1 The limitations of Hartree Fock approximation
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1 Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants cnstructed with a set f n rthnrmal paired spin rbitals ven then it des nt prvide chemical accuracy It can, at best, be the starting pint f mre refined calculatins What is the reasn f the inadequacy f the Hartree Fck (HF) wave functins? One way t underst the surce f inadequacy is t analyze the frm f the prbability density f finding the electrns in space, that the HF wave functin prduces Let us take the case f a tw electrn system in its spin singlet grund state ( S ) The tw rthnrmal spin rbitals used in the HF descriptin are φ () r φ () r given by the Stater determinant (α, β are rthnrmal spin eigenfunctins) HF = φ α( r) φ β( r) α β HF is = { φ α( r) φβ( r) φ α( r) φβ( r)} () φ α φ β have the same space part ( φ () r ( r ) φ = ( r ) ( s α φ α ) β r r s ), but different spin functins α β i e, φ ( )= φ ( ) β ( )) where r = spatial crdinate s = spin crdinate The tw electrn spin spatial prbability density Pr (, r, αβ, ) is easily generated (assuming real spin rbital) as fllws; HF HF Pr (, r, αβ, )= ( ) * = [( φ α( r) φβ( r)) + ( φ α( r) φβ( r)) φ α( r) φβ( r) φ α( r) φβ( r) φ ( r ) φ ( r) φ ( r) φ ( r )] () α β α β
2 If we integrate ver the spin variables ( s, s ), we get the spatial prbability density functin Pr (, r ) f the tw electrns (ie the prbability f finding ne electrn at r the ther at r whatever may the spin angular mmenta f the electrns are) T arrive at 03, we have used the usual rthnrmality cnditins fr the spin functins, α () s β () s quatin 03 immediately tells us that Pr (, r ) is a prduct f tw spatial prbability density functins which are mutually independent That is, the prbability f finding electrn at r = r is uncrrelatedwith the prbability f finding electrn at r = r But that is clearly unphysical lectrns are negatively charged repel each ther by Culmb frces (repulsin energy= ) They wuld naturally try t avid each ther nt be at the same pint in space ( r = r = r) They wuld exclude a certain vlume f space arund each f them (due t repulsin) where the prbability f finding the ther wuld be small In ther wrds the prbability density wuld be crrelated The tw-electrn HF spatial prbability density is, n the ther h, ttally uncrrelated a prperty that arises clearly frm the single determinant apprximatin used in the HF descriptin We suspect therefre the neglect f electrn crrelatin (mre precisely Culmb crrelatin) in the HF descriptin is at the rt f the inadequacy f HF wavefunctin in describing the real atms mlecules We nte here that vanishes when which means tw electrns having the same spin cannt be at same pint (r) in space The HF descriptin has Fermi r spin crrelatin built int the wave functin, but Culmb crrelatin is missing Pst-Hartree Fck methds attempt t take care f the deficiency by switching ver t many determinant descriptin f the state being prbed s that the (?hithert) neglected (Culmb) crrelatin appears in the spatial prbability density functin We wuld nw examine the issue thrughly find ut what can be dne t imprve the quality f the wave functin that the HF methd prvides Let us assume that we have generated all the HF rbitals { φ }, ccupied r unccupied, i i
3 fr the tw-electrn systems The exact tw electrn wave functin fr the singlet grund state f ur tw-electrn system ( ( x, x) ) can be written as a prduct f ( r, r) (space part f ) η ( s, s, S = 0) (spin part f the wave functin) s that s s S spin ( x, x ) = ( r, r ) η ( s, s, S = 0) (4) space spin η (,, = 0) is given by the cmbinatin eigenfunctins f the s z peratr as η ( s, s, S spin = 0) = { () () () ()} α β β α (5) T arrive at a cmplete descriptin f ( r space, r ) we start by assuming that we have kept r fixed at r s that ( r space, r ) can be exped in the cmplete set f rthnrmal HF rbitals { φ ( r )}: i ( r, r )= c ( r ) φ ( r) (6) space i i i= where ci ( r ) parametrically depends n r Nw we assume that r has been unfrzen, s that ci ( r ) can be regarded as s functin f the cmplete rthnrmal set { φ j ( r )} yielding, i ij j j= r (as a crdinate) which can therefre be exped in c ( r )= c φ ( r ) (7) Using 08 in 07, we have, fr the space part f the tw-electrn wave functin ( r, r )= c φ ( r) φ ( r ) (8) the ttal wave functin as space ij i j i j ( x, x ) = { c φ ( r) φ ( r )} { α() β() β() α()} (9) ij i j i j ( x, x ) as represented in equatin 00 suffers frm ne shrtcming, it des nt satisfy Pauli exclusin principle; ie it is nt antisymmetric with respect t interchange f space-spin crdinates f the tw electrns An inspectin f the right h side f equatin 00 immediately tells us that η ( s, s spin, S = 0) is antisymmetric with respect t the interchange f the spin crdinates ( s, s ) s that ( r space, r ) must be symmetric with respect t the interchange f spatial crdinates ( r, r ) f the tw electrns That means we must symmetrize
