2. Hartree-Fock formalism

Transcription

1 2. Hartree-Fock formalsm 2.1 Slater determnants One speaks of the Hartree-Fock (HF) approxmaton and of the HF method. The former s also called the self-consstent feld approxmaton, or mean feld approxmaton. Its meanng s that every electron moves due to the acton of an electrostatc feld created by the presence of all other electrons (plus external feld sources). In ths sence t s close to what we appled whle dervng the Thomas-Ferm formalsm. The dfference s that we won t deal wth the charge densty from the very begnnng. Instead, we ll try to trace what happens wth the wave functons. We ll come back to the concept of densty at the end of the route. The HF approxmaton, or the mean-feld approxmaton, s an essental part of the apparat of theory, often used to solve dfferent models. It ntroduces an approxmaton that sometmes allows to solve a certan physcal problem analytcally. The HF method s more about solvng complex many-body problems numercally. In dong so, one essentally apples the self-consstent feld approxmaton. But there are many approaches that pursue ths way start from a tral wavefuncton, calculate the potental generated by correspondng charge dstrbuton, etc. Essentally, the startng pont s agan the statonary Schrödnger equaton (1.1). What s specfc for the HF method s the Ansatz used to represent a many-body wavefuncton. It s, n the HF method the wave functon s searched for as a Slater determnant constructed from onepartcle wave functons. Lets dscuss what ths s. If we consder the smplest case of partcles whch do not nteract, the probablty to fnd each partcle near certan place n space s the same and ndependent on that of other partcles. One would expect the densty ρ) to be smply N tmes the one-partcle densty, ρ) = Nϕ). (2.1) On the other sde, the densty s related wth many-body wavefuncton va Eq.(1.3). A possble consstent Ansatz s to construct the many-body wavefuncton as a product of (normalzed) ndvdual wave functons, then (2.1) wll obvously hold: Ψ 1,...,r N ) = ϕ 1 1 )ϕ 2 2 )...ϕ N N ). (2.2) Ths Ansatz s known; t leads to the Hartree method, not n use any more, because t neglects an mportant property of electrons that they are Ferm partcles, and the wavefuncton of a system of fermons must be antsymmetrc wth respect to nterchangng any par of partcles: Ψ 1,...r k,...r l,...r N ) = Ψ 1,...r l,...r k,...r N ). (2.3) We emphasze here that the poston of an argument n Ψ fxes the number of partcular electron, and the argument value the actual coordnate of that electron. Obvously 8

2 enough, the wave functon constructed accordng to (2.2) does not possess ths property. Actually, t does not have any partcular symmetry property wth respect to nterchangng partcles. But we can easly force the wave functon to obey Eq.(2.3); for ths, we smply take products lke (2.2), nterchange there postons of all partcles parwse and construct the combnatons whch are a pror antsymmetrc. Ths s easy to do for two partcles: Ψ 1,r 2 ) = 1 2 ϕ 1 1 )ϕ 2 2 ) ϕ 1 2 )ϕ 2 1 ). (2.4) The factor 1/ 2 s ntroduced n order to keep the normalzaton. If our one-partcle wave functons are normalzed to 1,.e., the probablty to fnd one electron somewhere n space ϕ ) 2 dr = 1, (2.5) then the probablty to fnd two partcles anywhere n space must be 1, and ndeed Ψ 1,r 2 ) 2 dr 1 dr 2 = 1 ϕ 1 1 ) 2 dr 1 ϕ 2 1 ) 2 dr 2 2 }{{}}{{} =1 =1 ϕ 2 1)ϕ 1 1 )dr 1 ϕ 1 2)ϕ 2 1 )dr 2 }{{}}{{} = = ϕ 1 1)ϕ 2 1 )dr 1 ϕ 2 2)ϕ 1 2 )dr 2 + }{{}}{{} = = + ϕ 2 1 ) 2 dr 1 ϕ 1 2 ) 2 dr 2 = 1. }{{}}{{} =1 =1 In dong so, we assumed that ndvdual one-partcle wavefunctons are all orthogonal. Ths s ndeed the case f they are solutons of the same Hamlton operator. If they are not for some reason but stll form the complete bass of solutons to the correspondng one-partcle problem, they can be one by one orthogonalzed before proceedng further wth the constructon of the many-body wavefuncton. How do we proceed f there are more than 2 electrons? There are N! possbltes to nterchange them. We sum up over all of them, try all possbltes to nterchange every two electrons and wrte down antsymmetrc terms lke n Eq.(2.4). Ths results n a fully antsymmetrc wave functon: Ψ 1,...,r N ) = 1 sgn(p) ϕ P1 1 ) ϕ P2 2 )... ϕ PN N ). (2.6) N! P P runs over all permutatons of electrons, sgn(p)=1 for even permutatons and sgn(p)=- 1 for odd permutatons. It s common and convenent to wrte down the wavefunctons as determnants, because (2.6) s smlar to how the determnant of a matrx s determned, and that s why the wave 9

