AB INITIO DENSITY FUNCTIONAL THEORY FOR OPEN - SHELL SYSTEMS, EXCITED STATES AND RESPONSE PROPERTIES

Transcription

1 AB INITIO DENSITY UNCTIONAL THEORY OR OPEN - SHELL SYSTEMS, EXCITED STATES AND RESPONSE PROPERTIES By DENIS BOKHAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL O THE UNIVERSITY O LORIDA IN PARTIAL ULILLMENT O THE REQUIREMENTS OR THE DEGREE O DOCTOR O PHILOSOPHY UNIVERSITY O LORIDA 007

2 007 Dens Bohn

3 ACKNOWLEDGMENTS I would le to thn professors Hen Monhorst nd So Hrt for helpfull dsussons. I wnt to thn lso Dr. Igor Shwegert nd Dr. Norert loe for the helpng me to wrte OEP ode. My spel thns to Ttyn nd Thoms Alert for help wth the preprton of the dssertton text. 3

4 TABLE O CONTENTS pge ACKNOWLEDGMENTS LIST O TABLES LIST O IGURES ABSTRACT CHAPTER INTRODUCTION A Into Wvefunton-Bsed Methods Hrtree-o Method Eletron-Correlton Methods Kohn-Shm Densty untonl Theory Tme-Dependent Densty untonl Theory Tme-Dependent Densty untonl Theory Lner Response Theory 3..3 Prolems wth Conventonl untonls Ortl-Dependent untonls A Into Densty untonl Theory INTERCONNECTION BETWEEN UNCTIONAL DERIVATIVE AND E- ECTIVE OPERATOR APPROACHES TO AB INITIO DENSITY UNC- TIONAL THEORY Equtons for the Exhnge-Correlton Potentl n the untonl Dervtve Approh Equtons for the Exhnge-Correlton Potentl n n Effetve Opertor Approh Interonneton n Artrry Order Dgrmmt untonl Dervtves Dgrmmt untonl Dervtves n Seond-Order Mny-Body Perturton Theory Interonneton n Hgher Orders Interonneton n Infnte Order AB INITIO TIME-DEPENDENT DENSITY UNCTIONAL THEORY EM- PLOYING SECOND-ORDER MANY-BODY PERTURBATION OPTIMIZED EECTIVE POTENTIAL Dgrmmt Construton of the Exhnge-Correlton Kernels ormlsm An Exmple: Dgrmmt Dervton of Exhnge-Only Kernel.. 5 4

5 3. Kernel for the Seond-Order Optmzed Effetve Potentl Mny-Body Perturton Theory Correlton Potentl Propertes of the Correlton Kernel Numerl Testng Conlusons AB INITIO DENSITY UNCTIONAL THEORY OR SPIN-POLARIZED SYSTEMS Theory Results nd Dsusson Totl Energes Ionzton Potentls Dssoton Energes Snglet-Trplet Seprton n Methylene Conlusons EXACT-EXCHANGE TIME-DEPENDENT DENSITY UNCTIONAL THE- ORY OR OPEN-SHELL SYSTEMS Ext-Exhnge Densty untonl Theory Tme-Dependent Optmzed Effetve Potentl Theory nd Implementton Numerl Results Chrge-Trnsfer Exted Sttes Conlusons EXACT EXCHANGE TIME-DEPENDENT DENSITY UNCTIONAL THE- ORY OR HYPERPOLARIZABILITIES Theory Tme-Dependent Densty untonl Theory Response Propertes Dgrmmt Dervton of the Seond Ext-Exhnge Kernel Propertes of the Seond Ext-Exhnge Kernel Numerl Results Conlusons APPENDIX A B INTERPRETATION O DIAGRAMS O THE SECOND-ORDER MANY- BODY PERTURBATION THEORY OPTIMIZED EECTIVE POTENTIAL CORRELATION KERNEL INTERPRETATION O DIAGRAMS O EXACT-EXCHANGE SECOND KERNEL REERENCES

6 BIOGRAPHICAL SKETCH

7 Tle LIST O TABLES pge 3- Ortl energes nd zero-order pproxmtons to extton energes Extton energes of Ne tom usng OEP-MBPT( Kohn-Shm ortl energes Extton energes of Ne tom usng exhnge-only ortl energes Extton energes of Ne tom usng ortl energes nd ortls from OEP(s. All equtons for TDDT re the sme Totl energes Ionzton potentls (n e. v Dssoton energes (n J/mol Snglet nd trplet energes of methylene Totl (n. u. nd ortl (n e. v. energes of Ne tom Totl nd ortl energes of He tom Extton energes (V - vlene stte, R - Ryderg Ortl energes (n e. v. of Ne tom Ionzton energes (n e. v Stt polrzltes (n. u Isotrop C 6 oeffents (n. u Hyperpolrzltes of severl moleules (n. u

8 gure LIST O IGURES pge 4- Exhnge nd orrelton potentls of L tom (rdl prt. A Exhnge potentl. B Correlton potentl Exhnge nd orrelton potentls of O moleule ross the moleulr xs. A Exhnge potentl. B Correlton potentl LH potentl energy urve OH potentl energy urve H potentl energy urve Exhnge potentls of Ne tom, otned n dfferend ss sets A Π hrge-trnsfered exted stte of He... Be LUMO-HOMO ortl energy dfferene

9 Astrt of Dssertton Presented to the Grdute Shool of the Unversty of lord n Prtl ulfllment of the Requrements for the Degree of Dotor of Phlosophy AB INITIO DENSITY UNCTIONAL THEORY OR OPEN - SHELL SYSTEMS, EXCITED STATES AND RESPONSE PROPERTIES Chr: Rodney J. Brtlett Mor: Chemstry By Dens Bohn August 007 A nto densty funtonl theory (DT sed on the optmzed effetve potentl (OEP method s new pproh to study the eletron struture of tom, moleulr nd sold stte systems. It ontns elements of oth wve funton nd densty funtonl theores nd s free from lmttons of onventonl DT euse of usng ortldependent funtonls derved from systemt pproxmtons of the wve funton theory. A nto DT methods wth exhnge-orrelton funtonls sed on mny - ody perturton theory (MBPT hve een derved nd mplemented reently. The exhnge-orrelton potentls derved from MBPT hve omplted struture nd ther dervton n hgher-order of MBPT y the use of the hn-rule for funtonl dfferentton requres sgnfnt effort. To fltte suh dervtons, I developed spel dgrmmt formlsm for tng funtonl dervtves. An lterntve wy to onstrut OEP MBPT exhnge-orrelton potentls s to use the densty ondton. It mes t possle to otn potentls for dfferent prttonngs of the full moleulr Hmltonn wth MBPT. Usng the dgrmmt formlsm developed for tng funtonl dervtves, we show n order-y-order equvlene etween the funtonl dervtve nd the densty ondton pprohes to OEP MBPT for the se of the Kohn- Shm prttonng of the moleulr Hmltonn. or ny other prttonngs, dfferent 9

