Unit 3 Lecture Number 16

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr 6 Eltron Gas n Hartr Fo and Random Phas Aroxmatons Frst, a short rvst to Hartr Fo Formalsm, but from a dffrnt rout. PCD STTACS Unt 3 Eltron Gas n HF & RPA

W shall sulmnt and omlmnt that dsusson to qu ourslvs to buld th mahnry to s how th mthods of nd quantzaton dvlod n Unt an b xtndd to addrss th ltron COULOMB orrlatons that ar lft out of th HF mthod. PCD STTACS Unt 3 Eltron Gas n HF & RPA

H H + H f ( q ) v( q, qj) + j j Z f( q) r Many-Eltron Hamltonan n th notaton of FIRST QUATIZATIO H f j + j v l j j l j j l j v l dq dq φ q φ q v q, q φ q φ q HF SCF Mthod: STAP Unt 4 L Rfrnhtt://www.ntl.a.n/downloads/5657/ Many-Eltron Hamltonan n th notaton of SECOD QUATIZATIO * * j l PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

Ψ () t t () f j j+ j j v l l Ψ t j j l H f j + j v l j j l j j l Fttr & Wala (.8); Rams (.3;4) ot: Ordr j v l dq dq ψ q ψ q v q, q ψ q ψ q * * j l Th ordr dos not mattr for Bosons; for Frmons, t dos mattr. For ltrons, ψ ψ χ ζ ( q) ( r) sn orbtal χ ( ζ) s thr or PCD STTACS Unt 3 Eltron Gas n HF & RPA α for ms + β for ms 4

Fld orators dfnton ψ Lnar ombnaton of raton & dstruton orators ( q) : sngl artl wavfuntons.. sn-orbtals nd, : quantzaton dstruton & raton orators { }, ms or { n,,, } wth l j mj m s + or Fr ltron Hydrogn Potntal Sn-orbtals ψ ψ χ ζ ( q) ( r) whr χ( ζ ) or χ ζ for ms + or m s PCD STTACS Unt 3 Eltron Gas n HF & RPA ψˆ ( q) ψˆ ( q) ψ * ψ adjont sn orbtals ψ * ψ * χ ζ q q ( q) ( r) [ ] or [ ] χ ζ χ ζ for ms + or m s 5

H Φ E Φ -ltron Shrodngr quaton Φ ( q, q,.., q ) Φ ( q, q,.., q ) ( ) n, n,.., n,.., n a, a,.., a Ordrd st: a < a <.. < a <.. < a <.. < a j ( ) n, n,.., n,.., n Slatr dtrmnantal wavfunton ( q, q,.., q ) Φ! ψ ψ ( q ).... ψ ( q ) a a.... ( q j ).. ψ.. ( q ).... ψ ( q ) a a a * ψ ( q) ψ j q dx δj ( ') ( ') ( ') * ψ ψ δ δ δ q q q q r r ζζ PCD STTACS Unt 3 Eltron Gas n HF & RPA Orthonormal omlt st of on-ltron sn-orbtals ' 6

Fld Orators ψˆ ( q) ψˆ ( q) ψ * Mult-omonnt sn-orbtal wavfunton (j+) numbr of omonnts ψ q ψ Fld Orator ψˆ α( q) ψ α ( q) ψˆ ( q) ψ ( q) α,,3,...,( j + ) ψˆ ˆ α q, ψ ( q' ) β δαβδ ( q q' ) ψˆ ˆ α q ψβ ( q ) Frm + Bos - Fld orators q Inluson of sn: mult-omonnt sn-orbtals ± ( q) In gnral, for sn j : α,,...( j+ ) j : ntgr for Bosons, half-ntgr for Frmons ψ ψ ψ ψ ψ, α, α, α 3, α..., α j+ ( q) ( q) ( q) ( q) ( q) * α α, ' ( q) ψˆ ( q ) ψ ˆ, α β ' ± ± PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

Fld Orator ψˆ ( q) ψ H f j + j v l α α * α q ψ α ( q) ψˆ α,,3,...,( j + ) q, δ r, σ rσ ± r, σ rσ rσ rσ ± rr σσ ± j j l j j l δ ψ ψ ( q) ψ α ms + α ms sn : Hamltonan n trms of sngl artl raton and dstruton orators ψˆ ˆ α q, ψ ( q' ) β δαβδ ( q q' ) ± ψˆ ( q ) ˆ ( q ) α ψ β ψˆ ( q), ψˆ ( q' ) Hamltonan n trms of fld orators α β ±, ', α, α H ψˆ ( q) f ( q) ψˆ( q) dq + ψˆ ( q) ψˆ ( q ')v( q, q ') ψˆ( q ') ψˆ( q) dqdq ' That ths form s orrt an b sn asly as shown on nxt sld F + B - PCD STTACS Unt 3 Eltron Gas n HF & RPA ± ot: Ordr ( q) ( q) 8

H ψˆ ( q) f ( q) ψˆ( q) dq + ψˆ ( q) ψˆ ( q ')v( q, q ') ψˆ( q ') ψˆ( q) dqdq ' * ˆ( q) ˆ q ( q) q ψ ψ ψ ψ ( q) f( q) ψ * H ψ j q dqj j + + j * * j j l v(, ') ψ q ψ q q q ψ q ψ dqdq ' l l H f j + j v l j j l j j l Rams, Many Eltron Thory / Eq..7 /.4 PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

Comlt xrssons for th Hamltonan, nlusv of sn labls aσ aσ aa σσ aσ aσ aσ aσ H ψˆ ˆ ˆ ˆ v(, ') ˆ ( ') ˆ α q f q ψ β q dq + ψα q ψ β q q q ψδ q ψγ( q) dqdq ' PCD STTACS Unt 3 Eltron Gas n HF & RPA * ˆ ˆ α α α β jβ j α β j ψ q ψ q ψ q ψ q β ( q) f( q) ψ * H ψ j q dq α α β jβ j +, δ δ,, ± ± ± + α β j l α β δ γ v( * * ψ α q ψ jβ q q, q') ψl δ q ψγ dqdq ' γ l δ α jβ H α f jβ + α, jβ v lδ, γ α jβ α jβ γ lδ j j l α β α β δ γ Rams /.4 / Eq..7 nlusv of sn labls

W rognz that and ar Hrmtan onjugats. Ths orators wr ntrodud as dstruton & raton orators. Lt Proo f : Φ Φ + PCD STTACS Unt 3 Eltron Gas n HF & RPA a Φ Φ b (,,,,,,... ) (,,,,,,...) dτ * Φ a Φb and Φb Φ a dstruton orator :dstruton orator H all othr ouaton numbrs n + Φ a Φ Φ b Φ & bng sam H lt Hrmtan onjugat of H w must show that : by dfnton of * H * b adτ b adτ ( * H ) * a bdτ Φ Φ Φ Φ Hrmtan onjugat normalzaton ntgral H Φ b Φ a umbr of oud stats rdng th th stat: vn.. and ar Hrmtan onjugats raton orator Φ Φ normalzaton ntgral

Z rj ( ) (,,.., ) + r < j H q q q f( r) + ; j j rj (,,.., ) v(, ) ( ) H q q q f r + r rj ; j j add and subtrat q r ( ) r ( r ) ( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j Modfd on-ltron orator Modfd ntraton F? PCD STTACS Unt 3 Eltron Gas n HF & RPA

q r ( ) r ( r ) ( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j Z r H f( r) f( r) f Modfd onltron orator would ontan muh/most of th fft of th two-ltron trms. ( ) H ( q, q,.., q ) f H + F + F Cho of th orator F s to b so mad that th total nrgy s mnmsd. Modfd, rsdual, ntraton btwn ars of ltrons. Ths trm would b wa, and would b tratd rturbatvly. PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j ( ) Φ Z r PCD STTACS Unt 3 Eltron Gas n HF & RPA q r ( ) r ( r ) f ( r ) Modfd on-ltron orator ψ ψ! ψ ψ [ ] ()...... ψ ()...... ψ...... j ψ ( q ).. ()...... ψ ()...... ψ f( r) + F( r) ψ σ( r) εψ σ( r) wth ψ ( r) ψ ( r) or ψ ( r) ε σ : doubly dgnrat, wth on gnfunton ah for sn & j Modfd ntraton Whn th nd trm s ngltd, ths dtrmnant s th unrturbd ground stat wavfunton. 4

( ) Φ q r ( ) r ( r ) ( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j Z f ( r ) r Modfd on-ltron orator! ψ ψ ()...... ψ ψ ()...... ψ ψ...... j ψ ( q ).. ()...... ψ ()...... ψ j ε Modfd ntraton [ f( r) + F( r) ] ψ σ( r) εψ σ( r) wth ψ ( r) ψ ( r) or ψ ( r) σ : doubly dgnrat, wth on gnfunton ah for sn & as ψ, ψ, ψ,..., ψ, ψ 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA ψ ()...... ψ ψ ()...... ψ ( ) Φ j ψ qj........! ψ ()...... ψ ψ ()...... ψ 5

ψ ()...... ψ ψ ()...... ψ ( ) Φ j ψ qj........! ψ ()...... ψ ψ ()...... ψ [ f( r) + F( r) ] ψ σ( r) εψ σ( r),,3,, ε : Lowst / gnvalus Wav funtons of th EXCITED unrturbd stats ar also th ordr dtrmnants, mad u gnfuntons of [ ] ( ) H ( q,.., q ) f ( r ) + ; j j F( r ) v( r, r ) f ( r ) + F( r ) ψ ( r ) εψ ( r ) σ σ PCD STTACS Unt 3 Eltron Gas n HF & RPA ε : doubly dgnrat, wth on gnfunton ah for sn & but wth on or mor ε > ε / + j Fr 6

( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j Z r PCD STTACS Unt 3 Eltron Gas n HF & RPA q r ( ) r ( r ) f ( r ) H f( r) F F( r) Modfd onltron orator would ontan muh/most of th fft of th twoltron trms. Cho of th orator F s to b mad suh that th total nrgy s mnmsd. It turns out, as wll b shown rsntly, that ths hans whn: f Modfd, rsdual, ntraton btwn ars of ltrons. Ths trm would b wa, and would b tratd rturbatvly. qf qv qv 7

q r ( ) r ( r ) ( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j Z f ( r ) r It turns out that ths hans whn: Cho of th orator F s to b so mad that th total nrgy s mnmsd. qf qv qv Rmmbr th two ntr COULOMB & EXCHAGE ntgrals: q v dq dq ψ q ψ q v q, q ψ q ψ q j v l dq dq ψ q ψ q v q, q ψ q ψ q * * j l * * q q v dq dq ψ q ψ q v q, q ψ q ψ q sam * * q PCD STTACS Unt 3 Eltron Gas n HF & RPA 8

Lt th ground stat unrturbd wav funton dsrbd abov b: Lt an xtd stat wav funton, n whh only a sngl ltron from th abov stat s xtd, b: ( ) Φ Φ ψ In th ordrd st of th sngl artl stats : & q > ψ ()...... ψ.......... ψ ()...... ψ ()!.......... ψ ()...... ψ ψ ()...... ψ.......... ψ ()...... ψ ()!.......... ( ) q q q ()...... ψ PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

Z rj ( ) (,,.., ) + r < j H q q q Sam Slatr dtrmnant h ( q ) + H + H ; j j rj Φ H Φ α f α f Φ H Φ j v j j v j Φ H Φ f + PCD STTACS Unt 3 Eltron Gas n HF & RPA j j + v v [ ] [ j j j j ]

H ( ) ( ) arox ( q, q,.., q ) f( r) + Fr + v( r, r ) Fr PCD STTACS Unt 3 Eltron Gas n HF & RPA j ; j j H ( q, q,.., q ) f( r) + Fr f+ F ψ()...... ψ Φ.......... ψ ()...... ψ ()!.......... Φ ψ ()...... ψ Φ Usng sam thnqus dsussd n STAP Unt 4 L Rfrn htt://www.ntl.a.n/downloads/5657/ w an fnd ( ) ( ) Φq Harox( q, q,.., q ) Φ Φq f + F Φ? sld 4: [ f ( r) + F( r) ] ψ σ( r) εψ σ( r).. f ( r) + F( r) s dagonal n ψ ( r) Φq f + F Φ [ ] { } σ ot th OTATIO! ψ()...... ψ.......... ψ ()...... ψ ()!.......... ψ ()...... ψ ( ) q q q

H ( q, q,.., q ) f( r) + F( r) f + F ( ) arox Φ Φ ψ ()...... ψ.......... ψ ()...... ψ ()!.......... ψ ()...... ψ Φ orators n SIGLE COORDIATES ψ ()...... ψ.......... ψ ()...... ψ ()!.......... ( ) q q q ψ ()...... ψ Φ H ( q, q,.., q ) Φ Φ f + F Φ ( ) ( ) q arox q f + F: dagonal wth rst to on-ltron funtons and q But, H ( q, q,.., q ) H ( q, q,.., q ) + arox v( r, r ) F( r) j ; j j of whh th frst trm gvs Φ H Φ Φ f + F PCD STTACS Unt 3 Eltron Gas n HF & RPA Φ ( ) q arox q

Hn, f w hoos F suh that Φ Φ Φ Φ thn H ( q, q,.., q ) H ( q, q,.., q ) + arox of whh th frst trm gvs ( ) q F q v( r, rj) ; j j w shall g THUS, hoos F suh Φ H Φ t q Φ Φ Φ Φ q F q v( r, rj) ; j j q v q v that Φ H ( ) q arox j ; j j Φ v( r, r ) F( r) Matrx lmnts of th abov two trms would anl; qual & oost sgns n ordr to gt Φ H Φ q PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