3 functon of ths form s called Slater determnant: Ψ 1,...,r N ) = 1 ϕ 1 1 ) ϕ 1 N )... (2.7) N! ϕ N 1 ) ϕ N N ) The determnant form helps to llustrate mportant propertes of a many-partcle wavefuncton. Each lne corresponds to a certan one-electron state, and each column to a certan poston n space, among N postons r 1,...,r N of partcles we consder. The nterchange of ether two rows or two columns means that we nterchanged two partcles; the wave functon then changes sgn by constructon. Moreover, f there happen two dentcal lnes or two dentcal columns, t means that two partcles share the same spatal coordnates; the determnant then equals zero, meanng that such stuaton s physcally mpossble. Actually, the latter s only true f two electrons n queston have the same spn drecton; otherwse, they can well share the same cell n the same phase space,.e. have the same one-partcle wave functon and the same spatal coordnate. In order to allow for that, we ntroduce a generalzed coordnate ncorporatng poston and spn, x = {r, σ}, and we ll wrte dx...for σ dr.... then (2.7) must be wrtten as Φ(x 1,...,x N ) = 1 N! ϕ 1 (x 1 ) ϕ 1 (x N ).. ϕ N (x 1 ) ϕ N (x N ). (2.8) We shall keep for Slater determnants a specal notaton Φ, to dstngush them from general-form Ψ(x 1,...,x N ). So far, we concentrated on symmetry propertes of a many-body wavefuncton, and we dd not specfy the shape of one-partcle wavefunctons whch consttute a Slater determnant. They could be taken from the soluton of a correspondng one-electron problem (.e., neglectng the nteracton between the electrons), but t wll be hardly a good approxmaton to Ψ wth nteracton. What we ll do next s to search for the best one-electron wavefunctons ϕ(x) whch wll allow to construct the best approxmaton to the true many-body wavefuncton, keepng the determnantal form (2.8) for our approxmatons to Ψ. As a crteron for the best functons, we ll rely on the varatonal prncple. As n the course of dervng the Thomas-Ferm equaton, we ll search for those ndvdual ϕ s whch mnmze the total energy. How s all ths ustfed? How do we know that the wave functon of the form (2.8) s a reasonable one? So far, we only used Slater determnants as an Ansatz n order to acheve the antsymmetry of the many-body wave functon, wthout any reference to ts other propertes. The mportance of Slater determnants les n the fact that, for a gven number of electrons N, the Slater determnants (constructed from all possble oneelectron functons) form a complete bass set. In other words, f we have the Hlbert space of one-partcle wavefunctons H(1), then the Hlbert space of many-body wavefunctons s obvously a drect product of one-partcle Hlbert spaces: H(N) = H(1) H(1)...H(1) }{{} N (2.9) 1

4 It means e.g. that any many-body wavefuncton can be expanded nto a sum of products of approprate one-partcle functons: Ψ(x 1,...,x N ) = a ν1 ν 2...ν M ϕ ν1 (x 1 ) ϕ ν2 (x 2 )...ϕ νn (x N );. (2.1) ν 1 ν 2...ν N It can be shown that the bass of N-partcle Slater determnants (constructed from all possble one-electron wavefunctons n H(1) s a complete bass n H(N),.e. any antsymmetrc N-partcle wavefuncton can be expanded over t: Ψ(x 1...x N ) = µ C µ Φ(x 1...x N ). (2.11) Ths s the so-called theorem of completeness. It emphaszes the mportance of Slater determnants n the treatment of many-electron systems. Now, the HF approach assumes that from the expanson (2.11) one keeps only one term, but the possble best one. In the search for the best Slater determnant, we vary the consttuent ϕ (x ) functons. 2.2 Dervaton of the HF equatons We start from the Hamltonan of a many-body system, as n (1.1). We only wrte t down n a form that emphaszes the dfference between one-partcle and two-partcle operators: H = N =1 h2 2m 2 + u(x ) N v(x, x ). (2.12) dfferentates wth respect to coordnates of the th electron; u(x ) s one-electron operator that can typcally be the feld of a nucleus (or nucle) n an atom (or a molecule), stuated at R α : u(x ) = Z α e α r R α ndependently on spn, but t can be spn-dependent as well (e.g., an external magnetc feld). v(x, x ) ncorporates all nteractons whch may depend on the coordnates (and spns) wthn each partcle par; n the followng, t wll be everywhere the Coulomb nteracton between two electrons: v(x, x ) = e 2 r r. We construct the total energy as the expectaton value of ths Hamltonan, Φ H Φ, and mnmze t n a varatonal approach smlarly to how we have t done n the Thomas-Ferm part under consderng the normalzaton condton: δ Φ H Φ δϕ α(x) N ε dy ϕ (y) ϕ (y) =. (2.13) =1 11