10 exhnge-orrelton potentls re produed y the funtonl dervtve nd densty ondton pprohes. The tme-dependent extenson of OEP n the dt pproxmton wth exhnge-only potentls nd ernels ws reently mplemented nd ppled to some moleulr systems. The orrespondng extton energes nd polrzltes re n good greement wth tme-dependent Hrtree-o results. However, suh results nnot e used for omprson wth expermentl vlues due to l of desrpton of eletron orrelton effets. In order to ount for orrelton effets, orrelton ernel orrespondng to the MBPT( potentls hs een derved. Its struture nd propertes re desred n detl. The frst numerl results for extton energes wth pure nto potentls nd ernels re presented. The reently mplemented OEP MBPT( method hs een extended to the se of spn-polrzed open-shell systems. The totl energes otned for severl open-shell systems re very lose to the orrespondng vlues otned wth the hghly-orrelted oupled luster sngles nd doules wth perturtve trples (CCSD(T method. Comprson wth results otned wth the OEP MBPT( exhnge-orrelton potentls nd some densty funtonls hs shown qulttvely norret shpe for some wdely used exhnge-orrelton potentls. Hgher-order response propertes, suh s hyperpolrzltes, re desred very poorly wth onventonl funtonls. The typl reltve errors desred n the lterture re out 00 perent. The reson for the poor desrpton s n norret long-rnge symptot ehvor nd n nomplete nellton of the oulom selfnterton. The OEP method s free from those drws nd lulted vlues of stt hyperpolrzltes wth the exhnge-only potentl re lose to those derved from Hrtree-o theory. The seond ernel requred for lultons of hyperpolrzltes wthn the DT frmewor hs een derved for OEP potentls y usng the developed 0

11 dgrmmt tehnque. The struture nd propertes of the seond OEP ernel re dsussed.

12 CHAPTER INTRODUCTION The defnton of heml reton s trnsformton of one sustne to nother t the moleulr level mens regroupng of nule nd eletrons. In the dt pproxmton, suh nuler regroupngs n e presented s moton n the feld of some potentl, lso nown s potentl energy surfe. Potentl energy surfes n e otned y solvng the egenvlue prolem wth the so lled eletron Hmltonn ˆ H el = ele ele = = Nul A= ele Z A r R A + < r r ( Whole nuler oordntes re treted s set of prmeters. If there re more thn two prtles n the system, the egenvlue prolem ˆ H el (R...R M Ψ(r...r N, R...R M = E(R...R M Ψ(r...r N, R...R M ( n not e solved nlytlly. or the lulton of eletron energy levels, some pproxmtons hve to e used. Modern quntum hemstry nludes three lsses of methods for the soluton of the eletron prolem. The frst one, lled wve funton theory, uses dfferent pproxmtons for the lulton of the wve funton Ψ(r...r N, R...R M nd the orrespondng energy E(R...R M. The seond lss, nown s densty funtonl theory uses the densty s prmry oet. Wthn the DT pproh, the energy s wrtten s funtonl of densty, nd the onstruton of the wve funtons s not neessry. The thrd lss of methods uses whole densty mtrx. The densty mtrx renolmlzton group (DMRG nd the densty mtrx funtnl theory (DMT re typl methods of the thrd lss. A nto DT ontns elements of the frst two lsses. It uses lol multpltve potentl, typl for the DT pproh. However, the orrespondng exhnge-orrelton

13 potentls re derved from ortl-dependent energy funtonls, ten from wve funton theory. The mn dvntge of wve-funton sed methods s the posslty to otn systemtlly mprovle results. Ths mens tht wth the extenson of the ss set, t wll e possle to otn more urte energes nd wve funtons, nd n the omplete set t s possle to get the ext soluton of the Shrödnger equton. On the other hnd, rgorous nto methods usng wve funton methods re usully very ostly for omputtons on lrge moleules. A nto wve funton methods n usully e ppled only for systems wth 0-30 toms n resonle ss sets. The mn DT dvntge s smll omputtonl ost; t n hndle systems wth severl hundreds of toms. However, most of onventonl densty funtonls do not hve the pty to produe systemtlly mprovle results. A nto DT s ple of produng systemtlly mprovle results, ut t s omputtonlly more ostly thn DT wth onventonl funtonls. Despte ts omputtonl ost, nto DT n e used s method for the lrton of denstydependent funtonls. An lterntve wy of provdng suh nformton s Quntum Monte-Crlo (QMC method, however, QMC results re not vlle for moleules nd open-shell systems.. A Into Wvefunton-Bsed Methods.. Hrtree-o Method In the Hrtree-o method, wve funton s onsdered s Slter determnnt[] ϕ (r ϕ (r N Ψ H =.. ( 3 ϕ N (r ϕ N (r N The sngle-eletron wve funtons ϕ (r (or ortls re determned y the ondton tht the orrespondng determnnt mnmzes the expetton vlue of the eletron 3

14 Hmltonn E H = mn < Ψ H Ĥ Ψ H > ( 4 suet to the ondton of orthonormlty on the ortls < ϕ ϕ >=. Susttutng the explt form of the wve funton from Equton ( 3 nto n expetton vlue, t s possle to wrte the Hrtree-o energy n terms of ortls E H = funtonl ele < ϕ Nul A Z A r R A ϕ > + ele (< ϕ ϕ ϕ ϕ > < ϕ ϕ ϕ ϕ > To derve the Hrtree-o equtons t s onvenent to mnmze the followng, ( 5 I = E H, ε < ϕ ϕ > ( 6 where ε - re Lgrnge multplers. Mnmzton of the funtonl from Equton ( 6 requres tht the funtonl dervtves of I wth respet to the ortls vnsh I ϕ = 0, I ϕ = 0 ( 7 Tng nto ount Equton ( 3 ondton ( 7 n e presented s fϕ = ε ϕ ( 8 The opertor f hs the struture f = + ˆv ext + ˆv H + ˆv nlx ( 9 4

