A Scheme for Calculating Atomic Structures beyond the Spherical Approximation

Transcription

1 Jounal of Moden Physcs,,, 4-4 do:.46/jmp..55 Publshed Onlne May ( A cheme fo Calculatng Atomc tuctues beyond the phecal Appoxmaton Abstact Mtyasu Myasta, Katsuhko Hguch, Masahko Hguch Gaduate chool of cence and Engneeng, hnshu Unvesty, Ueda, Japan Gaduate chool of Advanced cence of Matte, Hoshma Unvesty, Hgash-Hoshma, Japan Depatment of Physcs, Faculty of cence, hnshu Unvesty, Matsumoto, Japan E-mal: myasta.mtyasu@gmal.com Receved Febuay, ; evsed Apl, ; Apl 4, We pesent a scheme fo calculatng atomc sngle-patcle wave functons and specta wth takng nto account the nonsphecal effect explctly. The actual calculaton s also pefomed fo the neutal cabon atom wthn the Hatee-Fock-late appoxmaton. As compaed wth the conventonal atomc stuctue of the sphecal appoxmaton, the degeneate enegy levels ae splt patally. The gound state values of the total obtal and spn angula momenta ae estmated to be both about unty, whch coesponds to the tem P n the L-multplet theoy. Ths means that the nonsphecal effect may play an essental ole n the descpton of the magnetzaton caused by the obtal polazaton. Keywods: Nonsphecal Dstbuton of Electons, phecal Appoxmaton, Obtal Polazaton, Atomc tuctue, Cabon Atom. Intoducton Let us stat wth evstng the conventonal atomc stuctues. We consde the solated neutal atom wth the atomc numbe Z. Neglectng the elatvstc effects, the chödnge Equaton fo the statonay state s gven by ĤE () wth Z ˆ Z Z H, j, () j j whee and stand fo the poston of the th electon and ts magntude, espectvely, and whee the atomc unt s used. Equaton () can be numecally solved only n small atomc systems, but n lage atomc systems we have to utlze the theoes to educe Equaton () nto the effectve sngle-patcle Equaton such as the Hatee [], Hate-Fock [] and Kohn-ham [,4] Equatons, etc. The sngle-patcle Equaton s geneally wtten by, () V whee denotes the up-spn o down-spn. In ode to solve Equaton (), we have usually used the sphecal appoxmaton,.e., the cental feld appoxmaton [5]. Unde such the appoxmaton, Equaton () s sepaable nto two Equatons, one of whch depends on the adal vaable and the othe on the angula vaables, and. If the effectve potental s sphecally symmetc and local [6], and f the solutons ae gven by nlm pnl Ylm,, (4) then two Equatons ae d l l V p nl nl pnl d ˆ Ylm, l l Ylm,, (5) l, (6) whee ˆl s the opeato of the obtal angula momentum, and Ylm, ae the sphecal hamoncs [7]. Radal wave functons pnl of Equaton (5) ae calculated easly by means of the numecal methods such as the Heman-kllman method [8,9]. Thus, the conventonal atomc stuctues, whee the egenstates ae specfed by the quantum numbes nl, n, mlm l and can be obtaned. Hee the queston s ased of whethe the sphecal Copyght cres.

