Ŝ z (s 1 ) m (s 1 ), m 1/2, m 1/2. Note that only ion states

Transcription

1 JOURAL OF CHEMICAL PHYSICS VOLUME 119, UMBER 4 22 JULY 2003 Physcal nterpretaton and evaluaton of the Kohn Sham and Dyson components of the I relatons between the Kohn Sham orbtal energes and the onzaton potentals O. V. Grtsenko, B. Braïda, a) and E. J. Baerends Secton Theoretcal Chemstry, Vrje Unverstet, De Boelelaan 1083, 1081 HV Amsterdam, The etherlands Receved 29 January 2003; accepted 24 Aprl 2003 Theoretcal and numercal nsght s ganed nto the I relatons between the Kohn Sham orbtal energes and relaxed vertcal onzaton potentals VIPs I j, whch provde an analog of Koopmans theorem for densty functonal theory. The Kohn Sham orbtal energy has as leadng term n I js ()n j I j, where I s the prmary VIP for onzaton ( ) 1 wth spectroscopc factor proportonal to the ntensty n the photoelectron spectrum n close to 1, and the set s () contans the VIPs I j that are satelltes to the ( ) 1 onzaton, wth small but non-neglgble n j.in addton to ths average spectroscopc structure of the there s an electron-shell step structure n from the contrbuton of the response potental v resp. Accurate KS calculatons for prototype second- and thrd-row closed-shell molecules yeld valence orbtal energes, whch correspond closely to the expermental VIPs, wth an average devaton of 0.08 ev. The theoretcal relatons are numercally nvestgated n calculatons of the components of the I relatons for the H 2 molecule, and for the molecules CO, HF, H 2 O, HC. The dervaton of the I relatons employs the Dyson orbtals the n are ther norms. A connecton s made between the KS and Dyson orbtal theores, allowng the spn-unrestrcted KS xc potental to be expressed wth a statstcal average of ndvdual xc potentals for the Dyson spn orbtals as leadng term. Addtonal terms are the correcton v c,kn, due to the correlaton knetc effect, and the response v resp,, related to the correcton to the energy of (1) electrons due to the correlaton wth the reference electron Amercan Insttute of Physcs. DOI: / I. ITRODUCTIO The fundamental mportance of the Kohn Sham densty functonal theory KS-DFT Refs. 1 3 s based on the fact, that t offers an exact ndependent-partcle approach n many-electron theory. It s exact n the sense, that the exact densty of the nteractng system s delvered wth the KS nonnteractng system. In the spn-densty verson of DFT SDFT Refs. 4 6 the exact spn-densty of the hghest state (MS) of a multplet SM s produced wth the lowest KS spn orbtals, r 1 r From a general pont of vew, the orbtals can be defned meanngfully as the Dyson orbtals of the KS nonnteractng system. Indeed, for the nteractng system the Dyson spn orbtals d are defned conventonally 7 wth the overlap of a pure electron ground state 0 SS we consder the top component of the multplet SM of degenerate states wth the spn S wth pure (1) electron states (1)SM, a Also at Laboratore de Chme Physque, Groupe de Chme Théorque, Unversté Pars-Sud, Orsay, France. Present address: ICMAB- CSIC, Campus de la UAB, Bellaterra, Barcelona, Span. d r 1 s 1 1SM* x2,...,x 0 SS x 1,...,x dx 2dx MSm. 1.2 ote, that n the lterature the Dyson orbtals are also called generalzed overlap ampltudes. Throughout the paper for any wave functon SM the frst superscrpt ndex s the number of electrons, the second one S s ts spn, and the thrd one M s ts egenvalue for the operator Ŝ z. The ndex runs over all the on states, each on state s characterzed by a specfc total spn S and z-component M. In Eq. 1.2 m s the egenvalue of Ŝ z for the functons (s 1 ), Ŝ z (s 1 )m (s 1 ), m 1/2, m 1/2. ote that only on states (1)SM for whch MS1/2 and S1/2 SS1/2 correspond to nonzero Dyson orbtals. In turn, the KS spn orbtals can be defned wth the overlap of the correspondng nonnteractng states, r 1 s 1 1* s,sm,x 2,...,x s,ms x 1,...,x dx 2dx. 1.3 In Eq. 1.3 s,(ms) s the KS determnant wth -electrons and -electrons, correspondng to the nteractng state SS, and the (1) electron states (1) result from removal of the orbtal from s,(msm ), /2003/119(4)/1937/14/$ Amercan Insttute of Physcs

2 1938 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Grtsenko, Braïda, and Baerends s,(ms). These states are not necessarly egenstates of Ŝ 2, therefore we only specfy the egenvalue of Ŝ z n the subscrpt. We note that n the nonnteractng system only prmary onzatons out of occuped orbtals wthout any other orbtal exctaton to vrtual orbtals have nonzero Dyson orbtals, whch are dentcal to the Kohn Sham orbtals. Furthermore, the orbtals d have mnus the onzaton energes or vertcal onzaton potentals VIPs of the nteractng system I S S E0 E (1)S as ther egenvalues n the one-electron Dyson equatons, v ext r 1 v Coul r 1 ˆ xc,i S d r 1 I S d r 1, 1.4 where v ext s the external potental, v Coul s the Hartree potental of the electrostatc electron repulson of the ground state electron densty, and ˆ xc, s the nonlocal xc selfenergy operator. In ther turn, the KS orbtals have onzaton energes of the nonnteractng system E s,(ms) (1) E s,(sm ), as ther egenvalues n the KS one-electron equatons, v ext r 1 v Coul r 1 v xc, r 1 r 1 r 1, 1.5 where v xc, s the local, state-ndependent xc potental. ote that, unlke n the fnte, ndependent-partcle representaton 1.1, the same spn-densty s expressed wth all nteractng Dyson orbtals d, r 1 d r The physcal meanng of the KS orbtal energes s provded by an analog of Koopmans theorem establshed recently n Refs. 8 and 9. It s based on the exact relatons between the energes of H/2 occuped orbtals and VIPs I j, M 1 PIM 1 resp, M j 2 r 2 j r 2 dr HH matrx, 1.7 r P j 2 r 2 d j r 2 dr H matrx. r These equatons were derved for closed-shell systems. The nfnte-dmensonal column vector I contans all onzaton energes, whch are ordered n the same way as the set of Dyson orbtals.e., the columns of P, namely the prmary onzatons come frst. The prmary onzatons are those that can be descrbed n good approxmaton by a Koopmans confguraton,.e., a sngle orbtal onzaton. When there s strong correlaton n the ground state or n the onzed state, t may not be possble to dentfy H onzatons unambguously as prmary ones. Such molecules are not ncluded n ths study. The H-dmensonal column vector resp contans the matrx elements resp j j (r 1 ) 2 v resp (r 1 )dr 1 of the response potental for the occuped KS orbtals. 