I. INTRODUCTION. Self-interaction correction: The transition-metal atoms

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1 PHYSCAL REVEW A VLUME 38, NUMBER 8 CTBER 15, 1988 Self-inteaction coection: The tansition-metal atoms Pieto tona Dipatimento di Fisica, Univesita degli Studi di Genova, Genova, taly (Recejved 10 Mach 1988) The method that we have ecently poposed to coect the local-density appoimation of the density-functional theoy fo spuious self-inteaction effects is used to calculate the inteconfiguational enegies and the ionization potentials of the tansition-metal atoms. The esults so obtained ae compaed with the coesponding quantities calculated by using the loca1- density appoimation and the Pedew and Zunge self-inteaction coection. n both cases we find vey lage impovements which ae essentially due to a bette desciption of the 3d electons. The esults of this pape show that the lage local-density eos fo these quantities ae mainly due to the self-inteaction tems. They also indicate that the latte ae pobably a majo souce of eos in the local-density calculations of the electonic popeties of the tansition-metal dimes and solids.. NTRDUCTN The calculation of the tansition-metal atoms inteconfiguational enegies (CEs) by using appoimate vesions of the density-functional theoy has been the object of a numbe of papes. Some yeas ago it was shown that the local-density appoimation (LDA) epoduces emakably well the tend of the CEs (3d )" (4s) -( 3d )" (4s) and (3d)"-(3d)" (4s), but that the numeical values of these enegies ae wong fo about 1 ev and 1.5 ev, espectively. t was also shown that these eos ae not sensibly modified by using diffeent local epessions of the coelation enegy. The ealie calculations wee pefomed without taking into account the multiplet stuctue contibutions. This was done by Gunnasson and Jones by means of the method suggested by Ziegle, Rauk, and Baeends and by von Bath. t was found that the multiplet contibutions ae impotant in ode to account fo the small deviations fom the tend of the epeimental data, but that they ae quite ineffective on the values of the discepancies fom these data. So, in ode to study these discepancies, it is about equivalent to include the multiplet stuctue effects into the theoetical calculations and to compae with the epeimental data o to pefom the usual cental-field calculations and to compae with spheically aveaged epeimental esults (obtained by taking a weighted aveage of the tems which have the same spin multiplicity as the lowest tem coesponding to each configuation). Nonlocal effects wee consideed by Gunnasson and Jones who calculated the CEs by using thei nonlocal echange-coelation functional. Although these calculations wee pefomed by intoducing a shell patitioning in ode to obtain the best pefomance of this method, only maginal impovements of the LDA esults wee found. Hatee-Fock (HF) data fo the CEs wee consideed by Hais and Jones. These authos used elativistic HF data by Kagawa and they pointed out lage discepancies between the tends of the theoetical and of the epeimental esults. This conclusion was contadicted by Gunnasson and Jones, who eeamined the HF CEs using HF data by Clementi and Roetti and who found a bette ageement between the tends of the epeimental and of the theoetical esults. The latte, howeve, pesented some esidual iegulaities. Moe ecently Baoni" has pefomed HF calculations of CEs. The tend of the Baoni esults agees vey well with the epeimental one and the discepancies ae of the same ode of those obtained fom LDA calculations including coelation. n the same pape Baoni has also consideed the coelation contibution as a fist-ode petubation and using the LDA to the CEs. He found that this contibution bings the HF CEs in a quite good ageement with the epeimental ones. n paticula, the sign of this coection is always such as to educe the discepancies between the theoetical and the epeimental data. CEs calculations wee also pefomed by using the self-inteaction-coected LDA in the fom poposed by Pedew and Zunge. t was found that the eos of this appoimation ae of the same ode of those of the LDA. This is quite diffeent fom what one obtains by teating the self-inteaction tems by fist-ode petubation theoy. n