15 where ˆv H (r = ele ˆv ext (r = ϕ (r ϕ (r dr = r r Nul A Z A r R A ρ(r r r dr ele ρ(r = ϕ (rϕ (r ( 0 The Hrtree-o exhnge opertor ˆv nlx s non-lol,. e. t nnot e presented s n nlytl funton of sptl vrles. However, t s possle to wrte ts ton on some ortl ϕ ˆvnlxϕ(r = ϕ (r ϕ (r ϕ (rdr ( r r Sne the o opertor ˆf s hermtn nd v H (r nd ˆv nlx ϕ (r re nvrnt wth respet to untry trnsformtons of ortls, Equton ( 8 n e rewrtten n the nonl form fϕ = ε ϕ ( Ths form of the Hrtree-o method s most ommon for prtl mplementtons. The Hrtree-o method s system of ntegro-dfferentl equtons, whh nnot e solved nlytlly. Itertve methods n e used for the pproxmte soluton of the Hrtree-o equtons wth gven ury. As frst step some guess of ortls should e ssumed nd susttuted nto the equtons. Durng the seond step the system of equtons s solved nd new guess for the ortls s otned. Then those new ortls should e susttuted nto the equtons gn, untl the totl energy nd densty t suessve tertons dffers less thn requred ury. Ths s nown s the self-onsstent feld (SC method. or prtl purposes moleulr ortls re usully presented s eng deomposed nto Gussn funtons, entered on the nule. In ths se the numer of solutons s muh lrger thn the numer of eletrons. To dede whh ortls should e nluded 5

16 nto the Slter determnnt, the Aufu prnple should e used. Consder the nresng ortl energes n the order ε < ε < ε 3... < ε N... < ε M ( 3 where N - numer of eletrons nd M - numer of solutons. Aordng to the Aufu prnple, the orrespondng frst N spn-ortls should onsttute the Slter determnnt. Suh ortls re lled ouped ortls nd the rest of the ortls lled vrtul ortls... Eletron-Correlton Methods The Hrtree-o method n reover up to 99% of the totl eletron energy. Yet, even the remnng error of % s too g from the pont of vew of hemstry nd n led to qulttvely wrong predtons of eletron struture. The dfferene etween the Hrtree-o nd the ext soluton s due to eletron - orrelton effets. The prmry onern of eletron-orrelton methods s to develop mny-ody tehnques for gong eyond the Hrtree-o pproxmton nd tng nto ount the smultneous eletron-eletron ntertons. These methods, ontrry to the reltvely smple Hrtree-o pproxmton, n e qute hllengng oneptully nd ostly, omputtonlly. The orrelton lmt n e otned from the ull Confgurton Interton (CI method. In ths pproh the totl eletron wve funton s expressed n the followng wy o vrt o vrt Ψ CI = Φ H + C Φ + C Φ +... ( 4 where Φ,Φ, et re Slter determnnts, formed y susttuton of ouped ortls,... y vrtul ortls,... wth the orrespondng reorderng of rows. The expnson oeffents re found from the vrtonl ondton on the expetton vlue of the Hmltonn E CI = < Ψ mn CI Ĥ Ψ CI > C,C,... < Ψ CI Ψ CI > ( 5 6

17 However, the numer of possle exted determnnts grows rpdly wth the numer of eletrons nd ss funtons n the system. Therefore, the full CI method s omputtonlly ntrtle for ny ut very smll systems. Among the pproxmte eletron-orrelton pprohes, the most ommon re the Coupled Cluster method[] nd Mny - Body Perturton Theory[3]. Any trunted verson of the CI method hs qulttvely wrong ehvor of the energes nd wve funtons whle nresng the numer of prtles n the system. Therefore, the CI methods wth lmted level of exttons nnot e used for hghly-orrelted systems. The Coupled Cluster method nd Mny - Body Perturton theores re free from ths l of extensvty flure nd re very ommon for the moleulr omputtons. In some ses perturton theory n provde n urte desrpton of eletron-orrelton effets t muh lower ost thn neessry for the Coupled Cluster method. The seond-order Rylegh-Shrödnger perturton theory s the smplest nd lest expensve nto method for tng nto ount eletron orrelton effets. In ths perturton theory the soluton of Shrödnger equton ĤΨ = EΨ ( 6 n e found usng the Slter determnnt s referene. Generlly, suh determnnt my e onstruted from the ortls, generted y some one-eletron opertor ĥϕ p = ( + ûϕ p = ε p ϕ p ( 7 The frst step of ny perturton theory s the prttonng of the Hmltonn nto zero-order H 0 nd perturton Ĥ = Ĥ0 + ˆV ( 8 where Ĥ 0 Φ = E 0 Φ = ele (h Φ = ( ε Φ ( 9 7

18 The perturton opertor s usully defned s the dfferene etween the full nd the zero-order Hmltonns ˆV = Ĥ ele ele Ĥ0 = ˆv ext (r û(r + ele r r ( 0 Introdung λ s the smll perturton prmeter, the Hmltonn nd the wve funton n e wrtten s Ĥ = Ĥ0 + λ ˆV ( Ψ = Φ + λψ ( + λ Ψ ( +... ( E = E 0 + λe ( + λ E ( +... ( 3 These order-y-order orretons n e found y susttutng expnsons (, ( nd ( 3 nto ( 6 nd olletng terms wth the orrespondng order n λ (E 0 Ĥ0 Ψ ( >= ( ˆV E ( Φ > ( 4 (E 0 Ĥ0 Ψ ( >= ( ˆV E ( Ψ ( > E ( Φ > ( 5 Choosng the perturtve orretons to e orthogonl to the referene determnnt < Ψ (n Φ >=0 (so lled ntermedte normlzton t s esy to get the expressons for perturtve orretons t ny order. Proetng the equtons ( 4 nd ( 5 onto the referene determnnt expressons for energy orretons re otned E ( =< Φ V Φ > ( 6 E ( =< Φ V Ψ ( > ( 7 Expressons for the order-y-order expnson of wve funtons n e wrtten usng the resolvent opertor[4] Ψ ( >= ˆR 0 V Φ > ( 8 Ψ ( >= ˆR 0 ( ˆV E ( Ψ ( >= ˆR 0 ( ˆV E ( ˆR 0 ˆV Φ > ( 9 8