4 the prduct φi( r) φ j( r) replace it by { φi( r) φj( r) + φi( r) φj( r)} With the impsitin f the symmetry cnstraint, we nw have ( x, x ) = c { φ ( r) φ ( r ) + φ ( r ) φ ( r)} { α() β() β() α()} (0) ij i j i j i j Intrducing spin rbitals φ, φ etc we have iα iβ (, ) = [ { φ () φ () φ () φ ()}] () x x cij iα jβ jα iβ ij where φi α () = φi( r ) α () φ () = φ ( ) β () () jβ j r are spin rbitals, similar definitin hlds fr φ (), φ () φi α() φjβ(), φj α() φ iβ() represent single Slater determinant wave functins, cnstructed frm the rthnrmal spin rbitals φ (), r φ () r Since the spin rbitals are already iα knwn (frm HF calculatin), the nly unknwns in equatin 0 are the cefficients cs ij We can try t determine them variatinally by invking the linear variatin principle But the prblem is that it will lead t an infinite number f secular equatins ( cs ij are infinite in number) T keep the prblem tractable, ne is frced t use a finite number f spin rbitals ( i, j =, N, say ) in the expansin 0 generated frm the n cc number f dubly ccupied HF rbitals n =( N n cc ) number f unccupied r virtual rbitals, slve a finite number f secular equatins arising ut f the linear variatinal principle The mst imprtant pint abut the expansin 0 is that the prbability density Pr (, r ) can n lnger be factrized int jα jβ iβ as was pssible in the HF descriptin (equatin 03) The many determinant wave functin f equatin 0 is therefre said t have electrn-crrelatin built int it All Pst-Hartree Fck methds try t incrprate the effects f the additinal determinants either variatinally (cnfiguratin interactin (CI), MCSCF) r perturbatively (many bdy perturbatin thery) r by nn-perturbative nn-variatinal (such as cupled cluster) methds Let us first cnsider the variatinal ptin
5 in general Prblem Shw that the tw-electrn density P(r,r ) functin is nt factrizable Linear Variatinal Principle Let the system be represented by the Hamiltnian H Let us suppse that we have carried ut a linear variatinal calculatin with tw rthnrmal basis functin φ φ let the tw rts f the secular equatin be () 0 () 0 with () () The subscript sts fr the grund state, fr the first excited state while the superscript ( ) st fr the tw- () () dimensinal nature f the basis-space Let the crrespnding wavefunctins be We nw add a third basis functin φ 3 (rthgnal t functin, () determinant reads () (), nrmalized) The () φ 3 frm an rthnrmal set In this 3-dimensinal basis space the secular ( ε ) 0 () () ( ε)= 0 ( ε) h * * h h H33 h ε () H =< φ H φ > h =< H φ >, h =< H φ > () () 3 3 ε is the variatinal apprximatin t energy which can be btained by setting ( ε )=0 By exping ( ε ) we have ( ε ) h 0 ( ε ) ( ) =0 () () () ε + h * * * h ( H33 ε ) h h A- is a cubic plynmial equatin in ε We can try t lcate the range in which the pssible rts can lie by applyingthe rule f sign change f the plynmial as ε is varied () Let us first nte that as ε, term n the LHS f eqn 3 ( ε )>0 as the sign will be cntrlled by the first
6 At () ε =, the first the 3rd terms are zer, while the nd term <0 (nte > ) () () Therefre ( ε ) changes sign between ε = () S the lwest rt f 3 3 secular equatin (Let us call it At ) lies n the energy axis t the left f () () ε =, the first term is nce again zer while the nd term als vanishes If we nte that <, we immediately see, that the 3rd term n the LHS f eqn is greater than 0 s () () that we get anther sign change f 3 () ( ε ) between ε = then be lwer than () () Similarly by letting ε +, we can shw that the 3rd rt () The nd rt wuld 3 wuld lie t the right f Nw yu can cnsider passage frm three dimensinal rthnrmal basic space furdimensinal space in the same vain t shw that, (4) 3 3 (4) (4) finally generalize the result fr increasing the basis space frm n t n + dimensin (See Mcdnald)
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