this latte way, in fact, one finds eos geate than the LDA eos. This is due to the impotant elaation effects involved in these tansitions, effects which seem to be bette taken into account by using the self-inteaction coection than the too soft LDA. t should also be noted that in a late pape Haison has shown that a modified vesion of the Pedew and Zunge theoy poduces CEs in bette ageement with epeiment. Recently we have poposed a new method of intoducing a self-inteaction coection into the LDA, which diffes fom the Pedew and Zunge one fo the teatment of the intashell echange tems. n that pape (heeafte efeed to as ) we have successfully tested ou appoimation by calculating a numbe of electonic popeties of light atoms. The pupose of the pesent wok is to in The Ameican Physical Society

2 SELF-NTERACTN CRRECTN: THE TRANSTN vestigate if that method educes the lage LDA eos also in the case of the CEs of the tansition metal atoms. This will be caied out in Sec., while in Sec. we shall compae biefly the main featues of ou coection with those of the othe moe stictly elated methods, and Sec. V will be devoted to the conclusions of this wok.. THE SELF-NTERACTN-CRRECTED A simple way to ty to impove LDA the LDA consists in intoducing a self-inteaction coection. Biefly, in this appoach one teats eactly the self-echange tems and one uses an appoimate epession fo the inteelecton echange and coelation. The one-electon equation which is obtained in this way is not a Kohn-Sham equation because the potential is obital dependent. Nevetheless, it can be consideed as an appoimation of anothe eact one-electon equation also poposed by Kohn and Sham which is obtained by sepaating the echange fom the coelation enegy and by using fo the fist one its HF epession. t should be noted that the usual self-inteaction-coected LDA potentials, at lage distance fom a finite neutal system, decease popotionally to This. is a majo diffeence fom the LDA and its main consequence is that the enegy eigenvalues ae a bette appoimation of the electonic ionization potentials. n a vey compehensive aticle on the self-inteaction coection we efe to this wok fo an etensive discussion of the theoetical aspect of this type of theoy as well as fo a eview of the woks on this subject up to 1981 Pedew and Zunge have pesented a lage numbe of esults obtained by using the following geneal pesciption in ode to deduce, fom an abitay appoimate epession E,of the echange-coelation enegy, a coesponding self-inteaction fee epession E,: ~pt pl l & [p whee a is the quantum numbe set chaacteizing (with 0) the one-paticle states, p is the total chage density of the electons of spin 0., and p is the chage density of one electon in the state ao. Pedew and Zunge justified this pesciption essentially by the fact that the echange-coelation enegy of a system containing only one electon should be zeo, and they veified that in a numbe of cases the esults given by this method impoved the LDA ones. Two yeas late, Haison pointed out that the spheical aveage which is commonly used in the atomic calculations, can be pefomed in two nonequivalent ways. n the fist one, which is the taditional pocedue fo the Hatee-like theoies and which was used by Pedew and Zunge, one eplaces the obital chage densities in Eq. (1) with thei spheical aveages and one deives the potential fom the esulting epession. n the second one, nonspheical obital chage densities ae used in Eq. (1). Then one deives the potential, which esults to be spheically symmetic, but to depend fom the quantum numbe m&. Finally, the cental-field appoimation is obtained by taking the aveage of the potentials fo the electons which have identical quantum numbes, but diffeent values of m&. Following this second pocedue one finds esults which depend on the choice of the epesentation fo the angula pat of the obitals. n paticula, Haison found that using spheical-hamonic o Catesian epesentations in echange-only calculations, one obtains lowe enegies in the fist case, while the second choice poduces esults in bette ageement with HF. Fo this latte eason Haison decided to use Catesian obitals in his calculations, and in this way he found that the Pedew and Zunge esults can be consideably impoved. At about the same time, we