3,10,11 A qualtatve nterpretaton of Eq. 1.7 made n Ref. 8 allows us to dentfy the KS orbtal energes as approxmate but rather accurate relaxed VIPs I, I. 1.8 The qualty of ths approxmaton appears to be hgh for outer valence orbtals and t becomes an exact dentty for the hghest occuped molecular orbtal HOMO. In ths paper a comparatve theoretcal and numercal study of the Kohn Sham and Dyson orbtals s carred out and calculaton of varous components of the relatons 1.7 are performed for prototype molecules. In Sec. II a connecton s establshed between the one-electron equatons for the Dyson spn orbtals d and for the Kohn Sham orbtals. The equatons for the Dyson spn orbtals d and the Schrödnger equaton for the square root of the spn-densty are derved as lmtng cases of unversal equatons for partal spn-denstes of an arbtrary subset of the Dyson spn orbtals. Comparson of these equatons wth the spn-unrestrcted KS one-electron equatons allows to establsh a relaton between the xc potentals operators for Dyson and KS orbtals. In Sec. III the orbtal energes are obtaned wth accurate KS potentals constructed from ab nto denstes for some prototype closed-shell molecules of elements of the frst three perods and they are compared wth the expermental VIPs I. The accurate of the valence orbtals provde a very good estmate of the correspondng I, wth average devaton of only 0.08 ev. Secton IV presents a benchmark calculaton for H 2 of the KS and Dyson components of the relatons 1.7, whch provdes a numercal confrmaton of these relatons. In Sec. V the ngredents of Eq. 1.7, the Dyson orbtals d, the matrces M and P, and the components M 1 PI and M 1 resp are obtaned wth ab nto and accurate KS calculatons for the molecules CO, HF, H 2 O, HC. In Sec. VI the mplcatons of the present results for DFT are dscussed and the conclusons are drawn. II. UIVERSAL EQUATIO FOR PARTIAL SPI-DESITIES The road to a meanngful comparson of the one-electron equatons for the KS and Dyson orbtals passes through the Schrödnger-type equaton for the square root of the spndensty, v eff, r 1 r 1 r In ths secton we shall show that both Eq. 1.4 for d and Eq. 2.1 for are just the lmtng cases of an unversal equaton for the square root of the partal spn-densty p, p r 1 jp d j r of any arbtrary subset d j, j p of Dyson spn orbtals. In order to derve ths equaton, we expand the electron SS ground state 0 n terms of (1) electron states (1)SM and the correspondng Dyson orbtals d (r 1 )(s 1 ),

3 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Kohn Sham orbtal energes SS x 1,...,x 1 S1/2, SSm d r 1 SSm s 1 1SSm x2,...,x, 2.3 where S(Sm ) means that the ndex only runs over those on states (1)S()M() for whch the total spn S() S and the Ŝ z -egenvalue M()Sm. It s also possble to expand the wave functon usng the spn-resolved condtonal probablty ampltudes x 2x r 1 r 1,x 2x / r 1 /, 2.4 SS 0 x 1,...,x, r 1 x 2x r 1 s 1, s an (1)-electron wave functon, the square of whch s the probablty to fnd electrons 2,..., wth spatal and spn coordnates x 2 r 2,s 2,...,x r,s, f electron 1 s at the poston r 1 wth spn. It follows from Eqs. 2.3 and 2.4, that for the mnor spn only the states (1)(S1/2)(S1/2) wth S(S1/2), M(S1/2) contrbute to, whle for the major spn the states (1)(S1/2)(S1/2) wth MS1/2 and both SS1/2 and SS1/2 contrbute to. Then, by analogy wth, we ntroduce the partal spn-ampltude p, p 1 x 2x r 1 1S jm j d p j r 1 j r 1 j p x 2x M jsm, 2.5 whch ncludes the Dyson orbtals d j, j p contrbutng p to of 2.2 and the parent (1) electron states (1)S( j),m( j) j,(m( j)sm ). Wth Eqs , one can derve an effectve oneelectron equaton for p from the Schrödnger equaton Ĥ SS 0 E S 0 SS 0 by parttonng of ts spn-free Hamltonan Ĥ nto the (1) electron Hamltonan Ĥ 1 and the remander, Ĥ v ext r 1 j2 1 r 1 r j Ĥ1. For each spn we also subtract from both sdes of the Schrödnger equaton the term E (1)() p0 SS 0, where (1)() (1)S( j),m( j) s the lowest energy for the states j E p0 whch contrbute to p. Insertng expanson 2.3 and the parttonng of the Hamltonan n the Schrödnger equaton wth E (1)() p0 SS 0 subtracted, and then multplyng by the partal spn-ampltude p (x 2,...,x r 1 )(s 1 ) and ntegratng over s 1,x 2,...,x, we obtan an exact equaton for the square root of the partal spn-densty p, p v ext r 1 v kn, r 1 v 1p p r 1 Ŵ cond, p r 1 p p r 1, 2.6 where p E S 0 E (1)() p 1(p) p0. In Eq. 2.6, v kn, and v are local potentals defned n terms of the partal ampltude p p. In partcular, v kn, s the knetc contrbuton, p v kn, r 1 p p 2.7 the brackets denote ntegraton over s 1,x 2,...,x, not 1(p) over r 1 ) and the potental v s determned wth the expectaton value of the (1) electron Hamltonan Ĥ 1, v 1p r 1 p Ĥ 1 p 1 E p0 jp 2 d j r 1 p r 1 1S E j j E 1 p (p) The nonlocal