 1159 y Fr( )s Fs(i)c FA(=)f Fs(h)p Fr( )s(;)14 b( )s Fs(i)p Fr(:)813 b FA(\(B.4\))599 1335 y(In)28 b(the)g(same)f(w)n(a)n(y)-7 b(,)27 b(it)h(turns)f(out)h(that)1475 1511 y Fs(h)p Fr(T)1556 1523 y Fo(D)1616 1511 y Fr( )s(;)g(V)1772 1523 y Ft(ne)1843 1511 y Fr( )s Fs(i)23 b FA(=)g Fs(h)p Fr( )s(;)28 b(V)2231 1523 y Ft(ne)2301 1511 y Fr( )s Fs(i)p Fr(:)688 b FA(\(B.5\))599 1687 y(The)28 b(exp)r(ectation)f(v)-5 b(alue)28 b(of)g Fr(V)1575 1699 y Ft(ne)1645 1687 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))29 b(is)e(explicit,)684 1868 y Fs(h)p Fr( )s(;)h(V)872 1880 y Ft(ne)943 1868 y Fr( )s Fs(i)p 684 1905 349 4 v 751 1981 a(h)p Fr( )s(;)14 b( )s Fs(i)1066 1924 y FA(=)22 b Fs(\000)p Fr(\020)6 b(\013)1347 1863 y FA(1)18 b(+)g(\()p Fr(\013R)q FA(\))1671 1833 y Fl(\000)p Ft(1)1761 1796 y Fp(\000)1799 1863 y FA(1)g Fs(\000)g FA(\(1)g(+)g Fr(\013R)q FA(\))p Fr(e)2305 1833 y Fl(\000)p Ft(2)p Fo(\013R)2487 1796 y Fp(\001)2544 1863 y FA(+)g(2\(1)g(+)g Fr(\013R)q FA(\))p Fr(e)3032 1833 y Fl(\000)p Fo(\013R)p 1347 1905 1835 4 v 1763 1985 a FA(1)g(+)g(\(1)g(+)g Fr(\013R)i FA(+)2310 1953 y Ft(1)p 2310 1967 34 4 v 2310 2014 a(3)2353 1985 y Fr(\013)2406 1961 y Ft(2)2444 1985 y Fr(R)2508 1961 y Ft(2)2545 1985 y FA(\))p Fr(e)2616 1961 y Fl(\000)p Fo(\013R)3191 1924 y Fr(;)3101 2085 y FA(\(B.6\))599 2184 y(whereas,)26 b(unfortunately)-7 b(,)27 b(not)g(all)g(in)n (tegrals)f(for)g(of)h Fr(V)2338 2196 y Ft(ne)2409 2184 y FA(\()p Fr(R)q(;)14 b(\020)6 b FA(\))2616 2154 y Ft(2)2681 2184 y FA(can)26 b(b)r(e)i(expressed)599 2284 y(in)g(terms)g(of)f (elemen)n(tary)g(functions,)623 2452 y Fs(h)p Fr( )s(;)h(V)830 2422 y Ft(2)811 2473 y(ne)882 2452 y Fr( )s Fs(i)p 623 2489 349 4 v 690 2565 a(h)p Fr( )s(;)14 b( )s Fs(i)1004 2508 y FA(=)1513 2452 y Fr(\020)1555 2422 y Ft(2)1593 2452 y Fr(\013)p 1079 2489 1002 4 v 1079 2569 a FA(1)k(+)g(\(1)g(+)g Fr(\013R)i FA(+)1626 2537 y Ft(1)p 1626 2551 34 4 v 1626 2598 a(3)1669 2569 y Fr(\013)1722 2545 y Ft(2)1759 2569 y Fr(R)1823 2545 y Ft(2)1860 2569 y FA(\))p Fr(e)1931 2545 y Fl(\000)p Fo(\013R)2114 2416 y Fp(\020)2173 2452 y Fr(\013)2226 2422 y Ft(2)p 2173 2489 91 4 v 2187 2565 a Fr(R)2288 2395 y Fp(Z)2371 2416 y Fl(1)2334 2584 y Ft(0)2455 2508 y Fr(dr)d(r)r(e)2630 2474 y Fl(\000)p Ft(2)p Fo(\013r)2809 2508 y FA(ln)2878 2441 y Fp(\000)2926 2452 y Fr(R)i FA(+)f Fr(r)p 2926 2489 205 4 v 2926 2565 a(R)h Fs(\000)f Fr(r)3141 2441 y Fp(\001)3179 2458 y Ft(2)1170 2749 y FA(+)h(4)p Fr(\013)1349 2715 y Ft(2)1386 2749 y Fr(R)1463 2636 y Fp(Z)1546 2656 y Fl(1)1509 2824 y Ft(1)1630 2749 y Fr(d\030)g(\030)t(e)1807 2715 y Fl(\000)p Fo(\013R\030)2002 2749 y FA(ln)2071 2682 y Fp(\000)2119 2693 y Fr(\030)k FA(+)18 b(1)p 2119 2730 184 4 v 2119 2806 a Fr(\030)23 b Fs(\000)18 b FA(1)2312 2682 y Fp(\001)2392 2749 y FA(+)g(2)p Fr(\013)g FA(+)2692 2693 y(2)p 2681 2730 64 4 v 2681 2806 a Fr(R)2755 2682 y Fp(\000)2793 2749 y FA(1)g Fs(\000)g Fr(e)2975 2715 y Fl(\000)p Ft(2)p Fo(\013R)3157 2682 y Fp(\001)3195 2657 y(\021)3245 2749 y Fr(:)3101 2895 y FA(\(B.7\))599 2995 y(An)28 b(upp)r(er)g(b)r(ound)f (on)g(the)h(\014rst)f(in)n(tegral)f(in)i(Eq.)f(\(B.7\))g(is)g(deriv)n (ed)g(b)n(y)g(estimating)599 3094 y(\(for)h Fr(R)c Fs(\025)e FA(1\))691 3420 y(ln)760 3328 y Fp(\020)819 3364 y Fr(R)e FA(+)e Fr(r)p 819 3401 205 4 v 819 3477 a(R)i Fs(\000)e Fr(r)1034 3328 y Fp(\021)1084 3345 y Ft(2)1167 3420 y Fs(\024)1278 3200 y Fp(8)1278 3274 y(>)1278 3299 y(>)1278 3324 y(<)1278 3474 y(>)1278 3499 y(>)1278 3524 y(:)1377 3269 y FA(ln\(2)p Fr(R)h Fs(\000)f FA(1\))1759 3239 y Ft(2)1796 3269 y Fr(;)610 b(r)26 b Fs(2)d FA([0)p Fr(;)14 b(R)19 b Fs(\000)f FA(1])1377 3369 y(ln\(2)p Fr(R)q FA(\))1616 3339 y Ft(2)1671 3369 y Fs(\000)g FA(ln\()p Fr(R)i Fs(\000)e Fr(r)r FA(\))2092 3339 y Ft(2)2130 3369 y Fr(;)276 b(r)26 b Fs(2)d FA([)p Fr(R)d Fs(\000)e FA(1)p Fr(;)c(R)q FA(])1377 3469 y(ln\(2)p Fr(R)19 b FA(+)f(1\))1759 3439 y Ft(2)1814 3469 y Fs(\000)g FA(ln\()p Fr(r)k Fs(\000)c Fr(R)q FA(\))2236 3439 y Ft(2)2273 3469 y Fr(;)133 b(r)26 b Fs(2)d FA([)p Fr(R)q(;)14 b(R)19 b FA(+)f(1])1377 3568 y(ln\(2)p Fr(R)h FA(+)f(1\))1759 3538 y Ft(2)2429 3568 y Fr(r)26 b Fs(2)d FA([)p Fr(R)d FA(+)e(1)p Fr(;)c Fs(1)p FA(\))p Fr(;)3101 3420 y FA(\(B.8\))599 3741 y(whic)n(h,)30 b(together)f(with)h(similar)e (estimates)h(of)h(the)f(exp)r(onen)n(tial)g(function)h(for)f Fr(r)g Fs(2)599 3840 y FA([)p Fr(R)20 b Fs(\000)e FA(1)p Fr(;)c(R)q FA(])26 b(and)i Fr(r)e Fs(2)d FA([)p Fr(R)q(;)14 b(R)19 b FA(+)f(1],)27 b(up)r(on)h(in)n(tegration)e(yields)713 3950 y Fp(Z)796 3971 y Fl(1)759 4139 y Ft(0)880 4063 y Fr(dr)17 b(r)s(e)1056 4029 y Fl(\000)p Ft(2)p Fo(\013r)1234 4063 y FA(ln)1304 3971 y Fp(\020)1363 4007 y Fr(R)i FA(+)f Fr(r)p 1363 4044 V 1363 4120 a(R)h Fs(\000)f Fr(r)1578 3971 y Fp(\021)1628 3988 y Ft(2)1711 4063 y Fs(\024)1877 4007 y FA(1)p 1832 4044 133 4 v 1832 4120 a(4)p Fr(\013)1927 4096 y Ft(2)1974 3971 y Fp(\020)2023 4063 y FA(ln\(2)p Fr(R)h Fs(\000)f FA(1\))2405 4029 y Ft(2)2461 4063 y FA(+)g(\(1)g Fs(\000)g FA(2)p Fr(\013)h FA(+)f(2)p Fr(\013R)q FA(\))1035 4268 y Fs(\002)g Fr(e)1157 4233 y Fl(\000)p Ft(2)p Fo(\013)p Ft(\()p Fo(R)p Fl(\000)p Ft(1\))1490 4268 y FA(ln)1560 4200 y Fp(\000)1650 4235 y Ft(2)p Fo(R)p 1608 4249 169 4 v 1608 4296 a Ft(2)p Fo(R)p Fl(\000)p Ft(1)1786 4200 y Fp(\001)1824 4217 y Ft(2)1880 4268 y FA(+)g(\(1)g(+)g(2)p Fr(\013R)q FA(\))p Fr(e)2368 4233 y Fl(\000)p Ft(2)p Fo(\013R)2564 4268 y FA(ln)2633 4200 y Fp(\000)2681 4235 y Ft(2)p Fo(R)p Ft(+1)p 2681 4249 168 4 v 2723 4296 a(2)p Fo(R)2859 4200 y Fp(\001)2897 4217 y Ft(2)2934 4175 y Fp(\021)1118 4433 y FA(+)g(\(2)p Fr(R)h Fs(\000)1450 4400 y Ft(1)p 1450 4414 34 4 v 1450 4461 a(2)1493 4433 y FA(\))p Fr(e)1564 4398 y Fl(\000)p Ft(2)p Fo(\013)p Ft(\()p Fo(R)p Fl(\000)p Ft(1\))1902 4433 y FA(+)f(\(2)p Fr(R)h FA(+)2234 4400 y Ft(1)p 2234 4414 V 2234 4461 a(2)2277 4433 y FA(\))p Fr(e)2348 4398 y Fl(\000)p Ft(2)p Fo(\013R)2531 4433 y Fr(:)3101 4533 y FA(\(B.9\))p eop %%Page: 36 36 36 35 bop 599 75 a FA(36)863 b Fz(BRIET)24 b(AND)f(HOGREVE)599 320 y FA(The)28 b(second)f(in)n(tegral)f(in)i(\(B.7\))g(can)f(b)r(e)h (b)r(ounded)g(analogously)-7 b(,)697 463 y Fp(Z)780 483 y Fl(1)743 652 y Ft(1)864 576 y Fr(d\030)18 b(\030)t(e)1040 