2 4 M. MIYAITA ET AL. appoxmaton s always appopate o not. The sphecal appoxmaton s easonable fo atoms havng the outemost shell that s fully o half occuped snce the electon denstes ae exactly sphecal. Howeve, n the othe atoms the electon denstes ae not necessaly sphecal, so that the conventonal pctue of the atomc stuctues s not goous but just an appoxmaton. To what extent the effect of the nonsphecal dstbuton of electons (whch s heeafte called the nonsphecal effect) modfes the conventonal pctue of the atomc stuctues seems to be nteestng and mpotant. Ths s because the electonc stuctues of molecules and solds have been fequently consdeed on the bass of the odnay atomc wave functons and specta. The typcal examples ae the late ntegals contaned n the model Hamltonans lke the Hubbad model [,], and n the LDA + U method [,]. In addton to the above, thee exsts an obvous flaw n the sphecal appoxmaton. The total obtal angula momentum becomes necessaly zeo n the conventonal atomc stuctue, because the sphecal appoxmaton concdes wth the fllng appoxmaton n whch electons ae unfomly dstbuted nto each state n the outemost shell [4]. Ths means that the obtal polazaton neve appeas n the atomc stuctues of the sphecal appoxmaton. The obtal polazaton s an ogn of the magnetsm of solds as well as the spn polazaton [5-], especally fo the 5f-electon systems [5-7]. o fa the obtal polazaton has been dscussed as a pat of the coelaton effects [7] o on the bass of the L-multplet theoy []. In ths pape, we shall dscuss the nonsphecal effect on the obtal polazaton fom the vewpont of the sngle-patcle pctue. It wll be shown n the followng sectons that the obtal polazaton appeas wthout the coelaton effects. As a fst attempt to take nto account the nonsphecal effect, late has poposed a scheme fo expandng the egenfunctons of Equaton () wth the sphecal hamoncs [9]. Howeve hs method s dffcult to be pefomed because an nfnte numbe of smultaneous equatons have to be solved. Afte late s poposal, thee have been two knds of appoaches to ths poblem. One s the vaatonal method whee the sngle-patcle wave functon s expanded by usng appopately chosen bass functons [,]. Anothe s the densty functonal scheme contanng the effect of the obtal cuent densty explctly.[4-] In ths pape, we adopt the fome appoach. As the bass functons, egenfunctons fo the sphecal pat of the sngle-patcle potental ae used and updated fo each teaton of the self-consstent calculatons. They ae appaently dffeent fom those of the pevous woks [,]. The am of ths pape s to pesent the tactable scheme fo calculatng the atomc stuctues beyond the sphecal appoxmaton, and to dscuss the nonsphecal effect. Oganzaton of ths pape s as follows. In ecton, we pesent a scheme fo dealng wth the nonsphecal effect explctly. In ode to check the valdty of the scheme, we apply t to the neutal cabon atom n ecton. The calculaton pocedue s also explaned. The esults ae shown n ecton 4, wth a focus on the dffeences between the pesent atomc stuctues and the conventonal one. The gound state values of the total obtal and spn angula momenta ae also estmated. Fnally concludng emaks ae gven n ecton 5.. A Vaatonal Method beyond the phecal Appoxmaton In ths secton, we pesent a vaatonal method fo calculatng atomc stuctues wth takng nto account the nonsphecal effect. Let us consde solvng the sngle-patcle Equaton (). The Hatee-Fock-late appoxmaton s utlzed [],.e., the effectve potental of Equaton () s gven n a local fom V. Fst, we expand the effectve potental wth the sphecal hamoncs: lm lm, lm lm, lm * * V v Y v Y, (7) whee vlm ae the adal components and the explct foms ae gven n Appendx. Fo the convenence of the subsequent dscusson, the opeato of the lefthand sde (LH) of Equaton (5) s defned as ˆ ˆ d l H : V, (8) d whee let V be the sphecally aveaged potental fo Equaton (7), whch s defned as V ()sn d d 4π V (9) v * v. 4π In the Expanson 7, the tem lm coesponds to the sphecal pat of the effectve potental as shown n Equaton (9), whle the othe tems coespond to the nonsphecal pats. Next, n a smla way to Equaton (7), we shall expand the soluton of Equaton () wth the set of known functons. As the known functons, we hee adopt ones gven by Equaton (4), the adal pat of whch s the egenfuncton fo the Hamltonan 8. Thus, the soluton of Equaton () s wtten as Cnl, lm pnl Ylm,. () nl lm Copyght cres.