19 where ˆR 0 = ˆQ E 0 Ĥ0 ( 30 nd ˆQ = - Φ >< Φ s the proetor onto the omplementry spe of Φ >. Sne Ĥ0 s dgonl n the ss of Slter determnnts, t s possle to wrte ˆR 0 = n 0 or the spel se of the Hrtree-o referene determnnt Φ n >< Φ n E 0 E n ( 3 E 0 + E ( =< Φ Ĥ0 Φ > + < Φ ˆV Φ >= E H ( 3 the seond-order orreton to the energy s E ( = o uno,, < > (< > < > ε + ε ε ε ( 33 or the se of more generl referene determnnt, onstruted from ortls of Equton ( 7, the seond-order orreton to the energy hs the followng struture o E ( = uno < ĥ ˆf > ε ε + o, uno, < > (< > < > ε + ε ε ε ( 34. Kohn-Shm Densty untonl Theory Densty untonl Theory s n lterntve pproh to the desrpton of the eletron struture of moleulr nd sold-stte systems. Ths method uses eletron densty nsted of the wve funton s the s oet of theory. The mthemtl ss of DT s provded y two theorems, ntrodued y Kohn, Hohenerg nd Shm. The frst one, nown s Hohenerg-Kohn theorem[5], estlshes one-to-one mppng etween the ground-stte eletron densty nd the externl potentl. The externl potentl defnes prtulr oet (tom, moleule, et nd, euse of the one-to-one mppng, the densty ontns ll the nformton out the system. In prtulr, the ground-stte energy n e wrtten s funtonl of the densty. To get the ground stte energy n 9

20 terms of the densty, the Kohn-Shm theorem[6] n e used. Ths theorem sttes, tht the ground-stte energy s funtonl of densty hs mnmum, f the densty s ext. Therefore, gven the energy funtonl, one n otn the ground-stte densty nd energy y vrtonl mnmzton of the funtonl. The forml defnton of DT does not tell how to onstrut suh funtonl. Severl pproxmte forms hve een suggested, however, whose ury vres gretly for dfferend propertes. The net energy of eletron moton s prtulrly dffult to pproxmte s densty funtonl. The s de of Kohn nd Shm ws to trnsform the vrtonl serh over the densty nto serh over the ortls tht ntegrte to gven trl densty ( + v s (rϕ p (r = ε p ϕ p (r ( 35 Suh trnsformton does not restrt the vrtonl spe, provded tht every physlly menngful densty orresponds to unque set of ortls (the v-representlty ondton. The use of the Kohn-Shm SC model ensures not only tht the vrtonl densty s fermon densty, ut t lso provdes good pproxmton for the net energy. If ortls ntegrte to the true densty t s nturly to expet tht T s = ele ounts for lrge prt of rel net energy. < ϕ ϕ > ( 36 The rest of the unnown terms re grouped nto the exhnge-orrelton funtonl E x [ρ] = E[ρ] T s E ext E H ( 37 where E H s Hrtree energy nd n e esly e lulted from gven set of Kohn- Shm ortls. The non-ntertng net energy T s s supposed to reprodue lrge prt of the ext net energy T. Therefore, E x s eser to pproxmte s densty funtonl thn the totl energy E. 0

21 The defnton of the exhnge prt of E x n te nto ount ts defnton n wve-funton theory, E x =< Φ s V Φ s > E H ( 38 Kohn-Shm ortls re defned y n effetve lol potentl v s. Trnsformng the vrtonl ondton on the energy funtonl nto ondton for the onstrned serh over the ortls, t s possle to wrte v s (r = (E[ρ] T s ρ = (E ext + E H + E x ρ = v ext + v H + v x ( 39 where the exhnge-orrelton potentl s defned s the funtonl dervtve of the exhnge-orrelton energy v x (r = E x ρ(r ( 40 One E x s pproxmted nd the Kohn-Shm equtons re solved, the totl energy n e found from the followng expresson E = = ε dr dr ρ(r ρ(r r r dr ρ r v x (r + E x ( 4 The soluton of the Kohn-Shm equtons s ompletely nlogous to the soluton for the Hrtree-o se, nd t s usully done y n SC proedure. After self-onssteny s rehed, the Kohn-Shm ortls re gurnteed to reprodue the true densty of the mny-eletron system. Vrtully ll modern mplementtons of DT use the Kohn-Shm sheme. However, the theory stll hs open questons s to how to onstrut the exhnge-orrelton funtonl. Thus, the mn hllenge for the theoretl development of DT remns the onstruton of urte exhnge-orrelton funtonls... Tme-Dependent Densty untonl Theory The Kohn-Shme sheme provdes the posslty to desre ground stte energes nd denstes. or the desrpton of extton energes tme-dependent generlzton of onventonl DT n e used. Ordnry tme-dependent DT s sed on the exstene

22 of n ext mppng etween denstes nd externl potentls. In the ground stte formlsm, the exstene proof reles on the Rylegh-Rtz mnmum prnple for the energy. A strghtforwrd extenson to the tme-dependent domn s not possle sne mnmum prnple s not vlle n ths se. The exstene proof for the one-to-one mppng etween tme-dependent potentls nd tme-dependent denstes, ws frst gven y Runge nd Gross[7]. We n strt from the tme-dependent Shrödnger equton Ψ(t t = Ĥ(tΨ(t ( 4 evolvng from fxed ntl mny-prtle stte Ψ(t 0 = Ψ 0 ( 43 under the nfluene of dfferent externl potentls v(r,t. or eh fxed ntl stte Ψ 0, the forml soluton of the Shrödnger equton ( 4 defnes mp A : v(r, t Ψ(t ( 44 etween the externl potentl nd the orrespondng tme-dependent mny-prtle wve funton nd seond mp B : Ψ(t ρ(r, t =< Ψ(t ˆρ(r, t Ψ(t > ( 45 Thus, the followng mppng n e estlshed: G : v(r, t ρ(r, t ( 46 The Runge-Gross theorem estlshes tht the G mppng s nvertle up to some ddtve, tme-dependent onstnt. In other words, two denstes ρ(r, t nd ρ (r, t evolvng from the ommon ground stte Ψ 0 under the nfluene of potentls v(r,t nd

23 v (r,t re lwys dfferent, f ntrodued v(r, t v (r, t + C(t ( 47 After nvertlty for the G mppng s estlshed, the ton funtonl n e A[ρ] = t nd the vrtonl ondton n e ppled, dt < Ψ[ρ](t Ĥ Ψ[ρ](t > ( 48 t 0 t A[ρ] ρ(r, t = 0 ( 49 Usng the sme mnpultons s for the ground stte set of one-prtle equtons n e derved where t ϕ (r, t = ( + v s [ρ(r, t]ϕ (r, t ( 50 v s [ρ(r, t] = v ext (r, t + dr ρ(r, t r r + v x[ρ](r, t ( 5 The gret dvntge of the tme-dependent Kohn-Shm sheme les n ts omputtonl smplty ompred to other eletron-orrelton methods. The tme-dependent Kohn-Shm sheme wth explt tme-dependene of the densty nd the potentl n e ppled for ny type of externl, tme-dependent potentls. However, when the tmedependent externl potentl s smll t n e treted wth tme-dependent perturton theory. If the ppled perturton s perod eletromgnet feld, t s more onvenent to use response theory... Tme-Dependent Densty untonl Theory Lner Response Theory Consder the N-eletron system eng ntlly,. e. t t < t 0 n ts ground stte. In ths se the ntl densty ρ 0 n e lulted from the ordnry ground-stte Kohn- Shm equton ( + v 0 (r + dr ρ 0 (r r r + v x[ρ 0 ](rϕ (0 (r = ε (0 ϕ (0 (r ( 5 3

