Lecture IV : The Hartree-Fock method I. THE HARTREE METHOD We have see the prevous lecture that the may-body Hamltoa for a electroc system may be wrtte atomc uts as Ĥ = N e N e N I Z I r R I + N e N e r r + N I N J I Z I Z J R I R J. () For the tme beg we take the fal term to be a costat ad we oly cosder the electroc Hamltoa, whch s ust Ĥ e = N e N e N I Z I r R I + N e N e r r () I Eq. the ketc eergy term ad the ucleus-electro teracto term are sums of sgle-partcle operators, each of whch act o a sgle electroc coordate. The electro-electro teracto term o the other had s a par teracto ad acts o pars of electros. To facltate the upcomg math, let s make the followg defto Ĥ e = ĥ ( x ) + ĥ ( x, x ). (3) where x s ow a geeralzed coordate that cludes spatal as well as sp degrees of freedom. The Hartree-Fock method s a varatoal, wavefucto-based approach. Although t s a may-body techque, the approach followed s that of a sgle-partcle pcture,.e. the electros are cosdered as occupyg sgle-partcle orbtals makg up the wavefucto. Each electro feels the presece of the other electros drectly through a effectve potetal. Each orbtal, thus, s affected by the presece of electros other orbtals. The startg pot of the Hartree-Fock method s to wrte a varatoal wavefucto, whch s bult from these sglepartcle orbtals. Oce we make a sutable asatz to the wavefucto, all that s left s the applcato of the varatoal prcple as descrbed Lecture. The smplest wavefucto that ca be formed from these orbtals s ther drect product Φ( x,, x N ) = φ ( x )φ ( x ) φ N ( x N ). (4) Ths s the Hartree approxmato ad t s a straghtforward task to calculate the varatoal lowest eergy from Eq. 4. However, the Hartree wavefucto has a very mportat shortcomg, whch s that t fals to satsfy atsymmetry, whch states that a fermo wavefucto chages sg uder odd permutatos of the electroc varables. The permutato operator s defed by ts acto o the wavefucto ˆP Φ( x,, x,, x,, x N ) = Φ( x,, x,, x,, x N ) = Φ( x,, x,, x,, x N ) (5) If a odd umber of such permutato operators are appled to the wavefucto, t pcks up a mus sg whle o chage sg occurs uder a eve umber of permutatos. I order to satsfy the atsymmetry codto, a more sophstcated form tha that of the Hartree wavefucto s eeded. II. THE SLATER DETERMINANT If, for example, we have a two-electro system wth orbtals φ ( x ) ad φ ( x ), the followg varatoal wavefucto satsfes the atsymmetry codto, at the same tme preservg the sgle-partcle pcture Φ( x, x ) = c [φ ( x )φ ( x ) φ ( x )φ ( x ) (6) where c s the ormalzato costat. For three electros, the equvalet atsymmetrzed wavefucto would be [ Φ( x, x, x 3 ) = c φ ( x )φ ( x )φ 3 ( x 3 ) φ ( x )φ ( x 3 )φ 3 ( x ) + φ ( x 3 )φ ( x )φ 3 ( x ) φ ( x )φ ( x )φ 3 ( x 3 ) + φ ( x 3 )φ ( x )φ 3 ( x ) φ ( x )φ ( x 3 )φ 3 ( x ). (7)
