Self-Study Notes on Soft self-consistent pseudopotentials (PP) in a generalized eigenvalue formalism PRB

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1 Sef-Study Notes on Soft sef-consstent pseudopotentas PP n a generazed egenvaue foras PB Soe dervaton and coents are descrbed n the footnote by Masahro Yaaoto Davd Vanderbt Departent of Physcs, Harvard Unversty, Cabrdge, Massachusetts 138 Modfed on February 4, A new approach to the constructon of frst-prncpes pseudopotentas s descrbed. The ethod aows transferabty to be proved systeatcay whe hodng the cutoff radus fxed, even for arge cutoff rad. Nove features are that the pseudopotenta tsef becoes charge-state dependent, the usua norconservaton constrant does not appy, and a generazed egenprobe s ntroduced. The potentas have a separabe for we suted for pane-wave sod-state cacuatons, and show prose for appcaton to frst-row and transton-eta systes. The deveopent of frst-prncpes norconservng pseudopotentas PP by Haann, Schüter, and Chang1 HSC and others,3 has paved the way to accurate cacuatons of sod-state propertes wthn the oca-densty approxaton usng pane-wave bass functons.4 However, the utty of ths approach to systes contanng hghy ocazed vaence orbtas e.g., for frst-row and transton-eta atos has been ted, because of the dffcuty of representng the pseudo-wave-functons n a pane-wave bass.... Here, a new approach to the constructon of frstprncpes PP s descrbed, n whch a fuy nonoca PP s generated drecty. It has the foowng desrabe propertes: It takes the for of a su of a few separabe ters. It becoes oca and vanshes outsde the core. The scatterng propertes and ther energy dervatves are, by constructon, correct at severa energes spannng the range of occuped states, and the transferabty can be systeatcay proved by ncreasng the nuber of such energes. v The norconservng constrant s reoved so that the pseudowave-functon can be constructed n such a way as to optze soothness. v The PP tsef becoes nvoved n the sef-consstent screenng process, thereby provng transferabty wth respect to changes n charge confguraton. Together, these features aow the cutoff radus to be ncreased wthout sacrfcng transferabty, even for probe cases such as p and d orbtas. The constructon of the new PPs w be descrbed n three stages. In the frst stage, I show that t s possbe to arrve at a fuy nonoca KB-type PP by workng wth the wave functon drecty, bypassng the constructon of a seoca potenta entrey. Moreover, ths can be done at an arbtrary energy ϵ, as suggested by Haann.1 1 As usua, an AE cacuaton s carred out on a free ato n soe reference confguraton, Ieadng to a screened potenta V AE r. Cutoff rad r c and rc oc are chosen for the wave functons and oca PP, respectvey, and a dagnostc radus s chosen arge enough that a pseudo- and AE quanttes agree at and beyond. Soe agorth s used to generate a sooth oca potenta V oc r whch approaches V AE r beyond rc oc. Now consder an AE wave functon ψ r of defnte anguar oentu I, whch s a souton of the Schrödnger equaton, reguar at t he orgn, at an arbtrary energy ϵ : T + V AE rψ r ϵ ψ r 1 Here s a coposte ndex, {ϵ }, T s the knetc energy operator 1 and V AE s the orgna reference screened potenta,.e., ψ s not deterned sef-consstenty. Atoc unts are used throughout. Despte the fact that ψ s, n genera, non-norazabe. I adopt a bracket notaton T + V AE ϵ ψ as a stand-n for the prevous equaton. Quanttes such as ψ ψ are -defned, but I sha ake use of the speca notaton ψ ψ to denote the ntegra of ψ rψ r nsde the sphere of radus. 1 Usuay the rada schrödnger equaton s soved usng Adas-Mouton ethod fro the orgn r and fro ψ, r, and atch the at the cassca turnng pont. In the Haann s paper ϵ s arbtrary chosen and usng the outward Adas-Mouton ethod the rada wavefuncton s soved. Thereby soetes the wavefuncton s dvergent at r. 1