3 M. MIYAITA ET AL. 4 ubsttutng Equatons (7) and () nto Equaton (), and wtng dstnctly the sphecal and nonsphecal pats of the effectve potental, we get ˆ H C p Y, v ( ) Y, NL N L L L N L L() * *,, v Y C p Y L L NL N L N L CNLpN YL,, N L () whee we use Equatons (8) and (9), and abbevate nl as N and lm as L fo ease of seeng. It can be easly shown that Equaton () s educed to the sphecal Equaton ncludng Equaton (5), f the second tem of the LH s neglected. Ths means that the second tem of the LH epesents the nonsphecal effect that has been dsegaded n the conventonal sphecal appoxmaton. Hee, fo smplcty, we shall use the common value of l fo N and L, and suppose that the egenvalues fo the Hamltonan 8 s denoted as N. Multplyng * * pn Y, L on both sdes of Equaton () and ntegatng ove the whole space, we have whee V LL () N N N LLO NN L, () * pn V d LL pn CNL * NN N N O p p d, () * v Y, Y, Y, sndd L L L L L() * * * v Y, Y, Y, sndd. L L L L L() (4) Equaton () s just the genealzed egenvalue poblem. If the matx elements, O NN and VNN, and the enegy specta of the sphecal appoxmaton, N, ae gven, then we can obtan the egenvalues,, and egenfunctons, C NL. It should be noted that the egenvalues ae guaanteed to be eal snce both matces of Equaton () ae hemtan. The angula ntegatons n Equaton (4) can be analytcally calculated by usng the Wgne j-symbols. Accodng to the popetes of the Wgne j-symbols, matx elements of Equaton (4) ae zeo unless l l l even, ll l l l and m m m [7]. These condtons also detemnes the uppe lmt of the summaton of Equaton (7). The egenfunctons thus obtaned yeld the new potentals by means of the expessons gven n Appendx. These potentals should concde wth the nput ones. Namely, the self-consstency s equed fo the potentals. The coespondng bass functons n Equaton () ae modfed fo each teaton snce the functon pnl s the adal pat of soluton fo the Hamltonan (8) wth the Potental 9. The teaton s contnued untl self-consstency fo the potentals s acheved. Let us show the detaled pocedue of the self-consstent calculatons. The flow chat of self-consstent calculatons s shown n Fgue. We fst gve a statng potental n some way, fo example va the LDA calculaton wthn the sphecal appoxmaton (tep n Fgue ). In ode to pepae the adal bass functons pnl, the sphecal pats of the potental ae deved. Usng these potentals, atomc stuctue calculatons ae pefomed (tep). Then, usng the bass functons and coespondng specta, the genealzed egenvalue poblem s solved (tep). The esultant egenfunctons povde the new potentals (tep 4). Hee we check whethe the potentals ae conveged o not (tep 5). Of couse, the checkng should be pefomed on both convegences fo the sphecal and nonsphecal pats of potentals. If the convegence s not yet obtaned, we etun to tep wth the sphecal potental calculated fom the new potental. The calculatons ae epeated untl the potentals ae conveged wthn some accuacy.. Applcaton to the Neutal Cabon Atom Compaed to the pevous ones [9,,], the pesent scheme seems to be moe tactable, but as to the effectveness actual calculatons have to be pefomed. Hee we apply t to the neutal cabon atom. In the Expanson, we choose the common value of l n both summatons fo nl and lm. That s to say, physcally meanngful functons ae pepaed fo bass functons of the expanson. In moe detal, we use fve functons havng the followng quantum numbes: nlm,,,,. Coespondngly, the uppe lmt of the potental gven by Equaton (7) s detemned fom the popetes of the Wgne j symbols, as aleady mentoned below Equaton (5). Ths tme, the expanson of the potental conssts of the followng tems:,,, ( lm),,,,,. The genealzed egenvalue poblem s educed to Copyght cres.

4 44 M. MIYAITA ET AL. nl n l * ll mm Onln l pnl Vlm, l m pn l dcnlm. (5) nlm Fo ease of undestandng the matces of the egen- value poblem, the explct foms ae shown below: O, O,,,, V V V V, V, O, O, C V, V, V, V, C V, C. C O, V, V, V, V, C V, O, V, V, V, V, V, O, V, V, V, V, V, (6) Fgue. Flow chat of the self-consstent calculatons. The detaled pocedue s shown n the text. olvng the above Equatons n a self-consstent way, we can obtan the atomc stuctue fo the neutal cabon atom. The concete steps of the calculatons ae shown n the flow chat of Fgue. Hee note that thee s a pos- Copyght cres.