Upo closer specto, we otce that the same permutatos of orbtals wth matchg sgs are obtaed by the followg determat φ ( x ) φ ( x ) φ 3 ( x ) Φ = c φ ( x ) φ ( x ) φ 3 ( x ) (8) φ ( x 3 ) φ ( x 3 ) φ 3 ( x 3 ) Geeralzg ths to a N-electro system where the orbtals are take to satsfy orthoormalty, we have Φ = φ ( x ) φ ( x ) φ N ( x ) φ ( x ) φ ( x ) φ N ( x )... φ ( x N ) φ ( x N ) φ N ( x N ) (9) where the factor frot esures ormalzato. For a arbtrary umber of electros the wavefucto form Eq. 9 ca be show to satsfy the desred atsymmetry codto. The determat, referred to as a Slater determat lterature, has terms each multpled by - or depedg o the party of the permutato. Each term has each orbtal φ oly oce ad each of the argumets x oly oce. Thus, each term may be wrtte as follows : ( ) P(,,,N) φ ( x )φ ( x ) φ N ( x N ) (0) where the dces,, take values betwee ad N ad the expoet of - frot refers to the order of appearace of the orbtal dces the term. The term pcks up a - frot f the correspodg permutato s odd ad + f t s eve. For ease of otato, we replace P(,,, N ) by the shorthad otato P(), where ow refers to a partcular arragemet (or sequece) of the N dces. The Slater determat may the be wrtte as ( ) P() φ ( x )φ ( x ) φ N ( x N ) () where the sum rus over the terms. Nothg has bee sad so far about the form of the orbtals φ ( x ) ad they are left to be foud as a result of the mmzato procedure assocated by the varato. I order to acheve that we ow calculate the expectato value of the Hamltoa for ths varatoal wavefucto E H = Φ Ĥe Φ. () III. ELECTRONIC ENERGY FOR THE SLATER DETERMINANT Frst, we tackle the sgle-partcle terms Eq.. Φ ĥ ( x ) Φ = ( ) P() ( ) P() φ ( x )φ ( x ) φ N ( x N ) ĥ( x ) φ ( x )φ ( x ) φ N ( x N ) (3), Because each orbtal makg up the Slater determat depeds oly o a sgle coordate, we ca par up those orbtals that have the same argumet ad separate them to dvdual er products, except for the oe that has the same argumet as the operator : Φ ĥ ( x ) Φ = ( ) P() ( ) P(), φ ( x ) φ ( x ) φ ( x ) φ ( x ) φ ( x ) ĥ( x ) φ ( x ) φ + ( x + ) φ + ( x + ) φ N ( x N ) φ N ( x N ) (4) Due to orthoormalty of the chose orbtals, all the er products yeld delta fuctos Φ ĥ ( x ) Φ = ( ) P() ( ) P() δ δ φ ( x ) ĥ( x ) φ ( x ) δ + + δ N N (5),
Due to the Kroecker delta s all k are equal to k except for. But because all terms appear exactly oce the products, = s also automatcally satsfed. Thus, the sequece of dces labeled s detcal to that labeled, makg the permutatos yeld detcal sgs. We have, as a result, Φ ĥ ( x ) Φ = φ ( x ) ĥ( x ) φ ( x ) (6) Now, for a gve sequece labeled by ad for a fxed, there are (N )! terms the sum. The sum over sequece dex may the be reduced to the sum of a sgle dex 3 Φ ĥ ( x ) Φ = (N )! φ ( x ) ĥ( x ) φ ( x ) (7) Because the expectato value s a tegrato over the varable x, each term yelds the same result ad there are N such terms. Fally we replace the ow arbtrary dex by a geerc dex. The fal expresso s the Φ ĥ ( x ) Φ = φ ĥ φ (8) Next, we deal wth the more complcated case of the two-partcle terms. The startg pot s detcal to the case of the sgle-body part of the Hamltoa where orbtals sharg the same argumet are pared up to er products except for the oes that have the same argumet as the operator. I the case of the two-body part of the Hamltoa there are two such orbtals. Assumg wthout loss of geeralty that < m we thus have Φ m ĥ ( x, x m ) Φ = ( ) P() ( ) P() δ δ δ + + δ m m δ m+ m+ δ N N m, Ths tme, there are two cases regardg the values ad m ca take :. = ad m = m. = m ad m =. φ ( x )φ m ( x m ) ĥ( x, x m ) φ ( x )φ m ( x m ) (9) The frst case s very smlar to the stuato the treatmet of the sgle-body term ad causes the sequeces ad to