Havng shown now that th ho F whh Φ Φ Φ Φ q F q v( r, rj) ; j j q v q v gvs w now show that th abov ho of gvsus: Φ H Φ ( ) ( ) q onurrntly gvs th bst sngl dtrmnantal ground stat wav funton aordng to th varaton rnl (Hartr-Fo SCF aroxmaton) OTE Φ Φ d r r F r r 3 * * : q F ψq ψ F, PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

orrt ground Lt us as: If Φ Φ wr not th stat wavfunton, ould any othr wav funton b th ground stat? Th most gnral form n whh just on of th onsttunt sn orbtal s dffrnt would b ψ q Φ + εφ, aart from an ovrall normalzaton... For ths wavfunton, th nrgy funtonal s: E ( ε ) Φ + εφ H Φ + εφ q Φ + εφ Φ + εφ q q q PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

E ( ε ) Φ + εφ H Φ + εφ q Φ + εφ Φ + εφ q q q Φ H Φ Φ H Φ Φ H Φ Φ H Φ ( ) ( ) + ε q + ε q + ε q q ε ε ε Φ Φ + Φ Φ + Φ Φ + Φ Φ q q q q { H q q H } Φ H Φ Φ H Φ ( ) ( ) + ε Φ Φ + Φ Φ + ε q q ε q Φ Φ + Φ Φ { H q q H } Φ H Φ Φ H Φ ( ) ( ) + ε Φ Φ + Φ Φ + ε q q + ε q PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

E ( ε ) { H H } Φ H Φ + ε Φ Φ q + Φq Φ + ε Φ H Φ ( ) ( ) ( ) ( ) + ε q q dffrntatng wth rst to ε d E ( ε ) dε d dε { } Φ H Φ + ε Φ H Φ + Φ H Φ + ε Φ H Φ q q q q + ε { } d ( ) q q q q + Φ H Φ + ε Φ H Φ + Φ H Φ + ε Φ H Φ + ε dε PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

d E ( ε ) dε { } Φ H Φ + Φ H Φ + ε Φ H Φ q q q q + ε d { Φ } ( ) q + Φ q Φ + Φq Φq H ε H H ε H ε dε + Φ Φ + Φ + d ε ε + ε + ε ε, 4 dε ( + ε ) ( + ε + ε ) whh gos to zro as ε d E ε dε ε { Φ H Φ q + Φq H Φ } PCD STTACS Unt 3 Eltron Gas n HF & RPA 8

d E dε { Φ H Φ + Φ H Φ } q q ε ε But w had sn that th ho F whh gvs Φ Φ Φ Φ q F q v( r, rj) ; j j q v q v ( ) Φ gav us : Φ ( )! Φ H Φ q SD d E ε dε ε E ( ε ): xtrmum... mnmum w gt th bst sngl dtrmnantal ground stat wav funton aordng to th varaton rnl PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

Thus, th ho F whh gvs Φ Φ Φ Φ q F q v( r, rj) ; j j PCD STTACS Unt 3 Eltron Gas n HF & RPA q v q v gvs us : ( ) ( ) Φ Φ and t gvs th bst sngl dtrmnantal ground stat wav funton aordng to th varaton rnl ε sn MIIMISES th varatonal nrgy funtonal: E ( ε ) Φ + εφ H Φ + εφ q Φ + εφ Φ + εφ q Qustons: d@hyss.tm.a. q q q H Hartr-Fo aroxmaton. 3

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr 7 Eltron Gas n Hartr Fo and Random Phas Aroxmatons HF SCF for Fr Eltron Gas PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

H ( ) ( ) arox ( q, q,.., q ) f( r) + Fr + v( r, r ) Fr ψ ()...... ψ.......... Φ ψ ()...... ψ () Φ!.......... ψ ()...... ψ PCD STTACS Unt 3 Eltron Gas n HF & RPA j ; j j H ( q, q,.., q ) f( r) + Fr f+ F Th varatonal funton w onsdrd s: ψ Φ + εφ Φ q ψ ()...... ψ.......... ψ ()...... ψ ()!.......... ( ) q q q ψ ()...... ψ Varaton onsdrd s n just on orbtal All othr orbtals FROZE Hartr Fo: FROZE ORBITAL APPROXIMATIO Sn / statstal / Frm orrlatons nludd Coulomb orrlatons gnord SCF: slf onsstnt fld STAP Unt 4 L Rfrn htt://www.ntl.a.n/downloads/5657/ 3

f( r) u ( r ) + * u j ( r) dv u r u r m m u r u r j j r ε u j δ ( s, s ) j ( r ) PCD STTACS Unt 3 Eltron Gas n HF & RPA 33

f( r) u ( r ) + * u j ( r) dv u r u r m m u r u r j j r ε u r Chang of ( ) H ( q, q,.., q ) notaton slghtly: ; j ; u( r) ψ ( r) r r; r r' ψ () r + m PCD STTACS Unt 3 Eltron Gas n HF & RPA j δ ( s, s ) j Z r * ψ ( r') dv ' ( r) ( r ') ( m, m ) ( r ') ( r) r r' ε ψ ( ψ ) ψ δ s s ψ ψ Z + m r < j rj f ( r) + H + H ; j j rj ( r) otaton hangd only to brng t losr to that n Rams: Many Eltron Thory (97; orth Holland) 34

m Z ψ ( r') ψ ( r) dv ' ψ ( r) ψ ( r') δ( m, m ) ψ ( r') ψ ( r) ε ψ * + s s r r r' ( r) Z m r ψ * ψ ( ξ') ψ( ξ') ψ ( ξ) + r r' 4 ( ξ) dv' * ψ ( ') ( ') 4 ξ ψ ξ ψ ξ δ ( ms, m ) ' s d V ε ψ ξ r r ' PCD STTACS Unt 3 Eltron Gas n HF & RPA 35

on-frromagnt systms: qual numbr of & ε : doubly dgnrat; on gnfunton ah for sn & Ground stat Slatr dtrmnant ontans th st of on-ltron orbtals: ψ, ψ, ψ,..., ψ, ψ ψ, ψ, ψ, ψ,..., ψ, ψ 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA 36

Z m r ψ * ψ ( ξ') ψ( ξ') ψ ( ξ) + r r' 4 ( ξ) dv' * ψ ( ') ( ') 4 ξ ψ ξ ψ ξ δ ( ms, m ) ' s d V ε ψ ξ r r ' Carryng out th dsrt sum ovr th sn varabls: Z m r * ψ ( r') ψ( r') ψ () r + dv ' ψ () r r r' * ψ ( r') ψ ( r') dv ' ψ r ε ψ r r r ' Hartr-Fo on ltron Slf onsstnt fld quaton. PCD STTACS Unt 3 Eltron Gas n HF & RPA 37

Z m r * ψ ( r') ψ( r') ψ () r + dv ' ψ () r r r' * ψ ( r') ψ ( r') dv ' ψ r ε ψ r r r ' v( r, r ') Coulomb ntraton r r' Z m r ψ () r dv ' ψ(') r v(, r r') + ψ () r * ψ( r) dv ' ψ ( r ') ψ ( r ')v( r, r ') ε ψ r Rams / Many Eltron Thory / Eq.3.3; ag 53 PCD STTACS Unt 3 Eltron Gas n HF & RPA 38

H H + H j Z j f( q) r ' f ( q ) v( q, qj) + Many-Eltron Hamltonan n th notaton of FIRST QUATIZATIO H f j + j v l j j l j j l j v l dq dq ψ q ψ q v q, q ψ q ψ q Many-Eltron Hamltonan n th notaton of SECOD QUATIZATIO * * j l PCD STTACS Unt 3 Eltron Gas n HF & RPA 39

H f( q) + v( q, qj) j j I Q H f j + j v l j j l j j l Hn, IF q r ( ) r ( r ) ( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j THE H f + F j j + j Rams / Many Eltron Thory / Eq.3.7; ag 55 PCD STTACS Unt 3 Eltron Gas n HF & RPA jv l F j j l j j l j II Q I Q II Q 4

Φ H f( q) + v( q, qj) ψ ()...... ψ.......... ψ ()...... ψ ()!.......... ( ) q q q ψ ()...... ψ j j Egnfuntons of th sngl artl orator q r ( ) r ( r ) ( ) H ( q, q,.., ) f + F r + v( r, j) F ; j j From sld # 4, U3L7: ( f + F) φ ( q) ε φ ( q) j j j f + F j ε j εδ j j j Φ Φ Φ Φ F F q q q v q v PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

Φq F Φ q v q v PCD STTACS Unt 3 Eltron Gas n HF & RPA 4 4 * * q F d ξ d ξ ψ ξψq ξ r r ψ ξ ψ ξ Φ Φ v (, ) d ξ d ξ ψ ξ ψ ξ r r ψ ξ ψ ξ v (, ) 4 4 * * q 4 * q F d ξψ q ξψ ξ r Φ Φ d 4 ( ξ ) d v (, r ) ψ ( ξ ) ξψ ξ ψ ξ 4 * d ξψ ( ξ ) v ( r, r ) ψ ( ξ ) 4 * q ntrhangng ξ ξ n th sond (xhang) trm: 4 * q F d ξψ q ξ d r r ψ ξ Φ Φ d 4 ξψ( ξ) v (, ) ξψ ( ξ ) d ξψ ( ξ ) v ( r, r ) ψ ( ξ ) ψ ( ξ ) 4 * 4 * q 4

4 * q F d ξψ q ξ d r r ψ ξ Φ Φ d 4 ξψ( ξ) v (, ) ξψ 4 * ( ξ ) d ξψ ( ξ ) v ( r, r ) ψ ( ξ ) ψ ( ξ ) 4 * q 4 d ξψ ( ξ) v ( r, r) ψ ( ξ) 4 * Φq F Φ d ξψ q ( ξ) 4 * d ξψ ( ξ) v ( r, r) ψ ( ξ) ψ( ξ) 4 d ξψ ( ξ) v ( r, r) ψ ( ξ) 4 * Φq F Φ d ξψ q ( ξ) 4 * d ξψ ( ξ) v ( r, r) ψ ( ξ) ψ( ξ) 4 d ξψ ( ξ) v ( r, r) ψ ( ξ) Fψ ( ξ ) 4 * d ξψ ( ξ) v ( r, r) ψ ( ξ) ψ( ξ) PCD STTACS Unt 3 Eltron Gas n HF & RPA Rams / Many Eltron Thory / Eq.3.9; ag 5 43

4 d ξψ ( ξ) v ( r, r) ψ ( ξ) Fψ ( ξ ) 4 * d ξψ ( ξ) v ( r, r) ψ ( ξ) ψ( ξ) Fψ ξ d ξ ψ ξ r r ψ ξ 4 v (, ) d ξψ ξ ψ ξ ψ ξ v ( r, r ) 4 * arryng out th summaton ovr th dsrt sn varabl: Fψ ψ ψ 3 ( r ) dr ( r ) v ( r, r ) ( r ) d 3 rψ ψ ψ ( r ) v ( r, r ) ( r ) ( r ) * PCD STTACS Unt 3 Eltron Gas n HF & RPA 44

arryng out th summaton ovr th dsrt sn varabl: Fψ ψ ψ 3 ( r ) dr ( r ) v ( r, r ) ( r ) d 3 rψ ψ ψ ( r ) v ( r, r ) ( r ) ( r ) * r r r' r Z m r * ψ ( r') ψ( r') ψ () r + dv ' ψ () r r r' * ψ ( r') ψ ( r') dv ' ψ( r) ε ψ r r ' PCD STTACS Unt 3 Eltron Gas n HF & RPA ( r) Z ψ () r + Fψ + m r HF-SCF Eq. ( r) [ f F ] ψ ( r) ε ψ ( r) Rams / Many Eltron Thory / Eq.3.3; ag 53 45

Rall, from Sal/Slt Tos n Atom Physs, Unt STAP: 4, Unt Ltur 4, Ltur 3, Sld 3, Sld Hartr-Fo Slf-Consstnt Fld formalsm Rfrn htt://www.ntl.a.n/downloads/5657 E ψ H ψ ε j sld 4: [ f r + F r ] ψ σ r εψ σ r [ f r + F r ] { ψ r }.. s dagonal n f + F ε f + [ j v j j v j ] F σ Rams, Eq.3.35 E f + F j PCD STTACS Unt 3 Eltron Gas n HF & RPA [ j v j j v j ] Rams, Eq.3.36 46

E Also, f + F j [ v v ] j [ j v j j v j ] j j j j E + f + F Φ H Φ E f + j v j j v j j E f E + f + F [ ] E f + f F f ε + + PCD STTACS Unt 3 Eltron Gas n HF & RPA Rams, Many Eltron Thory Eq.3.38 / ag 56 47

Hartr Fo Slf Consstnt Fld for th Fr Eltron Gas For FEG, th HF-SCF an b obtand AALYTICALLY FEG only many-ltron systm for whh HF-SCF an b obtand AALYTICALLY fr n V o ntraton wth any xtrnal fld PCD STTACS Unt 3 Eltron Gas n HF & RPA What about th fft of th ostv nul? Frm gas of ltrons whh ntrat only wth ah othr. 48

dsrt ostv hargs n th nul onsdrd smard out, l jlly bans nto a jllum. Whol systm: ltrally nutral. PCD STTACS Unt 3 Eltron Gas n HF & RPA 49