5 The Lagrange multplers ε take care of the normalzaton of each one-partcle functon ϕ (x) separately. Usng (2.6), Φ H Φ becomes: Φ H Φ = 1 { sgn (ν)sgn(µ) dx 1...x N ϕ ν1 N! (x 1)...ϕ νn (x N N) νµ =1 + 1 N v(x, x ) 2 ϕ ν1(x 1 )...ϕ νn (x N ). h2 2m 2 + u(x ) Let us dscuss the effect of one-partcle and two-partcle contrbutons n the Hamltonan separately. The one-partcle operator selects only functons wth the argument x ; the rest s ntegrated over varables dfferent from x. The ntegrals dx k ϕ νk (x k)ϕ µk (x k ) gve 1 f permuted ndces νk = µk and zero otherwse; n other words, ths condton works as δ νµ under the double sum over permutatons. Snce only dentcal permutatons ν = µ contrbute, sgn(ν) sgn(µ)=1 n all terms, ndependently on whether the permutaton n queston s odd or even; one sum over permutatons s lfted, and the remanng one contans N! terms, all permutng the ndex of the one-partcle wave functon. Taken together wth the summaton n, the sum over permutatons gves smply a sum over orbtal numbers 1 to N, each number appearng N! tmes. The result s: one-partcle contrbuton to Φ H Φ N = =1 dxϕ (x) h2 2m 2 + u(x) ϕ (x). (2.14) Smlarly n the two-partcle part, the ntegraton over varables not appearng n the two-partcle nteracton term gves 1 whenever the ndces of ϕ and ϕ, each obtaned by ts correspondng permutaton, turn out to be equal. Ths reduces the sum to: two-partcle to Φ H Φ contrbuton = 1 1 N sgn (ν)sgn(µ) N! 2 νµ dx dx ϕ ν(x )ϕ ν(x ) v(x, x ) ϕ µ (x )ϕ µ (x ). Snce all functons but two have been already used n non-zero terms, there are only two possbltes how the ndces ν, µ may relate: 1) ν = µ and ν = µ ν and µ are dentcal, sgn(ν) sgn(µ)=1; 2) ν = µ and ν = µ µ s dentcal to ν, wth subsequent nterchangng of functons and ; sgn(ν) sgn(µ)= 1. In both cases, one can ntroduce δ νµ and lft one summaton over permutatons, and the second summarton produces N! dentcal terms, resultng n: two-partcle contrbuton to Φ H Φ = 1 2 N dxdy ϕ (x)ϕ (y)ϕ (x)ϕ (y) ϕ (x)ϕ (y)ϕ (y)ϕ (x) v(x, y). (2.15) 12

6 Summarzng, the functon to be vared n (2.13) s: N N Φ H Φ ε dyϕ (y)ϕ (y) = dxϕ (x) h2 =1 =1 2m 2 + u(x) ϕ (x) N dxdy ϕ 2 (x)ϕ (y)ϕ (x)ϕ (y) ϕ (x)ϕ (y)ϕ (y)ϕ (x) v(x, y) N ε ϕ (x) ϕ (x) dx. (2.16) =1 Its varaton n δϕ α (x) gves: { δ... = dxδϕ α (x) h2 2m 2 + u(x) ϕ α (x)+ + dy ϕ (y) ϕ α (x) ϕ (y) ϕ (y) ϕ α (y) ϕ (x) v(x, y) ε α ϕ α (x) =. α The factor 1 n front of the two-partcle term dsappears because the one-partcle functon 2 beng varyed, ϕ α, may concde wth both ϕ and ϕ. The condton δ... = for arbtrary δϕ α (x) leads to the HF equaton(s): h2 2m 2 + u(x) + dy v(x, y) ϕ (y) ϕ (y) ϕ α (x) α dy v(x, y) ϕ (y) ϕ (x) ϕ α (y) = ε α ϕ α (x). (2.17) α The term α dy v(x, y)ϕ (y) ϕ (y) has a clear meanng of a Coulomb potental whch acts on the electron n r due to the presence of all other electrons. The next sum whch can be formally represented as α dy v(x, y) ϕ (y) ϕ α (x) ϕ α(y) ϕ (x) ϕ ϕ α (x) (2.18) α(x) ϕ α (x) ntroduces the exchange potental actng effectvely on the one-partcle functon n queston, ϕ α (x); t s a correcton to the Coulomb potental and s due to the antsymmetry of the many-partcle wave functon. Ths term depends on the unknown functon ϕ α (x) tself. The HF equatons s a system of ntegro-dfferental equatons, whch couple N functons. The soluton s typcally done by teratons. Wth all functons ϕ α (x) found, the one-determnant many-body wavefuncton can be constructed. 13