542 y Fl(\000)p Fo(\013R\030)1235 576 y FA(ln)1304 509 y Fp(\000)1352 520 y Fr(\030)23 b FA(+)18 b(1)p 1352 557 184 4 v 1352 633 a Fr(\030)23 b Fs(\000)18 b FA(1)1546 509 y Fp(\001)1630 576 y Fs(\024)45 b Fr(e)1779 542 y Fl(\000)p Fo(\013R)1942 463 y Fp(Z)2025 483 y Ft(4)p Fo(=)p Ft(3)1989 652 y(1)2144 576 y Fr(d\030)18 b(\030)g FA(ln)2364 509 y Fp(\000)2412 520 y Fr(\030)23 b FA(+)18 b(1)p 2412 557 V 2412 633 a Fr(\030)23 b Fs(\000)18 b FA(1)2605 509 y Fp(\001)2182 802 y FA(+)41 b(ln\(7\))2477 689 y Fp(Z)2560 710 y Fl(1)2523 878 y Ft(4)p Fo(=)p Ft(3)2644 802 y Fr(d\030)18 b(\030)t(e)2820 768 y Fl(\000)p Fo(\013R\030)1024 1021 y FA(=)1135 929 y Fp(\020)1211 988 y Ft(7)p 1194 1002 67 4 v 1194 1050 a(18)1284 1021 y FA(ln)q(\(7\))g(+)1571 988 y Ft(1)p 1571 1002 34 4 v 1571 1050 a(3)1614 929 y Fp(\021)1664 1021 y Fr(e)1703 987 y Fl(\000)p Fo(\013R)1870 1021 y FA(+)g(ln)q(\(7\)\()p Fr(\013R)q FA(\))2310 987 y Fl(\000)p Ft(2)2399 929 y Fp(\020)2449 1021 y FA(1)g(+)2602 988 y Ft(4)p 2602 1002 V 2602 1050 a(3)2645 1021 y Fr(\013R)2762 929 y Fp(\021)2812 1021 y Fr(e)2851 987 y Fl(\000)p Ft(4)p Fo(\013R=)p Ft(3)3100 1021 y Fr(:)3060 1146 y FA(\(B.10\))599 1245 y(In)29 b(the)g(pro)r(of)f(of)h(theorem)f(4.2,)g(w)n(e)h(c)n(ho)r (ose)e(the)i(in)n(tern)n(uclear)e(separation)g Fr(R)j FA(suc)n(h)599 1345 y(that)42 b Fr(\013R)k FA(=)g(2.)77 b(Moreo)n(v)n(er,)41 b(w)n(e)g(only)g(consider)f Fr(\013)46 b Fs(2)g FA(\(0)p Fr(;)14 b FA(1)p Fr(=)p FA(4].)75 b(Under)41 b(these)599 1444 y(prerequisites,)34 b(after)f(inserting)g(the)g (inequalities)g(\(B.9\))g(and)h(\(B.10\))e(in)n(to)h(\(B.7\),)599 1544 y(monotonicit)n(y)25 b(argumen)n(ts)f(can)g(b)r(e)i(in)n(v)n(ok)n (ed)e(to)h(estimate)g(further)g(the)g(terms)g(that)599 1644 y(arise)i(in)h(the)g(resulting)f(b)r(ound)h(on)f Fs(h)p Fr( )s(;)h(V)1954 1614 y Ft(2)1935 1664 y(ne)2006 1644 y Fr( )s Fs(i)p Fr(=)p Fs(h)p Fr( )s(;)14 b( )s Fs(i)p FA(,)28 b(viz.,)877 1828 y Fr(\013)p 877 1865 54 4 v 883 1941 a FA(4)955 1884 y(ln)1024 1792 y Fp(\020)1089 1828 y FA(4)p 1083 1865 V 1083 1941 a Fr(\013)1165 1884 y Fs(\000)18 b FA(1)1290 1792 y Fp(\021)1362 1884 y Fs(\024)1450 1767 y Fp(\024)1504 1828 y Fr(\013)p 1504 1865 V 1510 1941 a FA(4)1581 1884 y(ln)1650 1792 y Fp(\020)1716 1828 y FA(4)p 1710 1865 V 1710 1941 a Fr(\013)1791 1884 y Fs(\000)g FA(1)1916 1792 y Fp(\021)1966 1767 y(\025)2009 1967 y Fo(\013)p Ft(=1)p Fo(=)p Ft(4)2231 1884 y FA(=)2329 1828 y(ln\(15\))p 2329 1865 217 4 v 2395 1941 a(16)2578 1884 y Fr(<)2676 1828 y FA(10)p 2676 1865 84 4 v 2676 1941 a(59)2769 1884 y Fr(;)268 b FA(\(B.11\))599 2139 y(and,)28 b(with)g(the)g(follo)n(wing)f(expression)f(b)r(eing)h (regarded)f(as)h(co)r(e\016cien)n(t)h(of)f Fr(\013)p FA(,)722 2293 y Ft(1)p 722 2307 34 4 v 722 2355 a(4)765 2326 y FA(\(5)18 b Fs(\000)g FA(2)p Fr(\013)p FA(\))p Fr(e)1106 2292 y Fl(\000)p Ft(4+2)p Fo(\013)1336 2326 y FA(ln)1406 2259 y Fp(\000)1501 2293 y Ft(4)p 1454 2307 129 4 v 1454 2355 a(4)p Fl(\000)p Fo(\013)1592 2259 y Fp(\001)1648 2326 y FA(+)1741 2293 y Ft(5)p 1741 2307 34 4 v 1741 2355 a(4)1784 2326 y Fr(e)1823 2292 y Fl(\000)p