2 Now a pseudo-wave-functon φ s constructed, subect to the constrants that t on soothy to ψ at r c and that t satsfy the nor-conservng property φ φ ψ ψ. Snce the wave functon χ ϵ T V oc φ 3 s oca t vanshes at and beyond V AE V oc and φ ψ, the nonoca PP operator V NL χ χ χ φ 4 s we defned. It s straghtforward to verfy that φ s an egenvector of T + V oc + V NL, and that the scatterng propertes and ther energy dervatves are correct at ϵ n the usua way see beow. The second stage of the new PP schee s arrved at by generazng the prevous constructon to the case of two or ore energes ϵ at whch the scatterng propertes w be correct, as foows. For a gven anguar oentu, soe nuber usuay between one and three of energes whch span the energy of occuped states of a target e.g., buk crysta cacuaton are chosen. Now the set of pseudo-wavefunctons φ are constructed fro the AE wave functons ψ as before, except that they shoud satsfy the generazed nor-conservng condton Q, Q ψ ψ φ φ Forng the atrx B φ χ and defnng a set of oca wave functons β B 1 χ 6 whch are dua to the φ, the nonoca PP operator can be chosen as V NL B β β 7 Then t can easy be shown that φ satsfes the secuar equaton H ϵ φ, H T + V oc + V NL. 3 I now show that the atrx B, and therefore the operator V NL, are Hertan when Q. Takng u r/r to be the rada wave functon assocated wth φ r, B 4 dru r ϵ + 1 d + 1 dr r V oc r u r 8 The expresson for B s dentca except that ϵ s repaced by ϵ, and the dervatve d /dr acts to the eft. After one ntegraton by parts on each expresson, B B ϵ ϵ φ φ + 1 u u u u 9 6 A sar expresson can be derved for the AE wave functons; subtractng ths fro the above equaton, and notng that the pseudo- and AE wave functons and ther dervatves atch at, one obtans B B ϵ ϵ Q 1 whch vanshes.e, Hertan when Q. Agan one ay verfy that d n u/dr and ts energy dervatve atch the correspondng AE quanttes at each ϵ. Thus, by ncreasng the nuber of φ k χ φ χ φ k β then φ k β δ k V NL φ k φ χ β β φ k φ χ k β φ χ k B 1 χ B k B 1 χ δ k χ χ k ϵ k T V oc φ k 4 In the centra force fed rada part of Schrödnger equaton becoes h d u + 1 dr r u + V u ϵu B φ χ χ φ V NL φ χ ϵ T V oc φ then φ s an egenvector of T + V oc + V NL. Then t s shown that the fuy nonoca KB-type PP can be obtaned drecty fro V NL χ χ / χ φ. 3 Fro the defnton of β B k β B 1 B k χ χ δ k χ χ k φ χ B β 6 B fg fg drϵ u ru r + 1 dru ru r + dru r + 1 r V oc r u r f g B ϵ φ φ + 1 u ru

3 energes ϵ at whch the constructon s done, the scatterng propertes of the PP can be ade to reproduce those of the AE potenta wth arbtrary accuracy over the energy range of nterest. 13 One coud stop here, and st have a usefu schee. However, I now show that the constrant Q s unnecessary, f one s wng to adopt a generazed egenvaue foras n whch an overap operator appears. In ths thrd stage, I defne a nonoca overap operator S 1 + Q β β 11 and redefne the nonoca potenta operator to be V NL D β β 1 D B + ϵ Q 13 and Q defned as before. Note that wth these defntons, φ S φ ψ ψ 14 7 Then ψ s easy shown to be a souton of the generazed egenvaue probe H ϵ S φ. 8 Now t foows fro Eqs., 1, and 13 that Q and D are Hertan atrces, even though B s not. Thus H and S are Hertan operators. 9 7 φ S φ φ φ + Q k φ β k β φ k δ k δ 8 V NL φ 9 φ φ + Q ψ ψ B k + ϵ Q k β k β φ k B k β k k kn B kb 1 kn χn χ + ϵ Q k β k β φ k χ + ϵ Q k β k β φ δ k ϵ T V oc φ + ϵ Q k β k β φ δ k ϵ 1 + Q k β k β T V oc φ k ϵ S T V oc φ D B + ϵ Q B ϵ ϵ Q + ϵ Q D Moreover, t foows fro the dentty that fro varatona prncpe d dϵ φ ϵ T + V oc + V NL ϵs φ ϵ 1 ϵϵ that 1 d d u dϵ dr n u ϵr φ φ + Q ψ ψ 16 See Appendx so that the atchng of the AE and pseudoogarthc dervatves foows n the usua way. The reaxaton of the constrant Q eans that each ψ can be ade nto a pseudo-wave-functon φ ndependenty, wth the ony constrant beng the atchng of φr to ψr at the cutoff radus. Thus t becoes possbe to choose the cutoff radus to be we beyond the rada wave-functon axu, as ustrated n Fg.. A consequence of ths freedo s that a generazed egenvaue probe has to be soved n the target sod-state cacuaton. However, wthn teratve approaches to the egenvector probe, the te-donant step s the utpcaton of H ϵs by a tra vector φ nk. In ths case the operaton count need hardy ncrease at a, because the dentca for of the nonoca parts of S and H aows the to be consodated nto a snge operator. Incdentay, the current PP bears a fora resebance to the orgna Phps-Kenan PP.14 The atter can be cast n the for of Eqs. 11 and 1 wth the β beng ust the core orbtas, but does not have an adustabe cutoff radus. In a sef-consstent cacuaton, the defct of vaence charge n the core regon assocated wth a pseudo-wave-functon such as that of Fg.1 w have to be restored. The soutons of the generazed egenvaue probe shoud be norazed accordng to φ nk S φ n k δ nn, 17 whch s autoatc n the usua ethods of souton. Taken together wth Eq. 14. Eq. 17 ensures that the pseudosouton has the sae aptude as the AE one at and beyond. To ake up the charge defct, the vaence charge densty s defned to be n v r nk φ nkrφ nk r + ρ Q r 18 ρ β φ nk r φ nk r β, nk 19 Q r ψ rψ r φ rφ r It foows fro Eqs. 11 and 17 that d 3 rn v r N v, exacty, N v, s the nuber of vaence eectrons n the unt ce. ***** The treatent of D n the foowng chapter was corrected n ther 1993 paper. K. Laasonen et a. PB, 47, 114, ***** 3