operator Ŵ cond, represents n Eq. 2.6 the potental effect of the electron electron nteracton, t s defned n terms of both partal p and total condtonal probablty spn-ampltudes and ts acton on p s defned as follows: p Ŵ cond, p r 1 p 1 j2 r 1 r j r (p) Thus, Ŵ cond, can be consdered as a superexchange operator actng on the superorbtal p, t takes ths superorbtal and replaces t wth the square root of the total spndensty. Subtractng from Ŵ cond, the Hartree potental (p) of the electrostatc electron repulson v Coul, we defne the partal xc-hole operator Ŵ hole(p) (p) xc, Ŵ cond, v coul (r 1 ), so that the unversal equaton 2.6 for partal spn-denstes assumes the fnal form, p v ext r 1 v kn, r 1 v 1p r 1 v Coul r 1 Ŵ holep xc, p r 1 p p r If only a sngle Dyson spn orbtal d j s taken n Eq. 2.2, the potentals v 1(p) p and v kn, vansh n the correspondng Eq and t turns nto the Dyson equaton represented as follows: hole, v ext r 1 v Coul r 1 Ŵ xc, d j r 1 I j d j r hole, j In Eq the acton of the operator Ŵ xc, on d j s defned as hole, Ŵ j 1 xc, d j r 1 1 j k2 r 1 r k r 1 v Coul r 1 d j r Comparson of Eq wth the standard form 1.4 of the Dyson equaton allows to dentfy the acton of the operator hole, j wth that of the xc self-energy operator ˆ xc, Ŵ xc, hole, Ŵ j xc, d j, ˆ xci j d j, For the beneft of further analyss we rewrte Eq wth hole, the multplcatve state-dependent potental v j xc, (r 1 ),

4 1940 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Grtsenko, Braïda, and Baerends hole, v ext r 1 v Coul r 1 v j xc, r 1 d j r 1 I j d j r 1, whch s defned as 2.14 hole, v j xc, r 1 1 d j r 1 Ŵ hole, xc, j d j r When all Dyson spn orbtals d j are ncluded n Eq. 2.2, Eq turns to the Schrödnger equaton for, v ext r 1 v kn, r 1 v 1 r 1 v Coul r 1 v hole xc, r 1 r 1 r 1, 2.16 wth E S (1)() 0 E 0 and the local, state-ndependent potentals v kn,, v 1, v hole xc,. In partcular, the knetc contrbuton v kn, s 12 v kn, r , and the potental v 1, v 1 r 1 Ĥ 1 E 0 1 S1/2 SSm SS,MSm d r 1 2 r E 1S 1 E can be expressed as the statstcal average over the Dyson spn orbtals of exctatons (E (1)S (1)() E0 ) n the (1) electron system. The acton of the xc-hole operator Ŵ hole(p) xc, turns n ths case to that of the local potental of the hole xc-hole v xc, Ref. 13 famlar n spn-densty functonal theory SDFT, Ŵ hole xc, r 1 v hole xc, r 1, 2.19 v hole 1 xc, r 1 j2 r 1 r j v Coul r 1 dr 2 r 2 r 1 r 2 g r 1,r 2 1, where g s the par-correlaton functon. From Eqs , 2.12, 2.15, and 2.19 follows, that the xc-hole potental can be expressed as the statstcal average of ndvdual xc potentals v xc, for the Dyson hole, j orbtals, hole r 1 d r 1 2 v xc, r 1 v hole, xc, r The xc hole potental s only a part of the total xc potental v xc, that features n the KS equatons, v xc, r 1 v hole xc, r 1 v c,kn, r 1 v resp, r 1, 2.21 where v c,kn, represents the correlaton knetc effect, and v resp, the change of the energy of (1) electrons from the ground state energy of the on to the energy of the condtonal ampltude (x 2,...,x r 1 ) whch represents the redstrbuton of the (1)-electron system due to the correlaton wth the reference electron at r 1,. The potentals v c,kn, and v resp, have been dscussed elsewhere. 3,10,14,15 Then, from Eqs and 2.21 follows the relaton between the xc potentals for the KS and Dyson orbtals, v xc, r 1 d r 1 2 r 1 v resp, r 1. v hole, xc, r 1 v c,kn, r Wth Eq. 2.22, the xc potental v xc, for the KS orbtals does not only contan a local hole potental whch s a statstcal average of ndvdual xc potentals v xc, for the nter- hole, actng Dyson orbtals d representng the acton of the selfenergy operator, but has n addton the terms v c,kn, and v resp,. Equaton 2.22 provdes an explct relaton between the Kohn Sham potental and the potentals self-energy operator featurng n the equatons for the Dyson orbtals. ether Eq nor the result of Ref. 16 that the xc Kohn Sham potental can be consdered to be the best local approxmaton to the exchange-correlaton self-energy allow us to make an estmate of how close or remote the correspondence between Kohn Sham orbtals only occuped and Dyson orbtals nfnte number actually s, n partcular snce t s already well known that Hartree Fock orbtals, Kohn Sham orbtals, and Dyson orbtals at least those correspondng to prmary on states are usually all very smlar. We return to ths pont below. The response potental plays a crucal role n the I relatons Eq Ths potental has a steplke behavor when gong from one shell to another shell n an atom 10 and ndeed a rather accurate approxmaton n the exchange-only case has been provded by Kreger, L, and Iafrate KLI, 17 who represented the response part of the exchange potental as the statstcal average over the occuped KS orbtals of the orbtal steps w, v resp r 1 /2 2 r 1 2 w r 1, 2.23 whch exhbts the step structure of v resp. 3 Insertng Eq n Eq. 1.7, one obtans M 1 PIw We wll verfy n the next secton for a seres of prototype closed shell molecules that the KS orbtal energes ndeed approxmate qute closely the expermental vertcal IPs for valence levels. Then n the next sectons we wll further analyze the relatons 1.7 and In partcular the correspondence between occuped Kohn Sham orbtals and Dyson orbtals belongng to prmary onzatons characterzed by a sngle orbtal onzaton as leadng term n the wave functon, and Dyson orbtals of satelltes wll be dscussed n order to provde an understandng of the structure of the matrces M and P hence M 1 P). Ths s requred to arrve at a full explanaton of how the relatons 1.7 can lead to the emprcal fndng that I.