Ft(4)1926 2326 y FA(ln)1995 2259 y Fp(\000)2043 2293 y Ft(4+)p Fo(\013)p 2043 2307 128 4 v 2090 2355 a Ft(4)2180 2259 y Fp(\001)2237 2326 y FA(+)g(8)2362 2259 y Fp(\000)2399 2326 y FA(\()2458 2293 y Ft(7)p 2442 2307 67 4 v 2442 2355 a(18)2532 2326 y FA(ln\(7\))g(+)2818 2293 y Ft(1)p 2818 2307 34 4 v 2818 2355 a(3)2861 2326 y FA(\))p Fr(e)2932 2292 y Fl(\000)p Ft(2)784 2473 y FA(+)877 2440 y Ft(11)p 877 2454 67 4 v 877 2501 a(12)966 2473 y FA(ln)q(\(7\))p Fr(e)1181 2439 y Fl(\000)p Ft(8)p Fo(=)p Ft(3)1337 2406 y Fp(\001)1393 2473 y FA(+)g(3)g Fs(\000)g Fr(e)1658 2439 y Fl(\000)p Ft(4)1793 2473 y Fs(\024)1914 2440 y Ft(5)p 1914 2454 34 4 v 1914 2501 a(4)1957 2473 y Fr(e)1996 2439 y Fl(\000)p Ft(4+2)p Fo(\013)2226 2473 y FA(ln)2295 2406 y Fp(\000)2390 2440 y Ft(4)p 2343 2454 129 4 v 2343 2501 a(4)p Fl(\000)p Fo(\013)2481 2406 y Fp(\001)2537 2473 y FA(+)2630 2440 y Ft(5)p 2630 2454 34 4 v 2630 2501 a(4)2673 2473 y Fr(e)2712 2439 y Fl(\000)p Ft(4)2815 2473 y FA(ln)2884 2406 y Fp(\000)2932 2440 y Ft(4+)p Fo(\013)p 2932 2454 128 4 v 2979 2501 a Ft(4)3070 2406 y Fp(\001)1116 2619 y FA(+)g(8)1241 2552 y Fp(\000)1278 2619 y FA(\()1337 2587 y Ft(7)p 1320 2601 67 4 v 1320 2648 a(18)1410 2619 y FA(ln)q(\(7\))g(+)1697 2587 y Ft(1)p 1697 2601 34 4 v 1697 2648 a(3)1740 2619 y FA(\))p Fr(e)1811 2585 y Fl(\000)p Ft(2)1919 2619 y FA(+)2012 2587 y Ft(11)p 2012 2601 67 4 v 2012 2648 a(12)2102 2619 y FA(ln\(7\))p Fr(e)2316 2585 y Fl(\000)p Ft(8)p Fo(=)p Ft(3)2472 2552 y Fp(\001)2528 2619 y FA(+)g(3)g Fs(\000)g Fr(e)2793 2585 y Fl(\000)p Ft(4)1784 2753 y Fs(\024)46 b FA([)p Fr(:)14 b(:)g(:)p FA(])2038 2778 y Fo(\013)p Ft(=1)p Fo(=)p Ft(4)2283 2753 y Fr(<)45 b FA(3)18 b(+)2546 2720 y Ft(89)p 2546 2734 V 2546 2781 a(69)2622 2753 y Fr(;)3060 2863 y FA(\(B.12\))599 2962 y(and)28 b(corresp)r(ondingly)d(the)j(co)r (e\016cien)n(t)g(of)f Fr(\013)2039 2932 y Ft(2)2077 2962 y Fr(=)p FA(2)687 3154 y(\(4)19 b Fs(\000)873 3121 y Fo(\013)p 873 3135 44 4 v 878 3183 a Ft(2)926 3154 y FA(\))p Fr(e)997 3120 y Fl(\000)p Ft(4+2)p Fo(\013)1232 3154 y FA(+)f(\(4+)1463 3121 y Fo(\013)p 1463 3135 V 1468 3183 a Ft(2)1516 3154 y FA(\))p Fr(e)1587 3120 y Fl(\000)p Ft(4)1722 3154 y Fs(\024)46 b FA(4)p Fr(e)1914 3120 y Fl(\000)p Ft(4+2)p Fo(\013)2148 3154 y FA(+)18 b(\(4)g(+)2416 3121 y Fo(\013)p 2416 3135 V 2421 3183 a Ft(2)2469 3154 y FA(\))p Fr(e)2540 3120 y Fl(\000)p Ft(4)1476 3295 y Fs(\024)46 b FA([)p Fr(:)14 b(:)g(:)p FA(])1730 3320 y Fo(\013)p Ft(=1)p Fo(=)p Ft(4)1974 3295 y FA(=)46 b(4)p Fr(e)2166 3261 y Fl(\000)p Ft(7)p Fo(=)p Ft(2)2340 3295 y FA(+)2433 3262 y Ft(33)p 2433 3276 67 4 v 2449 3324 a(8)2509 3295 y Fr(e)2548 3261 y Fl(\000)p Ft(4)2683 3295 y Fr(<)2804 3262 y Ft(11)p 2804 3276 V 2804 3324 a(56)2880 3295 y Fr(:)3060 3229 y FA(\(B.13\))599 3496 y(Com)n(bing)30 b(the)g(preceding)g(inequalities)f(\(B.7-13\),)h (w)n(e)g(arriv)n(e)e(at)i(a)f(rather)g(simple)599 3596 y(b)r(ound,)1118 3738 y Fs(h)p Fr( )s(;)f(V)1325 3708 y Ft(2)1306 3758 y(ne)1377 3738 y Fr( )s Fs(i)p 1118 3775 349 4 v 1184 3851 a(h)p Fr( )s(;)14 b( )s Fs(i)1522 3794 y Fr(<)1732 3738 y(\020)1774 