24 At t = t 0 the perturton s ppled so tht the totl potentl s gven y v(r, t = v 0 (r + v (r, t ( 53 where v (r, t=0 for t t 0. The oetve s to lulte the lner densty response ρ (r, t to the perturton v (r, t. Conventonlly, ρ (r, t s omputed from the full lner response funton χ s ρ (r, t = dr dt χ(r, t, r, t v (r, t ( 54 t 0 Sne the tme-dependent Kohn-Shm equtons ( 50 provde formlly ext wy of lultng the tme-dependent densty, t s possle to ompute ext densty response ρ (r, t s the response of the non-ntertng system ρ (r, t = dr dt χ KS (r, t, r, t v s ( (r, t ( 55 t 0 where v s ( (r, t s the effetve tme-dependent potentl evluted to frst order n the perturng potentl,.e., v s ( (r, t = v (r, t + dr ρ (r, t r r + dr dt f x (r, t, r, t ρ (r, t ( 56 The exhnge-orrelton ernel f x s gven y the funtonl dervtve of v x f x (r, t, r, t = v x[ρ](r, t ρ(r, t ( 57 t ρ = ρ 0. Whle the full response funton χ s very hrd to lulte, the non-ntertng χ KS n e omputed frly esly. In terms of the stt Kohn-Shm ortls the ourer trnsform of χ KS (r, t, r, t wth respet to (t t n e expressed s χ KS (r, r, ω = lm 0+ o uno ϕ (rϕ (rϕ (r ϕ (r ω (ε ε ( 58 4

25 Equtons ( 55 nd ( 56 re the ss of lner response theory. Sne equton ( 55 s not lner wth respet to ρ the soluton should e otned y some tertve proedure. However, for ll prtl purposes dret use of equton ( 55 s not onvenent. To get more onvenent form, the followng mtrx element should e onsdered < ϕ ρ (ω ϕ >=< ϕ v (ω ϕ > + drϕ (r( dr ρ (r, ω r r ϕ (r + + drϕ (r( dr f x (r, r, ωρ (r, ωϕ (r ( 59 On the other hnd, < ϕ ρ (r, ω ϕ > n e expressed n terms of response funton o ρ (r, ω = uno P (ωϕ ϕ + P (ωϕ ϕ ( 60 where P (ω = < ϕ v (ω ϕ >, P (ω = < ϕ v (ω ϕ > ω + (ε ε (ε ε ω Susttutng equton ( 6 nto equton ( 59 we hve ( 6 o uno P (ω(ω + ε ε =< ϕ v (ω ϕ > + ([< ϕ ϕ ϕ ϕ > + < ϕ ϕ f x (ω ϕ ϕ >]P + [< ϕ ϕ ϕ ϕ > + < ϕ ϕ f x (ω ϕ ϕ >]P o uno P (ω(ε ε ω =< ϕ v (ω ϕ > + ([< ϕ ϕ ϕ ϕ > + < ϕ ϕ f x (ω ϕ ϕ >]P + [< ϕ ϕ ϕ ϕ > + < ϕ ϕ f x (ω ϕ ϕ >]P ( 6 Introdung the nottons A, = (ε ε + < ϕ ϕ ϕ ϕ > + < ϕ ϕ f x (ω ϕ ϕ > B, =< ϕ ϕ ϕ ϕ > + < ϕ ϕ f x (ω ϕ ϕ > ( 63 5

26 equton ( 6 n e rewrtten n the followng mtrx form A B B + ω 0 P A 0 P + = v (ω v + (ω ( 64 Ths equton provdes posslty to lulte the lner response of the densty, used y the externl perturton v (r, ω. If the frequeny of the ppled eletromgnet feld s equl to some resonne frequeny (.e. frequeny, whh orrespond to trnston to some exted stte even n nfntesml perturton uses fnte response of the densty. In ths se equton ( 64 n e rewrtten s [8 ] A B B P = ω P A P + P + ( 65 Equton ( 65 s ommon for the lulton of extton energes wth tmedependent densty funtonl theory (TDDT. Most modern mplementtons of TDDT use the frequeny-ndependent exhnge-orrelton ernels, defned s smple funtonl dervtve f x (r, r = V x(r ρ(r Suh n pproh s nown s the dt pproxmton. If the exted stte orresponds to the promoton of sngle eletron, the dt pproxmton s pplle nd lulted trnston frequenes re qute urte. However, f there s promoton of two eletrons tng ple, the dt pproxmton wors very poorly euse the ( 66 underlyng equtons hve no solutons, tht orrespond to the promoton of two prtles...3 Prolems wth Conventonl untonls The onventonl pproh s to pproxmte the energy funtonl y some nlytl expresson of the densty nd ts grdents. The effetve potentl n then e wrtten n nlytl form, nd the Kohn-Shm equtons n e solved redly. There re severl levels of pproxmton for exhnge-orrelton funtonls. The strtng pont for the 6

27 onstruton of pproxmte funtonls s the Lol Densty Approxmton (LDA, where the energy s gven through n ntegrl of lol funtonl of the densty. The seond level s Generlzed Grdent Approxmton (GGA funtonls, whh mprove upon LDA y the ddton of the dependene on the grdent of the densty. The thrd level, nown s met-gga, ontns lso dependene upon the grdents of the Kohn-Shm ortls. or the GGA nd the met-gga there s the posslty to hoose form of funtonl. A numer of dfferent forms hve een suggested. A stndrd wy of hoosng the form of GGA nd met-gga funtonls s to stsfy set of theorems, whh re nown to e stsfed y the ext funtonl. Some funtonls ontn set of prmeters to reprodue expermentl dt. Suh funtonls re lled sememprl. There re lso funtonls whh do not ontn prmeters ut stsfy n extended set of theorems (non-emprl funtonls. Kohn-Shm DT wth onventonl funtonls n produe very urte results for some propertes. Vlues of the totl energy, otned wth GGA re usully omprle wth some nto orrelted results, le MBPT( or CCSD. Ths s gret dvntge of onventonl DT. However, the ury of GGA funtonls s restrted y ts nlytl form. Generlzed grdent pproxmton exhnge funtonls nnot provde the ext elmnton of oulom self-nterton of the Hrtree energy. Sne the exhnge energy s muh lrger thn the orrelton energy, n nomplete elmnton of self-nterton n sgnfntly redue the ury of the totl energy. Prtulrly, the mposslty to desre wely-ntertng systems, ounded mostly y dsperson fores, rses from n nomplete elmnton of self-nterton nd the wrong ehvor of the exhnge-orrelton funtonls wth the seprton of the system onto severl frgments. Although the exhnge-orrelton energes re qute urte wth GGA funtonls, the orrespondng potentls re not nerly urte, espelly n the nter-shell nd symptot regons. As onsequene, the ury of the densty s muh lower thn the ury for the energy. urthermore, the qulttvely norret potentls redue the 7