7 2.3 Formulaton n terms of densty and densty matrx Note that the requrement α can be dropped n (2.17), because for = α the Coulomb and exchange terms exactly cancel. The summaton n s over occuped orbtals. Wth the condton α lfted, the sum n the Coulomb term ϕ (x) ϕ (x) gves the partcle densty defned by Eq. (1.3). Let us show t. In dong so, we ll generalze (1.3) as dependng on x,.e., our densty wll be r-dependent and have a certan spn ndex. Usng the determnant form of the many-body wave functon, Ψ (x, x 2,...,x N )Ψ(x, x 2,...,x N ) dx 2...dx N = = 1 sgn(ν) sgn(µ) ϕ ν1 N! (x) ϕ µ1(x) ϕ ν2 (x 2)ϕ µ2 (x 2 )dx 2... ϕ νn (x N)ϕ µn (x N )dx N. νµ The ntegrals over x 2... x N are non-zero only f all ν = µ (and are =1 n ths case), due to orthonormalty of one-electron functons. Ths demands n non-zero terms to be ν1 = µ1 as well and hence the permutatons ν and µ to be dentcal, that lfts one sum n permutatons. Then Ψ (x, x 2,...,x N )Ψ(x, x 2,...,x N ) dx 2...x N = 1 ϕ N! ν(x) ϕ ν (x). ν The sum over permutatons ncludes N! terms, among them N that permute the frst ndex and, for each of them, (N 1)! possbltes to permute other ndces, whch are however not anymore explctly present. Hence ϕ ν1 (x) ϕ ν1(x) = (N 1)! ϕ (x) ϕ (x). ν Consderng the factor N n the defnton of ρ), Eq. (1.3), one arrves at ρ(x) = ϕ (x) ϕ (x) (2.19) n the HF formalsm. So, the Coulomb term n the HF equaton (2.17) can be transformed to explctly nclude the (spn)-densty. The spn component s mplctly present n x = {r, σ}. If we wsh to express the densty rrespectvely of spn, t wll suffce to sum up over spn components n x,.e., ρ) = ρ, ) + ρ, ). Now, f we do a smlar trck wth the exchange term, t wll be reduced to the form contanng the densty matrx γ(x; y). The defnton of the latter s γ(x; y) = N Ψ (y, x 2,...,x N ) Ψ(x, x 2,...,x N ) dx 2...dx N, (2.2) so that ρ(x) = γ(x; x). (2.21) 14

8 We follow exactly the same arguments as for ρ(x), ust keepng y dfferent from x, and arrve at γ(x; y) = ϕ (y) ϕ (x). (2.22) The HF equaton (2.17) then transforms nto: h2 2m 2 +u(x)+ dy v(x, y) ρ(y) ϕ α (x) dy v(x, y) γ(x; y) ϕ α (y) = ε α ϕ α (x). (2.23) Ths can be looked at as an (ntegro-dfferental) operator acrtng on each one-partcle functon ϕ α (x): ĥ HF ϕ α (x) = ε α ϕ α (x) ; (2.24) ĥ HF = h2 2m 2 + u(x) + v(x, y) ρ(y) dy v(x, y) γ(x; y) }{{} dy (2.25) s called the Fock operator. It s the same for all orbtals and s hermtan. In order to show that, we construct matrx elements of t between any functons f and g from the same Hlbert space as one-electron functons: f ĥhf g = h2 dxf (x) 2 g(x) + dxf (x)u(x)g(x) + 2m + dxf (x)g(x) dy v(x, y)ρ(y) dxf (x) dy v(x, y)γ(x; y) g(y) ; on the other hand, usng ρ (x) = ρ(x), γ (x; y) = γ(y; x), v (x, y) = v(x, y) = v(y, x), ( g ĥ HF f ) = h2 dxg(x) 2 f (x) + dxg(x)u(x)f (x) + 2m + dxg(x)f (x) dy v(x, y)ρ(y) dxg(x) dy v(x, y)γ(x; y) f (y). Last terms are dentcal after renamng ntergraton varables x y n one of them, and the frst terms (those wth 2 ) become dentcal va ntegraton by parts for the functons f, g whch are zero at nfnty. Hence the Fock operator s hermtan and st egenvalues real. 2.4 Meanng of HF egenvalues; Koopmans theorem We dscuss now the meanng of ε α, whch have been ntally ntroduced as Lagrange multplers takng care of the normalzaton condton n the varatonal approach. We construct the expectaton value of the total energy of the system wth all N electrons and wth an electron n the state α removed. E HF (N) = Φ H Φ, that s (2.16) wthout the last term. N E HF (N 1 α ) = α ϕ (x) h2 2m 2 + u(x) ϕ (x) dx + 15