pointed out that the inteelecton echange enegy F. "" (fo unit volume) of a homogeneous gas containing N electons of spin c had been calculated by Rae and that it can be witten in the following fom: 13 Z"." = p."y(n. ) = y(n. )E"., whee E is the total echange enegy of the homogeneous gas and y(n ) is a function of the numbe of electons N defined by the two equations: N =~ 6&+ ~~& (3) y(n )=1,P+, P -,, P -. (4) n that pape we also used these equations in a simple way in ode to appoimate the inteelecton echange potential of the self-inteaction-coected one-electon equation and in this way we obtained some impovements with espect to LDA. t is inteesting to compae Eqs. (2) (4) with the analogous quantities calculated by the Pedew and Zunge theoy. Using the latte one obtains the following value ofy: y(n )=1 (N ) which shows that this theoy does not epoduce Raes coect esult fo this limiting case. n pape we have discussed in consideable moe detail the poblem of applying the Rae equations to the inhomogeneous case. We found that a easonable way to do this consists in patitioning the total electon density of the inhomogeneous system in a numbe of pieces which have small ovelaps. Equations (2) (4) can then be used fo the inteelecton echange of each single piece and the echange inteaction between the diffeent pieces (which obviously does not contain self-inteaction contibutions) can be taken into account by the usual local epession of the echange enegy. n the case of an atomic system, fo eample, the natual units fo patitioning the total chage density ae the electonic shells. Assuming this type of patition, one obtains the following epession of the inteelecton echange potential V & fo one electon belonging to the shell of quantum numbe nl: Pnlo N,~1 (2) y(n,)]

3 3852 PETR CRTNA 38 and fo the inteelecton echange enegy E"": n these epessions N i is the numbe of electons of spin c pesent in the shell nl. n pape we eplaced N i in the coefficient N i [ y( N i ) j of the potential with the degeneacy (21+1) of the same shell. This makes only a little diffeence on the total enegies (only the electons belonging to open shells ae teated diffeently) and ensues a moe effective compensation of the eos when one is concened with total enegy diffeences of two diffeent systems. We emphasize that the scheme just descibed is stictly a local scheme, ecept fo the self-echange tems which ae teated eactly. n we have indicated this method with D-SC. We will adopt this notation in this pape as well and, following the common use, we will indicate with SC the Pedew and Zunge coection.. THE TRANSTN-METAL ATMS NTERCNFGURATNAL ENERGES AND NZATN PTENTALS t is clea that a self-inteaction coection can solve only in pat the poblems of the LDA and that fo a numbe of popeties one can epect only a patial impovement o, in some cases, no impovement at all. A typical quantity which cannot be bette estimated by intoducing the self-inteaction coection is the spin-flip enegy of the 4s electon in the (3d)" (4s ) configuation of the tansition-metal atoms. n fact, the self-inteaction contibutions to the total enegies of the two configuations (3d)" (4s)& and (3d)" (4s)& ae about the same, and they cancel out when one takes the diffeence. n the othe hand, the spin-flip enegy is entiely due to the inteaction of the 4s electon with the electons belonging to the moe inne shells. This is a typical nonlocal inteaction that cannot be coectly descibed by the LDA. So, it is quite natual to find lage discepancies between the LDA spin-flip enegies and the epeimental ones. Unfotunately the self-inteactioncoected methods, as SC and D-SC, descibe this inteaction eactly in the same way as LDA and one epects to find and this is indeed confimed by the actual calculations about the same esults fom the thee appoimations. n the case of the s-d pomotion enegies, the selfinteaction contibution is not the same fo the two configuations and does not cancel out in taking the diffeence. So, it is possible that intoducing a selfinteaction coection one finds impoved values of these quantities. Howeve, it is physically clea that puely nonlocal inteactions, as those discussed fo the spin-flip enegies, should give an impotant contibution to the CE*s as well