5 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Kohn Sham orbtal energes 1941 TABLE I. Comparson of the KS, HF, and GGA-BP orbtal energes (ev) wth expermental vertcal onzaton potentals the correspondng references are ndcated n the table. AAD are the average absolute dfferences between the KS orbtal energes and the VIPs for ether the upper valence levels, AADval or lower valence and core levels, AADnner. For BP n parentheses are the sum ( H I H ) for the HOMO, the orbtal energes shfted by ths amount for the other orbtals, and AADs for the shfted orbtal energes. MO HF GGA-BP KS Expt. Molecule MO I CO a e AADval AADnner SO d AADval b,c,e 2 3 g u u AADval g u g AADnner f P 2 2 u g AADval u g g u u g u g u g HF b,g,h AADval AADnner HCl l AADval H 2 O b,c,d 1b a b AADval a a AADnner HC k

6 1942 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Grtsenko, Braïda, and Baerends TABLE I. Contnued. MO HF GGA-BP KS Expt AADval FC l AADval a Reference 42. b Reference 43. c Reference 44. d Reference 45. e Reference 46. f Reference 47. g Reference 48. h Reference 49. Reference 50. j Reference 51. k Reference 52. l Reference 53. III. OF ACCURATE KS POTETIALS VERSUS HF AD EXPERIMETAL VIPs The orbtal energes can be obtaned wth KS potentals constructed from hghly accurate ab nto denstes. In our prevous publcatons the KS solutons have been produced for molecules of the elements of the frst and second perods. 8,18 20 In ths paper the KS energes are obtaned for the closed-shell datomc molecules SO, P 2, and HCl of the elements S, P, Cl of the thrd row. Table I presents these together wth those for ther second-row analogs CO, 2, and HF obtaned at the expermental equlbrum geometry. In addton, for the tratomc molecules H 2 O, HC, and FC are calculated and they are also presented n Table I. They are compared wth the expermental valence VIPs I determned wth HeI UV photoelectron spectroscopy.e., I 21.2 ev) and, n some cases, wth the expermental VIPs for the deep valence and core levels. The comparson s also made wth calculated wth the Hartree Fock HF method as well as wth the potental of the standard DFT generalzed gradent approxmaton GGA, the combnaton BP of the exchange functonal of Becke B88 Ref. 21 and the correlaton functonal of Perdew P The teratve local updatng scheme of van Leeuwen and Baerends LB Ref. 23 has been used to get the KS soluton from the ab nto densty. The molecular KS xc potental v xc s constructed n the bass of the Gaussan functons. The correct Coulombc asymptotcs 1/r together wth the requrement, that the HOMO orbtal energy H should be close to I H, s mposed on v xc wthn the LB scheme, as was descrbed n Refs. 8 and 24. The ab nto confguraton nteracton CI calculatons have been performed by means of the ATMOL package. 25 The constructon of the KS soluton has been performed wth a Gaussan orbtal densty functonal code 12,26 based on the ATMOL package. For the secondrow elements the quadruple-zeta correlaton-consstent polarzed core-valence aug-cc-pcvqz bass sets 27 of contracted Gaussan functons have been used wth all f-, g-, and the most dffuse d-functons excluded. For CO, HF, and H 2 O the bass sets dffer from those used n our Ref. 8, so that the correspondng are slghtly dfferent. For H and the thrdrow elements smlar quadruple-zeta correlaton-consstent polarzed valence aug-cc-pvqz bass sets 28,29 wthout f-, g-, and the most dffuse d-functons have been used. The KS orbtal energes closely match the expermental VIPs I for the valence levels see Table I. The average over all consdered molecules devaton of valence s only 0.08 ev. In partcular, for the HF molecule not only the frst, but also the second VIP practcally concdes wth. For molecules of the thrd-row elements a typcal devaton appears to be even smaller, than that for molecules of the second-row elements. For example, for SO and HCl s only 0.03 ev. On the other hand, the largest 0.21 ev s found for FC. The KS orbtal energes reproduce the trend of the expermental VIPs, namely, the valence levels of the thrd-row-element molecules SO, P 2, and HCl are hgher than the correspondng levels of ther second-row-element analogues CO, 2, and HF. The Hartree Fock Koopmans theorem produces, on the average, an order of magntude worse estmate of valence VIPs see Table I. The average over all consdered molecules devaton for HF amounts to 1.27 ev, the smallest s 0.43 ev for P 2, whle the largest s 1.85 ev for FC. As a rule, Koopmans theorem consderably overestmates VIPs, due to the neglect of electron relaxaton n the catonc states. ote, that for the HF HOMOs of the thrdrow-element molecules SO, P 2, HCl and larger molecules HC, FC ths lack of relaxaton seems to be compensated wth the neglect of electron Coulomb correlaton n the HF method. Due to ths, H HF for SO and HCl are only somewhat hgher than I H, for FC these quanttes practcally

7 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Kohn Sham orbtal energes 1943 concde, whle for P 2 and HC H HF are even smaller than I H. Another excepton s HF for the 1 u MO of 2, whch s smaller than the correspondng I, because of the wrong orderng of onzatons, whch Koopmans theorem provdes for 2 see Table I. As has been already mentoned n Ref. 8, the agreement between and I s less precse for the lower valence and core levels see the data n Table I for CO, 2,HF,H 2 O), wth the absolute devaton ncreasng wth the depth of the level. In ths case the HF orbtal energes HF are consstently lower than the KS, so that Koopmans theorem consstently overestmates VIPs of the deep valence and core levels, whle ts KS analog underestmates them. It s nterestng to note, that for these levels a typcal HF devaton s always close to 12 ev, whle a typcal devaton s close to 14.5 ev for CO, 2, HF, and only for H 2 Ots ev, whch s somewhat less than the HF devaton of ev. The BP column of Table I dsplays, at frst glance, just the well-known feature of the GGA potentals, namely, the systematc underestmaton on absolute value of the KS orbtal energes. BP greatly underestmates and I n all cases, wth the average errors wth respect to I ) rangng from 3.44 ev for P 2 to 6.44 ev for HF. However, a closer look at these errors reveals, how remarkably systematc they are for a partcular molecule. For each molecule we present n the parentheses for the HOMO the sum ( H BP I H ), and for the other orbtals the energes shfted downward by ths sum, as well as ther average error devaton from VIP. The surprsng concluson s, that the ( H BP I H )-shfted BP orbtal energes reproduce the valence VIPs remarkably well see Table I. The correspondng vares from 0.02 ev for 2 to 0.25 ev for CO, so that the average error over all consdered molecules of 0.09 ev s practcally dentcal to the 0.08 ev average error for the KS orbtals. For the lower valence and core levels see the data n Table I for CO, 2,HF,H 2 O) the BP values are by ev larger than the KS ones. Thus, judgng from the calculated orbtal energes, the ( H BP I H )-shfted BP xc potental s close to the accurate KS one, especally n the valence regon. Therefore, n spte of the well-known defcency of the long-range asymptotcs of the GGA potentals, GGA-BP may reproduce rather accurately the form of the KS