Postv harg dnsty smard out unformly. PCD STTACS Unt 3 Eltron Gas n HF & RPA ltrons n a ubal box. Eah sd has lngth L ρ 3 V Volum of th box V L nxλx L π nx L; x x π n L π n ˆ + n ˆ + n ˆ L x x y y z z x Box normalzaton wth Born von Karmann boundary ondtons How many wavlngths ft n th box? r ψ (r) χ σ 3 σ ζ L orbtal art sn art 5

Z Fr ltron s wav vtor r θ dθ φ dϕ Y X PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

X dω Y ê x Construt a grd of onts labld by ntgrs sad at unform dstans along th X, Y, Z axs. Z PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

E + + m m ( ) x y z π E n + n + n m L E π ml n ( ) x y z Y ê x X stats wth dffrn t n, n, n x y z n + n + n n x y z ar dgnrat dω Z 3-dmnsonal orthogonal sa of ndndnt ntgrs n, n, n. x y z PCD STTACS Unt 3 Eltron Gas n HF & RPA 53

ψ () r m + V( r) ψ ( r) dv ' ψ( r ') v( r, r ') + ψ ( r) ψ ψ ψ * ( r) dv ' ( r') ( r')v( r, r') ε ψ ( r) m * ψ () r ψ() r d ' ψ (') ψ (')v(, ') ε ψ V r r r r r 54 PCD STTACS Unt 3 Eltron Gas n HF & RPA

m * ψ () r ψ() r d ' ψ (') ψ (')v(, ') ε ψ V r r r r r Rall, from Sal/Slt Tos n Atom Physs, STAP Unt 4, Ltur 3, Sld 8 HF SCF formalsm Rfrn htt://www.ntl.a.n/downloads/5657 ψ * ( q') ψ ( q') xhang V ( q) ψ ( q) ψ ( q) dq' r r ' * ψ ( q') ψ ( q') xhang V ( q) ψ ( q) ψ ( q) dq' r r ' m ζ ' ζ ' m δm, m ζ ' s s s s xhang 3 * ψ ψ ' ψ ( ') ψ ( ')v, ' V q q r d r r r r r PCD STTACS Unt 3 Eltron Gas n HF & RPA 55

m * ψ () r ψ() r d ' ψ (') ψ (')v(, ') ε ψ V r r r r r xhang 3 * ψ ψ ' ψ ( ') ψ ( ')v, ' V q q r d r r r r r xhang ψ () r V ( q) ψ ( q) ε ψ m m ( r) V q q m xhang.. ψ ε ψ + F g( q) ψ ( q) ε ψ xhan ( r) ( r) xhang F ( q) V ( q) xhang PCD STTACS Unt 3 Eltron Gas n HF & RPA Rams, Many Eltron Thory Eq.3.44 / ag 58 56

m * ψ () r ψ() r d ' ψ (') ψ (')v(, ') ε ψ Exhang Trm Sld 57 PCD STTACS Unt 3 Eltron Gas n HF & RPA V r r r r r 3 * ψ ( r) d rψ ( r ' ) ψ ( r)v( r, r) ψ ( ' ) ε ψ m ' L ot sgn ( ') ' 3 3 L dr 3 ' r { r } ( r ) r ψ (r) χ σ 3 σ ζ r r L orbtal art sn art ET, S57 57

m 3 * ψ ( r) d rψ ( r ' ) ψ ( r)v( r, r) ψ ( r ' ) ε ψ r ' L ( ') ' 3 3 L dr 3 ' r r r ET, S57 L 3 3 L dr 3 ' r PCD STTACS Unt 3 Eltron Gas n HF & RPA 3 L r 3 L ( ' ) r ' r r 58

L 3 3 L dr 3 ' r 3 L r 3 L ( ' ) r ' r r ( ' ) r{ } ' r dr 3 L 3 ' r L 3 r PCD STTACS Unt 3 Eltron Gas n HF & RPA 59

ET, S 57 ( ' ) r{ } ' r 3 r dr 3 L 3 ' r L ' ( r r) 3 dr ψ r ' r 3 L L PCD STTACS Unt 3 Eltron Gas n HF & RPA 3 ' ( ' ) r ( ) r ψ 3 L ' 3 d r r φ ψ ' r r r ' r 6

ET, S 57 L ( ' ) r 3 ' φ r ψ r φ r ' r ' r 3 4π d r r ' Th Wav Mhans of Eltrons n Mtals by Stanly Rams, ag 7, Eq.7.4 ET, S 57 ' r ( ' ) r 4π 3 L ' π 4 3 L ' ' ψ ' r ψ r ε ψ r PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

r r 3 * ψ ( r) d rψ ( r ' ) ψ ( r)v( r, r) ψ ( ' ) ε ψ m ' m ε ψ r ψ ( ) r + ε ψ ε ψ m K.E. + ε ot sgn ε xt: alulaton of ( r ) r whr ε ε Qustons: d@hyss.tm.a. 4π 3 L ' ' Hartr-Fo Eq for Fr Eltron Gas Rams / Wav Mhans of Eltrons n Mtals / Eq.7.4, ag 7 PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr 8 Eltron Gas n Hartr Fo and Random Phas Aroxmatons Eltron-Eltron Exhang Enrgy PCD STTACS Unt 3 Eltron Gas n HF & RPA 63

Hartr-Fo Eq for Fr Eltron Gas m K.E. ψ ( ) r + ε ψ ε ψ ( r ) r m + ε Dtrmnaton of ε PCD STTACS Unt 3 Eltron Gas n HF & RPA ε whr ε 4π 3 L ' ' Rams / Wav Mhans of Eltrons n Mtals / Eq.7.4, ag 7 ltron gas n jllum otntal K + xhang HF orrlaton E E E 64

Postv harg dnsty smard out unformly. PCD STTACS Unt 3 Eltron Gas n HF & RPA ltrons n a ubal box. Eah sd has lngth L ρ 3 V Volum of th box V L nxλx L π nx L; x x π n L x Box normalzaton wth Born von Karmann boundary ondtons How many wavlngths ft n th box? π ( n ˆ ˆ ˆ ) xx + nyy + nzz L In th -sa π 'volum' of ah stat L 3 65

ε ε Sum ovr all stats ε 4π 3 L ' ' ' π L PCD STTACS Unt 3 Eltron Gas n HF & RPA 3 In th -sa π 'volum' of ah stat L ' : ntgraton n sa 3 d L π ' L 4π 3 d' 3 3 Intgraton u to th Frm lvl ' ' 'sn F π π π ' θ ϕ ' d d d θ θ ϕ Rams / Wav Mhans of Eltrons n Mtals / Eq.7.4nxt, ag 7 66 ' 3

ε ' ' 'sn F π π π ' θ ϕ ' d d d Rams / Wav Mhans of Eltrons n Mtals / Eq.7.4nxt, ag 7 d d d d θ θ ϕ d d ' ε ' d' ' sn d d F θ θ ϕ π π ε π ' θ ϕ PCD STTACS Unt 3 Eltron Gas n HF & RPA ( ' ) ( ' ) ' F π π ' d'sn d d π ' θ ϕ ' ' θ θ ϕ 67

ε ' F π π ' d'sn d d π ' θ ϕ ' ' θ θ ϕ ntgratng ovr ϕ ' ' 'sn ( ) F π d θdθ ε π π ' θ ' ' ε ε PCD STTACS Unt 3 Eltron Gas n HF & RPA ' F π ' θ ' d 'sn d π ' F μ+ ' μ θ θ + ' 'os ' d ' d os θ μ ;.. sn θdθ dμ π μ + ' ' μ θ 68

ε ' F μ+ ' d ' d π ' μ μ + ' ' μ ε π ' F μ+ ' d' ' μ dμ + ' ' μ ε + π ' F μ ' d' ln ' ' μ + ' μ ( ') ε + π ' ' F ' d' ln ' ' μ + ' PCD STTACS Unt 3 Eltron Gas n HF & RPA 69

ε + π ' ' F μ + ' d' ln ' ' μ ' μ ε π ' F ' d' ' ( ) ( + + + ) ln ' ' ln ' ' ' ' ε ' F ' d ' ( ) ( ) ' ln + ' ' ln + ' + ' π ε ' F ' d ' ln ( ') ln ( ') ' + π PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

ε ' F ' d ' ln ( ') ln ( ') ' + π ε ' F ' π ' ' d ' ln + ' ε π ' F ' ' ' d ' ln + ' ε π ' F ' d ' ' ln + ' ' PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

ε ε ' F ' ' ln d' π ' + ' π { ' ' } ' ln ' d' ' ln ' d ' F F + ' ' f x a ( x a) xln( x+ a) dx ln( x+ a) 4 ( ' + ) ' ( ' ) ' ε ln ' ln ' + + π 4 4 f ε ( ' + ) ( ' ) ' ' ln + π + ' 4 4 ' f ' PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

ε ( ' + ) ( ' ) ' ' ln + π + ' 4 4 ' f ' ε π f f f f + ln + + 4 4 + ln + 4 4 f ε f f f + f ln + π + f 4 4 PCD STTACS Unt 3 Eltron Gas n HF & RPA 73

ε f f f + f ln + π + f 4 4 ε f f ln f π + f ε f π f f ln + f + f Exhang Trm ε + f + ln εxhang f f f f π PCD STTACS Unt 3 Eltron Gas n HF & RPA 74

ε + ε m & ε xhang xhang f f + f + ln π f f ε f + ln m π f f + f f lt ρ f ε f ρ + ρ + ln m π ρ ρ ε f ρ + ρ ln m π + ρ ρ dfn: ρ + ρ F( ρ) + ln ρ ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA 75

ε ε + ε m f ρ + ρ ln m π + ρ ρ ρ f f ρ ρ + ρ ε : EXCHAGE TERM IS EGATIVE Snglt Stat Trlt Stat Slt/Sal Tos n Atom Physs htt://ntl.a.n/ourss/5657/ Unt 4 Trlt Stat s lss unshd by th oulomb ntraton - Landau & Lfshtz PCD STTACS Unt 3 Eltron Gas n HF & RPA 76

Snglt : χ( ζ, ζ) χ( ζ, ζ) ant-symmtr sn art φ( r, r) + φ( r, r ) + ( ) ϕ r ϕ r ϕ r ϕ r snglt : orbtal art doubl as r r Frm orrlaton * ltrons wth antaralll sns to lum togthr, * as f n a ha of ltral harg * Ths auss ICREASED rulson lss stabl PCD STTACS Unt 3 Eltron Gas n HF & RPA 77

Trlt : χζ (, ζ) + χ( ζ, ζ) ant-symmtr sa art φ( r, r) φ( r, r ) ( ) ( ) ϕ r ϕ r ϕ r ϕ r trlt: orbtal art as r r Frm orrlaton * ltrons wth aralll sns hav an EXCLUSIO rgon of sa * as f a shral avty s gnratd around t n whh anothr ltron wth a aralll sn annot ntr * DECREASED rulson mor stabl PCD STTACS Unt 3 Eltron Gas n HF & RPA 78

ψ () r m + V( r) ψ ( r) Sld o. 54, Prvous ltur dv ' ψ( r ') v( r, r ') + ψ ( r) ψ ψ ψ * ( r) dv ' ( r') ( r')v( r, r') ε ψ ( r) m * ψ () r ψ() r d ' ψ (') ψ (')v(, ') ε ψ V r r r r r PCD STTACS Unt 3 Eltron Gas n HF & RPA 79

8 PCD STTACS Unt 3 Eltron Gas n HF & RPA 3 ' 3 : nt ' graton n sa L d π 3 'volum' of ah stat L π 3 ' 3 : ntgraton n sa ' d L π 3 'volum' of ah stat L π n th sa n th sa

Slt/Sal Tos n Atom Physs htt://ntl.tm.a.n/ourss/5657/ Unt 4 / Sld # & E ψ H ψ atom HF f + j g j j g j j [ ] Th orator f ontans th K.E. orators and th nular attraton orators Eltron gas n jllum otntal attratv jllum otntal anls th ltron-ltron drt Coulomb rulson trms ltron gas n * ntgraton nstad of th abov dsrt sum jllum otntal EHF 3 L f θ π ϕ π d sn θdθ dϕ + ε 3 xhang θ ϕ π m PCD STTACS Unt 3 Eltron Gas n HF & RPA 8

ltron gas n jllum otntal EHF 3 L f θ π ϕ π d sn θdθ dϕ + ε 3 xhang θ ϕ π m ltron gas n jllum otntal HF K + xhang orrlaton E E E whr 3 L f θ π ϕ π EK d snθdθ dϕ 3 θ ϕ π and m 3 L f θ π ϕ π Exhang d sn θdθ dϕ ε 3 θ ϕ orrlaton π xhang PCD STTACS Unt 3 Eltron Gas n HF & RPA 8

ltron gas n jllum otntal HF K + xhang orrlaton E E E whr 3 L f θ π ϕ π EK d snθdθ dϕ 3 θ ϕ π m E K 3 L 4π ( π ) m f 3 4 d E K 3 L ( π ) 3 4π m 5 5 f E K 3 L π m K: K.E. art of th HF nrgy of th dgnrat fr ltron gas f 5 f : Frm lvl E K V π m Rams / Many Eltron Thory / Eq.3.64, ag 63 f 5 PCD STTACS Unt 3 Eltron Gas n HF & RPA 83

umbr of ltrons n th sa 'volum' of ah stat Tw th umbr of sngl-ltron stats n th ' volum' ( n - sa) sannd by th Frm shr 4 3 whos volum s π f 3 4 π 3 π 3π π 4π 3 L 3π f V 3 f 3 3 L 3 L 4 3 3 f π f 3 3 4 3 3 L 4 π 3 4π 3 V 4 π π 3 f 3 f π L 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA 84