5 M. MIYAITA ET AL. 45 sblty to exst the multple self-consstent solutons. In ode to cove such solutons, vaous knds of nput potentals should be aanged. Ths tme, we pepae the obtals of the sphecal appoxmatons,.e. s, s and p obtals, and take the lnea combnaton of them so as to constuct the nput potentals. The s and s obtals ae used as they stand, whle p obtals ae tansfomed nto the followng obtals: cos sn sn cos, (7) cos sn sn cos whee, and ae p obtals of the sphecal appoxmaton. The nput potentals ae constucted fom the s, s and two obtals chosen among thee ones gven by Equaton (7). The angles and ae changed by 5, espectvely. A total of 864 knds of dffeent potentals ae taken as the statng potentals ( C 864 ). As shown n a subsequent secton, sx knds of self-consstent solutons can be obtaned coespondngly to the statng potentals. 4. Results and Dscussons In ths secton, we wll gve the esults of the atomc stuctue of the neutal cabon atom. Fgue shows the enegy specta of the pesent scheme, togethe wth those of the conventonal sphecal appoxmaton. It should be noted that the conventonal atomc specta can be specfed by the quantum numbes nlm, but n the pesent scheme they ae specfed only by the odnal numbes because of lack of the sphecal symmety. Fo nstance, s states of the sphecal appoxmaton coespond to the st and nd states of the pesent scheme. The conventonal p states ae splt nto doubly degeneate levels and sngle one due to the nonsphecal effect. Thee exst two types of splttng. In othe wods, two types of the conveged self-consstent solutons (C) can be found fom the vewpont of the splttng of enegy levels. One s that the doubly degeneate levels ae hghe than the sngle one, and anothe s the opposte. They ae denoted as C-A and C-B, espectvely, n Fgue. On the othe hand, conventonal s and s states ae lttle nfluenced by the nonsphecal effect. Ths s because the p states (5th and 6th states) ae dectly nfluenced by the nonsphecal denstes of electons, whle the wave functons fo s and s states (fom st to 4th states) ae well localzed nea the nuclea whee the sphecal potental manly caused by the nuclea s domnant. In ode to dscuss the gound-state popetes n moe detal, we shall nvestgate the components of the egenfunctons,.e. the expanson coeffcents of Equaton (). The C-A and C-B ae classfed nto two and fou types, espectvely, accodng to the components of the Fgue. Enegy specta fo the neutal cabon atom. The fst column shows the esults fo the conventonal sphecal appoxmaton. The second and thd columns ae the self-consstent solutons fo the pesent scheme, whch ae denoted as C-A and C-B, espectvely. The up- and down-aows denote the occuped states, and open ccle the unoccuped states. All values ae gven n Rydbeg Unt. Copyght cres.

6 46 M. MIYAITA ET AL. wave functon. They ae denoted as A-, A- and B-, B-, B-, B-4 espectvely. The expanson coeffcents fo the st, nd, d and 4th states ae shown n Table. Fo all of the coveged Cs, the components ae same as those of the sphecal appoxmaton wthn the accuacy of. These esults ae consstent wth the fact that the states fom st to 4th fo the pesent scheme ae n a good ageement wth the conventonal s and s states, espectvely (Fgue ). Concenng the 5th and 6th states, thee s a lage dffeence between the pesent and conventonal schemes, whch s shown n Table. In the conventonal sphecal appoxmaton, electons ae dstbuted nto each shell n an equal weght, so that the coespondng coeffcents ae all. Meanwhle, components of each the conveged C ae patal to some of them. Ths patalty causes the polazaton of the obtal angula momenta, whch s so-called obtal polazaton. In ode to vefy t, we calculate the gound state values of the total obtal and spn angula momenta. The gound state of the Hatee-Fock-late appoxmaton s gven by a sngle late detemnant that s wtten as 4 x,, x (8) 6! whee x denote the coodnates fo th electon ncludng spatal coodnate and spn coodnate, and whee () s the soluton of Equaton (6), and whee ( ) and ( ) ae wave functons fo up- and down-spns, espectvely. Usng Equaton (8), the total obtal angula momentum and ts z-component, L and L, ae espectvely calculated by z whee 6 6 ( x, x ) L ˆ x, x L L, (9), ˆ, x x L x x L, () 6 z 6 6 Lˆ l ˆ, and l ˆ s the opeato of the obtal angula momentum fo the th electon. mlaly, the total spn angula momentum and ts z-component, and z z, ae espectvely gven by ˆ x, x x, x, () whee 6 6, ˆ, x x x x, () 6 z 6 6 ˆ ˆ s, and s ˆ s the opeato of the spn an- gula momentum fo the th electon. The esults ae shown n Table. In a nonelatvstc many-electon system, LL, z, and z ae the conseved quanttes. All of the coveged Cs yeld L and wthn the ac- cuacy of. Ths means that the gound states of the pesent scheme coespond to the tem P that s known to be the gound state of the L-multplet theoy. Futhemoe, t s notced that the pesent scheme obvously z Table. The expanson coeffcents of Equaton () fo the st, nd, d and 4th states. Both C-A and C-B gve the same esults, so we don t lable C-A and C-B dstnctly. (nlm) Enegy [Ryd.] () () (+) () (-).54. phecal appoxmaton Pesent esults fo C-A and C-B Copyght cres.