3708 y Ft(2)1812 3738 y Fr(\013)p 1620 3775 357 4 v 1620 3855 a FA(1)k(+)1773 3822 y Ft(13)p 1773 3836 67 4 v 1789 3884 a(3)1849 3855 y Fr(e)1888 3831 y Fl(\000)p Ft(2)2010 3727 y Fp(\000)2058 3761 y Ft(10)p 2058 3775 V 2058 3823 a(59)2153 3794 y FA(+)2246 3761 y Ft(296)p 2246 3775 100 4 v 2262 3823 a(69)2355 3794 y Fr(\013)h FA(+)2536 3761 y Ft(11)p 2520 3775 V 2520 3823 a(112)2629 3794 y Fr(\013)2682 3760 y Ft(2)2720 3727 y Fp(\001)2758 3794 y Fr(;)279 b FA(\(B.14\))599 3998 y(v)-5 b(alid)28 b(for)f Fr(\013R)d FA(=)f(2)k(and)h(0)22 b Fr(<)h(\013)g Fs(\024)g FA(1)p Fr(=)p FA(4.)1409 4263 y Fu(A)m(CKNO)m(WLEDGMENTS)682 4375 y Fz(The)f(second)h(author)f (\(H.H.\))f(is)f(grateful)i(for)e(the)i(w)n(arm)e(hospitalit)n(y)i(of)f (Ph.)30 b(Briet,)21 b(P)-6 b(.)21 b(Duclos)599 4454 y(and)g(other)g (colleagues)g(at)g(the)g(CPT,)e(and)i(for)e(tec)n(hnical)j(assistance)f (b)n(y)f(the)h(Rec)n(henzen)n(trum)g(der)599 4533 y(Univ)n(ersit\177) -35 b(at)24 b(Hildesheim.)p eop %%Page: 37 37 37 36 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(37)1619 320 y Fu(REFERENCES)621 432 y Fz(1.)33 b(A.)28 b(Messiah,)f("Quan)n(tum)h(mec)n(hanics)g(V)-6 b(ol.)27 b(I)h(and)h(I)r(I",)f(North-Holland)g(Publishing,)f(Ams-)709 511 y(terdam,)c(1961)621 625 y(2.)33 b(K.)23 b(Rueden)n(b)r(erg,)i Fb(R)l(ev.)g(Mo)l(d.)i(Phys.)d Fa(34)e Fz(\(1962\))k(326-376)621 738 y(3.)33 b(M.)22 b(J.)h(F)-6 b(ein)n(b)r(erg,)24 b(K.)e(Rueden)n(b)r (erg,)i(and)g(E.)f(L.)f(Mehler,)h(in:)f("Adv)l(ances)j(in)e(quan)n(tum) g(c)n(hem-)709 817 y(istry")h(V)-6 b(ol.)23 b Fa(5)o Fz(,)g(ed.)h(P)-6 b(.)23 b(O.)g(L\177)-35 b(owdin,)24 b(Academic)f(Press,)g(New)h(Y)-6 b(ork,)23 b(1970,)h(pp.)g(27-98)621 931 y(4.)33 b(I.)c(V.)f(Komaro)n(v,)f(L.)h(I.)h(P)n(onomarev,)f(and)h (S.)f(Y)-6 b(u.)28 b(Sla)n(vjano)n(v,)i("Spheroidal)f(and)g(Coulom)n(b) 709 1010 y(spheroidal)24 b(functions")g(\(in)g(Russian\),)g(Nauk)l(a,)g (Mosco)n(w,)f(1976)621 1124 y(5.)33 b(Y)-6 b(u.)24 b(N.)f(Demk)n(o)n (v,)g Fb(JETP)j(L)l(ett.)d Fa(7)g Fz(\(1968\))i(,)f(76-79)621 1238 y(6.)33 b(C.)23 b(A.)g(Coulson)h(and)g(A.)f(Joseph,)i Fb(Int.)g(J.)h(Quant.)f(Chem.)f Fa(1)g Fz(\(1967\),)h(337-)621 1352 y(7.)33 b(B.)25 b(M)r(\177)-37 b(uller,)25 b(J.)g(Rafelski,)g(and) h(W.)g(Greiner,)f Fb(Z.)i(Physik)f Fa(257)f Fz(\(1972\),)i(183-211;)f Fb(Phys.)i(L)l(ett.)709 1431 y(B)c Fa(47)f Fz(\(1974\),)i(5-7)621 1545 y(8.)33 b(B.)23 b(M)r(\177)-37 b(uller)23 b(and)h(W.)g(Greiner,)f Fb(Z.)i(f.)g(Naturfor.)f Fa(31)o Fz(a)g(\(1976\),)h(1-30)621 1659 y(9.)33 b(I.)26 b(P)-6 b(.)25 b(Gran)n(t)h(and)g(H.)f(M.)f(Quiney) -6 b(,)26 b Fb(A)l(dv.)i(A)n(t.)e(Mole)l(c.)i(Phys.)e Fa(23)e Fz(\(1988\),)j(37-86;)f Fb(Phys.)i(R)l(ev.)709 1737 y(A)c Fa(62)f Fz(\(2000\),)i(022508)586 1851 y(10.)33 b(W.)24 b(Kutzelnigg,)g Fb(Phys.)i(Scripta)e Fa(36)f Fz(\(1987\),)i(416-431)586 1965 y(11.)33 b(I.)17 b(Labzo)n(wsky)-6 b(,)19 b(G.)e(Klimc)n(hitsk)l(a)n(y)n(a,)f(and)i(Y)-6 b(u.)17 b(Dmitriev,)f("Relativistic)h(e\013ects)i(in)e(the)h(sp)r (ectra)709 2044 y(of)24 b(atomic)f(systems",)g(Institute)i(of)f(Ph)n (ysics)f(Publishing,)g(Bristol,)f(1993)586 2158 