E K E V π m 5 3π f r 3 s K PCD STTACS Unt 3 Eltron Gas n HF & RPA f 3 V 4 3 3π π rs V 3 3 f rs : radus of a shr whos volum s qual to th avrag volum r ltron. 4 3 5 π r s 3 3 3 3π 3 9π π m 4 3 m 4 r π rs 3 K.E. ontrbuton to th avrag HF ground stat nrgy r ltron n a fr-ltron-gas Rams / Many Eltron Thory / Eq.3.68, ag 63 ( 9π ) ( 9π ) /3 /3 9π 4 r 4 4 m 3 s f f v f s 3 EK 3 9π m 4 r. Ryd r s s 4 Ryd m 3.6569... V ; Bohr unt.59a m 85

ltron gas n jllum otntal HF K + xhang orrlaton E E E K.E. E K. Ryd r s V f Exhang d θ π sn d ϕ π d 3 θ θ ϕ ε xhang θ ϕ orrlaton π ε xhang f π f + f + ln f f Qustons: d@hyss.tm.a. PCD STTACS Unt 3 Eltron Gas n HF & RPA 86

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr 9 Eltron Gas n Hartr Fo and Random Phas Aroxmatons Fr Eltron Gas n Jllum Baground Potntal PCD STTACS Unt 3 Eltron Gas n HF & RPA 87

ltron gas n jllum otntal HF Knt + Exhang Enrgy Corrlaton E E E whr 3 L f θ π ϕ π EKnt d snθdθ dϕ 3 θ ϕ Enrgy π m E K 3 L 4π ( π ) m 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA f 4 d L V E π m π m 3 5 5 K f f K: K.E. art of th HF nrgy of th dgnrat fr ltron gas Rams / Many Eltron Thory / Eq.3.64, ag 63 E K 3 L ( π ) f : Frm lvl 3 4π m 5 5 f 88

E r K s V π m 5 f ( 9π ) ( 9π ) /3 /3 4 4 mv f E K f 3 3 9π m 4 r s 4 3 3π π rs V 3 r s 3 f : radus of a shr whos volum s qual to th avrag volum r ltron. r s : Bohr unts r s : Stz aramtr K.E. ontrbuton to th avrag HF ground stat nrgy r ltron n a fr-ltron-gas Rams / Many Eltron Thory / Eq.3.68, ag 63 PCD STTACS Unt 3 Eltron Gas n HF & RPA s 3 EK 3 9π m 4 r. Ryd r 4 Ryd m 3.6569... V ; Bohr unt.59a m s 89

ltron gas n jllum otntal HF K + xhang orrlaton E E E K.E.. Ryd r? V f Exhang d θ π sn d ϕ π d 3 θ θ ϕ ε xhang θ ϕ orrlaton π E K s ε xhang f + + ln f f f f π E xhang orrlaton V f θ π ϕ π f f f + d sn d d ln 3 θ θ ϕ θ + ϕ π π f f PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

E xhang orrlaton V f θ π ϕ π f f f + d sn d d ln 3 θ θ ϕ θ + ϕ π π f f E xhang orrlaton 3 d d V f ( 4 ) f + π d ln 3 8 f + f π π f E xhang orrlaton V f ( ) f + d ln 3 4 f + f π f PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

E xhang orrlaton V f ( ) f + d ln 3 4 f + f π f E xhang orrlaton V 4 f 3 4π PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

E xhang orrlaton V 4 f 3 4π 4 4 3 f π 3 9 4π 4π 4 r 3 3 4 s 3π 4 3 f ; & π rs V V 3 from : sld 85, last lass 4 3 4 3 9π Exhang π rs 3 4 orrlaton 3 4π 4 rs PCD STTACS Unt 3 Eltron Gas n HF & RPA 9π rs 4 3 3π 4 E xhang orrlaton.96 r s Ryd 93

ltron gas n jllum otntal HF KE + xhang orrlaton E E E 3 L f θ π ϕ π whr EKE d snθdθ dϕ 3 θ ϕ π m 3 L f and Exhang d 3 orrlaton ( π ) Addng both th trms θ π ϕ π θ ϕ sn θdθ dϕ εxhang For fr ltron gas n SCF jllum otntal : E HF..96 Ryd rs r s Avrag HF nrgy r ltron PCD STTACS Unt 3 Eltron Gas n HF & RPA r s : Bohr unts H f( q) + v( q, qj) j j 94

ltron gas n jllum otntal HF K + xhang orrlaton E E E Avrag HF nrgy r ltron E HF..96 Ryd rs r s ltron-ltron ntraton, rdus th nrgy BELOW that of th Sommrfld gas (of ours n th ostv jllum otntal) ψ () r m + V( r) ψ ( r) dv ' ψ( r ') v( r, r ') + ψ ( r) ψ ψ ψ ε ψ * ( r) dv ' ( r') ( r')v( r, r') ( r) 95 PCD STTACS Unt 3 Eltron Gas n HF & RPA

Avrag HF nrgy r ltron ltron-ltron ntraton, rdus th nrgy BELOW that of th Sommrfld gas (of ours n th ostv jllum otntal) FEG n HF-SCF jllum otntal : E HF..96 Ryd rs rs : Bohr unts r s 4 Ryd m 3.6569... V Bohr unt.59... A m Frst Ordr Prturbatv tratmnt of th xhang trm SAME RESULT (nxt lass) Sond (and hghr) Ordr Prturbatv tratmnt of th ltron-ltron Coulomb ntraton howvr dvrgs. PCD STTACS Unt 3 Eltron Gas n HF & RPA 96

For fr ltron gas n jllum otntal : E HF..96 Ryd rs r s Bohm & Pns: md-ffts D.Pns (963) Elmntary xtatons n solds (Bnjamn, Y) Random Phas Aroxmaton E BP..96 3.96 β 4 + β 3 β rs rs r rs 48 s f PCD STTACS Unt 3 Eltron Gas n HF & RPA β d! Many-body thory byond rturbaton mthods : Ur bound to wav numbr of lasma osllatons Lowr bound to wav lngth; sn osllatons gt damd by th random thrmal moton of th ltrons. 97

Frst, th lassal modl ρ : avrag volum harg dnsty ξ Postv and gatv harg n balan Dslamnt of all th ltrons to th rght PCD STTACS Unt 3 Eltron Gas n HF & RPA 98

Frst, th lassal modl ρ : avrag volum harg dnsty ξ Dslamnt of all th ltrons to th rght nt ostv surfa harg r unt ara + ρ ξ nt ngatv surfa harg r unt ara ρ ξ surfa harg dnsty : σ ρξ PCD STTACS Unt 3 Eltron Gas n HF & RPA nt fld n-btwn E ρξuˆ ε 99

nt fld n-btwn E ρξuˆ ε CGS unts ; 4π 4πε ε 4πρ ω m PCD STTACS Unt 3 Eltron Gas n HF & RPA d ξ m ρξ dt ε d ξ dt ρ mε ξ ω SI unts ρ mε Frquny of lasma osllatons Thrmal moton of ltrons: gnord xt that thrmal flutuatons would hav ausd th onst of lasma osllatons Thrmal moton dsrson Eq. of moton whn dsrson s rsnt: ω E F ω m

dsrt ostv hargs n th nul onsdrd smard out, l jlly bans nto a jllum. Unform harg dnsty Whol systm: ltrally nutral. PCD STTACS Unt 3 Eltron Gas n HF & RPA

Postv harg dnsty l b ρ V PCD STTACS Unt 3 Eltron Gas n HF & RPA ltrons n volum V togthr wth a unform ostv harg baground jllum dstrbuton. Jllum baground H H + H + H st trm n th Hamltonan μ r Hl + m r r nd trm j j j l b l b Mathmatal dv to avod dvrgns. Latr, w ta th lmt: μ x x ' 3 3 ρ + xρ + x' Hb dx dx' x x' 3 rd trm n th Hamltonan 3 H d x r j μ x r ρ x + x r μ ltrons and th baground: EUTRAL systm

ρ nd trm ρ + + PCD STTACS Unt 3 Eltron Gas n HF & RPA μ x x ' 3 3 ρ + xρ + x' Hb dx dx' x x' V x x' (unform dnsty) x' x z dx ' dz... at onstant x μ x x ' 3 3 Hb d x d x' V x x' 3 3 H dx dz V z b μz μz 3 μz dz 4π zdz 4π z dz z z 4π 4π H b V V μ V μ Rfrn: Fttr & Wala - Eq.3.7 n Quantum Thory of Many-Partl Systms; ag H b μ dvrgs 4π μ μz Contrbuton of ths trm (r ltron) dvrgs as μ μ dvrgn 3

3 rd trm 3 H d x l b ρ V 4π Hl b V μ μ x r ρ x + x r μ x r 3 Hl b d x V x r 4π Hb V H b μ μ μ dvrgn Contrbuton of ths trm (r ltron) dvrgs as μ μ dvrgn Rfrn: Fttr & Wala - Eq.3.8 n Quantum Thory of Many-Partl Systms; ag PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

H Hl + Hb + Hl b H 4π 4π Hl + V μ V μ H H l 4 V π μ Contrbuton of ths trm (r ltron) dvrgs as μ μ dvrgn Dos th dvrgng trm anl wth any art of H l? Produr to ta lmts: FIRST: L (.. V ) and thn μ Rfrn: Eq.3.9 n Fttr & Wala - Quantum Thory of Many-Partl Systms; ag 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

PCD STTACS Unt 3 Eltron Gas n HF & RPA H H H f( q ) v ( q, q ) IQ C l + + j j j II Q H f j + j v l st trm H IQ l C j j l j j l C * * C v φ φj v, φ φl j l dq dq q q q q q q μ r r + m r r j j j Hn H + II Q s l j j j j v l l j m j l μ r r * * j v s l dq dqφ ( q) φj ( q) φ ( q) φl ( q) r r 6 j v s

H j + j l II Q s l j j v l j m j l μ r r * * v s φ φj φ φl r r j l dq dq q q q q Showng th summaton ovr sn varabls xltly: H II Q l σ σ σ m σ σ + σ σ σ σ 3 3 4 4 σ σ σ v σ σ s σ σ 3 3 4 4 σ σ 4 4 3 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

H II Q l σ σ σ σ σ m + σ σ σ σ 3 3 4 4 σ Frst, xamn th K.E. trm. σ σ v σ σ s σ σ 3 3 4 4 σ σ 4 4 3 3 3 x x σ σ δ σ, σ d x V V m m ( π ) δ σ, σ x x σ σ 3 d x 3 m mv PCD STTACS Unt 3 Eltron Gas n HF & RPA m δ σ, σ ( ) δ ( ) 3 x dx V 3 ( ) x dx Dra δlta funton 8

Postv harg dnsty smard unformly ρ V π n ˆ + n ˆ + n ˆ L π ( π ) 3 ltrons n a ubal box. Eah sd has lngth L 3 x Volum of th box V L V Box normalzaton wth Born von Karmann boundary ondtons π π nx nxλ x L; nx L; x L x x y y z z ( K ) x dx δ K ( ) δ ( ) 3 x dx x In th -sa 'volum' of ah stat π L 3 ( ) x dx δ 3, L Eq.3.; ag 3; F&W x 3 3 σ σ, PCD STTACS Unt 3 Eltron Gas n HF & RPA δ d x m mv σ σ x ( ) δ mv σ σ δ, V, Eq.3.; ag 3; F&W 9

H II Q l σ σ σ σ σ m + σ σ σ σ 3 3 4 4 σ Frst, xamn th K.E. trm. σ σ v σ σ s σ σ 3 3 4 4 σ σ 4 4 3 3 m m σ σ σ σ δ, δ, H II Q l σ m σ, σ σ σ σ + σ σ σ σ 3 3 4 4 δ δ, σ σ v σ σ s σ σ 3 3 4 4 σ σ 4 4 3 3 H II Q l σ m σ σ + σ σ σ σ 3 3 4 4 σ σ v σ σ s σ σ 3 3 4 4 σ σ 4 4 3 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA

H II Q l σ σ m σ + σ σ σ σ 3 3 4 4 σ σ v σ σ s σ σ 3 3 4 4 σ σ 4 4 3 3 nd trm δ δ σ, σ σ, σ 3 4 σ σ v s 3σ 4σ μ r r 3 3 * * δσ, σδ 3, dr σ σ4 drφ r φ r φ r φ r σ σ 3σ 4σ r r x x V μ r r s 3 3 σ σ v 3σ3 4σ 4 δ σ, σ δ 3 σ, σ 4 V r r dr dr + r + r 3 4 PCD STTACS Unt 3 Eltron Gas n HF & RPA

μ r r s 3 3 σ σ v 3σ3 4σ 4 δ σ, σ δ 3 σ, σ 4 V r r d r d r r r y + r + r 3 4 r x r y + r y + x μ y s 3 3 σσ v 3σ34σ 4 δσ, σδ 3 σ, σ d y d x 4 V y + y+ x + x 3 4 σ σ v σ σ δ δ V μ y s y x y 3 3 + 3 + 4 + 4 3 34 4 σ, σ3 σ, σ d y d x 4 + + y+ x y + + x + y 3 4 4 3 4 3 y μ y s 3 3 σσ v 3σ34σ 4 δσ, σδ 3 σ, σ d y d x 4 V y + + x + y 3 4 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA

μ y s 3 3 σσ v 3σ34σ 4 δσ, σδ 3 σ, σ d y d x 4 V y s x σ σ v 3σ34σ 4, δ, d y V y + + x + y 3 4 3 μ y ( + ) ( ) 3 + 3 4 3 + 3 δσ d x σ3 σ σ4 y Consrvaton of lnar momntum n homognous sa s σ σ v 3σ 3 4σ 4 δ, σ, σδ 3 σ, σ Vδ 4 3 4 d y V y μ y 3 + ( 3 ) ( + + ) s σσ v 3σ34σ 4 δσ, σ δ 3 σ, σ δ 4 +, 3 + 4 d y V y y μ y 3 + ( 3 ) ( ) y Fourr transform of th Srnd Coulomb Potntal PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

Small dgrsson: Fourr transform of th Coulomb Potntal Fourr transform g of f( r ) : 3 g r f( rdr ) Whn th ntgral dos not onvrg: r r 3 g lm μ f ( rdr ) + μ PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

Gvn g, howdo w rovr f( r)? 3 + r 3 f( r) g d π PCD STTACS Unt 3 Eltron Gas n HF & RPA Whn th ntgral dos not onvrg: r 3 f ( r) lm + μ g d + μ rotatonal symmtry : ( ) Whn f r f ( r ), thn g ( ) g ( ); & v vrsa ( ) Inthasof rotatonal symmtry, f r f ( r ) f ( r): 4π g g g drrf( r)sn( r) ( ) FT of Coulomb otntal, V r V ( r ) V ( r) r 4π g g( ) g dr r sn( r) 4π r dr sn ( r) 5

( ) FT of Coulomb otntal, V r V ( r ) V ( r) r 4π 4π g g g dr r r r dr r sn sn Th abov ntgral dos not onvrg μr SC ( SC ) SC FT of Srnd Coulomb otntal, V r V ( r ) V ( r) lm+ μ r μr 4π 4π g g g dr r r dr r μ r μ μr lm sn lm sn + + 4π 4π g lm dr sn r lm dr Im + μ + μ 4 r r g π r μr μ lm Im dr ( ) + μ 4π lm Im + μ μ μr μr r PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

g g r μr 4π 4π lm Im lm Im + μ μ + μ μ 4π lm Im [ ] + μ μ r μr [ ] 4π 4π lm Im lm Im + + μ μ μ μ 4π μ+ 4π 4π lm Im lm + + μ μ + μ μ + FT of FT μr 4π r μ + of SC C 4π r FT of FT of μr 4π μ + r 4π r PCD STTACS Unt 3 Eltron Gas n HF & RPA C SC 7

H s σσ v 3σ34σ 4 δσ, σ δ 3 σ, σ δ 4 +, 3 + 4 d y V y FT of SC μr 4π r μ + 3 4 μ y 3 + ( 3 ) ( ) ( + + ) y Fourr transform of Srnd Coulomb Potntal s σ σ v 3σ3 4σ 4 δσ, σδσ, σ δ, 3 4 σ m σ σ, σ, σ, σ, σ 3 3 4 4 σ, σ σ, σ 3 4 V 3 ( +, + ) II Q l δ 3 4 + Rfrn: Fttr & Wala - Quantum Thory of Many- Partl Systms; ag 4 / Eq.3.5 δ Rfrn: Fttr & Wala Quantum Thory of Many-Partl Systms ag 4 / Eq.3.4 δ V 4π + μ σ σ σ σ 4 4 3 4π + μ 3 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA Rarrang th trms Canllatons wth trms from th baground. 8

H Contrbuton of H Hl + Hb + Hl b H σ H m l σ σ, σ, σ, σ, σ 3 3 4 4 4 3 V σ, σ σ, σ 4 π μ ths trm (r ltron) dvrgs as μ μ dvrgn ( +, + ) II Q l δ 3 4 + Qustons: d@hyss.tm.a. δ δ V 4π σ σ 4σ4 3σ3 3 + μ ' ' δ + + q 3 4 4 3 Momntum transfr For fr ltron gas n jllum E otntal : H I ordr PT? PCD STTACS Unt 3 Eltron Gas n HF & RPA xt lass: Rarrang th trms Canllatons wth th dvrgn trms from th baground. 9

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr Eltron Gas n Hartr Fo and Random Phas Aroxmatons Plasma Osllatons n Fr Eltron Gas Rfrn: Fttr & Wala Quantum Thory of Many-Partl Systms Rfrns: Th thory of lasma osllatons n mtals - by S Rams 957 R. Prog. Phys. & Many Eltron Thory by Stanly Rams PCD STTACS Unt 3 Eltron Gas n HF & RPA

H Hl + Hb + Hl b H H l 4 V π μ Fr Eltron Gas n Postv Jllum Baground Potntal μ dvrgn Dos th dvrgng trm anl wth any art of H l? H σ m σ σ, σ, σ, σ, σ 3 3 4 4 σ, σ σ, σ 3 4 ( +, + ) II Q l δ 3 4 + δ δ V 4π 3 + μ 3 4 4 3 Rarrang th trms Canllatons wth trms from th baground. ' ' δ + + q σ σ σ σ 4 4 3 PCD STTACS Unt 3 Eltron Gas n HF & RPA 3 Momntum transfr

H PCD STTACS Unt 3 Eltron Gas n HF & RPA σ m σ σ, σ, σ, σ, σ 3 3 4 4 ' ' δ ( ), 3 + + 4 + 3+ 4 4 3 q : momntum transfr 3 + q 3 + q + q q 4 3 q q 4 σ, σ σ, σ 3 4 3 ( +, + ) II Q l δ 3 4 + δ δ V Rfrn: Fttr & Wala - Quantum Thory of Many-Partl Systms; ag 4 / Eq.3.5 4π + μ σ σ σ σ 4 4 3 3 4

H II Q l H H PCD STTACS Unt 3 Eltron Gas n HF & RPA II Q σ m σ σ ( +, + ) II Q l δ 3 4 + V 4π σ σ 3 4 σ σ 4σ 3σ 3 + μ ( + q) K.E. trm + qσ σ q q m + + σ 4π + qσ qσ σ + σ V q σ σ q + μ + q; q; ; 3 4 sarat th q trm n th ntraton + + 3 4 4π q + qσ σ σ σ V q σ q σ μ + 4π + σ σ σ σ V σ σ μ q μ q + μ for q + q 3

H II Q 4π q + qσ σ σ σ V q σ q σ μ + 4π + ( σ σ σ q σ ) trm V μ σ σ saratd q, δ δ,, ± ± ± rσ rσ rr σσ rσ rσ rσ rσ σ σ σ σ σ σ σ σ δ δ σ σ σ σ σ σ σ, σ, H IIQ 4π q + qσ σ σ σ V q σ σ q μ + 4π + ( σ σ σ σ, ) σ σ δσ σ δ, V μ σ σ q PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

H 4π + H V μ 4π II Q l II Q + σ H σ V μ + σ m H II Q Rfrn: Fttr & Wala - Quantum Thory of Many-Partl Systms; ag 4 / Eq.3.5 q trms 4π q + qσ σ σ σ V q σ σ q μ + 4π ( + ) σ σ σ σ σ σ δ σ, σ δ, V μ σ σ q q trms PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

H IIQ 4π q + qσ σ σ σ V q σ σ q μ + 4π + ( σ σ σ σ, ) σ σ δσ σ δ, V μ σ σ q W now wrt ths two trms saratly 4π q + qσ σ σ σ V q σ σ q μ + II Q 4π H + σ σ σ σ V μ σ σ q 4π σ σ δ σ, σ δ, V μ σ σ q PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

7 PCD STTACS Unt 3 Eltron Gas n HF & RPA 4 4 4 q q q q q II Q V q H n n V n V σ σ σ σ σ σ σ σ σ σ σ σ π μ π μ π μ + + +,, 4 q q q q II Q q V q H V V σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ π μ δ δ + + +

H H H II Q l, ST 4π q + qσ σ σ σ V q σ σ q μ + 4π 4π + n n σ n σ σ V μ σ V σ μ σ q q 4π q + qσ σ σ σ V q σ σ q μ +, ST 4π 4π + n n σ n σ σ V μ σ σ V μ σ q q Th abov summatons gv th total numbr orator II Q l II Q l, ST V 4 q σ σ 4π ˆ V μ V + PCD STTACS Unt 3 Eltron Gas n HF & RPA q trms 4π ˆ μ q π + μ + qσ qσ σ σ q trms 8

H II Q V 4 + qσ qσ σ σ q σ σ 4π 4π ˆ V μ V μ + ˆ q π + μ W now rla th numbr orators by thr gnvalus H II Q V 4 q σ σ q trms 4π 4π V μ V μ + PCD STTACS Unt 3 Eltron Gas n HF & RPA q π + μ + qσ qσ σ σ 9

q trms 4π 4 π V μ V μ + H C-numbr ontrbutons to H and hn to H H + H + H From sld : H l l b l b 4 V π μ Rfrn: Fttr & Wala Quantum Thory of Many-Partl Systms; ag 5 PCD STTACS Unt 3 Eltron Gas n HF & RPA ontrbuton to E HF 4 V : r artl π μ Frst : V, nxt : μ 3 L V : L; μ μ L 3

H H V 4 + qσ qσ σ σ q σ σ 4π 4π V μ V μ + anl q π + μ H H + H + H l b l b q trms E ] Lm HF V Hamltonan for a bul ltron gas n a unform ostv baground jllum otntal ] μ m σ Rfrn: Fttr & Wala σ Quantum Thory of Many-Partl Systms; ag 5 / σ Eq.3.9 4π + qσ qσ σ + σ V q σ q σ PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

Hamltonan for a bul ltron gas n a unform ostv baground jllum otntal 4π H q q m σ + σ V +,, q q σ σ σ σ σ σ σ Rf: F & W; ag 5 / Eq.3.9 4 3 π rs V rs : radus of a shr whos 3 volum s qual to th avrag volum r ltron. dmnsonlss : lngth sal : Bohr radus rs r a V s 3 s s rs salng: r; V ; r ; q rq m?? V q a m Rfrn: Fttr & Wala PCD STTACS Unt 3 Eltron Gas n HF & RPA Quantum Thory of Many-Partl Systms; ag 5 / Eq.3.4 3

lngth sal : Bohr radus a m dmnsonlss : V salng: r ; V ; r ; q rq s 3 s s rs r rs a r s m m r s m m m ar m r r m ar Vq s 3 rvq s arvq ar r r Vq PCD STTACS Unt 3 Eltron Gas n HF & RPA 33

Hamltonan for a bul ltron gas n a unform ostv baground jllum otntal 4π H q q m σ + σ V +,, q q σ σ σ σ σ σ σ H m ar σ σ σ ar r + V q σ σ Rfrn: Fttr & Wala PCD STTACS Unt 3 Eltron Gas n HF & RPA Quantum Thory of Many-Partl Systms; ag 5 / Eq.3.4 34 r Vq ar Vq 4π q qσ + qσ σ σ

H σ σ σ ar + r V q σ σ 4π q qσ + qσ σ σ Rfrn: Fttr & Wala Quantum Thory of Many-Partl Systms; ag 5 / Eq.3.4 r st : "hgh dnsty" ordr rturbatv tratmnt ossbl vn f th rturbaton s not wa. 4π H q q m σ + σ V +,, q q σ σ σ σ σ σ σ ( unrturbd art ) H ( rturbaton) H + Rfrn: Fttr & Wala Quantum Thory of Many-Partl Systms; ag 5 / Eq.3.4 PCD STTACS Unt 3 Eltron Gas n HF & RPA 35

Φ H Φ Φ H Φ + Φ H Φ Φ Φ Φ Φ H σ m σ σ m m σ F ' π L 3 d 3 STTACS ' : ntgraton n sa Unt 3 Ltur 8 Sld o. 8 V f Φ H Φ 4π d 3 8π m PCD STTACS Unt 3 Eltron Gas n HF & RPA 36

4π H q q m σ + σ V +,, q q σ σ σ σ σ σ σ ( unrturbd art ) H ( rturbaton) H + Φ H Φ Φ H Φ + Φ H Φ Φ H Φ Φ σ Φ σ σ m σ m m F V f Φ H Φ 4π d 3 8π K.E. ontrbuton to th m avrag HF ground stat V f 4 4π d nrgy r ltron n a 3 m 8π fr-ltron-gas. 5 V F V 5 4 π 3 () F m 8π 5 mπ E HF. Ryd r s Rf: F & W QToMPS; 7 Eq.3.3 PCD STTACS Unt 3 Eltron Gas n HF & RPA Rf: F & W QToMPS; 5 Eq.3.9 37

4π H q q m σ + σ V +,, q q σ σ σ σ σ σ σ ( unrturbd art ) H ( rturbaton) H + Φ H Φ Φ H Φ + Φ H Φ 4π H q q + σ σ σ σ V, σ, σ q q Φ Φ Φ Φ, σ, σ q Frst ordr Prturbaton Thory 4π Φ Φ q V Φ Φ + qσ qσ σ σ + qσ qσ would b + qσ qσ σ σ unlss, so that annhlat ltrons n thos stats and f σ σ Rf: F & W QToMPS; 7 Eq.3.3 rat artls n th sam/orrsondng mty stats. PCD STTACS Unt 3 Eltron Gas n HF & RPA 38