28 usefulness of the Kohn-Shm ortls nd ortl energes for the lulton of groundstte propertes. Extton energes nd response propertes re lso very senstve to the form of the potentl nd ther desrpton wth GGA re sometmes very nurte. Some of the prolems n e fxed wthout hngng the funtonl form. Severl post-sc orretons hve een suggested to prtlly remove the self-nterton error. or exmple, fter the Kohn-Shn equtons hve een solved, dsperson orreton to the energy s ntrodued to fx the long-rnge symptots of the exhnge potentl. However, suh orretons re very spef for the prtulr funtonl nd the lss of the systems, nd they re nomptle wth eh other. The extenson of GGA to met- GGA does not fx ll these prolems. Ther systemt nd onsstent resoluton s needed to go eyond GGA...4 Ortl-Dependent untonls An lterntve pproh to DT s to ompletely dsmss the onventonl wy of expressng the exhnge-orrelton energy s n explt funtonl of densty. Insted, the exhnge-orrelton energy s expressed n terms of ortls nd ortl energes, yet, the potentls remn mplt funtonls of the densty. In Kohn-Shm DT there s one-to-one mppng etween the densty nd potentl. Sne ortls re generted y n effetve potentl through Kohn-Shm equtons, there s one-to-one mppng etween the set of ortls nd the densty. Therefore, ortls nd ortl energes n e onsdered s funtonls of densty. In the se of ortl-dependent energy funtonls orrespondng exhngeorrelton potentls n not e lulted dretly s funtonl dervtves wth respet to densty. Insted, the hn rule should e used for the dervton of orrespondng potentls. Suh potentls re equvlent to the orrespondng potentls from optmzed effetve potentl (OEP method. Therefore, the OEP-sed methods onsttute the ss of DT wth ortl-dependent funtonls. 8

29 The Hrtree-o exhnge funtonl provdes n exhnge prt of ortldependent exhnge-orrelton potentls. Kohn-Shm DT wth OEP exhnge potentls provdes superor results to ll nown densty-dependent exhnge funtonls. To me OEP potentls useful for prtl lultons t s neessry to hve the orrelton potentl, whh n e omned wth t. Conventonl GGA orrelton funtonls re developed together wth the orrespondng exhnge potentls, whh re very nurte. Beuse of ths orrelton potentl of GGA ompenstes errors n the exhnge prt. In most ses GGA orrelton potentls hve n opposte sgn to nerly ext QMC potentls. The orretng terms norported nto orrelton potentls, so t s relly dffult to extrt the rel orrelton prt. So fr, t s not surprsng tht omnng GGA orrelton potentls wth OEP exhnge potentls the ssoted results re nferor to exhnge-only ses. Sne onventonl orrelton funtonls re not omptle wth OEP, the development of ortl-dependent orrelton funtonls s neessry. A nto DT uses ortl-dependent orrelton energy funtonls from the rgorous wve-funton methods. Sne orrespondng orrelton potentls re derved from nto methods, they re omptle wth OEP exhnge potentls. Unle onventonl ones, nto funtonls re systemtlly mprovle, sne one n lwys use hgher-level pproxmton to otn more urte funtonl. They lso hve well-defned lmt represented y the CI method...5 A Into Densty untonl Theory The mn dvntge of ortl-dependent energy funtonls s the ft, tht suh funtonls re nown n nlytl form. Prtulrly, the exhnge funtonl n e wrtten n followng wy E x =< Φ V ee Φ > E H = < ϕ ϕ ϕ ϕ > ( 67, 9

30 Here ortls onsdered s mplt funtonls of the densty nd the orrespondng funtonl dervtve wll produe ext-exhnge (EXX potentl. The most mportnt property of EXX funtonl s tht, unle onventonl densty funtonls, t ompletely nels the self-nterton omponent of Hrtree energy, so the orrespondng EXX potentl wll ompletely elmnte the self-nterton terms n Kohn- Shm equtons. Thus, usng the EXX funtonl nd potentl wll vod mny prolems, used y the self-nterton error n onventonl, pproxmted DT for oth energy nd densty. However, ortl dependene of EXX funtonl uses one omplton. Sne there s no explt dependene upon densty, potentls n not e wrtten dretly. Insted, one must rely on the hn rule, whh tes nto ount the mplt dependene of ortls on the densty, expressng the dervtve of nterest through the produt of nown dervtves. A nto ortl-dependent orrelton energy funtonls suh s MBPT( n e used for the onstruton of orrelton potentls. Suh potentls exht orret C/r 4 symptot ehvor. It ws lso shown tht MBPT( potentls re ext n the hghdensty lmt nd lso very lose to nerly ext QMC potentls. Alterntvely, orrelton potentls n e onstruted from the so lled densty ondton. 30

31 CHAPTER INTERCONNECTION BETWEEN UNCTIONAL DERIVATIVE AND EECTIVE OPERATOR APPROACHES TO AB INITIO DENSITY UNCTIONAL THEORY The use of the Kohn-Shm ortls nd ortl energes to onstrut n mplt densty-dependent energy funtonl usng,e.g perturton theory[, 3] s strghtforwrd pproh to the nto DT. The orrespondng Kohn-Shm potentl n then e otned y tng the funtonl dervtve of the fnte-order energy v the hn rule, tht trnsforms the dervtve wth respet to the densty nto the dervtve wth respet to ortls nd ortl energes[4].ths leds to the Optmzed Effetve Potentl[3, 5] equtons whh n e omplted lredy n seond order[3] Alterntvely, one n determne the frst nd seond order Kohn-Shm potentl y requrng tht the orrespondng frst nd seond order perturtve orretons to the referene densty vnsh.unle the funtonl dervtve pproh suh ondton on the densty n e desred wth stndrd mny-ody tehnques.the reent wor[7] uses dgrms to derve the seond-order OEP equton n systemt nd ompt fshon,whle seond pper [6] does so lgerlly. Both the funtonl dervtve nd densty(effetve opertor pprohes led to extly the sme equton n the frst order[7].however,the funtonl dervtve of the seond energy nvolves ertn type of denomntors tht re not present n the densty ondton(effetve opertor pproh.stll the terms nvolvng suh denomntors n e trnsformed to mth the densty-sed equton extly. Ths vet rses the queston of whether the two pprohes re equvlent n hgher orders,or for dfferent prttonngs of the hmltonn.. Equtons for the Exhnge-Correlton Potentl n the untonl Dervtve Approh To derve the equtons n oth pprohes we frst splt the full eletron hmltonn nto perturton nd zero-order hmltonn: H = H 0 + V ( 3