be equal. For the secod case however, the sequeces dffer by a sgle par, where oe has m ad terchaged. Ths term the pcks up a mus sg. We may ow reduce oe of the sums over the sequeces ad obta Φ m ĥ ( x, x m ) Φ = [ φ ( x )φ m ( x m ) ĥ( x, x m ) φ ( x )φ m ( x m ) m φ m ( x )φ ( x m ) ĥ( x, x m ) φ ( x )φ m ( x m ) (0) By a smlar argumet to the oe that leads to Eq. 8, we otce that for a fxed sequece ad for fxed ad m, there are (N )! terms the sum. We do ot dvde by two because ths factor s already cluded. We do however, make sure that m because each term shows up oly oce the sum. Eq. the becomes Φ ĥ ( x, x m ) Φ = (N )! [ φ ( x )φ m ( x m ) ĥ( x, x m ) φ ( x )φ m ( x m ) m m m φ m ( x )φ ( x m ) ĥ( x, x m ) φ ( x )φ m ( x m ) () If s ot equal to m, for each there are N dfferet values for m. Thus the umber of (, m) pars s N(N ) ad we may replace the sum over such pars wth ths factor. Fally, we replace ad m by more covetoal dces such as ad ad the two-body term reduces to Φ ĥ ( x, x m ) Φ = m, [ φ φ ĥ φ φ φ φ ĥ φ φ (),
where we have oce aga removed the depedece o orbtals. Notce the mportat fact that the term = does ot gve ay cotrbuto to the sum because the two terms exactly cacel that case. Ths s gog to be a maor dfferece betwee the Hartree-Fock method ad desty fuctoal theory. Puttg together Eq. 8 ad Eq., the expectato value of the Hamltoa for the Slater determat s Φ Ĥe Φ = φ ĥ φ +, [ φ φ ĥ φ φ φ φ ĥ φ φ Next, we would lke to apply the varatoal prcple to the value Eq. 3 to obta the best possble estmate. 4 (3) IV. THE HARTREE-FOCK EQUATIONS The varatoal prcple that we wll apply here s rather dfferet from the lear varato that we saw the frst lecture. There the form of our approxmate wavefucto was wrtte as a expaso over a collecto of predetermed fuctos ad we mmzed the expectato value (at the same tme obeyg the ormalzato costrat) wth respect to the coeffcets of the bass fuctos. Here however we employ a much more geeral treatmet where we mmze wth respect to the bass fuctos themselves! Needless to say, ths requres fuctoal dfferetato where ay chage affected the expectato value Eq. 3 due to a ftesmal chage ay of the orbtals φ k should be zero. φ k φ k + δφ k δ Φ Ĥe Φ = 0 (4) I addto, we demad through Lagrage multplers that the set of ortals φ k rema orthogoal throughout the mmzato process. The codto to be satsfed s the δf δ Φ Ĥe Φ λ ( φ φ δ ) = 0 (5), where the umber of the Lagrage multplers s ow N ad we demad that the dervatves th respect to all the Lagrage multplers vash. Frst, let s see how the sgle-body term chages uder a ftesmal chage oe of the orbtals, φ k. δ Φ ĥ ( x ) Φ = δφ k ĥ φ k + φ k ĥ δφ k = δφ k ĥ φ k + δφ k ĥ φ k (6) The last equalty of Eq. 6 has bee wrtte makg use of the hermtcty of the sgle-partcle operator. Smlarly, the varato of the two-body term yelds δ Φ ĥ ( x, x ) Φ = δ { } φ φ ĥ φ φ φ φ ĥ φ φ, = { } φ δφ k ĥ φ φ k + φ φ k ĥ φ δφ k δφ k φ ĥ φ φ k φ k φ ĥ φ δφ k + { } δφ k φ ĥ φ k φ + φ k φ ĥ δφ k φ φ δφ k ĥ φ k φ φ φ k ĥ δφ k φ (7) where t s uderstood that the secod le ĥ carres dces ad k ad the thrd le t carres dces ad k. We ca smplfy the equato Eq. 7 by makg use of the hermtcty of ĥ ad the detty f f  f 3f 4 = f f  f 4f 3 (8) whch holds f both of the fuctos o ether sde of the two-body operator are terchaged smultaeously. Iterchagg the order of orbtals o the thrd le yelds the same terms as the secod le except for the summato dces ad whch are arbtrary ad terchagable. Havg two detcal sums, the factor of cacels. Applyg fally the Hermtcy, we reduce the two-body term to δ Φ ĥ ( x, x ) Φ = [ φ δφ k ĥ φ φ k + φ δφ k ĥ φ φ k δφ k φ ĥ φ φ k δφ k φ ĥ φ φ k (9)