Φ Φ + qσ qσ σ σ + qσ qσ would b unlss, so that annhlat ltrons n thos stats and f σ σ rat artls n th sam/orrsondng mty stats. () + q, σ, σ & q, σ, σ q or () + q, σ, σ & q, σ, σ sond ossblty must b orrt, not frst. Φ Φ δ δ Φ Φ + qσ qσ σ σ + q, σ, σ + qσ σ + qσ σ a, a δ r s rs σ + q qσ qσ σ + + q Φ Φ δ δ Φ Φ qσ qσ σ + σ + q, σ, σ + qσ + σ σ σ PCD STTACS Unt 3 Eltron Gas n HF & RPA 39

q Φ Φ δ δ Φ Φ qσ qσ σ + σ + q, σ, σ + qσ + σ σ σ 4π w had : Φ H Φ Φ Φ Φ H Φ + qσ qσ σ σ, σ, σ q q V PCD STTACS Unt 3 Eltron Gas n HF & RPA 4π, q σ Φ Φ q V σ Φ H Φ { } δ δ q σ, σ qσ + + + qσ σ σ numbr orators 4π δ δ n n q σ Φ Φ q V σ q, σ, σ + + qσ σ Φ n n Φ, for + q q, F and + σ σ F for + q > or > (or both > ) F F f 4

Φ H Φ 4π δ δ Φ n n Φ + q V q σ σ q, σ, σ + q, σ, σ Φ n n Φ, for + q q, F and + σ σ F for + q > or > or both > F F F Φ n n Φ for + q F and ( F) + qσ σ for + > > ( q ) or ( ) F F Havsd st funton Φ n n Φ ( + q.. ) ( ) + q F F σ σ θ θ PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

Φ H Φ 4π δ δ Φ n n Φ + q V q Φ H Φ σ σ q, σ, σ + q, σ, σ Φ n n Φ θ + q θ σ σ + q,, F F 4π δ δ q, σσθ F + θ + F q V q q σ σ 4π θ F + F q V ( q ) θ( ) q q ( q ) ( ) Rf: F & W QToMPS; 8 Eq.3.33 4π Φ H Φ θ F + q F V θ PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

4π Φ H Φ θ F + q F V q q θ From Unt 3, Ltur 8, Sld umbr 8: ' L π 3 3 d ' 3 3 L 3 L 3 4π Φ H Φ d dq θ F + q F π π q V θ 3 q now nludd: d q q dqsn d d 4π V 3 3 Φ H Φ d dq θ 6 F + q θ F ( π ) q θ θ φ PCD STTACS Unt 3 Eltron Gas n HF & RPA Rf: F & W QToMPS; 8 Eq.3.34 43

4π V 3 3 Φ H Φ d dq θ 6 F + q θ F ( π ) q hang varabl, + q P.. P q onsquntly: ( + q) P+ q Φ H Φ Rf: F & W QToMPS; 8 Eq.3.34 4π V 3 3 d q dpθ 6 F P q θ F P q ( π ) + q d dp 3 3 ot th symmtry W hav to valuat ths volum n th -sa. PCD STTACS Unt 3 Eltron Gas n HF & RPA 44

Φ H Φ W hav to valuat ths volum n th -sa. 4π V 3 3 d q dpθ 6 F P q θ F P q ( π ) + q P q P q P+ q q q PCD STTACS Unt 3 Eltron Gas n HF & RPA 45

two rls of radus F F P q P q P+ q q q F ot whr th ntrs of th rls ar hosn PCD STTACS Unt 3 Eltron Gas n HF & RPA 46

F : radus of th rls F Evaluat th volum of th ntrston of th two rls n th -sa. n th rgon of ntrston of th two rls, w hav P+ q < F and also P q < P q P q P+ q q q F F Rf: F & W QToMPS; 8 Fg.3. 3 dpθ F P+ q θ F P q PCD STTACS Unt 3 Eltron Gas n HF & RPA 47

Φ H Φ 4π V 3 3 dq dpθ 6 F P+ q θ F P q ( π ) q 3 4π 3 3 3 dpθ ( ) F P+ q θ F P q F x+ x θ x, 3 F&W: QToMPS; 8 Eq.3.35 q wth x Φ H Φ F 4π V 4π q 6 ( π ) whol sa wth F dx dq 4π dq 3 3 3 ( ) F x+ x θ x q 3 4πV 4π 3 3 3 4π 6 Fdx F x x θ x ( π ) + 3 whol sa 48 PCD STTACS Unt 3 Eltron Gas n HF & RPA

Φ H Φ 4π V 3 3 dq dpθ 6 F P+ q θ F P q ( π ) q 4πV 4π 3 3 3 4π 6 Fdx F x x θ x ( π ) + 3 whol sa x 3 3 4πV 4π 3 6 F ( 4π F) dx x x ( π ) 3 + x wth x q F r s ( π ) /3 9 4... atom unts ( 4πε )...from sld 89, STTACS, Unt3, Ltur 9 f nrgy atom unts 4 m 4πε dstan a atom unts m rmttvty 4πε of vauum nrgy atom unts a a E ( 4πε ) Rydbrgs au Hartr of nrgy I ordr PT.96 r s rs Rydbrs PCD STTACS Unt 3 Eltron Gas n HF & RPA 49

For E HF-SCF fr ltron gas n jllum otntal : HF..96 Ryd rs r s As r (low dnsty) s E.P.Wgnr Phys Rv 46: (934) E For fr ltron gas n jllum otntal : Prturbaton thory gvs th sam rsult E I ordr PT..96 Ryd rs r s rs Wgnr sold.79.66 + +.. a rs r s Mnmum At ngatv nrgy Systm: bound Rfrn: Fttr & Wala Quantum Thory of Many- Partl Systms; Fg.3./ag 9 EXT CLASS: RPA PCD STTACS Unt 3 Eltron Gas n HF & RPA Qustons: d@hyss.tm.a. 5

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr Eltron Gas n th Random Phas Aroxmatons Plasma Osllatons n Fr Eltron Gas Rfrns: Th thory of lasma osllatons n mtals - by S Rams 957 R. Prog. Phys. Also: Chatr 4 n Many Eltron Thory by Stanly Rams PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

htt://www.hyss.ws.du/musum/ Exhbts- /Modrn/PlasmaTub/ndx_lasma. html htt://s.hys.ut.du/astr6/lt/arth/atmoshr.html PLASMA: 4 th stat of mattr.. hghly onzd rgon.. ostv hargd ons and vrtually fr ltrons PCD STTACS Unt 3 Eltron Gas n HF & RPA htt://www.rdorbt.om/duaton/rfrn_lbrary/sa_/solar_systm/5746/van_alln_radaton_blt/ 5

Ignor moton of th ons. as f thy ar frozn. Ions: rlatvly far mor massv and hav larg nrta. Mtal lasma Whol systm: ltrally nutral. PCD STTACS Unt 3 Eltron Gas n HF & RPA 53

Postv and gatv harg n balan ξ Dslamnt of all th ltrons to th rght nt ostv harg r unt ara + ρ ξ nt ngatv harg r unt ara ρ ξ surfa harg dnsty : σ ρξ nt fld n-btwn E ρξuˆ ε PCD STTACS Unt 3 Eltron Gas n HF & RPA 54

Eq. of moton nt fld n-btwn d ξ E ρξuˆ m ε dt ρξ ε ρ ω mε SI unts d ξ ρ CGS unts ξ CGS unts dt mε 4πρ ; 4π ω 4πε m ε Frquny of lasma osllatons Thrmal moton of ltrons: gnord xt that mltly w assumd that thrmal flutuatons would hav ausd dartur from qulbrum n lasma dnsty and thrby aus an onst of lasma osllatons. PCD STTACS Unt 3 Eltron Gas n HF & RPA 55

nt fld n-btwn E ρξuˆ ε ω ρ 3 CGS unts 4πρ m 4 3 π r 4 s ω 3 π r 3 3 s 3 mrs PCD STTACS Unt 3 Eltron Gas n HF & RPA Eq. of moton d ξ m ρξ dt ε d ξ dt ρ mε ξ Frquny of lasma osllatons ω Thrmal moton dsrson whn dsrson s rsnt: ω 4π 3 4 π r m 3 s E m F ω 56

For fr ltron gas n jllum otntal HF EPT..96 Ryd rs r s E BP β + + f 4. 3.96 β β β 3 rs r rs 48 s ; : ur bound to th wav numbr osllatons gt damd by random thrmal moton of th ltrons ω ( 3 )( ) 3/ m r s : Bohm & Pns: md-ffts D.Pns (963) Elmntary xtatons n solds (Bnjamn, Y) zro ont nrgy of th lasma osllatons 3 ω whr ω Ryd 3 r Random Phas Aroxmaton s PCD STTACS Unt 3 Eltron Gas n HF & RPA 57

Fld Orators ψˆ ( q) ψˆ ( q) ψ * ψ q q Rfrn: STTACS / Unt 3 / ltur 9 / H ψˆ ( q) f ( q) ψˆ( q) dq + ψˆ ( q) ψˆ ( q ')v( q, q ') ψˆ( q ') ψˆ( q) dqdq ' quvalnt H f j + j v l j j l j j l Comlt xrssons for th orators, nlusv of sn labls PCD STTACS Unt 3 Eltron Gas n HF & RPA 58

Comlt xrssons for th orators, nlusv of sn labls aσ aσ aa σσ aσ aσ aσ aσ H ψˆ ˆ ˆ ˆ v(, ') ˆ ( ') ˆ α q f qψβ q dq+ ψα qψ β q q q ψδ q ψγ( q) dqdq' PCD STTACS Unt 3 Eltron Gas n HF & RPA * ˆ ˆ α α α β jβ j α β j ψ q ψ q ψ q ψ q β ( q) f( q) ψ * H ψ j q dq α α β jβ j +, δ δ,, ± ± ± boms, nlusv of th xlt sn labls: + α β j l α β δ γ v( * * ψ α q ψ jβ q q, q') ψl δ q ψγ dqdq ' γ l δ α jβ H α f jβ + α, jβ v lδ, γ α jβ α jβ γ lδ j j l α β α β δ γ Rams /.4 / Eq..7 nlusv of sn labls 59

q r, ζ sa + sn oordnat ψ ψ ρ ζ q q ( q) artl dnsty orator 3 3 drρ( q) drψ qψ q : numbr of ltrons n th rgon ψ δ ψ ζ ' ζ ( q') ( q q') ( q) dq' ψ ( r') χ ζ ' δ( r r') ψ( r) χ ζ d r' ζ ' δ 3 ζ, ζ ' χ ζ ' χ ζ ψ ( r') δ( r r') ψ( r) d r' δ 3 ζ, ζ ' χ ζ χ ζ ψ ( r) ψ( r) ψ q ψ q ρ( q) PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

ψ ψ ρ q q ( q) artl dnsty orator ρ( r ) δ r r PCD STTACS Unt 3 Eltron Gas n HF & RPA 3 3 drρ( r) drδ r r 3 δ Rams, Many Eltron Thory Eq. 4.4; ag 7 ρ : dmnsonlss dr r r Fourr xanson: ρ( r) ρ V r 6

Postv harg dnsty ρ smard out unformly. ltrons r unt volum: ρ/v Fourr xanson of th ltron-ltron Coulomb ntraton [ ] [ ] harg L r j r rj ( ) V PCD STTACS Unt 3 Eltron Gas n HF & RPA Th abov sum s a trl sum, ovr th thr omonnts of. 6

r rj ( ) V multlyng both r onsdr frst ' j ( rj r) sds by ' r r ' r r r r r V j ( j ) ( j ) ( j) ' r r r r r V j j ' ( j ) j ' ( j) dr dr 3 ' r r 3 r r j j rj V PCD STTACS Unt 3 Eltron Gas n HF & RPA Th Wav Mhans of Eltrons n Mtals by Stanly Rams, ag 85 63

4π 4π ' PCD STTACS Unt 3 Eltron Gas n HF & RPA 3 ' ( r rj) d r j V Dra δ ( ' ) r δ ( ') ' 3 ' rj r 3 ' r r j drj drj rj V ' ( rj r) 3 3 r r j j rj V d r d r ' from sld 7, L9 : FT of C 4π r '.. ( ') ( j ) 4π xt whn [ ] [ ] harg L 64

r j Intgratng r rj ( ) V 3 3 j rj V 4π xt whn What s whn { } r r j j d r d r? 3 ' ( r rj) now, d r ( ') j δ V 3 r r j.. for ' : d r j δ δ V PCD STTACS Unt 3 Eltron Gas n HF & RPA Eq.3.; ag 3; F&W 65

3 3 j rj V 3 ( r r ) r r r j { } r r j j d r d r j V dr 3 d r δ r j δ Th Wav Mhans of Eltrons n Mtals by Stanly Rams, ag 85 3 d r r Potntal nrgy of th th ltron du to on ltron harg unformly smard throughout th box. PCD STTACS Unt 3 Eltron Gas n HF & RPA 66

Potntal nrgy of th th ltron du to th j th : r j r rj ( ) V Potntal nrgy of th th ltron du to all th ltrons: Pr V r r j rj j j j ( j) 4π xt whn 3 d r r PCD STTACS Unt 3 Eltron Gas n HF & RPA 67

Potntal nrgy of th th ltron du to all th ltrons: Pr Potntal nrgy of th th ltron du to all th ltrons and th ostv baground: V r r j rj j j j 4π xt whn Ur j j ( j) Sld 3 (rvous lass) trm anls th ostv jllum 4π V 3 d r r r r ( j) PCD STTACS Unt 3 Eltron Gas n HF & RPA 68