and that fo this eason one can only epect a patial impovement of the LDA values. n the following we shall give the esults of selfconsistent spin-polaized calculations. As in pape we have taken into account the coelation contibutions by using the Pedew and Zunge paametization of the Cepeley and Alde Monte Calo data fo the coelation enegy of the homogeneous gas and the small nonothogonality effects by a Schmidt othogonalization of the obitals afte each iteation. n all the figues of this pape we epot the diffeences between theoetical and epeimental quantities. The latte, which have been deduced fom Mooes spectoscopic data tables, have been spheically aveaged in ode to obtain values which can be diectly compaed with the esults of ou cental-field theoetical calculations. n Figs. 1 and 2 we show the eos of the diffeent appoimations fo the CEs (3d)" (4s )-(3d)" (4s) and (3d)"-(3d)" (4s), espectively. As it was noted in pevious woks, the LDA and SC desciptions of these quantities ae about equivalent and SC is a little bette in the second case. n the othe hand, D-SC intoduces systematic impovements. n the case of Fig. 1 the eos ae educed fo about 50% in the fist half of the seies and even moe in the second half. t should also be noted that D-SC gives the coect gound-state configuation in all cases, while SC and LDA fail in the cases of Ti and of. The impovements ae even moe evident in Fig. 2. n this case, in fact, the eos of D-SC ae smalle than ev (ecept fo Ca), while LDA undeestimate these CEs fo about 1.5 ev and the SC eos ae included between 0.8 and 1.9 ev. t is also inteesting to note that the LDA does not pedict the coect ode of the configuations fo Ni: indeed, in this case, the LDA ene- Ld 1.0 (D Q5~ Ca (3(j ) (g g )~ (3g ) ~(g )~ FG. 1. Diffeences between the vaious theoetical values fo the (3d)" (4s)-(3d)" (4s) inteconfiguational enegies and the coesponding epeimental data. D-SC;,LDA;..., SC. C

4 38 SELF-NTERACTN CRRECTN: THE TRANSTN (3d) (3d)" (4S ) (3d) (4s) ( 3d) (4g ) Q) - LL CL L LLJ 0 5 l.0 lw 0 L. ~ & -1.5 ] i q p ~ LL 0 ~ i i Ca Fe Ni Sc Cu Ca Sc Tj C Fe Mn FG. 2. Diffeences between the vaious theoetical values fo the (3d)"-(3d)" (4s) inteconfiguational enegies and the coesponding epeimental data., D-SC;,LDA; SC. gy of the (3d)" configuation is lowe than that of the (3d)" (4s) one. Moe infomation about the natue of the impovements intoduced by using D-SC can be obtained by analyzing the ionization potentials (P s). These quantities ae geneally studied by consideing the enegy diffeences between the vaious ionic configuations and the (3d)" (4s) atomic configuation o, altenatively, the diffeences between the enegies of the ionic and the atomic gound states, ignoing the (possible) diffeences in the gound-state configuations given by self-consistent calculations and by epeiment. Poceeding in these ways, howeve, a pat of the discepancies is due to the eos in the s-d pomotion enegies, which have been sepaately eamined. %e ae instead inteested in the en- FG. 3. Diffeences between the vaious theoetical values fo the (3d)" (4s)-(3d)" (4s) ionization potentials and the coesponding epeimental data.,lda and D-SC; SC egy equied to take away a 4s o a 3d electon fom an atom without having to tansfe at the same time any electon between these two shells. t is then possible, fo eample, to conside the (3d)" (4s)-(3d)" (4s) CE as the diffeence between the (3d)" (4s)-(3d)" (4s) and the (3d)" (4s)-(3d)" (4s) Ps. All the possible Ps which do not involve electon tansfes between the 4s and 3d shells ae epoted in Figs n Fig. 3 the D-SC data ae not shown because they ae pactically indistinguishable fom the LDA ones. Figues 3 and 4 show that the LDA and D-SC values of the Ps fo the 4s electons ae quite simila, while SC gives a little bette esults. Thee is some analogy between these esults and what is found fo the aveage adius of the 3d and 4s obitals. ndeed, in the case of the (3d)" (4s) configuation, the LDA and D-SC adius agee geneally bette than 1%, while SC gives values which ae sma11e fo the 3d electons and geate fo the 4s. f couse this cannot be consideed the eason fo the X (D LL LLj ~ ~ ~ ~ ~ ~ Ca Cf Fe Nj Sc V Mn CU FG. 4. Diffeences between the vaious theoetical values fo the {3d)" -{3d)" {4s) ionization potentials and the coesponding epeimental data. D-SC;,LDA; - -,SC. ~