potental n the bulk regon. The results of the calculatons presented n ths secton provde further support for the nterpretaton of the KS orbtal energes as approxmate relaxed VIPs Eq In the next secton ndvdual components of the I relatons wll be evaluated and the condtons of the type 4.13 wll be analyzed n order to get a theoretcal understandng of the correspondence between and I establshed n ths secton. IV. STRUCTURE OF I RELATIOS AD UMERICAL COFIRMATIO FOR H 2 In vew of the mportant role of Dyson orbtals n the I relatons, we frst make a few comments on the Dyson orbtals n relaton to the theory of photoonzaton. Aganst that background the matrces M and P wll then be dscussed. When the Hartree Fock determnant s a good zeroorder approxmaton of the wave functon, and f there would be no electron relaxaton after an orbtal onzaton, the Dyson orbtal of that onzaton would be practcally dentcal to the Hartree Fock orbtal. The same reasonng holds when one takes the Kohn Sham orbtals as one-partcle bass whch are qute close to Hartree Fock orbtals anyway 30. Of course correlaton effects n the ground state, and relaxaton plus correlaton effects n the onzed state, leadng to admxture of substtuted determnants, wll cause devaton from precse equalty between Dyson orbtals and Hartree Fock or Kohn Sham orbtals. otably, the norms of the Dyson orbtals wll start to devate from 1. ote, that the d, defned n Eq. 1.2, are nonorthogonal, non-normal, and generally lnearly dependent orbtals. The norms n d r 1 2 dr 1, whch obey the sum rule n / are called spectroscopc factors and are related to the ntensty of the correspondng on state n the photoelectron spectrum. As we wll see, typcally as, for nstance, for the molecules nvestgated n the prevous secton the overlap of the normalzed Dyson orbtal of a prmary onzaton wth the correspondng Kohn Sham orbtal s very close to The norms of these Dyson orbtals correspondng to prmary on states, typcally dffer less than 10% from 1. As can be seen from Eq. 1.2, these condtons wll arse when the ndependent partcle pcture s a good approxmaton, n the sense that the ground state wave functon s well approxmated by a sngle determnant of Kohn Sham orbtals, and when the Koopmans or frozen-orbtal approxmaton would be reasonable for the prmary on state. These statements can be made for Hartree Fock as well as for Kohn Sham orbtals, the overlaps of the normalzed prmary Dyson orbtals wth the occuped Hartree Fock orbtals are as close to 1 as they are for the occuped Kohn Sham orbtals. So we cannot conclude that Kohn Sham orbtals are closer to Dyson orbtals than to Hartree Fock orbtals. There are many more on states than the prmary ones. Each s characterzed by a Dyson orbtal d c (ch). These Dyson orbtals have n general qute small norms, and the ntensty n the photoelectron spectrum s neglgble. It occasonally happens that the norm of such a Dyson orbtal s non-neglgble, and the ntensty n the photoelectron spectrum the pole strength of the onzaton s then sgnfcant usually such peaks n the photoelectron spectrum occur as satelltes to the large ntensty peaks of the prmary onzatons. 31 We pont to two stuatons where such behavor can occur. Suppose that electron relaxaton upon a partcular orbtal onzaton, from say, s descrbed by admx-

8 1944 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Grtsenko, Braïda, and Baerends ng to the frozen-orbtal determnant descrbng the orbtalonzaton, a certan one-electron exctaton j a. Ths may for nstance happen when the orbtal s localzed on the left atom of a datomc, and j s a bondng orbtal and a s an antbondng orbtal. Then the j a exctaton leads to partal replacement of the densty correspondng to a doubly occuped j orbtal by a densty j a 2, whch s localzed on the left atom. Ths s the expected relaxaton of electron densty, whch moves screenng charge towards the onzed atom. There s also a hgher lyng on state whch s the j a excted state shake-up of the prmary on state ( ) 1. Ths on state wll now have, f only for orthogonalty reasons, some admxture of the determnant wth only the sngle ( ) 1 onzaton the frozen orbtal determnant for the prmary onzaton. In the overlap Eq. 1.2 ths sngly onzed determnant appearng n the CI expanson of the shake-up on state wll yeld, from the overlap wth the leadng fully occuped determnant n the neutral ground state wave functon, the orbtal as major contrbuton to the Dyson orbtal. It wll however be multpled wth the relatvely small mxng coeffcent of the prmary onzaton determnant n the shake-up wave functon, hence the norm of ths Dyson orbtal wll not be close to 1 but wll be small. The correspondng ntensty n the photoelectron spectrum of ths shake-up satellte to ( ) 1 wll be much smaller than that of the prmary onzaton, but not neglgble. ote that n ths case the satellte s an exctaton shake-up of just the prmary onzaton that s responsble for the man peak. A second case where ntensty of satellte peaks appears s when an on state that contans an exctaton n addton to a prmary onzaton, becomes accdentally near-degenerate wth a dfferent prmary on state. The near-degeneracy may lead to mxng, and n that case the admxng of that prmary on state causes the Dyson orbtal to be lke the onzed orbtal of that prmary on state, and the dfferent-from-zero norm leads to ntensty of the shake-up on state. In ths case the shake-up on state steals ntensty from the dfferent prmary on state and appears on account of the ntal neardegeneracy as a satellte to that prmary on state. Of course, much more ntrcate CI mxngs wll occur n many cases, and we have notced many Dyson orbtals whch after normalzaton could not be dentfed wth a sngle occuped Kohn Sham orbtal. However, we do fnd n our present molecules a very large overlap of the normalzed Dyson ampltude of the mportant satelltes those wth nonneglgble ntensty wth a sngle occuped Kohn Sham orbtal. In that case we can unambguously defne the set s () of satelltes to the prmary on state ( ) 1, n the sense that ther Dyson orbtals have nonneglgble norm and have, after normalzaton, large overlap wth. Let us consder the structure of the matrces P and M 1 P aganst ths background. As shown below we collect n the frst H columns of P the weghted wth 1 ) overlaps of 2 wth dj 2, and n the remanng columns those wth the Dyson orbtals of the shake-up onzatons see Eq When the normalzed Dyson orbtals n 1/2 d of the prmary onzatons are close to the occuped Kohn Sham orbtals, we can substtute d n 1/2 n the leadng HH block of P and obtan that each Structure of matrx P column j of ths part of the P matrx let us denote t as P s multpled by the correspondng n j,.e., PM f s dagonal, j n j. So M 1 P, a dagonal matrx wth the pole strengths of the prmary onzatons n as the dagonal elements. Smlarly, f n c 1/2 d c of a satellte s close to an occuped KS orbtal, say, the correspondng column of the P matrx becomes a weghted column of the row, (M 1 P) jc j n c. Ths entry ndcates the strength of the satellte, Approxmate structure of matrx M 1 P the M matrx, P jc n c M j. The correspondng column of M 1 P, the Dyson orbtal of whch belongs to an on state that s a satellte to the prmary onzaton from, wll then only have an entry (n c ) at the -? Wth ths, the I relatons 1.7 n ther smplfed form 2.24 assume the form, n I cs n c I c cs M 1 P c I c w. 4.3 The frst term n the r.h.s. of Eq. 4.3 s the contrbuton to from the prmary onzaton, the second term dsplays the satellte structure, the thrd term collects contrbutons from other onzatons, and the fourth term s the response step. For outer valence orbtals there s neglgble or no satellte structure, so the pole strengths n of the prmary onzatons are close to 1, whle the last three terms are expected to be much smaller than the frst one. Ths leads to the Koopmans nterpretaton Eq. 1.8 of the I relatons 1.7, I. More specfcally, by wrtng Eq. 4.3 n the form, TABLE II. Components ev of the I relatons for H 2. H 2 H I H (calc.) H resp nh S HH calc. estm