Potntal nrgy of th th ltron du to all th ltrons and th ostv baground: For xrtd on th th ltron: PCD STTACS Unt 3 Eltron Gas n HF & RPA Ur 4π V j j mr mv U ( r ) war magnt fors gnord 4π r v U( r) m V j m j alraton of th th ltron j j ( ( r )) rj 4π j V m ( ( r r )) r r ( j) 69

4π r v U( r) m V m j j ( ( r )) rj Du to th symmtral dstrbuton of th vtors th summand on th 4π RHS for ( j ) s ( ) m Hn no nd to xlud j trm.. 4π r v U( r) m V m PCD STTACS Unt 3 Eltron Gas n HF & RPA j ( ( r )) rj 7

4π r v U( r) m Vm alraton of th th ltron ltron harg dnsty ρ( r ) δ r r j r r ( j ) 3 3 drρ( r) dr r r δ ( ) Fourr xanson of harg dnsty ρ( r) ρ V r ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA : dmnsonlss 7

ρ Fourr xanson of harg dnsty ρ( r) ρ V 3 r drρ( r ) ρ ρ r r 3 r drδ r r ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA ρ( r ) δ r r 3 r dr δ ( r r ) total numbr of ltrons omonnts: dnsty flutuatons ovr th avrag ρ : dmnsonlss 7

r 4π r v U( r) m Vm alraton of th th ltron r 4π v mv j 4π ρ mv j r PCD STTACS Unt 3 Eltron Gas n HF & RPA j ρ r r ( j ) r d r r v ρ ρ dt ρ r ρ ( ) r d dt r ( r) 73

ρ ( ) r r d d ρ ρ r r dt dt r ρ r r + r r ρ r ( ) r r PCD STTACS Unt 3 Eltron Gas n HF & RPA 74

ρ r ( ) r PCD STTACS Unt 3 Eltron Gas n HF & RPA r r ρ r r from Sld 7: ρ 4π v ρ mv r r r 4 π ' ' r r r ρ ' mv ' ' ' ρ r 4π ' ( ' ) r r ρ ' m V ' ' ' 75 r

ρ r 4 π ' ( ' ) r r ρ ' V m ' ' ' ( ) r 4π ρ V m π ρ r 4π ( r ) r ρ 4 ' ρ ' V m ' ' ' V 4 π ' ρ ' V m ' ' ' m ρ ( ' ) r ( ' ) r ' PCD STTACS Unt 3 Eltron Gas n HF & RPA ' trm trms 76

ρ ρ r 4π r V m Eq. of moton for dnsty flutuatons 4 π ' ρ ' V m ' ' ' 4 π ' ρ ' ' ' ' ρ ( ' ) r r 4π r ρ V m V m ( ' ) r ow, rmmbr that r ρ ρ Qustons: d@hyss.tm.a. PCD STTACS Unt 3 Eltron Gas n HF & RPA r 4π r V m V m 4 π ' ρ ' ' ' ' ' ρ ( ρ ) 77

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr Eltron Gas n th Random Phas Aroxmatons QUATUM THEORETICAL TREATMET Plasma Osllatons n Fr Eltron Gas Rfrns: Th thory of lasma osllatons n mtals - by S Rams 957 R. Prog. Phys. Also: Chatr 4 n Many Eltron Thory by Stanly Rams PCD STTACS Unt 3 Eltron Gas n HF & RPA 78

Fourr xanson of harg dnsty ρ( r) ρ V ρ 3 r drρ( r ) r ρ : dmnsonlss ρ( r ) δ r r ρ 3 r drδ r r ρ r ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA total numbr of ltrons omonnts: dnsty flutuatons ovr th avrag 79

ρ ρ r 4π r V m Eq. of moton for dnsty flutuatons ( r ) 4 π ' ρ ' V m ' ' ' 4π ρ ( ' ) r r ρ V m ρ 4 π ' ( ' ) r ρ ' ' ' ρ ' ' V m Smlar to r ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA r 4π r V m 4 π ' ρ ' ' V m ' ' ' ρ ( ρ ) 8

ρ r 4π r V V m π Eq. of moton for dnsty flutuatons y PCD STTACS Unt 3 Eltron Gas n HF & RPA 4 ' ρ ρ ' ' mv ' ' ' r ρ ρ ' ρ Quadrat trms n dnsty flutuatons ( ' ) r θ x Phas fators of modulus unty Sum of vtors, n random drtons, n th omlx lan zx+y Random Phas Aroxmaton: glt quadrat trms n dnsty flutuatons omard to th lnar trms. OTE: LIEARIZATIO Bohm & Pns (95,53) 8

ρ ρ r 4π r V m π Eq. of moton for dnsty flutuatons RPA 4 ' ρ ρ ' ' m ' ' ' ( r ) r 4π V m ρ ρ Random Phas Aroxmaton LIEARIZATIO from Sld o.5; L : ρ RPA ρ ( r ) V r? ρ 4πρ m r ρ 8

ρ RPA ( r ) r 4πρ m ρ Ths trm dos not hav any alraton trm. It has only vlots: du to thrmal moton; t s not du tm-ndndnt to - ntraton st : trm Ο gnorabl for small valus of not gnorabl f would gt larg byond som lmt. must hav an ur lmt RPA + 4πρ ρ ρ ω ρ m PCD STTACS Unt 3 Eltron Gas n HF & RPA 83

ρ RPA ( r ) r 4πρ m ρ RPA + 4πρ ρ ρ ω ρ m ρ ω ρ + Th Fourr omonnts of th ltron dnsty osllat at th lasma frquny. PCD STTACS Unt 3 Eltron Gas n HF & RPA 84

ρ RPA ( r ) r 4πρ m ρ ρ ω ρ + RPA + Th Fourr omonnts of th ltron dnsty osllat at th lasma frquny. Colltv osllatons PLASMOS of th ltron gas Quantzd olltv xtatons lmntary xtatons W shall now xamn th ur lmt on PCD STTACS Unt 3 Eltron Gas n HF & RPA 85

ρ V r r 4 πρ m ρ ρ ω ρ r r V ρ r ρ r r ω r ρ + r ω r PCD STTACS Unt 3 Eltron Gas n HF & RPA 86

ρ + r ω glt of trm rqurs: r ( r ) ω st avrag r ρ v ω... for all, nludng for ltrons at th Frm surfa v(max) v v Frm f v f ω max ω v must hav an ur lmt f dnotd by Ur bound to wav numbr of lasma osllatons Lowr bound to wav lngth PCD STTACS Unt 3 Eltron Gas n HF & RPA 87

Quantum tratmnt H ψ Eψ Hamltonan for a bul ltron gas n a unform ostv baground jllum otntal H Hl + Hb + Hl b H 4π + m V j j r r ( j) H + m V j j PCD STTACS Unt 3 Eltron Gas n HF & RPA π r r ( j) 88

H + m V j j Mthod: transform th abov Hamltonan suh that lasma osllatons aar xltly as solutons of a st of Hamltonans for sml harmon osllators for varous valus of ω wth max v PCD STTACS Unt 3 Eltron Gas n HF & RPA π f r r H ( j) Quantum tratmnt ψ D. Bohm and D. Pns Phys. Rv. 8 65 (95) D. Pns and D. Bohm Phys. Rv. 85 338 (95) D. Bohm and D. Pns Phys. Rv. 9 69 (953) D. Pns Rvws of Modrn Physs 8 84 (956) S Rams 957 R. Prog. Phys. Th thory of lasma osllatons n mtals Eψ 89

H π + m V j j PCD STTACS Unt 3 Eltron Gas n HF & RPA r r ( j) Mthod: transform th abov Hamltonan suh that lasma osllatons aar xltly as a st of Hamltonans for sml harmon osllators for varous valus, ω wth max h' v SHO + q + mω q f m m ω ; mω m h' hsho + ω q m H P P + ω Q Q Hrmtan q, : Hrmtan Q, P : Hrmtan? anonally onjugat orators 9

H π + m V j j r r ( j) π H r + m V j j PCD STTACS Unt 3 Eltron Gas n HF & RPA ρ * r ρ j + rj r Inlud th j trm, and thn subtrat ts fft! H? j trms would gv : +++.+ π + ( * ρ ρ ) m V j 9

H π + ( * ρ ρ ) m V Transformaton Mthod: start wth a modl Hamltonan 4π H P P M P M ρ wth V Q, P : OT Hrmtan H P P + ω Q Q P P ; Q Q ρ r * + r ρ ρ ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA H Hrmtan 9

4π H P P M P M ; ρ V H P P M P ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA H P P M P ρ * H P P M P ρ H P P M P ρ H ρ * + r ρ ρ ρ P P ; P P r MP H sa symmtry ρ Hrmtan MP Q, P : OT Hrmtan ρ 93

4π H P P M P M ; ρ V max ω v f ω max v f Th ur lmt on lmts th total dgrs of frdom so that th total numbr of dgrs rmans fxd at 3 H ψ Eψ Th wavfunton must b a funton only of th ltron oordnats. PCD STTACS Unt 3 Eltron Gas n HF & RPA 94

ω max v H PCD STTACS Unt 3 Eltron Gas n HF & RPA f Th ur lmt on lmts th total dgrs of frdom so that th total numbr of dgrs rmans fxd at 3 ψ Eψ Th wavfunton must b a funton only of th ltron oordnats. ( ; f ) ψ funton Q ψ ψ Q funton q : ltron oordnats for < [, ] P Q Subsdary ondton Rams: Many Eltron Thory; Eq.4., ag 76 W must not ntrodu any addtonal dgrs of frdom Pψ for < Q P δ ', ' anonal onjugaton 95

H ψ ψ Q Eψ for < ;.. Pψ Hψ PCD STTACS Unt 3 Eltron Gas n HF & RPA 4π H P P M P M ; ρ V H + H ψ Eψ ow, w fft a UITARY TRASFORMATIO of H + H th Hamltonan S M Q ρ ; S * ; ; M Q ρ U M Q ρ S S U S UITARITY S U U Rams: Many Eltron Thory; ag 76,77 96

U S S M Q ρ ; Transformaton of all orators and th wavfunton undr th untary transformaton S * ; ; M Q ρ M Q ρ S Ω Ω Ω U nw U U U U S S S U U ψ nw U ψ ψ r U r U r nw Q U Q U Q nw U ρ ρ U ρ nw PCD STTACS Unt 3 Eltron Gas n HF & RPA sn ρ r r, Q, ρ : nvarant HOWEVER :, P : hang undr th transformaton 97

P Q P U P U nw q F r [, ] δ [, ] ', ' [, ] δ [, ] P Q P F Q ', ' [ P, U] U Q U P U UP nw + Q U P U? U S Fr q FQ Q U PU + UP Q S MQρ ; [ ] Q P, P nw + U P U PCD STTACS Unt 3 Eltron Gas n HF & RPA 98

P P + U [ P, U] nw P P nw + U U Q S Q S S Q S U Q PCD STTACS Unt 3 Eltron Gas n HF & RPA U Q S [ P, U] U wth S M Qρ U Q U M ρ ; U Q P P nw + U U Mρ P + U UM ρ P P + M ρ nw Rams: Many Eltron Thory; Eq.4.35, ag 77 99

Transformaton of th x omonnt of th momntum orator for th th ltron: U U x nw x q F r [, ] δ [, ] ', ' Fr q U U U U q [, ] U ( ) U U U x nw x x qx U q x PCD STTACS Unt 3 Eltron Gas n HF & RPA

U ( ) U U U x nw x x qx U q x [, ] [ U], x x nw x U U U U U q + U [ U], x nw x x PCD STTACS Unt 3 Eltron Gas n HF & RPA

+ U [ U], x nw x x ( ) U x nw x q x PCD STTACS Unt 3 Eltron Gas n HF & RPA U S ; ow : U wth S M Q ρ ; sn, U x U q [ ] U S ρ q q q U U M Q x x x ρ x nw x U U M Q q x ; ρ ( x ) nw x + M Q ; q x x

ρ x nw x M Q q x ρ + ; rj qx qx j q x r ρ r + { r } M Q x nw x x ; r ( x ) x nw x x ; PCD STTACS Unt 3 Eltron Gas n HF & RPA M Q r Smlar rlatons for y and z omonnts 3

M Q nw ; r Rams: Many Eltron Thory; Eq.4.38, ag 78 Smlar rlatons for y and z omonnts r, Q, ρ : nvarant undr th transformaton HOWEVER,, P : hang undr th transformaton PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

H ψ ( ; f ) ψ funton Q < ψ Eψ Th wavfunton must b a funton only of th ltron oordnats. funton q : ltron oordnats W must not ntrodu any addtonal dgrs of frdom ψ Q for < [, ] ' Subsdary ondton PCD STTACS Unt 3 Eltron Gas n HF & RPA Pψ for < P Q P δ, ' Q anonal onjugaton 5

ψ Q for < P Q Pψ for < P ψ < for ( )( ψ ) nw nw U PU U for < P P + M ρ from sld 95: nw ( ) for + < P M ρ ψ nw Qustons: d@hyss.tm.a. Subsdary ondton PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr 3 Eltron Gas n th Random Phas Aroxmatons Plasma Osllatons n Fr Eltron Gas Rfrns: Th thory of lasma osllatons n mtals - by S Rams 957 R. Prog. Phys. Also: Chatr 4 n Many Eltron Thory by Stanly Rams PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

H π + ( * ρ ρ ) m V H P P M P ρ wth M 4π V Hamltonan for a bul ltron gas n a unform ostv baground jllum otntal Ω Ω Ω nw U U U U nw ; PCD STTACS Unt 3 Eltron Gas n HF & RPA U M Q S S ; r, Q, ρ : nvarant undr th transformaton W now as: H nw r M Qρ H U H + H U P P + M ρ nw? 8

Transformaton of all orators and th wavfunton undr th untary transformaton Ω Ω Ω nw U U U U S ψ nw U ψ ψ Subsdary ondtons ψ Q S U ; for < Pψ for < S M Qρ PCD STTACS Unt 3 Eltron Gas n HF & RPA P ψ < for nw nw 9

H + M m H P P M P ρ H * ( ρ ρ ) + H H + H ψ Eψ * + M P P M P m + ρ ρ ρ H Hl + Hb + Hl b Eltrons + Postv Baground Auxlry Hamltonan π M V Our quston: PCD STTACS Unt 3 Eltron Gas n HF & RPA H H U H + H U nw?