y(12.)33 b(T.)f(Saue,)h(V.)f(Bakk)n(en,)i(T.)e(Enev)n(oldsen,)h(T.)f(Helgak)n (er,)g(H.)g(J.)h(Aa.)f(Jensen,)h(J.)f(Laerdahl,)709 2237 y(K.)e(Ruud,)g(J)g(.Th)n(yssen,)g(and)h(L.)e(Vissc)n(her,)h("DIRA)n(C,) g(a)g(relativistic)g(ab)g(initio)g(electronic)709 2316 y(structure)25 b(program)d(",)h(Release)i(3.2)e(\(2000\))j(\(h)n (ttp://dirac.c)n(hem.sdu.dk\))586 2430 y(13.)33 b(I.)24 b(P)-6 b(.)23 b(Gran)n(t,)h Fb(A)l(dv.)i(A)n(t.)f(Mole)l(c.)h(Phys.)e Fa(32)f Fz(\(1994\),)i(169-186)586 2544 y(14.)33 b(M.)c(S.)h(Marino)n (v)g(and)g(V.)g(S.)f(P)n(op)r(o)n(v,)i Fb(Sov.)h(Phys.)f(JETP)g Fa(38)f Fz(\(1974\),)h(1069-1075;)g Fb(Sov.)h(J.)709 2623 y(Nucl.)26 b(Phys.)e Fa(23)f Fz(\(1976\),)i(251-255)586 2736 y(15.)33 b(W.)24 b(P)n(o)r(esc)n(hl)g(and)g(K.)f(Dietric)n(h,)g Fb(Z.)i(Phys.)h(A)e Fa(351)e Fz(\(1995\))k(271-)586 2850 y(16.)33 b(J.)24 b(C.)f(W)-6 b(ells,)23 b(B.)g(Segev,)h(and)h(J.)e(Eic) n(hler,)g Fb(Phys.)j(R)l(ev.)f(A)e Fa(59)g Fz(\(1999\))j(346-357)586 2964 y(17.)33 b(V.)g(I.)g(Matv)n(eev,)i(D.)e(U.)g(Matrasulo)n(v,)g(and) h(H.)f(Y)-6 b(u.)33 b(Rakhimo)n(v,)g Fb(Phys.)i(A)n(tom.)g(Nucl.)e Fa(63)709 3043 y Fz(\(2000\),)25 b(318-322)586 3157 y(18.)33 b(O.)23 b(Kullie)g(and)h(D.)f(Kolb,)g Fb(Eur.)j(Phys.)g(J.)g(D)d Fa(17)g Fz(\(2001\),)i(167-173)586 3271 y(19.)33 b(M.)23 b(Klaus,)g Fb(Helv.)i(Phys.)h(A)l(cta)d Fa(53)g Fz(\(1980\),)j(463-482) 586 3385 y(20.)33 b(L.)28 b(P)n(auling,)g("The)i(nature)f(of)f(the)h(c) n(hemical)f(b)r(ond",)i(Cornell)d(Univ)n(ersit)n(y)i(Press,)e(Cornell,) 709 3464 y(1948)586 3578 y(21.)33 b(J.)27 b(C.)g(Slater,)h("Quan)n(tum) f(theory)i(of)e(molecules)g(and)h(solids,)e(V)-6 b(ol)27 b(1:)g(Electronic)h(structure)709 3657 y(of)c(molecules",)e(McGra)n(w)i (Hill,)e(New)i(Y)-6 b(ork,)23 b(1963)586 3771 y(22.)33 b(C.)c(Cohen-T)-6 b(annoudji,)31 b(B.)e(Diu,)g(and)h(F.)f(Lalo)n(\177) -33 b(e,)30 b("Quan)n(tum)g(mec)n(hanics,)f(V)-6 b(ol.)30 b(2",)f(Wiley)-6 b(,)709 3849 y(New)24 b(Y)-6 b(ork,)23 b(1977)586 3963 y(23.)33 b(J.)24 b(Dolb)r(eault,)g(M.)e(J.)i(Esteban,)g (and)g(E.)f(S)n(\023)-33 b(er)n(\023)g(e,)24 b Fb(J.)i(F)-5 b(unct.)25 b(A)n(nal.)f Fa(174)f Fz(\(2000\),)i(208-226)586 4077 y(24.)33 b(M.)23 b(Griesemer)f(and)i(H.)f(Sieden)n(top,)i Fb(J.)h(L)l(ondon)h(Math.)f(So)l(c.)e Fa(60)f Fz(\(1999\),)i(490-500) 586 4191 y(25.)33 b(M.)23 b(Griesemer,)f(R.)h(T.)g(Lewis,)g(and)h(H.)f (Sieden)n(top,)i Fb(Do)l(c.)h(Math.)e Fa(4)f Fz(\(1999\),)i(275-283)586 4305 y(26.)33 b(B.)23 b(Thaller,)g("The)h(Dirac)f(equation",)i (Springer,)e(Berlin,)g(1992)586 4419 y(27.)33 b(J.)24 b(W)-6 b(eidmann,)23 b Fb(Math.)j(Z.)d Fa(119)f Fz(\(1971\),)k(105-115) 586 4533 y(28.)33 b(U.W.)23 b(Sc)n(hminc)n(k)n(e,)h Fb(Math.)i(Z.)d Fa(126)f Fz(\(1972\),)j(71-81)p eop %%Page: 38 38 38 37 bop 599 75 a FA(38)863 b Fz(BRIET)24 b(AND)f(HOGREVE)586 320 y(29.)33 b(J.J.)24 b(Landgren)g(and)h(P)-6 b(.A.)22 b(Reit\177)-35 b(o,)24 b Fb(J.)i(Math.)g(Phys.)e Fa(20)f Fz(\(1979\),)i(2204-2211;)g(J.J.)f(Landgren,)709 399 y(P)-6 b(.A.)23 b(Reit\177)-35 b(o,)24 b(and)g(M.)f(Klaus,)f Fb(J.)k(Math.)g(Phys.)e Fa(21)f Fz(\(1980\),)i(1210-1217)586 510 