9 + 1 2 N α α = E HF (N) N α N α ϕ (x)ϕ (y)ϕ (x)ϕ (y) ϕ (x)ϕ (y)ϕ (y)ϕ (x) v(x, y) dxdy = ϕ α (x) h2 2m 2 + u(x) ϕ α (x) dx ϕ α(x)ϕ (y)ϕ α (x)ϕ (y) ϕ α(x)ϕ (y)ϕ α (y)ϕ (x) v(x, y) dxdy ϕ (x)ϕ α (y)ϕ (x)ϕ α (y) ϕ (x)ϕ α (y)ϕ (y)ϕ α (x) v(x, y) dxdy. Wth and x y two last sums become dentcal, hence E HF (N 1 α ) = E HF (N) dxϕ α (x) h2 N 2m 2 + u(x) + dy v(x, y) ϕ (y) ϕ (y) ϕ α (x) α N dxϕ α (x) dy v(x, y) ϕ (y) ϕ α(y) ϕ (x) α Comparng ths wth (2.17) we see that the underlned terms together gve exactly ε α ϕ α (x), then E HF (N 1 α ) = E HF (N) ϕ α (x)ε αϕ α (x) dx ; ε α = E HF (N) E HF (N 1 α ). (2.26) Ths s the theorem of Koopmans. It s mportant to menton that n the course of dervng t, we assumed that one-electron wavefunctons remaned unchanged, as one electron was removed. In realty ths would not be the case. In a practcal calculaton, one would estmate the exctaton energy E HF (N) E HF (N 1) (e.g., n photoemsson) as dfference of total energes n ntal and fnal states wth relaxaton of one-electron wavefunctons taken nto account,.e. from two self-consstent calculatons correpondng to dfferent electron confguratons of the system. The Hamltonan operator of a non-relatvstc atom commutes wth ˆL 2, ˆL z, Ŝ 2, Ŝ z ; the exact solutons of many-body Schröderger equaton are supposed to do the same. The Fock operator by ts constructon does not have any specal reason to commute wth any of these operators; hence the most general form of the HF solutons does not have any partcular symmetry propertes. A HF orbtal of general form looks lke where χ(σ) are Paul spnors χ + (σ) = ϕ α (x) α ) χ +(σ) + ϕ ( ) α ) χ (σ) (2.27) { { 1 for σ = 1/2 for σ = 1/2 for σ = 1/2 ; χ (σ) = 1 for σ = 1/2. (2.28) 16

10 The general form of the soluton (2.27) allows to treat systems wth non-collnear magnetc densty. From the general form (2.27) one can demand that the soluton commutes wth Ŝz,.e. the spn drecton wll be fxed for each orbtal, and no mxng of σ = 1/2 and σ = 1/2 s possble. Ths s known as unrestrcted Hartree-Fock (UHF), although ths s actually a restrcton wthn a more general formalsm. The one-partcle wavefunctons n UHF may be numbered as ) ϕ UHF α (x) = ϕ UHF α ( 1 α ) ( ) α 1 (x) = ϕ ( ) ) for α = 1,..., N (+), for α = N (+) +1,..., N, f α runs through all orbtals. In prncple, one could number orbtals correspondng to both spn drectons separately, because they are now completely decoupled. Only the total number of partcles must be conserved: ) ϕ UHF α (x) = α ) ( 1 ( ϕ UHF β (x) = ϕ ( ) β ) 1 Ŝ z ϕ UHF α (x) = ) for α = 1,..., N (+), for β = 1,..., N ( ), N (+) + N ( ) = N. (2.29) These functons are now egenfunctons of the spn proecton operator, + h 2 ϕuhf α (x) for α { 1,..., N (+)}, h 2 ϕuhf α (x) for α { 1,..., N ( )}, and the acton of Ŝz on the Slater determnant gves: Ŝ z Φ UHF (x 1,..., x N ) = h N (+) N ( ) Φ UHF (x 1,..., x N ). 2 So, the essence of UHF s that t allows to treat magnetc systems. Not only the orbtals, but the densty matrces are now labeled by spn ndex: γ (±) ;r ) = N (±) α ϕ (±) α ) ϕ (±) α ) ; (2.3) ρ (±) ) = γ (±) ;r), (2.31) and the HF system splts nto two parts, labeled by spn: h2 2m 2 + u) + dr ρ ) v,r ) α ) dr γ (+) ;r ) v,r ) α ) = = ε (+) α ϕ(+) α ) ; h2 2m 2 + u) + dr ρ ) v,r ) ϕ ( ) α ) dr γ ( ) ;r ) v,r ) ϕ α ( ) ) = 17 = ε ( ) α ϕ( ) α ), (2.32)