4 In order to ake a varatona theory, the tota energy E tot φ nk T + V on oc + β β φ nk D on + E H n v + E xc n c + n v 1 s to be nzed subect to the constrant 17. Here n c s a frozen-core densty ncuded to prove transferabty.1 Defnng V Hxc r V nv nv+nc H r + V xc r D Hxc drv Hxc rq r 3 the secuar equaton becoes T + V oc + V NL ϵ nk S φ nk 4 wth V NL and S gven by Eqs , V oc Voc on + V Hxc, and D D on + D Hxc. The Voc on and D on ust be obtaned by unscreenng the V oc and D of the generatng atoc confguraton n the usua way... The dependence of D upon n v through V Hxc pes that the PP tsef ust be updated as part of the sef-consstent screenng process.... In concuson, t s hoped that the present ethod w aow PPs to be apped to frst-row ato and transton-eta systes usng odest pane-wave cutoffs for the frst te. 1 Appendx Dervaton of ogarthc dervatveharrson, State Theory Sod 1.1 Nor-concervng pseudopotenta case ϵ h 1 r ϵ + δϵ h 1 r r + V r + h r + V r + h + 1 r + 1 r If we appy to hs of the frst equaton and to hs of the second equaton and ntegra fro to δϵ dr4πr 4π h r r Here we assue V r had no ϵ dependence. Usng parta ntegra δϵ dr4πr 4π h r dr r δϵ δϵ 4π h r r + δϵ dr4πr + δϵ 4π h dr r r + δϵr r + δϵ dr4πr + 4π h δϵ δϵ r r If we pck up the frst-order of δϵ drr h h 1 h h 1 n 1. In the case of utra-soft PP φ Y, u r H ϵ S φ φ ϵs φ φ T + V oc + V NL φ φ ϵ + δϵs φ φ T + V oc + V NL φ φ S φ φ S φ S 1 + φ VNL φ φ V NL φ D φ β β φ Q β β D φ β β φ δϵ φ S φ φ T φ φ T φ ϵ S Y h 1 r ϵs Y h 1 r φ S φ φ 1 + r Y + V φ h r Y + V oc r Y + h + 1 Y r + V NL Y φ φ + Q β β φ Q φ β β φ δϵ φ S φ π h r r Here we assue V r had no ϵ dependence. Usng parta ntegra δϵ π h φ S φ r + dr r Λ r Y 4

5 + r r + δϵ r dr r hs δϵ dr4πr + δϵ δϵ Q φ β β φ + φ /δϵ rhs π h π h δϵ r If we pck up the frst-order of δϵ + δϵr r + r δϵ hs dr4πr + Q δ δ dr4πr + Q φ φ + Q ψ ψ rhs π h π h π h π h eferences 1 n 1 1. D.. Haann, M. Schuter and C. Chang, Phys. ev. Lett. 43, G. Kerker, J. Phys. C 13, L A. Zunger and M. L. Cohen, Phys. ev. B 18, ; and,, W. E. Pckett, Coput. Phys. ep. 9, D. Vanderbt, Phys. ev. B 3, A. M. appe, K. M. abe, E. Kaxras and J. D. Joannopouos, Phys. ev. B 41, Car and M. Parrneo, Phys. ev. Lett., Nataraan and D. Vanderbt, J. Coput. Phys. 8, I. Stch,. Car, M. Parrneo and S. Baron, Phys. ev. B 39, X. Gonze, J. P. Vgneron and J.-P. Mchenaud, J. Phys. Condens. Matter 1, L. Kenan and D. M. Byander, Phys. ev. Lett. 48, D.. Haann, Phys. ev. B 4, For an aternatve approach, see E. L. Shrey, D. C. Aan,. M. Martn and J. D. Joannopouos, Phys. ev. B 4, J. C. Phps and L. Kenan, Phys. ev. 116, S. G. Loue, S. Froyen and M. L. Cohen, Phys. ev. B 6, G. B. Bacheet, D.. Haann and M. Schuter, Phys. ev. B 6,