7 M. MIYAITA ET AL. 47 Table. The expanson coeffcents of Equaton () fo the 5th and 6th states. (nlm) Enegy [Ryd.] () () (+) () (-) phecal appoxmaton.78 C A- C A- C B- C B- C C- C C Table. The gound state values of the tptal obtal angula momentum and td z-component, L and L Z, ae showm n the st and nd cplumns. Also, the gound state values of the total spn angula momentum and ts z-component, and Z, ae shown n the d and 4th columns. tate L L Z Z C A-.... C A-.... C B-.... C B-.... C B-.... C B causes the obtal polazaton. nce the sphecal appoxmaton neve bngs t, we may say that the nonsphecal effect s one of the keys to the appeaance of the obtal polazaton. Hee note that the Cs ae classfed nto thee knds of states, whch yeld Lz, and, espectvely. Ths s not supsng because the tem P ae tply degeneate wth espect to the obtal angula momentum. These thee states, by the natue, should be completely degeneate and the total eneges should be same as each othe. Actually, the total eneges of these states whch ae also evaluated by takng the expectaton values of the Hamltonan wth espect to Equaton (8) concde wth each othe. 5. Concludng Remaks In ths pape, we pesent a scheme fo calculatng the atomc stuctues beyond the sphecal appoxmaton and nvestgate to what extent the sngle-patcle pctue of atomc systems needs to be modfed. It s confmed that the obtal polazaton can appea only by consdeng the nonsphecal effect explctly. Compaed to the conventonal atomc stuctues, we fnd that the atomc levels ae patally splt. The magntude of splttng fo p states s about 5%, whch s neve neglected because the splttng tself causes the obtal polazaton. Also, such a debacle of the conventonal atomc stuctues seems to be conceptually mpotant. Although the pesent scheme shows the necessty of modfyng the sngle-patcle pctue of atomc systems, we have to consde the followng effects that ae neglected n the pesent calculatons: ) enhancement of the expanson bass functons n Equaton (); ) teatment of the exchange enegy beyond the Hatee-Fock-late appoxmaton; ) coelaton effects. Concenng the fst effect, we hee adopt only s, s and p obtals n the expanson of the egenfunctons. Howeve we had bette take moe functons as the bass functons. Especally fo the neutal cabon atom, d obtals should be added to the expanson bass functons snce the nondagonal elements between the p and d states would not be neglgbly small n Equaton () o (5). mlaly, the second and thd effects seem to be ndspensable fo descbng the nonsphecal effect n moe detal. But anyway, we can say wthn the knowledge obtaned n ths pape that the obtal polazaton cetanly emeges by takng nto account the nonsphecal effect even f the coelaton effects ae not explctly consdeed. Futhemoe, the effect of the nonsphecal dstbuton of electons cannot be neglected not only conceptually but also quanttatvely n the study on the sngle-patcle pctue of atomc systems. 6. Acknowledgements Ths wok was patally suppoted by Gant-n-Ad fo centfc Reseach (No ) and fo centfc Reseach n Poty Aeas (No. 7646) of The Mnsty of Educaton, Cultue, pots, cence, and Technology, Japan. Copyght cres.