11 where the external feld u) may be also spn-dependent (e.g., n the presence of a magnetc feld). The couplng between two parts s acheved by the fact that the total densty ρ) = ρ (+) ) + ρ ( ) ) enters both equatons; moreover, the total number of partcles s constant, N = N (+) + N ( ). The numbers N (+) and N ( ) however are not fxed and may vary (from teraton to teraton). In a practcal calculaton, one would expand the orbtals over a bass set whch s larger than N, and as n the course of soluton on each teraton one gets egenvalues, separately for (+) and ( ), the N lowest n energy among them all wll be occuped, that determnes N (+) and N ( ), ρ (+), ρ ( ) etc. for ths teraton. 6 It s noteworthy that the densty matrces γ (±) ;r ) are labeled by spn. Ths s so because the exchange nteracton only nvolves the one-electron states wth the same spn state. The effect of exchange can be looked at as the Coulomb nteracton of an orbtal searched for, say, α ) wth a correspondng exchange densty ρ (+) rα ): dr γ (+) ;r )v,r ) α ) ρ (+) dr γ(+) ;r )v,r ) α ) α ) α α ) ; rα ) = γ(+) ;r ) α ). (2.33) ) The exchange densty s non-local (t depends on both r and r ), and for every orbtal ndex α t ntegrates over r exactly to 1: dr ρ (+) r,α ) = dr =1 N (+) ) α ) } {{ } δ α N ) (+) α ) = =1 δ α = 1, that means that each electron s surrounded by ts correspondng exchange hole, from where the charge densty f exactly one electron of the same spn s excluded. We proceed dscussng symmetry aspects. If one demands that ndvdual orbtals do possess a good angular momentum value, for nstance can be casted as ϕ ν (x) = R nν )Y lνm ν (θ, φ)χ σ (σ) (n an atom), then one arrves at the so-called restrcted Hartree-Fock (RHF) formalsm. For a multatomc system, one can generalze ths demand by takng nto account approprate (by symmetry) combnatons of atom-centered functons wth the same angular momentum value. Then for each orbtal ˆlz ϕ ν (x) = hm ν ϕ ν (x), and for the Slater determnant ( ) ˆL z Φ(x 1,...,x N ) = h m ν Φ RHF (x 1,...,x N ). ν 6 ths scheme s refered to as the aufbau prncple. 18

12 The more severe the symmetry constrant s, the more restrcted the varatonal space for one-electron wavefunctons and the hgher the calculated ground-state energes. The ultmately best energes (the HF lmt) can be obtaned only wth freely varable orbtals whch do not possess any partcular symmetry propertes. Ths stuaton s known as the symmetry dlemma n the HF formalsm. The exchange densty ρ (+) r,α ) ntroduced n Eq.(2.33) s non local, as t should be from physcal consderatons, but the fact that t depends on the orbtal ndex s not physcally motvated. Slater (1951) proposed 7 to weght t over occuped orbtals, accordng to ther correspondng partal denstes: ρ (+) r ) = ρ (+) X,r ) = ρ (+) r ) ϕ )ϕ ) ϕ )ϕ ) = ϕ )ϕ )ϕ )ϕ ). (2.34) ϕ )ϕ ) After summaton, the exchange densty ρ (+) X,r ) does not depend on the orbtal ndex anymore. We perform now ths summaton analtcally for the easest case of free partcles, wth one-electron egenvalues ϕ k ) = 1 V e kr, where V s the volume of the box ncludng the N electrons, and perodcal boundary condtons are assumed. The summaton wll be substtuted by the ntegraton n the momentum space up to k F = p F / h, so that the lowest states wth energes up to E F = h 2 kf 2/2m are occuped. k F s related to densty as defned by Eq. (1.7), k F = (3π 2 ρ) 1/3, and kf dn ; dn = 2V k = V (2π) 3d3 4π 3 k2 dk sn θ k dθ k dφ k. When the electron gas s not spn polarzed, N (+) = N ( ) =, the summaton over orbtals of each spn component runs over The denomnator of Eq.(2.34) gves: The numerator of Eq.(2.34) dn (±) = V 8π 3 k2 dk sn θ k dθ k dφ k. ϕ )ϕ ) V 1 8π 3 V 4π 3 k3 F = k3 F 6π 2 = ρ 2. (2.35) ϕ )ϕ )ϕ )ϕ ) 7 Phys. Rev. 81, 385 (1951) ( ) V 2 kf k 2 dk 1 r) sn θ 8π 3 k dθ k dφ k V ek } {{} kf k 2 dk sn θ k dθ k dφ k 1 ) V ek r } {{}. 19

13 Each of the ntegrals n k yelds: kf Fnally k 2 dk sn θ k dθ k dφ k e kr kf π = 2π k 2 dk sn θ k dθ k e kr cos θ k = = 4π R kf k dk sn(kr) = 4π R 3 sn(k FR) k F R cos(k F R). ϕ )ϕ )ϕ )ϕ ) 1 sn (kf r r ) k F r r cos (k F r r 2 ) (8π 3 ) 2 (4πk3 F )2 (k F r r ) 3. (2.36) Substtutng (2.35) and (2.36) nto (2.34), we get ρ (+) X,r ) = 3k3 F 2π 2 sn (kf r r ) k F r r cos (k F r r 2 ) (2.37) = 9 2 ρ 1 (k F r r ) k F r r 2. (k F r r ) 3 The plot of the functon nvolved s shown n Fg ((sn(x)-x*cos(x))/x**3)**2.1 ((sn(x)-x*cos(x))/x**3)** ) 2 Fgure 2.1: Plot of ( 1 (x) x We now come back to the HF equatons (2.32) where the exchange term was transformed accordng to Eq.(2.33) dr γ (+) ;r ) v,r ) α ) dr ρ (+) rα )v,r ) α ) so t took a form of a potental actng on the orbtal we search for, v (+) X ) α ), 2