9 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Kohn Sham orbtal energes 1945 I 1n I cs cs M 1 P c I c w, n c I c 4.4 we note that the last four terms on the rght-hand sde of Eq. 4.4 should add up to zero n order to make the relaton I hold. For upper valence onzatons, where the satellte structure s very weak.e., n 1, n c 0) and the KLI constant very small, w 0, we mght expect ths to happen. However, we have already ndcated n Ref. 8 that the relaton, 1n I cs n c I c cs M 1 P c I c w may actually depend on cancellaton of these terms rather than them beng ndvdually very close to zero. A numercal TABLE III. Components ev of the I relatons for CO, HF, H 2 O, and HC. CO I (calc.) ,40.12,40.35 resp (M 1 resp ) (M 1 P) ,0.159,0.092 n ,0.083,0.171 S ,0.990,0.982 calc estm HC I (calc.) resp (M 1 resp ) (M 1 P) n S calc estm H 2 O 1b 1 3a 1 1b 2 2a 1 1a 1 I (calc) resp (M 1 resp ) (M 1 P) n S calc estm HF I (calc.) resp (M 1 resp ) (M 1 P) n S calc estm

10 1946 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Grtsenko, Braïda, and Baerends nvestgaton of these relatons wll be gven n ths secton for H 2 ) and the next secton for some other prototype molecules. Tables II and III present ndvdual components of the I relatons 1.7 calculated for the closed-shell molecules H 2, CO, HF, H 2 O, HC. The Dyson orbtals have been obtaned wth the MELD package, 32 n partcular, the program CISTAR of ths package has been used to generate the wave functons 00 0, (1)1/21/2 0 and to calculate the onzaton energes, and the program PES has been used for the subsequent calculaton of d accordng to Eq For each rreducble representaton of the molecular symmetry group MELD allows us to calculate up to 50 Dyson orbtals and the hghest possble symmetry group s D 2h. The matrces M, P, resp and ther combnatons M 1 P, M 1 PI, and M 1 resp n Eq. 1.7 have been calculated wth the abovementoned Gaussan orbtal densty functonal code 12,26 based on the ATMOL package. 25 In partcular, the response potental v resp has been calculated on a numercal grd by the drect subtracton 3.9 of the potentals v c,kn and v hole xc from the constructed KS potental v xc wth subsequent numercal ntegraton to get the matrx elements resp. To do ths, the potental v hole xc of Eq has been constructed from the secondorder densty matrx wth the codes. 12,26 The MELD calculatons have been performed n the same bass sets as the ATMOL ones descrbed n the prevous secton. The same bass sets have been used for and (1) electron systems, whch s requred to calculate the Dyson orbtals wth the PES program. Wthn these bass sets, the summaton over the Dyson orbtals, whch s requred n order to get the matrx M 1 PI, certanly surpasses even for small molecules the lmtatons of MELD. The problem, however, s greatly smplfed for two-electron systems. In ths case the (1) electron wave functons turn nto the Hartree Fock orbtals of a sngleelectron system, so that calculaton of all Dyson orbtals n the chosen bass becomes feasble. Thus, one can use benchmark two-electron calculatons to check numercally the valdty of the I relatons. For a closed-shell two-electron system we have just one occuped KS orbtal H. Snce 2( H ) 2, M 11 1, and all matrx elements (M 1 P) H, turn exactly to the pole strengths n. Then, the I relatons 1.7 are reduced to a sngle equaton, H n I H resp, 4.6 n Remarkably, the frst term n the r.h.s. of Eq. 4.6 has an average spectroscopc structure n the sense, that ndvdual onzatons appear n t wth weghts n that correspond to the ntenstes n the photoelectron spectrum of H 2. Takng nto account the exact property H I H of the HOMO, we can rewrte Eq. 4.6 n the form, 1n H I H n I H resp 0, 4.8 where the prme on the summaton ndcates omsson of the H term. Table II presents the ngredents of the equalty 4.8 as well as the overlap S HH, S HH 1 n H H rd H rdr 4.9 between the KS orbtal H and the normalzed Dyson orbtal n H 1/2 d H of the prmary onzaton calculated for the H 2 molecule. Ths calculaton has been performed n the 6-zeta augcc-pv6z bass 28 wthout f-, g-, and h-functons. The VIP of ev calculated wth the full CI n ths large bass s taken as I H. Though more than an order of magntude smaller than I H, the calculated contrbuton H resp 1.25 ev of the response potental s stll an apprecable quantty. 33 Judgng from the overlap ntegral S HH , the form of the Dyson orbtal of the prmary onzaton d H s very close to that of the g KS orbtal H. Even though the d H after normalzaton s very smlar to H, ts norm clearly devates from 1, wth the correspondng pole strength n H Wth ths, the frst term n the r.h.s. of Eq. 4.8 amounts to 0.77 ev whch, together wth H resp, yelds 2.02 ev. As follows from Eq. 4.8, ths quantty should be exactly compensated wth the contrbutons from hgher onzatons, n order that the I relatons 1.7 would provde the exact property H I H for the HOMO of H 2. Thus, the value of 2.02 ev of the prmed sum estmated from I H, H resp, and n H s placed n Table II n the entry estm. The largest pole strength n n the prmed sum n Eq. 4.8 belongs to the onzaton wth VIP I ev to the 2 g state of the caton H 2. The next two largest contrbutons come from hgher onzatons wth I ev and I ev, however, the correspondng pole strengths n and n are an order of magntude smaller than n 2. The summed contrbutons of these satelltes to the sum 1 n I s 1.21 ev, whch amounts to 60% of the requred 2.02 ev. The rest of the sum s scattered over many other onzatons. Remarkably, the drect calculaton of the total sum over hgher onzatons wth all onzatons wthn the bass set ncluded yelds just 2.02 ev the entry calc. n Table II, so that these onzatons provde the requred compensaton of 1 n I to (1n H )I H H resp to wthn 0.01 ev. Apparently the completeness whch requres that summaton s extended over the contnuum of on states wth the second electron also onzed s approxmated to ths level of precson by our dscrete sum over the on states descrbed n the fnte bass. Ths means, that n the case of H 2 the I relatons agree perfectly wth the well establshed condton H I H. Thus, the present benchmark calculatons for H 2 provde a numercal confrmaton for the I relatons 1.7. We note that the H I H relaton n ths case does not at all arse from smallness of the ndvdual terms n 4.8. We wll nvestgate n the next secton whether n general the remarkable correspondence between KS orbtal energes and VIPs arses from cancellaton n the terms of Eq. 4.5 rather than from them ndvdually beng neglgble.