32 Let us frst onsder the hoe of H 0 s sum of Kohn-Shm ortl energes: H 0 = K ɛ p { + p p } ( p= where K - s the numer of ortls.the rets mens tht the produt of seond quntzton opertors re wrtten n norml order. In the funtonl dervtve method we defne the exhnge-orrelton potentl of the n-th order s: V x (n (r = E(n ρ(r ( 3 where E (n - s the energy of the n-th order of mny-ody perturton theory. In snglereferene mny-ody perturton theory we hve the energy of the nth order s n explt funtonl of the ortls.in the Görlng-Levy pproh ths funtonl s onsdered to e funtonl for the ortl energes lso [4] : E (n = E (n (ϕ...ϕ n, ɛ...ɛ n Another possle onsderton[3],when ortl energes re onsdered to e funtonls over orrespondng ortls E (n = E (n (ϕ...ϕ n, ɛ [ϕ ]...ɛ n [ϕ n ] ( 4 s equvlent to the Görlng-Levy[4] pproh. To onstrut the hn rule we should te nto ount the ft tht the Kohn-Shm ortls re funtonls of V s : E (n = E (n (ϕ (V s...ϕ n (V s ( 5 Then the Kohn-Shm potentl n e onsdered to e funtonl of ρ(r: E (n = E (n [ϕ (V s (ρ(r...ϕ n (V s (ρ(r] ( 6 3

33 Vrton of the energy n e presented n followng form: E (n = p E (n dr ϕ p (r ϕ p(r ( 7 We n wrte down vrtons of the ortls nd the Kohn-Shm potentl too: ϕ p (r ϕ p (r = dr V s (r V s(r ( 8 V s (r = dr V s(r ρ(r ( 9 ρ(r Usng equtons ( 8 nd ( 9 we n rewrte the vrton of the energy n the followng form: E (n = p E (n dr ϕ p (r ϕ p (r dr V s (r dr V s(r ρ(r ( 0 ρ(r Now we n wrte down the hn-rule[3, 4] for the lulton of V (n x : E (n ρ(r = p E (n dr ϕ p (r ϕ p (r V s (r 3 dr 3 V s (r 3 ρ(r = E (n dr 3 V s (r 3 V s (r 3 ρ(r ( Tng nto ount the ft tht[4]: ϕ p (r V s (r = ϕ p (r ϕ q (r ϕ q (r ( ɛ p ɛ q p,q p ρ(r V s (r = X(r, r =, t s possle to wrte n explt expresson for V (n x (r [3] V x (n (r = p,q p ε p V s (r = ϕ p(rϕ p (r ( 3 E (n dr ϕ p (r ϕ q(r ϕ (r ϕ (r ϕ (r ϕ (r ε ε ( 4 dr 3 ϕ p (r 3 ϕ q (r 3 ε p ε q X (r, r 3 ( 5 Usng the ft tht dr X(r, r X (r, r 3 = (r r 3 ( 6 33

34 we n rewrte equton ( 5 s:, ϕ (r ϕ (r < ϕ V n x ϕ > ε ε = p,q p < E(n ϕ p ϕ q > ϕ p(r 3 ϕ q (r 3 ε p ε q ( 7 Usng the ft tht ortls p nd q n e ouped or unouped,we n seprte the rght-hnd ste of equton ( 7 nto four prts:, ϕ (r ϕ (r < ϕ V n x ϕ > ε ε =, < E(n ϕ ϕ > ϕ (r 3 ϕ (r 3 ε ε + +, < E(n ϕ > ϕ (r 3 ϕ (r 3 + < E(n ϕ > ϕ (r 3 ϕ (r 3 + ϕ ε ε ϕ ε ε, +, < E(n ϕ ϕ > ϕ (r 3 ϕ (r 3 ε ε +. ( 8 Ths form of equton ( 8 wll e the ss for the formulton of the dgrmmt rules for tng funtonl dervtves nd for mng the onneton wth n effetve opertor pproh to the OEP-MBPT exhnge - orrelton potentl.. Equtons for the Exhnge-Correlton Potentl n n Effetve Opertor Approh Let us onsder one-prtle densty opertor n seond-quntzed form: ρ(r = p,q < ϕ p (r r ϕ q > + p q ( 9 Usng W theorem,we rewrte ths opertor n norml form wth the Kohn-Shm determnnt s the erm-vuum: ρ(r = p,q < ϕ p (r r ϕ q > { + p q }+ < Φ KS (r r Φ KS >= = p,q N o < ϕ p (r r ϕ q > { + p q } + ϕ (r ( 0 = The frst memer of the prevous expresson wll e lled the densty orreton: ρ(r = p,q < ϕ p (r r ϕ q > { + p q } ( 34

35 Sne the onverged Kohn-Shm sheme gves n ext densty, ll orretons to ths densty must e equl to zero[7].ths mens, tht f we onstrut n effetve opertor of the densty usng MBPT, the orreton to the densty must vnsh n ny order.ths s the mn de of the effetve opertor pproh. or the exhnge-orrelton potentl of frst order we wll hve: < Φ KS Ω (+ ρ(rω ( Φ KS >= (Ω (+ ρ(r S + ( ρ(rω ( S = 0 ( Ths ondton n e presented y dgrm ( 3. = 0 ( ( 3 Usng the ft tht f pq = ε p pq + < ϕ p V H x V x (ρ V ϕ q >, we n extrt our desrle exhnge potentl.the orrelton prt s exluded to mntn frst order, s the orrelton potentl ontns expressons of seond nd hgher orders. Then the equton for V x n e dgrmmtlly represented y dgrms nd 3. - = X 3 or the seond-order effetve opertor for the densty orreton we wll use the expresson ( 4 < Φ KS (Ω (+ + Ω (+ ρ(r(ω ( + Ω ( Φ KS >= = (Ω (+ ρ(r S + ( ρ(rω ( S + +(Ω (+ ρ(r S + ( ρ(rω ( S + (Ω (+ ρ(rω ( S = 0 ( 4 35

36 Hene,the dgrmmt representton for the seond-order equton s presented y the dgrms (3-(8 of equton( 5. C 3 = ( 5 Dgrmmt representton of potentl, derved from the densty ondton wll e used for the estlshng nteronneton wth the funtonl dervtve pproh. 36