5 R 3854 PETR CRTNA 38 (3d ) (4S) (3d) (4s) (3d ) (3d) Q ~j CL CD LLj ~ LLj - Sc V C Mn Fe Nj Cu -05 Cg Ti C 5c Fe FG. 5. Diffeences between the vaious theoetical values fo the (3d)" (4s)-(3d)" (4s ) ionization potentials and the coesponding epeimental data., D-SC,,LDA; 0 ~ SC similaity of the LDA and D-SC Ps, which indeed seems athe casual. t is only anothe indication of the ough equivalence of the desciptions of the 4s electons given by these two appoimations. ming back now to the analysis of Figs. 3 and 4, it can be noted that the eo on the Ps inceases with the CL (D LLJ (3d ) (g s) -(3d ) (4s ) - Ca C FG. 6. Diffeences between the vaious theoetical values fo the (3d)" (4s)-(3d)" (4s) ionization potentials and the coesponding epeimental data., D-SC;,LDA; ~ ~ o Fe Ni Cu FG. 7. Diffeences between the vaious theoetical values fo the (3d)" -(3d)" ionization potentials and the coesponding epeimental data., D-SC;,LDA; SC. numbe of 3d electons of the same spin as the 4s electon concened in the ionization, while this eo is oughly constant in that pat of the seies whee the numbe of 3d electons of opposite spin is inceasing. So it is quite natual to attibute this effect to the echange inteaction between the 4s and the 3d electons, inteaction which seems to be, in paticula, the main souce of eos in the case of the (3d)" (4s) configuation. mpletely diffeent is the analysis of the 3d electon Ps (Figs. 5 7). n all the thee possible cases D-SC systematically impoves LDA as well as SC. n the othe hand, the quality of the esults obtained by these latte two appoimations ae about equivalent. LDA is bette fo the Ps of the 4s electon ich configuation (Fig. 5), while SC is to be pefeed in the opposite case (Fig. 7). Finally the two appoimations ae about equivalent in the intemediate case (Fig. 6). n conclusion, the bette desciption of the 3d Ps given by D-SC seems to be the oigin of the impovements that we have found fo the CEs. V. CNCLUSNS This wok shows that D-SC is successful not only in descibing the electonic popeties of the atoms of the two fist ows of the Peiodic Table, but also in the case of the thid ow tansition metal atoms. n paticula, we have shown that D-SC gives CEs and Ps (fo d electons) which agee consideably bette with epeiment than those calculated by using LDA o SC. This last point confims that the use of the self-inteaction coection pemits one to bette take into account the elaation contibutions to these quantities and, besides, that D-SC gives an impoved desciption of the inteelecton

6 38 SELF-NTERACTN CRRECTN: THE TRANSTN echange tems inside each shell. Nevetheless, the eos of D-SC ae still impotant. They oiginate fom inteactions which ae essentially nonlocal and that D-SC teats in the same way as LDA. Fo these inteactions, and paticulaly fo the inteshells ones, which ae pobably the main cause of the esidual eos, appopiate appoimations should be elaboated. This wok shows also that an obital dependent scheme can be vey useful in ode to obtain appoimations which accuately descibe the chemical bond. t should be pointed out, howeve, that the Kohn and Sham theoy states that this type of appoach is not necessay: n pinciple, one can obtain eact esults by using a unique effective potential fo all the electons. f couse, the difficulty is to find a good appoimation of this potential, but, in the last few yeas, some impotant pogess has been made in this 6eld. These new nonlocal functionals have not yet been tested by calculating citical popeties such as the CEs and it will be inteesting to compae thei esults with those epoted in this pape. n any case, an obital dependent scheme has a geate Aeibility, which pemits not only the use of the selfinteaction coection, but also, fo eample, a diffeent teatment of the inte- and inta-shells nonlocal contibutions. This last point, in ou opinion, could be quite impotant in ode to fomulate appoimate functionals which take into account the diffeent physical featues of these inteactions. Finally, we would like to point out that some methods fo applying the self-inteaction coection to solid state, molecula, and cluste calculations have been ecently elaboated