y(30.)33 b(U.W.)23 b(Sc)n(hminc)n(k)n(e,)h Fb(Math.)i(Z.)d Fa(119)f Fz(\(1971\),)j(349-373)586 621 y(31.)33 b(R.)23 b(W)r(\177)-37 b(ust,)24 b Fb(Math.)i(Z.)d Fa(131)g Fz(\(1973\),)i (339-349;)f Fa(141)f Fz(\(1975\),)i(93-98)586 732 y(32.)33 b(R.)23 b(W)r(\177)-37 b(ust,)24 b Fb(Math.)i(Z.)d Fa(152)g Fz(\(1977\),)i(259-271)586 843 y(33.)33 b(G.)24 b(Nenciu,)g Fb(Commun.)j(Math.)g(Phys.)d Fa(48)f Fz(\(1976\),)j(235-247;)f Fb(Helv.)g(Phys.)h(A)l(cta)e Fa(50)g Fz(\(1977\),)709 922 y(1-3)586 1033 y(34.)33 b(M.)24 b(Klaus)h(and)g(R.)f(W)r(\177)-37 b(ust,)26 b Fb(Commun.)i(Math.)f(Phys.)e Fa(64)f Fz(\(1979\),)i (171-176;)g Fb(J.)h(Math.)g(A)n(nal.)709 1111 y(Appl.)e Fa(72)d Fz(\(1979\),)k(206-214)586 1222 y(35.)33 b(M.)f(Tinkham,)f ("Group)i(theory)h(and)f(quan)n(tum)g(mec)n(hanics",)f(McGra)n(w-Hill,) e(New)j(Y)-6 b(ork,)709 1301 y(1964)586 1412 y(36.)33 b(W.)k(Greiner,)f("Relativistic)h(quan)n(tum)g(mec)n(hanics:)f(W)-6 b(a)n(v)n(e)38 b(Equations",)f(Third)f(Edition,)709 1491 y(Springer,)23 b(Heidelb)r(erg,)h(2000)586 1602 y(37.)33 b(T.)26 b(Kato,)g("P)n(erturbation)h(theory)g(for)e(linear)h(op)r (erators",)g(2nd.)g(edition,)g(Springer,)g(Berlin,)709 1681 y(1976)586 1792 y(38.)33 b(E.)23 b(M.)g(Harrell)f(and)j(M.)d (Klaus,)h Fb(A)n(nn.)j(Inst.)f(H.)h(Poinc)l(ar)n(\023)-35 b(e)25 b Fa(38)e Fz(\(1983\),)i(153-166)586 1903 y(39.)33 b(P)-6 b(.)23 b(Av)n(en)n(tini)i(and)f(R.)f(Seiler,)g Fb(Commun.)j(Math.)g(Phys.)e Fa(41)f Fz(\(1975\),)i(119-134)586 2014 y(40.)33 b(J.)24 b(M.)e(Com)n(b)r(es)h(and)i(R.)e(Seiler,)f Fb(Int.)k(J.)g(Quant.)f(Chem.)f Fa(14)f Fz(\(1978\),)i(213-229)586 2125 y(41.)33 b(J.)24 b(Morgan)f(and)h(B.)g(Simon,)e Fb(Int.)j(J.)h(Quant.)g(Chem.)e Fa(17)e Fz(\(1980\),)k(1143-1166)586 2236 y(42.)33 b(W.)d(Thirring,)f("A)h(course)g(in)g(mathematical)g(ph)n (ysics)g(3:)g(Quan)n(tum)g(mec)n(hanics)g(of)g(atoms)709 2315 y(and)24 b(molecules",)f(Springer,)g(New)h(Y)-6 b(ork,)23 b(1981)586 2426 y(43.)33 b(M.)17 b(Reed)i(and)f(B.)f(Simon,)g ("Metho)r(ds)h(of)g(mathematical)f(ph)n(ysics)h(iv:)f(Analysis)g(of)h (op)r(erators",)709 2505 y(Academic)24 b(Press,)e(New)i(Y)-6 b(ork,)23 b(1978)586 2616 y(44.)33 b(B.)40 b(Simon,)f("Quan)n(tum)i (mec)n(hanics)g(for)f(Hamiltonians)f(de\014ned)j(as)f(quadratic)g (forms",)709 2695 y(Princeton)25 b(Univ)n(ersit)n(y)e(Press,)g (Princeton,)h(1971)586 2806 y(45.)33 b(M.)23 b(Klaus,)g Fb(Helv.)i(Phys.)h(A)l(cta)d Fa(53)g Fz(\(1980\),)j(453-462)586 2917 y(46.)33 b(M.)25 b(Reed)h(and)g(B.)f(Simon,)g("Metho)r(ds)h(of)g (mathematical)e(ph)n(ysics)i(I:)g(F)-6 b(unctional)27 b(analysis",)709 2996 y(Academic)d(Press,)e(New)i(Y)-6 b(ork,)23 b(1972)586 3107 y(47.)33 b(P)-6 b(.)19 b(Duclos)g(and)h(H.)e (Hogrev)n(e,)i(in:)e("Rigorous)h(results)g(in)g(quan)n(tum)g(dynamics,) g(Liblice)f(1990",)709 3186 y(eds.)24 b(J.)f(Dittric)n(h)h(and)g(P)-6 b(.)23 b(Exner,)h(W)-6 b(orld)23 b(Scien)n(ti\014c,)i(Singap)r(ore)f (\(1991\),)h(63-70)586 3297 y(48.)33 b(H.)23 b(Hogrev)n(e,)h Fb(Phys.)i(R)l(ev.)f(A)f Fa(48)f Fz(\(1993\),)i(3382-3385)586 3408 y(49.)33 b(H.)23 b(R.)g(Alarcon)h(and)g(R.)f(D.)g(Benguria,)h