H + H + M + P P M P (T ) nw * m ρ ( ρ ρ ) T T Our quston: + ; ρ x nw x M Q q x Hnw U H + H U? r j MQ ( j + ) m m j m nw j M M Q Q m j ( + ) r Rams: Many Eltron Thory; Eq.4.48, ag 79 T 3 (T ) nw ( * M ρ ρ ) PCD STTACS Unt 3 Eltron Gas n HF & RPA ; sn : nvarant ρ

H + H + M + P P M P * m ρ ( ρ ρ ) T (T ) ( * M ) nw ρ ρ ; sn : nvarant ρ sarat th summaton n two arts: for () > and () < ; ω max v f (T ) nw H sr.. * M + ; ( * ρ ρ M ρ ρ ) ; > < Short rang long rang PCD STTACS Unt 3 Eltron Gas n HF & RPA

H + H + M + P P M P * m ρ ( ρ ρ ) U P P M P ρ U (T 3 ) nw P P + M ρ nw ( ) ( ρ.. ) P P + M P P + M nw nw T 3 ( P ) P ( P + M ρ )( P + M ρ ) nw ( + ) P P P P + M P ρ ρ P + M ρ ρ nw ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

( + ) P P P P + M P ρ ρ P + M ρ ρ nw * + r & P P ρ ρ ρ ( ) ( * M + ) P ρ ρ P M P ρ M ρ P + < < < shral symmtry of vtors ( ) ( * M + ) P ρ ρ P M P ρ M ρ P + < < < < Hn: + ( ) ( ρ ρ ) M P M P < < ( ρ + ρ M P ρ ) M P P < PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

U P P M P ρ U (T 3 ) nw P P U P P U nw ( + ) P P P P + M P ρ ρ P + M ρ ρ nw ( P ) P P P + M ( P ρ+ ρ P) nw < < < ( ρ + ρ ρ ) M P P M P < < + < M ρ ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

(T 3 ) nw U P P M P ρ U nw ( P ) P P P + M ( P ρ+ ρ P) < < < ( ρ ) ( ρ ) MP U MP U? nw ( ρ ρ ρ ) w hav sn that: M P + P M P + < < P P P P M P nw < < < + + < < ρ M M ρ ρ ρ ρ * 6 PCD STTACS Unt 3 Eltron Gas n HF & RPA

(T 3 ) nw U P P M P ρ U ( ρ ) ( ρ ) MP U MP U? nw MPρ MPρ + M ρ ρ nw * + r & P P ρ ρ ρ ( ) ( ρ.. ) P P + M ρ nw P P + M P P + M nw nw ( ) ( MPρ M P ) ρ nw nw M P + M ρ ρ ρ 7 PCD STTACS Unt 3 Eltron Gas n HF & RPA

U P P M P ρ U (T 3 ) nw MPρ MPρ + M ρ ρ nw * + r & P P ρ ρ ρ MPρ MPρ + M ρ ρ * nw MP ρ MP ρ M ρ * ρ nw Earlr, w showd that: P P P P M P M ρ ρ + ρ + (T 3 ) nw has * nw < < < < 8 PCD STTACS Unt 3 Eltron Gas n HF & RPA

H + H * + M P P M P m + ρ ρ ρ T T T 3 W had asd: H H U H + H U nw? PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

H nw (T ) nw H ( ) j + (T ) nw m m j m j MQ r M M Q Q j ( + ) r ( * ) ( * + M ρ M ) ρ + ρ ρ ; ; > < Short rang long rang j (T 3 ) nw + P P + M P ρ + M * < < < ρ ρ (T 3 ) nw has MP ρ M * PCD STTACS Unt 3 Eltron Gas n HF & RPA ρ ρ

H nw H sr.. H ( ) j + m m j m j MQ r M M Q Q j ( + ) r ( * ) ( * + M ρ M ) ρ + ρ ρ ; ; > < Short rang long rang + P P + M P ρ + M * < < < ρ ρ j > * Hsr.. M ; ( ρ ρ ) MP ρ M ρ ρ * PCD STTACS Unt 3 Eltron Gas n HF & RPA

H nw H ( ) j + Th thr trms shown by th arrows togthr anl ah othr. m m j m j MQ ; r M M Q Q ( ρ ρ ) * + H sr.. + M > < + P P + M ρ ρ * < < j ( + ) r j PCD STTACS Unt 3 Eltron Gas n HF & RPA (T 3 ) nw has * M ρ ρ

H nw H ( ) j + m m j m j + P P + < MQ H sr.. r M M Q Q ; M j ( + ) r j n th nxt st, w us: < < P P P P M 4π 4π V V.. M PCD STTACS Unt 3 Eltron Gas n HF & RPA 3

H nw m m H ( ) j + j m j MQ π sr.. ; V < r M M Q Q + H + P P j ( + ) r j PCD STTACS Unt 3 Eltron Gas n HF & RPA m m j j M M Q Q M M Q Q ( + ) r ( + ) r j j 4

m m m m j j j j M M Q Q M M Q Q shral symmtry of ( + ) r ( + ) r M M Q Q ( + ) M M Q Q ( + ) j j vtors r r j j PCD STTACS Unt 3 Eltron Gas n HF & RPA 5

m m j j M M Q Q r ( + ) M M Q Q r ( + ) j M j 4π V M m + m j M Q Q M M Q Q r ( + ) j PCD STTACS Unt 3 Eltron Gas n HF & RPA 6

m m m + m j j j M Q Q M M Q Q r ( + ) M M Q Q r ( + ) M M Q Q r ( + ) j M j j 4π M V shral symmtry of vtors PCD STTACS Unt 3 Eltron Gas n HF & RPA 7

H H nw ( j + ) HH nw j m m j m j MQ sr.. ; V < r M M Q Q π + H + P P sr.. ; V ( + ) PCD STTACS Unt 3 Eltron Gas n HF & RPA j ( + ) r M Q ( j + ) + M Q Q m m m rj + M M Q Q m j Qustons? Wrt to: d@hyss.tm.a.n π + H + < P P r j j 8

Slt/Sal Tos from Thory of Atom Collsons and Strosoy P. C. Dshmuh Dartmnt of Physs Indan Insttut of Thnology Madras Chnna 636 Unt 3 Ltur umbr 4 Eltron Gas n th Random Phas Aroxmatons Plasma Osllatons n Fr Eltron Gas Rfrns: Th thory of lasma osllatons n mtals - by S Rams 957 R. Prog. Phys. Also: Chatr 4 n Many Eltron Thory by Stanly Rams PCD STTACS Unt 3 Eltron Gas n HF & RPA 9

H H nw Hnt M Q ( j + ) + M Q Q m m m j + m ( H ) nt MQ j + m j PCD STTACS Unt 3 Eltron Gas n HF & RPA j sr.. ; V rj π + H + M M Q Q < r j r ( + ) P P K K M M Q Q m j j r j ( + ) 3

H H H M Q Q nw + nt + m m + K + H π + P P sr.. ; V < ( H ) nt MQ j + m K M M Q Q m j j r j M M r j ( + ) PCD STTACS Unt 3 Eltron Gas n HF & RPA 4π V 4π V 3

M M H H H M Q Q nw + nt + m m π sr.. ; V < + K + H + P P 4π V 4π V 4πρ ω ; ρ m V M ω m P M M M 4 mω ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA π V P mω ρ P V V 3

H Hnw + Hnt + Q Q m P + K + H π + P P sr.. ; V < ω H π H + P P + Q Q nw ( ω ) m ; V + H + H + K sr.. nt PCD STTACS Unt 3 Eltron Gas n HF & RPA 33

H π H + P P + Q Q nw ( ω ) m ; V + H + H + K sr.. nt EXACT Rams: Many Eltron Thory Eq.4.58, ag 58 > * H sr.. M ; ( ρ ρ ) ( H ) nt MQ j + m j r j K M M Q Q m j Random Phas Aroxmaton LIEARIZATIO j ( + r r ) PCD STTACS Unt 3 Eltron Gas n HF & RPA j 34

H Hnw ( P P + ω Q Q) + + m ; Quas artls ntratng va H s.r. π + H + H nt s.. r ; V > H M * sr.. ; short rang ntraton ( ρ ρ ) M 4π V H > π V sr.. ; ( * ρ ρ ) PCD STTACS Unt 3 Eltron Gas n HF & RPA 35

Potntal nrgy of th th ltron du to all th ltrons and th ostv baground: trms anl th ostv jllum Ur 4π V j j r r ( j) Total otntal nrgy du to Coulomb ntratons of all th ltrons and th ostv baground: π Ur V j j Sum ovr all th ltrons,,,.. r r ( j) PCD STTACS Unt 3 Eltron Gas n HF & RPA 36

Total otntal π nrgy du to r rj Ur ( ) Coulomb V j j ntratons of all th ltrons and th ostv add and subtrat j trms baground: π ( r rj) π Ur ( ) V V ρ PCD STTACS Unt 3 Eltron Gas n HF & RPA j V slf r ; π nrgy π π * Ur ( ) ρ V ρ V π ( * Ur ρ ρ ) V 37

Total otntal nrgy du to Coulomb ntratons of all th ltrons and th ostv baground: FT of FT μr 4π r μ + of SC C 4π r π ( * Ur ρ ρ ) H V > π V sr.. ; > μ + κ κ μ ( * ρ ρ ) Srnd Coulomb PCD STTACS Unt 3 Eltron Gas n HF & RPA H sr.. total otntal nrgy du to SHORT RAGE ntratons 38

H nt MQ + m j ( ) j r K M M Q Q m j { } r ( + ) j j Bohm and Pns: FURTHER transformaton of th Hamltonan HH nw PCD STTACS Unt 3 Eltron Gas n HF & RPA an b arrd out to aount for H nt. 39

( H ) nt MQ j + m j ; r j H Hnw + P P + Q Q m ( ω ) π V + + + H sr.. H nt K Ths two trms gt modfd as a rsult of ths furthr transformaton Bohm and Pns: FURTHER transformaton of th Hamltonan an b arrd out to aount for H nt. PCD STTACS Unt 3 Eltron Gas n HF & RPA Rams: Many Eltron Thory; ag 8 4

H Hnw + P P + Q Q m F ( s Sld 56, L) ( ω ) π V + + + H sr.. H nt K Ths two trms gt rlad, on ; aount of furthr β transformaton, + + by m 6 wth β ω ω + E ω ω m and F wa dsrson. max ( P P ω Q Q) ω v f PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

H Hnw + P P + Q Q m F ( s Sld 56, L) ( ω ) π V + + + H sr.. H nt K Ths two trms gt rlad, on ; aount of furthr β transformaton, + + by m 6 wth β ω ω + E ω ω m and F wa dsrson. max ( P P ω Q Q) ω v f PCD STTACS Unt 3 Eltron Gas n HF & RPA 4

H Hnw P P + Q Q + ( ω ) π + H + H + K m ; V sr.. nt π H H P P Q Q H nw + ( + ω ) + s.. r m ; V Subsdry ondton: ( ) for + < P M ρ ψ nw What nd of a systm dos ths Hamltonan dsrb? PCD STTACS Unt 3 Eltron Gas n HF & RPA 43

H Hnw + P P + Q Q m R-arrang th trms: ( ω ) π + V ; H s.. r H Hnw P P + Q Q ; ( ω ) + π + Hsr.. m ; V PCD STTACS Unt 3 Eltron Gas n HF & RPA 44

Rams: Many Eltron Thory; Eq.4.63, ag 8 π H H P P Q Q H nw ( + ω ) + + s.. r m ; V ; Subsdry ondton: for P ψ < nw nw What nd of a systm dos ths Hamltonan dsrb? SHO Hamltonan H + mω x m Plasma osllatons Quas artls ntratng va H s.r. A onstant trm that s art of th ltron slf-nrgy whh not aountd for n th lasma osllatons. Long rang ntraton s aountd for by PLASMOS, and th short rang art that rmans s a srnd Coulomb ntraton. PCD STTACS Unt 3 Eltron Gas n HF & RPA 45

Random Phas Aroxmaton LIEARIZATIO j K M M Q Q m j ( + r r ) Bohm and Pns Transformaton of th Hamltonan Othr aths to RPA Equaton of Moton mthod Row (968) Grns funton mthod Thoulss (96) Dagrammat rturbaton thory.. Lnarzd Tm Dndnt Hartr/Dra Fo Alx Dalgaarno.. Waltr Johnson RRPA PCD STTACS Unt 3 Eltron Gas n HF & RPA 46 j Qustons: d@hyss.tm.a.