11 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Kohn Sham orbtal energes 1947 V. EVALUATIO OF IDIVIDUAL COMPOETS OF THE I RELATIOS FOR CO, HF, H 2 O, AD HC Calculaton of the components of the I relatons for CO, HF, H 2 O, and HC produces a coherent pcture for all molecules consdered see Table III. In all cases the Dyson orbtals of the prmary onzatons from the core levels are not among the frst 50 d of the correspondng rreducble representaton allowed by MELD. Thus, the related entres n Table III are empty. Table III presents the calculated valence VIPs I of prmary onzatons and, n the case of deep valence levels, also n parentheses the VIPs of the most mportant satelltes. In most cases, the calculated VIPs are larger than the expermental ones compare Tables I and III, though the qualty of the calculated VIPs s reasonable. A possble reason for ths overestmaton s the use of the same bass sets for neutral and catonc systems. The bass s optmzed for the neutral molecule and may be less well adapted for the on. Also the correlaton treatment of the on s possbly less effectve. The only data n Table III dsplayed for both core and valence levels are the KS orbtal matrx elements resp of the response potental and the matrx elements (M 1 resp ), whch are components of the I relatons. These have been calculated by the ATMOL based DFT code as was descrbed above. Both resp and (M 1 resp ) represent the characterstc step structure of v resp, wth lower values for valence levels and hgher values for core levels. ote, that the resp values for deep valence levels are not much dstngushed from resp those for outer valence levels. In partcular, for H 2 O the value of 6.27 ev for the deep valence KS level 2a 1 s very close to that of 6.21 ev for the HOMO 1b 1, whle for HF the resp value 7.01 ev for the 2MO s even lower than that of 7.16 ev for the HOMO 1. Thus, for each molecule all the valence resp values form knd of a plateau, above whch stand the core resp values see Table III. Remarkably, the acton of the matrx M 1 makes the calculated (M 1 resp ) values for outer valence levels consstently smaller compared to the correspondng resp and t rases (M 1 resp ) for deep valence levels. As a result, only the outer valence (M 1 resp ) values form a plateau, wth the deep valence (M 1 resp ) standng apprecably hgher and the core (M 1 resp ) stll hgher see Table III. (M 1 resp ) do not necessarly follow the order of the orbtal energes. For example, for both H 2 O and HF the smallest (M 1 resp ) values correspond not to the HOMOs, but to the 1b 2 and 3MO, respectvely. Though much smaller than VIPs I, the elements (M 1 resp ) are apprecable even for outer MOs, rangng n ths case from 1.55 ev for the 1b 2 MO of H 2 O to 5.11 ev for the 4MO of CO. Thus, n the context of I relatons, the close correspondence between outer valence and I establshed n Sec. III has to be acheved through the compensaton of (M 1 resp ) wth the contrbutons k (M 1 P) k I k from other onzatons, as happens for the benchmark case of H 2 consdered n the prevous secton. We proceed wth the dscusson of the calculated Dyson orbtals d and the ngredents of Eqs. 1.7, whch nclude d. For all presented orbtals d of the prmary onzatons ther overlap ntegrals S wth the KS orbtals are very close to 1 after normalzaton of the d ) so that, judgng from ths crteron, the form of d s close to that of. Ths confrms the antcpaton of Refs. 16 and 34, although we should cauton that also the Hartree Fock orbtals have smlarly large overlaps wth the normalzed Dyson orbtals. It s nterestng to consder cases where Hartree Fock orbtals dffer essentally from the Kohn Sham orbtals and Dyson orbtals. In one such case the Cu 3 Cl 3 molecule t was observed 35 that the Kohn Sham orbtals do correspond more closely to the Dyson orbtals. Ths molecule exhbts the breakdown of Koopmans theorem for transton metal complexes observed long ago by Vellard and co-workers; 36,37 the metal d electrons are more loosely bound accordng to the photoonzaton experment, but the predomnantly 3d Hartree Fock levels le below the predomnantly Cl 3p ones. Ths s not a correlaton effect, but t s a consequence of the tght nature of the 3d orbtals and the ensung strong stablzng effect of the self-energy correcton part of the Hartree Fock exchange operator for the 3d orbtals. The Kohn Sham orbtals, wth ther sngle local potental for all orbtals, do not exhbt ths reversal n the order of metal and lgand levels. The Dyson orbtal of the 3d onzaton also exhbts the 3d character and t s demonstrated ncely n Ref. 35 that t corresponds to the upper Kohn Sham orbtal wth 3d character, not to the upper Hartree Fock orbtal wth Cl 3p character. It would be nterestng to extend such nvestgatons nto the nature of Kohn Sham and Dyson orbtals to molecules wth strong correlaton effects. In our molecules, there s no ambguty n the dentfcaton of the d that belong to the prmary onzatons. The ampltudes of the outer valence d onzaton energes below 22 ev are also farly close to that of, wth the correspondng pole strengths n hgher than 0.9. The n for onzaton from 4 MO of CO s the lowest, beng precsely equal to In contrast, the pole strength dstrbuton for onzatons from the deep valence levels exhbts an mportant satellte structure. In ths case, n of the prmary onzaton s consderably smaller than 1 rangng from for the 3 level of CO to for the 2 level of HF. evertheless, the overlap of the normalzed Dyson orbtal wth the correspondng Kohn Sham orbtal remans close to , so the shape of the Dyson orbtal remans smlar to that of the KS orbtal, only the ampltude s dmnshed n accordance wth the n factor. When n dffers from 1 we expect strong satelltes. The satellte pole strengths (n c ), whch are close to or larger than 0.1 are presented n Table III n parentheses. For CO there are three such satelltes for the 3 orbtal, wth the pole strengths of 0.177, 0.083, and 0.171, and VIPs 37.35, 40.12, and ev, respectvely. The overlaps of the normalzed Dyson orbtals n 1/2 c d c wth the correspondng KS orbtal, 3, are also gven n parentheses, below the overlap of the Dyson orbtal of the prmary on state wth the 3 orbtal. They are also very close to 1, dentfyng these on states as satelltes to (3) 1. The nature of these satelltes can be deduced from the on wave functons. The frst, at ev, s a clear-cut example of the second type of satellte men-