The fuctoal dervatve of the Lagrage multplers proceeds exactly the same way wth the Hermcty cosdered aga at the ed. Puttg everythg together the varato turs out to be 5 δf = δφ k ĥ φ k + δφ k ĥ φ k + [ φ δφ k ĥ φ φ k + φ δφ k ĥ φ φ k δφ k φ ĥ φ φ k δφ k φ ĥ φ φ k [λ k δφ k φ + λ k δφ k φ (30) For reasos whch wll become clear shortly, we shall evaluate the varato wth respect to φ k ad ot φ k. (For a proof that varato wth respect to φ k ad φ k are equvalet, see the appedx at the ed.) Wrtte as a tegral, terms Eq. 30 would read, for stace, δφ k φ ĥ φ φ k = δφ k ( x )φ ( x )ĥ( x, x )φ ( x )φ k ( x )d x d x. (3) Moreover, sce we are cosderg the varato th respect to oly δφ k, we would completely dsregard terms wth δφ the ket. As we saw the secto o fuctoal tegrals, evaluatg δf δφ bascally amouts to gettg rd of δφ k the bras k Eq. 30 ad also the tegral over ts argumet. Ths leaves ĥ φ k ( x ) + { } φ ( x )ĥ[φ ( x )φ k ( x )d x φ ( x )ĥ[φ ( x )φ k ( x )d x = λ k φ ( x ) (3) It s customary to wrte Eq. 3 usg the orbtal-depedet operators Ĵ ad ˆK as [ ĥ + (Ĵ ˆK ) φ k = λ k φ (33) where the frst ad the secod term are straghtforward, sgle-body operators ad the thrd term s a tegral operator. Ths s ow a set of terdepedet sgle-partcle egevalue equatos. The operator Ĵ correspods to the classcal teracto of a electro dstrbutos gve by φ ad φ k ad s called the drect term whle ˆK, called the exchage term, has o classcal aalogue ad s a drect result of the atsymmetry property of the wavefucto. The Fock operator s defed as ad usg ths defto Eq. 33 takes the smple form ˆF = ĥ + ˆFφ k = (Ĵ ˆK ) (34) λ k φ (35) = There are several dfferet solutos to the equatos Eq. 35 each correspodg to a dfferet set of λ k. We have the freedom to cocetrate upo those λ k whch satsfes λ k = δ k ǫ k (36) where ǫ k s essetally a ew ame for the Lagrage multplers[. Wth ths, Eq. 35 may be wrtte as ˆFφ k = ǫ k φ k. (37) I ths form, Eq. 37 s a tradtoal egevalue equato. For each k there s a equvalet equato defg a system of Schrödger-lke, oe-partcle equatos. Although t s temptg to terpret the egevalues ǫ k as the eergy levels of a teractg system, ths s fact ot ustfed because the sgle-electro pcture s ot correct. However, f terpreted correctly the Hartree-Fock egevalues do correspod to certa physcal ettes. V. TOTAL ENERGY AND KOOPMANS THEOREM The total Hartree-Fock eergy, E HF, of the system of N electros dscussed above may be wrtte as E HF = φ ĥ φ + [ φ φ ĥ φ φ φ φ ĥ φ φ (38),