8 48 M. MIYAITA ET AL. 7. Refeence [] D. R. Hatee, The Wave Mechancs of an Atom wth a Noncoulomb Cental Feld. PatI: Theoy and Method. Pat II: ome Results and Dscussons, Poceedngs of Cambdge Phlosophcal ocety, Vol. 4, No., 98, -. [] V. Fock, Z. Physk, Näheungsmethode zu Lösung des quantenmechanschen Mehköpepoblems, Zetschft Fü Physk, Vol. 6, No. -, pp do:.7/bf494 [] P. Hohenbeg and W. Kohn, Inhomogeneous Electon Gas, Physcal Revew, Vol. 6, No. B, 964, pp do:./physrev.6.b864 [4] W. Kohn and L. J. ham, elf-consstent Equatons Includng Exchange and Coelaton Effects, Physcal Revew, Vol. 4, No. 4A, 965, pp. -8. do:./physrev.4.a [5] The sphecal appoxmaton means that the effectve potental n Equaton () s appoxmated nto the cental feld,.e., V V. Fo nstance, see, J. C. late, Quantum Theoy of Atomc tuctue, McGaw-Hll, NY, Vol., 96. [6] Ths means that V V. [7] A. Messah, Quantum Mechancs, Dove Publcatons, NY, 999. [8] F. Heman and. kllman, Atomc tuctue Calculatons, Pentce-Hall Inc., New Jesey, 96. [9] J. C. late, The Calculaton of Molecula Obtals, John Wley & ons, NY, 979. [] N. F. Mott, The Bass of the Electon Theoy of Metals, wth pecal Refeence to the Tanston Metals, Poceedngs of the Physcal ocety, London, ecton A, Vol. 6, No. 7, 949, p. 46. do:.88/7-98/6/7/ [] P. W. Andeson, New Appoach to the Theoy of upeexchange Inteactons, Physcal Revew, Vol. 5, No., 959, pp. -. do:./physrev.5. [] V. I. Ansmov, J. Zaanen and O. K. Andesen, Band Theoy and Mott Insulatos: Hubbad U nstead of tone, Physcal Revew B, Vol. 44, No., 99, pp do:./physrevb [] A. I. Lechtensten, V. I. Ansmov and J. Zaanen, Densty-Functonal Theoy and tong Inteatons: Obtal Odeng n Mott-Hubbad Insulatos, Physcal Revew B, Vol. 5, No. 8, 995, R5468-R547. [4]. E. Koonn, Computatonal Physcs, Addson-Wesley, NY, 986. [5] O. Eksson, B. Johansson, R. C. Albes, A. M. Bong and M... Books, Obtal magnetsm n Fe, Co, and N, Physcal Revew B, Vol. 4, 99, do:./physrevb.4.77 [6] O. Eksson, M... Books and B. Johansson, Phys. Rev. B 4, 7 (99). do:./physrevb.4.77 [7] M... Books, O. Eksson, L. even and B. Johansson, pn and Obtal Magnetzaton Denstes n Itneant Magnets Physca B, Vol. 9, No. -, 99, pp do:.6/9-456(9)96-g [8] T. hshdou, T. Oguch and T. Jo, Hatee-Fock tudy on the 5f Obtal Magnetc Moment of U, Physcal Revew B, Vol. 59, No., 999, pp do:./physrevb [9] M. R. Noman, Obtal polazaton and the nsulatng gap n the tanston-metal oxdes, Physcal Revew Lettes, Vol. 64, No., 99, pp do:./physrevlett.64.6 [] G. H. Daaldeop, P. J. Kelly and M. F. H. chuumans, Magnetocystallne Ansotopy and Obtal Moments n Tanston-Metal Compounds, Physcal Revew B, Vol. 44, No., 99, pp do:./physrevb [] A. Nata and M. Hguch, Expessons of Enegy and Potental due to Obtal Polazaton, Jounal of the Physcal ocety of Japan, Vol. 75, No., 6, pp do:.4/jpj.75.4 [] J. F. Janak and A. R. Wllams, Method fo Calculatng Wave Functons n a Nonsphecal Potental, Physcal Revew B, Vol., No., 98, pp do:./physrevb..6 [] F. W. Kutzle and G.. Pante, Eneges of Atoms wth Nonsphecal Chage Denstes Calculated wth Nonlocal Densty-Functonal Theoy, Physcal Revew Lettes, Vol. 59, No., 987, do:./physrevlett [4] A. D. Becke, Local Exchange-Coelaton Appoxmatons and Fst-Row Molecula Dssocaton Eneges, Intenatonal Jounal of Quantum Chemsty, Vol. 7, No. 5, 985, pp do:./qua [5] A. D. Becke, Cuent Densty n Exchange-Coelaton Functonals: Applcaton to Atomc tates, Jounal of Chemcal Physcs, Vol. 7, No. 5,, pp do:.6/.577 [6] E. Oestes, T. Macasso and K. Capelle, Densty-Functonal Calculaton of Ionzaton Eneges of Cuent-Cayng Atomc tates, Physcal Revew A, Vol. 68, No.,, 5. do:./physreva.68.5 [7] E. Oestes, A. B. F. da lva and K. Capelle, Enegy Loweng of Cuent-Cayng ngle-patcle tates n Open-hell atoms due to an Exchange-Coelaton Vecto Potental, Intenatonal Jounal Of Quantum Chemsty, Vol., No. 5, 5, pp do:./qua.575 [8] G. Vgnale and M. Rasolt, Densty-Functonal Theoy n tong Magnetc Felds, Physcal Revew Lettes, Vol. 59, No., 987, pp do:./physrevlett.59.6 [9] G. Vgnale and M. Rasolt, Cuent- and pn-densty- Functonal Theoy fo Inhomogeneous Electonc ystems n tong Magnetc Felds, Physcal Revew B, Vol. 7, No. 8, 988, do:./physrevb [] M. Hguch and A. Hasegawa, A Relatvstc Cuent- Copyght cres.