12 1948 J. Chem. Phys., Vol. 119, o. 4, 22 July 2003 Grtsenko, Braïda, and Baerends toned n the dscusson above. The satellte on state corresponds to (5) 1 onzaton accompaned wth * exctaton. The satellte at ev s rather more mxed, t contans shake-ups to the (5) 1 prmary onzaton both 5 * and * shake-ups and also to the (4) 1 prmary onzaton 5 * shake-up. The thrd satellte, at ev, conssts of shake-ups to the (4) 1 prmary onzaton, ths tme mostly the * exctaton wth some admxture of 5 * exctaton. In all these satellte wave functons the determnant of the (3) 1 prmary onzaton has a sgnfcant coeffcent, doubtless due to neardegeneracy mxng of the shake-up states of the (5) 1 and (4) 1 onzatons wth the (3) 1 prmary onzaton. Ths lends the Dyson orbtals of these on states the 3 shape, whch n turn causes the correspondng spectroscopc factors to appear on the row of the M 1 P matrx for the 3 KS orbtal energy. It s nterestng to consder the structure of the M 1 P matrx n the case of CO: Approxmate structure of matrx M 1 P for the symmetry of CO We note that the quasdagonal structure of the leadng H H block s evdent, as well as the structure of the satellte columns, whch only exhbt a sgnfcant element n the row for the KS orbtal to whch the column s on state s a satellte. For other molecules just one satellte has a pole strength hgher than 0.1, wth the largest satellte pole strength n c beng calculated for the satellte of the 2a 1 level of H 2 O wth VIP I c ev, whle the calculated pole strength of the prmary onzaton from 2a 1 s n wth VIP I ev. Ths satellte corresponds to onzaton to the (1) state (1)1/2,1/2, whch s a mxture of the prmary electron confguraton wth a sngle hole (1h) n the 2a 1 MO wth a (2h1p) confguraton wth a double hole n the 1b 2 MO whch represents the -electron lone par of O and a sngle electron excted to the O H antbondng MO. Ths s a shake-up of the prmary 1b 1 2 onzaton, wth onzaton from 1b 2 accompaned wth exctaton 1b 2 *(O H). We are therefore agan dealng wth the second case we dscussed n the ntroducton to the prevous secton,.e., a shake-up state (1b 2 4a 1 ) of the 1b 1 2 prmary onzaton steals ntensty from the 2a 1 1 prmary on state wth whch s nearly degenerate. There s also some mxng nto ths on state of a shake-up of the 3a 1 1 prmary on state, namely the 3a 1 *(O H) exctaton. The large contrbuton n ths wave functon of the 2a 1 1 determnant lends the Dyson orbtal ts 2a 1 shape note the overlap of of the normalzed Dyson orbtal wth the 2a 1 Kohn Sham orbtal n Table III. The structure of the M 1 P matrx n the A 1 block of H 2 O s calculated to be: Approxmate structure of matrx M 1 P for the A 1 symmetry of H 2 O Agan the strongly dagonal nature of the P(HH) matrx, wth dagonal elements close to the n 3a1 and n 2a1 cf. Table III s evdent. The calculated VIPs and pole strengths agree reasonably wth the expermental estmates 38 n exp 0.58 and I exp 32.2 ev for the prmary onzaton and n c exp 0.18, I c exp 35.0 ev for the satellte. Prevous calculaton of H 2 Ona smaller bass 31 gave n 0.44, I 33.1 ev for the prmary onzaton and n c 0.22, I c 33.6 ev for the satellte. From the comparson of these data one can see, that our calculaton has produced VIPs and n, whch are closer to experment, whle n Ref. 31 an n c value closer to the experment has been obtaned. So the calculated M 1 P matrces have, ndeed, the smple structure that was antcpated n Sec. III. They have quasdagonal HH blocks for prmary onzatons wth small off-dagonal elements and the satellte columns have the unt-vector tmes n c ) lke structure. Table III presents the dagonal elements (M 1 P) for the prmary onzatons and n parentheses the elements (M 1 P) c for major satelltes. Comparson of these elements wth the pole strengths shows that n all cases (M 1 P) and (M 1 P) c are close to the correspondng n and n c. The only excepton s the 2 level of HF, for whch the n value s apprecably larger than (M 1 P). Thus, ndeed, the elements of the leadng HH block of M 1 P can be represented wth the approxmate equalty (M 1 P) j j n, and for the satellte columns the approxmate equalty (M 1 P) jc j n c (c s ()) holds true. From ths follows, that for each the term (M 1 PI) n Eq. 1.7 can be farly represented wth the spectroscopc average j n j I j over the prmary onzaton and the related satelltes. Just as was done for H 2 n the prevous secton, for all valence levels of the molecules consdered n ths secton the contrbuton to of Eq. 1.7 from other than I ) VIPs has been calculated. As follows from Eq. 1.7, the full sum over other VIPs should be equal to the followng combnaton of other ngredents: M 1 P k I k M 1 P I M 1 resp. k 5.1 In Table III, the sum of the l.h.s. of Eq. 5.1 s calculated for each level over the frst 50 terms the nfnte sum has to be restrcted to a fnte number of terms, and 50 corresponds to the abovementoned program lmtaton. The result s placed n the entry calc., whle the combnaton n the r.h.s. of Eq. 5.1 wth the KS orbtal energy and the expermental VIP I s placed as an estmate of the prmed sum n the entry estm.. Snce the quanttes, resp k, and M 1 are obtaned from a rather accurate KS soluton, we expect estm. to be a reasonable estmate of the prmed sum.