Fb(L)l(ett.)g(Math.)i(Phys.)e Fa(35)f Fz(\(1995\),)i(281-289)586 3519 y(50.)33 b(M.)f(B.)g(Rusk)l(ai,)g Fb(L)l(ett.)g(Math.)i(Phys.)f Fa(18)e Fz(\(1989\))k(121-132;)e Fb(Commun.)i(Math.)f(Phys.)e Fa(137)709 3598 y Fz(\(1991\))26 b(553-566)586 3709 y(51.)33 b(J.)24 b(Ph.)f(Solo)n(v)n(ej,)h Fb(Commun.)i(Math.)g(Phys.)e Fa(130)f Fz(\(1990\),)i(185-204)586 3820 y(52.)33 b(J.)24 b(Ac)n(k)n(ermann)f(and)h(H.)f(Hogrev)n(e)i Fb(J.)g(Phys.)h(B)e Fa(25)f Fz(\(1992\),)i(4069-4098)586 3931 y(53.)33 b(H.)23 b(Hogrev)n(e,)h Fb(J.)i(Chem.)g(Phys.)e Fa(98)f Fz(\(1993\),)i (5579-5594)586 4042 y(54.)33 b(H.)e(R.)f(Alarcon,)h(R.)g(D.)f (Benguria,)h(P)-6 b(.)31 b(Duclos,)g(and)h(H.)e(Hogrev)n(e,)i Fb(Helv.)g(Phys.)h(A)l(cta)e Fa(72)709 4121 y Fz(\(1999\),)25 b(386-407)586 4232 y(55.)33 b(F.)23 b(Sc)n(h)n(w)n(abl,)h("Adv)l(anced) i(quan)n(tum)e(mec)n(hanics",)g(Springer,)f(Berlin,)f(1999)586 4343 y(56.)33 b(F.)27 b(W.)g(J.)g(Olv)n(er,)g("Asymptotics)g(and)h(sp)r (ecial)g(functions",)f(Academic,)g(Press,)f(New)i(Y)-6 b(ork,)709 4422 y(1974)586 4533 y(57.)33 b(H.)23 b(Hogrev)n(e,)h(w)n (ork)g(in)f(progress)p eop %%Page: 39 39 39 38 bop 1118 75 a Fz(TW)n(O-CENTRE)23 b(DIRA)n(C-COULOMB)f(OPERA)-6 b(TORS)436 b FA(39)586 320 y Fz(58.)33 b(J.)18 b(M.)f(Com)n(b)r(es,)h (P)-6 b(.)17 b(Duclos,)h(and)h(R.)e(Seiler,)h(in:)f("Rigorous)h(atomic) g(and)h(molecular)e(ph)n(ysics",)709 399 y(eds.)24 b(A.)f(S.)g(Wigh)n (tman)h(and)g(G.)g(V)-6 b(elo,)23 b(Plen)n(um)g(Press,)f(New)i(Y)-6 b(ork,)23 b(1981,)h(pp.)g(185-212)586 511 y(59.)33 b(V.)d(V)-6 b(ogelsang,)30 b Fb(Math.)i(Z.)e Fa(196)f Fz(\(1987\),)i(517-521;)g Fb(Int.)h(Eq.)f(and)i(Op.)e(The)l(or.)g Fa(10)f Fz(\(1987\),)709 589 y(841-858)586 702 y(60.)j(E.)23 b(H.)g(Lieb)h(and)g(B.)f(Simon,)g Fb(J.)i(Phys.)h(B)e Fa(11)f Fz(\(1978\),)i(L537-L542)586 814 y(61.)33 b(E.)23 b(H.)g(Lieb,)h Fb(J.)h(Phys.)h(B)e Fa(15)f Fz(\(1982\),)i(L63-L70)586 926 y(62.)33 b(T.)23 b(Ho\013mann-Ostenho\013,)h Fb(J.)i(Phys.)g(A)d Fa(13)g Fz(\(1980\),)i(417-424)586 1038 y(63.)33 b(H.)23 b(Hogrev)n(e,)h Fb(J.)i(Phys.)g(A)p Fz(,)d Fa(26)g Fz(\(1993\),)i(159-170)586 1150 y(64.)33 b(I.)24 b(Daub)r(ec)n(hies)h(and)f(E.)f(H.)g(Lieb,)g Fb(Commun.)k(Math.)f(Phys.)e Fa(90)f Fz(\(1983\),)i(497-510)586 1262 y(65.)33 b(J.)24 b(Dolb)r(eault,)g(M.)e(J.)i(Esteban,)g(and)g(E.)f (S)n(\023)-33 b(er)n(\023)g(e,)24 b Fb(J.)i(Calc.)g(V)-5 b(ar.)23 b Fa(10)g Fz(\(2000\),)i(321-347)586 1374 y(66.)33 b(C.)g(Tix,)g Fb(Phys.)i(L)l(ett.)e(B)g Fa(405)g Fz(\(1997\),)i (293-296;)f Fb(Bul)t(l.)h(L)l(ondon)h(Math.)e(So)l(c.)g Fa(30)f Fz(\(1998\),)709 1453 y(283-290)586 1565 y(67.)g(W.)18 b(D.)f(Ev)l(ans,)g(P)-6 b(.)17 b(P)n(erry)-6 b(,)17 b(and)h(H.)f (Sieden)n(top,)i Fb(Commun.)i(Math.)f(Phys.)e Fa(178)e Fz(\(1996\),)j(733-746)586 1677 y(68.)33 b(A.)23 b(A.)g(Balinsky)h(and) g(W.)f(D.)g(Ev)l(ans,)h Fb(Commun.)j(Math)f(Phys.)e Fa(202)e Fz(\(1999\),)k(481-500)586 1789 y(69.)33 b(G.)24 b(Ho)r(ev)n(er)g(and)g (H.)f(Sieden)n(top,)i Fb(Math.)h(Phys.)g(Ele)l(ctr)l(onic)g(J.)e Fa(5)f Fz(\(1999\))j(pap)r(er)e(6,)f(1-11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303091646680--