9 M. MIYAITA ET AL. 49 and pn-densty Functonal Theoy and a ngle-patcle Equaton, Jounal of the Physcal ocety of Japan, Vol. 66, No., 997, p. 49 (997). do:.4/jpj [] M. Hguch and A. Hasegawa, ngle-patcle Equaton of Relatvstc Cuent- and pn-densty Functonal Theoy and Its Applcaton to the Atomc tuctue of the Lanthande ees, Jounal of the Physcal ocety of Appendx Expessons fo the Potentals In ths appendx, we pesent the expessons fo the sphecal and nonsphecal pats of the effectve potental. The effectve potental conssts of thee tems,.e., the nuclea, Hatee and exchange potentals, whch ae gven by Z V d 6, (A) 4π whee the exchange potental s smplfed wth the ad of the late appoxmaton [], and whee and denote the electon densty and electon densty wth -spn, espectvely. Usng Equaton, they ae wtten as occ. occ. nl lm OCC. spn OCC. spn nl lm C p Y nl, lm nl lm nl, lm nl lm,, C p () Y,, (A) (A) Japan, Vol. 67, No. 6, 998, pp do:.4/jpj.67.7 [] J. C. late, A mplfcaton of the Hatee-Fock Method, Physcal Revew, Vol. 8, No., 95, pp do:./PhysRev.8.85 whee the sum of Equaton A s ove only the occuped states wth -spn. What we need ae ) sphecal pat of Equaton A,.e., V * 4π v v, that appeas n Equaton 8, and ) nonsphecal components of Equaton A,.e., vkq, that appea n Equaton o 4. The above ) s ndspensable fo devng the bass functons pnl and coespondng specta nl, whch ae also used n Equaton o 4. Now let us show the explct foms of ) and ) by consdeng each tem of Equaton A. As fo the fst tem, we have no poblem because t s exactly sphecal. Also, the second tem can be easly sepaated nto the sphecal and nonsphecal pats by means of the multpole expanson of the Coulomb potental. Concenng the thd tem, we have to use an appoxmaton so as to deve one thd powe of. Usng the composton elaton fo the sphecal hamoncs, s fomally sepaated nto sphecal and nonsphecal pats as follows: wth N, (A4) m OCC. * * Cnl, lm C nl, l m pnl p nl l l spn nl l m n l lm 8 l l l l CC.., m m (A5) and OCC. OCC. N * () () Cn l, l m Cn l, l m spn spn nl l m nl lm m l l k l l k * pnl () p () nl 4π kll qk l l k l l k kq m Y (, ) CC m... q (A6) Copyght cres.

10 N nce s constucted fom electons of the unflled outemost shell alone, t s qute smalle than the sphecal pat of the electon densty. That s to say, N. Usng ths fact, we can get an appoxmate fom of the one thd powe of as follows M. MIYAITA ET AL. occ. Z * * nl, lm nl, l d m nl nl V C C p p nl lm nl l m N, (A7) The fst and second tems tun to the sphecal and nonsphecal pats of the exchange potental, espectvely. Usng these elatons, the esultant foms of ) and ) ae, espectvely, gven by m l l l l l l 6, m m 4 (A8) v C C p p occ. k kq nl l m nl l m nl nl nl lm nl l m * *,, d k l l 4π l l k l l k m k m m q OCC. * * Cnl, lm C nl, m nl nl ( spn) nl lm nl lm 4π m (l ) l k l l k l l k 4π m m q p p. (A9) Hee l l l m m m s the Wgne j symbol [7], and s gven by Equaton A5. Copyght cres.