makg use of Eq. 8 a Eq.. O the other had the egeerges, ǫ k, of the Hartree-Fock equatos gve Eq. 37 may be summed to gve 6 ǫ k = k φ ĥ φ + [ φ φ ĥ φ φ φ φ ĥ φ φ. (39), Eq. 38 ad Eq. 39 demostrate that the total eergy caot be gve as a smple sum over the egeeerges of the Fock operator but are closely related by E HF = k ǫ k [ φ φ ĥ φ φ φ φ ĥ φ φ. (40), Ths dscrepacy stems from the descrpto of a true may-body system terms of sgle-partcle orbtals. Next, let s compare the eergy of two systems, oe wth N electros ad oe wth N, where a electro has bee removed from the th level from the system wth N electros to obta the system wth N electros. The eergy requred to remove the electro s gve by E N E HF (N ) E HF (N) = φ ĥ φ + [ φ φ ĥ φ φ φ φ ĥ φ φ, φ ĥ φ [ φ φ ĥ φ φ φ φ ĥ φ φ, = φ ĥ φ [ φ φ ĥ φ φ φ φ ĥ φ φ [ φ φ ĥ φ φ φ φ ĥ φ φ. (4) whch s obtaed by evaluatg the dfferece a straghtforward maer. I the fal sum ca be substtuted by sce the dex s artrary. We the obta E N = φ ĥ φ [ φ φ ĥ φ φ φ φ ĥ φ φ [ φ φ ĥ φ φ φ φ ĥ φ φ. (4) I Eq. 4, t s permssble to exchage the dces ad f doe smultaeously o ether sde of the operator, we have E N = φ ĥ φ [ φ φ ĥ φ φ φ φ ĥ φ φ [ φ φ ĥ φ φ φ φ ĥ φ φ The two sums at the ed are detcal. Thus E N = φ ĥ φ [ φ φ ĥ φ φ φ φ ĥ φ φ (43) By comparg Eq. 43 ad Eq. 35, we see that the rght had sde s othg but ǫ E HF (N ) E HF (N) = ǫ (44) We thus arrve at Koopmas theorem, whch states that each egevalue of the Fock operator gves the eergy requred to remove a electro from the correspodg sgle-electro state. Smlarly the eergy requred to add a electro to orbtal m ca be prove to be gve by E HF (N + ) E HF (N) = ǫ m (45) I Eq. 44 ad Eq. 45, level s assumed to be occuped ad level m s assumed to be empty. Therefore ǫ k may be terpreted as orbtal eerges the approxmato that the chages the orbtals whe the electro s removed or added s small.
7 VI. HARTREE-FOCK-ROOTHAAN EQUATIONS Solvg the Fock equatos gve Eq. 33 volves the evaluato of the orbtals at every sgle pot space, whch s a rather dautg task. Istead, Roothaa proposed a lmted varato stead where the orbtals are expaded terms of a kow bass set ad the varato s coducted over the coeffcets of the bass fuctos. Ths s slghtly dfferet from the usual varatoal prcple that we dscussed Lecture sce there the approxmate wavefucto was drectly expaded terms of the kow bass wthout the extra step of the Slater determat betwee. I Roothaa s expaso, each orbtal s wrtte as N b φ l ( x) = c pl χ p ( x) (46) p where N b s the umber of bass sets used. Most of the tme, the bass that we use s gog to be a trucated set ad N b s gog to be fte. Whe substtuted to Eq. 5, ths expaso yelds ˆF = N b = m,=, N b m,,p,q c mc χ m ĥ χ + c mc p c c q [ χ m χ p ĥ χ χ q χ p χ m ĥ χ χ q [ N b ǫ,m c mc χ m χ (47) where we oce aga make use of dagoal Lagrage multplers. Sce we coduct the varato over the coeffcets rather tha the orbtals, we have δf = 0 F c rk The frst term wll yeld a ozero result oly f m = r ad k =, whch reduces to N b = 0 (48) c k χ r ĥ χ. (49) The secod term brgs two cotrbutos, oe for each of the c coeffcets. By prevous experece (ad also by a qute straghforward yet tedous calculato) we foresee that these two cotrbutos wll be equal. Let s the look at the case where m = r ad k = ad multply by two to obta the etre cotrbuto from the two-body term N b c pk c c q p,q [ χ r χ p ĥ χ χ q χ p χ r ĥ χ χ q (50) For further use, let s exchage the summato dces p ad to obta N b c k c pc q [ χ r χ ĥ χ p χ q χ χ r ĥ χ p χ q p,q (5) Fally the term wth the Lagrage multplers ca be evaluated qute smply to result c k χ r χ (5) ǫ k N b The mmzato the yelds, puttg everythg together ad collectg all terms uder a sgle sum N b χ N b [ r ĥ χ + c p c q χ k χ ĥ χ p χ q χ χ k ĥ χ p χ q c N b k = ǫ k χ r χ c k (53) p,q If ow, we call the term the large curly braces F r ad recogze the er product o the rght-had sde as a elemet of the overlap matrx, amely S r the Eq. 53 turs to the followg matrx equato F C = ǫc (54)