14 and now that after statstcal averagng ρ (+) rα does not depend on the orbtal ndex α anymore, the exchange potental s the same for all orbtals. Usng the result (2.37) for the exchange densty, (+) ρ X ) = e X,r ) dr = r r (centerng the coordnate system at r) = 4πe r 2 dr 1 3k 3 F sn(kf r ) (k F r ) cos(k F r 2 ) r 2π 2 (k F r ) 3 ( 3 1/3 ( ) sn x x cosx 2 = 6e ρ π) 1/3 xdx = e 3 ( ) 3 1/3 ρ 1/3. (2.38) }{{ x 3 } 2 π 1/4 v (+) We compare ths result wth (1.26) from the Thomas-Ferm secton and see that the exchange potental s ndeed proportonal to ρ 1/3, as was argued there based on dmensonalty consderatons. The prefactor must be dependent on the spatal dstrbuton of densty; t s constant n ths case because we assumed homogeneous densty dstrbuton n the dervaton. Otherwse, v (+) X ) would have got an explcte dependence on r. We note that, although the exchange densty ρ (+) X,r ) ntegrates over r exactly to 1 for every r, the value of the ntegral dr ρ (+) X,r )/ r r would of course depend on the shape of ρ (+) X,r ). However, snce the Coulomb nteracton 1/ r r has sphercal symmetry, only sphercally averaged part of ρ (+) X,r ),.e. dependent only on r r, wll contrbute to the value of the exchange potental at any gven poston r. Fnally, we note that one can cast the exchange energy n the HF scheme, after a statstcal averagng, n the form of a spatal ntegral obver exhange energy densty, E X = dr ɛ X ) ; obvously ɛ X ρ 4/3 for the homogeneous electron gas. Ths result wll be later on used and generalzed for the case of a slowly varyng densty. 2.5 Par correlaton functon The exhange densty ρ X ) has the meanng of the partcle densty expelled from the vcnty of each electron as a consequence of the Paul prncple. The same effect can be characterzed by a dmensonless property, a par correlaton functon, whch shows how any two partcles tend to avod each other. We start from some general defntons. In addton to the 1st order densty matrx prevously defned n Eq. (2.2), we ntroduce now the two-partcle densty matrx, γ 2 (x 1, x 2 ; y 1, y 2 ) = N(N 1) Ψ (y 1, y 2, x 3,..., x N )Ψ(x 1, x 2, x 3,...,x N ) dx 3...dx N. 2! (2.39) 21

15 2! n the denomnator stands for generalty, showng how to ntroduce hgher-order densty matrces. Obvously γ 2 s related to the 1st order matrx γ by γ(x; y) = 2 γ 2 (x, x 2 ; y, x 2 )dx 2. (2.4) N 1 Takng a dagonal of γ 2 we obtan another property dependng on the coordnates of two partcles, that s the par densty ρ 2 : ρ 2 (x, y) = 2γ 2 (x, y; x, y). (2.41) From ths, two dfferent par correlaton functons are derved: g(x, y) = ρ 2(x, y) ρ(x)ρ(y) ; (2.42) h(x, y) = ρ 2 (x, y) ρ(x)ρ(y). (2.43) For large spatal dstances between partcles, g 1 and h. These defntons are qute general. In order to conclude the HF part, we calculate par densty n the HF approxmaton for the same model case, a homogeneous electron gas. Frst we construct the two-partcle densty matrx for Slater determnants. Smlarly to how we proceeded for the one-partcle densty resultng n Eq. (2.19), N(N 1) γ 2 (x 1, x 2 ; y 1, y 2 ) = sgn(ν) sgn(µ) 2N! νµ N ϕ ν1 (y 1) ϕ ν2 (y 2)ϕ µ1 (x 1 ) ϕ µ2 (x 2 ) =3 dx ϕ ν (x ) ϕ µ (x ). In analogy wth consderng two-partcle contrbutons to Ψ H Ψ, Eq. (2.15), we conclude that ν = µ for 3, that leaves the combnatons ν1 = µ1, ν2 = µ2; sgn(ν) sgn(µ) = 1; ν1 = µ2, ν2 = µ1; sgn(ν) sgn(µ) = 1 and (N 2)1 possbltes to permute other ndces. Fnally γ 2 (x 1, x 2 ; y 1, y 2 ) = 1 2 N ϕ (x 1 )ϕ (x 2 )ϕ (y 1)ϕ (y 2), ϕ (x 1 )ϕ (x 2 )ϕ (y 1 )ϕ (y 2 ). (2.44) Ths can be further on wrtten down as γ 2 (x 1, x 2 ; y 1, y 2 ) = 1 N N ϕ 2 (y 1) ϕ (x 1 ) ϕ (y 2) ϕ (x 2 ) N N ϕ (y 2) ϕ (x 1 ) ϕ (y 1) ϕ (x 2 ) = 1 γ(x 1 ; y 1 ) γ(x 1 ; y 2 ) 2 γ(x 2 ; y 1 ) γ(x 2 ; y 2 ). (2.45) 22