and that in all these cases the utility of this type of coection has been poved. t would be inteesting to apply the coection used in this pape to molecula and to band-stuctue calculations: The esults we have epoted seem indicate that in this way one could eliminate a pat of the anomalies of the LDA esults fo the tansition dimes and metals. ACKNWLEDGMENTS This wok was suppoted in pat by the Cento nteunivesitaio di Stuttua della Mateia, Ministeo della Pubblica stuzione and Guppo Nazionale di Stuttua della Mateia, Unita di Genova, and in pat by a gant of the talian Reseach uncil (nsiglio Nazionale delle Richeche) unde the Fench-talian Scientific llaboation Ageement. Stimulating discussions with G. Bobel and F. G. Fumi ae gatefully acknowledged. J. Hais and R.. Jones, J. Chem. Phys. 68, 3316 (1978). 2J. Hais and R.. Jones, J. Chem. Phys. 70, 830 (1979).. Gunnasson and R.. Jones, Solid State mmun. 37, 249 (1981). 4R.. Jones, J. Chem. Phys. 76, 3098 (1982). 50. Gunnasson and R.. Jones, J. Chem. Phys. 72, 5357 (1980). T. Ziegle, A. Rauk, and E. J. Baeends, Theo. Chim. Acta 43, 261 (1977). 7U. von Bath, Phys. Rev. A 20, 1693 (1979).. Gunnasson and R.. Jones, Phys. Sc. 21, 394 (1980). T. Kagawa, Phys. Rev. A 12, 2245 (1975). E. Clementi and C. Roetti, At. Data Nucl. Data Tables 43, 261(1977). S. Baoni, J. Chem. Phys. 80, 5703 (1984). The iegulaities in the HF CEs epoted in Ref. (5) have been attibuted in this pape to an eo in the Clementi and Roetti data fo. J. P. Pedew and A. Zunge, Phys. Rev. B 23, 5048 (1981). n passing fom the (3d )" (4s) to the (3d)" (4s) configuation, the chage distibution of the 3d electons, as well as that of the 4s electons, epands. The same happens to the 3d electons in passing to the (3d )" configuation. This is a consequence of the inceased electostatic epulsion between these electons, while the epansion of the 4s electons is due to the moe effective sceening of the nuclea field. J. G. Haison, J. Chem. Phys. 79, 2265 (1983). P. tona, Phys. Rev. A 34, 769 (1986). W. Kohn and L. J. Sham. Phys. Rev. 140, A1133 (1965). J. G. Haison, J. Chem. Phys. 78, 4562 (1983). The noninvaiance fo unitay tansfomations of the obitals is a geneal poblem of the self-inteaction coection. Pedew and Zunge suggested that a possible solution consists in looking fo the obitals which minimize the total enegy. Howeve, fom a pactical standpoint, this is a quite complicated citeion and, on the othe hand, it is not sue that in this way one gets the best solutions, because the total enegy is calculated by an appoimate functional. So, the Haison choice can be citicized because it does not espect the Pedew and Zunge citeion, but it is also possible to adopt the opposite standpoint and to asm that the Haison calculations veify that this citeion is not always the bette one. t should be noted that to use the fist method of intoducing the cental-field appoimation is a simple way to avoid this poblem in the atomic case. G. Bobel and P. tona, J. Phys. B 16, 349 (1983). A.. M. Rae, Mol. Phys. 29, 467 (1975). D. M. Cepeley and B. J. Alde, Phys. Rev. Lett. 45, 566 (1980). C. E. Mooe, Atomic Enegy Levels, Natl. Bu. Stand. (U.S.) Cic. No. 467 (U.S. GP, Washington, D.C., 1949 and 1952), Vols. and. n the case of V, the thee methods give a (3d)" (4s) gound-state configuation, while the (3d)" (4s) is the epeimental one. Howeve, the spheically aveaged epeimental data do not show this discepancy fom the theoetical esults. ndeed, the almost identical Ps fo the (3d)" (4s) configuation deive fom the sum of contibutions, due to the diffeent components of the total enegy (kinetic, electonnucleus, electon-electon ulombic enegy, etc.), which have athe diffeent values in the two appoimations. 25D. C. Langeth and M. J. Mehl, Phys. Rev. B 28, 1809 (1983); 29, 2310 (1984). C. D. Hu and D. C. Langeth, Phys. Sc. 32, 391 (1985).

7 3856 PETR CRTNA 38 J. P. Pedew, Phys. Rev. Lett. 55, 1665 (1985); see also J. P. Pedew and W. Yue, Phys. Rev. B 33, 8800 (1986). S. K. Ghosh and R. G. Pa, Phys. Rev. A 34, 785 {1986). R. A. Heaton, J. G. Haison, and C. C. Lin, Phys. Rev. B 28, 5992 (1983). M. R. Pedeson, R. A. Heaton, and C. C. Lin, J. Chem. Phys. 80, 1972 (1984). F. R. Redfen and R. C. Chancy, Phys. Rev. B 33, 3823 (1986).