where C s the usual vector of coeffcets. Eq. 54, although very smlar to the matrx equato that we obtaed whe dealg wth the varatoal prcple, dffers that the represetato of the matrx F ay bass depeds aga o the coeffcets that we are tryg to calculate. Ths s aga a result of the descrpto of the may-body wavefucto as formed by oteractg sgle-partcle orbtals where the effect of the orbtals come out drectly the operator that defes the egevalue equato. A very smlar problem ll be ecoutered whe we deal wth DFT ad stadard methods of tacklg wth such so called self-cosstet feld problems wll be llustrated. 8 VII. SPIN PROPERTIES Because the Slater determat s a product of magato, t s o guarateed to satsfy all the propertes of the exact wavefucto. Let us, for stace, look at the sp propertes of the Slater determat. We shall cosder two operators : Ŝ z ad Ŝ. The total sp ay oe drecto s the sum of the sps of dvdual electros Ŝ x = ŝ,x Ŝ y = ŝ,y ad Ŝ z = ŝ,z (55) ad the total sp s gve by Ŝ = Ŝ x + Ŝ y + Ŝ z (56) I order to fd out about the sp propertes of the approxmate fucto, we eed to explctly specfy the sp depedece of the dvdual orbtals. Let s assume that of the orbtals have α sp whle m of them have β sp. The the determat may be wrtte shorthad otato as Φ( x, x,, x N ) = φ + α, φ+ α, φ+ α, φ βφ β φ mβ (57) where we have allowed the posto-depedet φ + ad φ orbtals to be dfferet. If the fucto, Φ Eq. 57 s wrtte dow explctly as a sum, we have Φ = ( ) P() φ + ( r )α() φ + ( r )α(), φ + ( r + )β( + ) φ + N ( r N ) (58) = Usg the Ŝz operator Eq. 55, the wavefucto Eq. 58 s clearly see to be a egestate of the Ŝz operator wth Ŝ z Φ = m Φ (59) Oe ca prove, however, through some straghforward (but somewhat legthy) algebra that Φ s a egefucto of S oly f m = ad φ = φ +. If we mpose these costrats o the Slater determat, ths amouts to the restrcted Hartree-Fock method. Alteratvely, we may ether gore the problem ad use urestrcted Hartree-Fock or resort to the proected Hartree-Fock method where the lear combato of more tha oe Slater determat s take as the approxmate wavefucto, whch s also a egefucto of the operator Ŝ. VIII. APPENDIX Cosder a real fucto, f(z, z ), of the complex varable z ad ts complex cougate z. The real ad magary parts of the varable z ca be wrtte as lear combatos of z ad z as z r = z + z z = z z. (60) If we cosder ow the dervatves wth respect to z ad make use of the cha rule, we obta f = f z r z z r z + f z z z = ( f f ) z r z z (6)
where the last equalty has bee wrtte usg Eq. 60. Smlarly the dervatve wth respect to z yelds f z = f z r z r z + f z z z = ( f + f ). (6) z r z z Sce f s real, the dervatves f z r ad f z Eq. 6 ad Eq. 6 are also real. Equatg both of these equatos to zero at the mmum the yelds the codtos 9 f z r = 0 ad f z = 0 (63) ad are thus equvalet. [ Oe ca prove that ths t s possble to move betwee the dagoal form of the Lagrage multplers ad the more geeral o-dagoal form through a utary trasform