16 Note that ths does not hold not n the general case anymore, but only for Slater determnants. Usng (2.41), we obtan for ρ 2 (x, y): ρ 2 (x, y) = 2 1 γ(x; x) γ(x; y) 2 γ(y; x) γ(y; y) = ρ(x) ρ(y) γ(x; y) 2. (2.46) Now, n order to specfy spatal dependences n the densty matrces and two-partcle densty, we must decde how are occuped one-partcle states dstrbuted over two spn components. Assumng paramagnetc electron gas, N (+) = N ( ) =, γ,r ) = γσ;r σ ) = σσ = From ths t follows +ϕ ( ) + ϕ ( ) =1 =1 ) ( 1 ) ( ) 1 }{{} ϕ (+) = ) ( 1 ) ( ) 1 ) ) ( 1 ) ( 1 ) )+ ) + ) ( 1 ) ( 1 ϕ ( ) ) = ) + ϕ ( ) )ϕ ( ) ) = ) } {{ } = ϕ ( ) ) + = γ (+) ;r ) + γ ( ) ;r ). (2.47) ρ) = ρ (+) ) + ρ ( ) ), as t should be. For γ 2 1,r 2 ;r 1,r 2 ) we have to consder 24 =16 terms wth dfferent spn attrbuton. We keep only those whch do not contan (...)ϕ( ) (...) or (...)ϕ( ) (...) and hence are not orthogonal n spn space. What s explctly left n the paramagnetc (PM) case: γ2 PM 1,r 2 ;r 1,r 2 ) = γ 2 1 σ 1,r 2 σ 2 ;r 1 σ 1,r 2 σ 2 ) = σ 1 σ 2 σ 1 σ 2 = 1 2 = 1 2,=1 + + ϕ ( ) + ϕ ( ),=1 ϕ ( ) 1 ) 2 ) 1 ) 2 ) + 1 ) ϕ ( ) 2 ) 1 ) ϕ ( ) 2 ) + 1 ) 2 ) ϕ ( ) 1 ) 2 ) + 1 ) ϕ ( ) 2 ) ϕ ( ) 1 ) ϕ ( ) 2 ) 1 ) 1 ) ϕ ( ) 2 ) 1 ) 2 ) 2 ) ϕ ( ) 1 ) ϕ ( ) 2 ) = 4 ϕ 1 ) ϕ 2 ) ϕ 1 ) ϕ 2 ) 2 ϕ 1 ) ϕ 2 ) ϕ 2 ) ϕ 1 ). (2.48) 23

17 From ths, ρ PM 2 1,r 2 ) = = 4,=1 4 ϕ 1 ) ϕ 2 ) ϕ 1 ) ϕ 2 ) 2 ϕ 1 ) ϕ 2 ) ϕ 2 ) ϕ 1 ) = ϕ 1 ) 2 ϕ 2 ) 2 2 ϕ 1 )ϕ 2) ϕ 2 )ϕ 1) = = ρ 1 ) ρ 2 ) 1 2 γ 1;r 2 ) 2. (2.49) The par correlaton functon g 1,r 2 ) s then g PM 1,r 2 ) = ρ 2 1,r 2 ) ρ 1 )ρ 2 ) = 1 1 γ 1 ;r 2 ) 2 2 ρ 1 )ρ 2 ). (2.5) For the gas of non-nteractng electrons wth constant densty, γ 1 ;r 2 ) = 2 ϕ 2 ) ϕ 1 ) V k 2 e k 1 r 2 ) dk sn θ 4π 3 k dθ k dφ k V and calculated as has been already done above, leadng to Eq.(2.36). Fnally 2 sn(kf r) (k F r) cos(k F r). (2.51) g r r ) = (k F r) 3 For fully spn-polarzed ( ferromagnetc ) electron gas, one would assume γ FM ;r ) = σσ γσ;r σ ) = γ FM 2 1,r 2 ;r 1,r 2 ) = 1 2 N (+) = N; N ( ) = ; ρ) = ρ (+) ); =1,=1 ) ( 1 ) ( ) 1 1 ) 2 ) ) = γ (+),r ) ; 1 ) 2 ) 1 ) 2 ) 1 ) 2 ) ; ρ FM 2 1,r 2 ) = ρ 1 ) ρ 2 ) γ 1 ;r 2 ) 2 ; (2.52) g FM 1,r 2 ) = ρfm 2 1,r 2 ) ρ 1 )ρ 2 ) = 1 γ 1;r 2 ) 2 ρ 1 )ρ 2 ). (2.53) Comparng wth (2.5), one can see that the correlaton functon s zero at the orgn, snce all electrons have the same spn now. k becomes larger and hence the fluctuatons n the correlaton functon have smaller perod than n the case of non-magnetc electron gas of the same densty. 24