37 .3 Interonneton n Artrry Order.3. Dgrmmt untonl Dervtves or formulton of the rules we wll use equton ( 8. The - funton needs to e ntrodued ordng to equton ( 6: = p,q p ϕ p (rϕ q (r ε p ε q = ( 6 Brets on the lst two dgrms denote denomntors. Expressons for the energy n MBPT onsst of lner omntons of terms, whh hve produts of moleulr ntegrls n the numertors nd produt of dfferenes of one-prtle energes n the denomntors, so t s esy to me dgrmmt representton for suh expressons.to te the funtonl dervtves from numertors, ordng to equton ( 8, ll lnes onneted to some vertex must e dsonneted from the orrespondng ple of lnng, nd when ths s done, new lne (orrespondng to ouped or unouped ortls must e nserted. In the fnl step ontrton wth the orrespondng - funton must e provded. All lnes orrespondng to denomntors re stll unhnged. Ths proedure must e provded for ll vertes,euse when we te funtonl dervtves from produts of funtons, we hve sum of produts, ordng to rules for tng dervtves. If new dgrm hnges ts sgn, mnus sgn must e ssgned to ths dgrm. When funtonl dervtves from denomntors re ten, t s more sutle to use equton ( 3. The most generl form for the denomntor n e represented y the followng formul: Den = α Kα ( = ε α o Kα = ε α uno ( 7 37

38 Usng equton ( 3 t s possle to wrte down the funtonl dervtves from some denomntor n the generl form: Den K = ( ϕ V o s = Kβ... + ( = K = ϕ uno ϕ β o Kβ = α,α Kα ( = ɛ α o ϕ β uno α,α β Kα = Kα ( = ɛ α uno ɛ α o Kα = ɛ α uno ( 8 Now t s possle to wrte down term whh nludes the funtonl dervtve from the denomntor: = β Nom Den = (Den V s Nom( Kβ = ϕβ o Kβ = ϕβ uno ( Kβ = ɛβ o Kβ = ɛβ uno α,α β ( Kα = ɛα o Kα = ɛα uno ( 9 We now hve the posslty to formulte how to te funtonl dervtves from denomntors. To te funtonl dervtves from denomntors, ll dgrms, where one of the lnes s douled nd etween these lnes whh rse from doulng, the orrespondng dgonl -funton s nserted. Ths proedure must e provded for ll horzontl lnes on the dgrm nd for ll the ontours the lnes ross..3. Dgrmmt untonl Dervtves n Seond-Order Mny-Body Perturton Theory Dret nteronneton n frst order s ler from dgrm s nd 3.The left-hnd sde of equton ( 7 s equl to dgrm, the rght-hnd ste s equl to dgrm 3 when the funtonl dervtve hs een ten from the expresson for the exhnge energy. The dgrmmt expresson for the seond-order energy n the generl se hs the form[]: E ( = + + ( 30 38

39 Consder the seond nd thrd dgrm n the energy expresson. or the se of smplty only non-equvlent dgrms wll e gven. Tng funtonl dervtves wth respet to ouped ortls we wll hve equton ( 3 + ( 3 Insertng the lnes of unouped ortls nd mng ontrtons wth the - funton, dgrms 4 nd 8 of equton ( 5 wll e otned ( 3: ( 3 When funtonl dervtves re ten wth respet to unouped ortls nd ouped ortls re nserted, dgrms 5 nd 9 of equton ( 5 wll e otned.when ths proedure s done for the lower vertex, omplex onugte dgrms wll e otned nd we wll hve the sme numer of dgrms of ths sort s n the effetve opertor pproh. When funtonl dervtves re ten wth respet to ouped ortls for the upper nd lower vertexes nd n ouped lne s nserted, we hve the dgrms, gven y equton ( 33 ( 33 Bret-type denomntor mens the dfferene ε ε. Summton of the fst two dgrms ordng to the rntz-mlls theorem[8] nd the sme proedure for the seond 39

40 two dgrms gves the dgrms of equton ( 34 + ( 34 On the two dgrms we mpose the restrton. To me dret orrespondene wth dgrms 7 nd we need to dd dgrms tht rse from tng funtonl dervtves from denomntors ( 35 ( 35 After these dgrms re dded to the prevous two, we wll hve dret orrespondene wth dgrms 7 nd. When funtonl dervtves re ten wth respet to the unouped ortls nd n unouped lne nserted, the sme proedure wll gve dret orrespondene wth dgrms 6 nd 0 of equton ( 5. Now onsder the frst dgrm for the seond-order energy expresson. The proedure for tng funtonl dervtves from the rght sde of the vertex, whh do not ontn losed -rng re solutely the sme s n the se of other seond nd thrd dgrms n the energy expresson. When suh funtonl dervtves re ten, we hve four dgrms orrespondng to dgrms -5. To te funtonl dervtves from the -rngs, the frst dgrm n the energy expresson must e represented wth more detl, tng nto ount tht f = h + J - K,n followng the form ( 36 X X = X X + + ( 36 40

41 When rules for tng funtonl dervtves dgrmmtlly re ppled to ths set of dgrms, we wll hve the next set of dgrms ( ( 37 The frst two dgrms re equl to dgrm 6; the thrd nd fourth equl to dgrms 7 nd 6 respetvely. Sne the left-hnd ste of eq. n e represented y dgrm 3, we hve n ext equvlene etween the funtonl dervtve nd the effetve opertor pprohes n the frst nd seond order for the Kohn-Shm prttonng of the hmltonn..3.3 Interonneton n Hgher Orders Equtons for the effetve-opertor pproh n the n-th order of MBPT hve the form ( 38 < Φ KS (Ω ( Ω (n+ ρ(r(ω ( +...+Ω (n Φ KS >= n (Ω (+ Ω (n S = 0 ( 38 =0 Ths equton tells s tht we need to ollet nd equte to zero the sum of ll possle omntons of wve opertors nd densty orreton opertors, nd these omntons must e of n-th order. The sme result should orrespond to tng funtonl dervtves from ll energy dgrms of n-th order. Sttement : When funtonl dervtves from some vertex of dgrm re ten wth respet to n ouped (unouped ortl, nd lne, whh orresponds to n unouped (ouped ortl nserted, we lwys wll hve dgrmmt expressons whh orrespond to prt of n =0 (Ω(+ Ω (n S. Proof: When funtonl dervtves re ten wth respet to n ouped (unouped ortl, nd lne whh orresponds to n unouped (ouped ortl s nserted, we lwys hve ontrton of the delt funton wth some fully onneted 4

42 dgrmmt expresson. All lnes whh orrespond to denomntors re stll unhnged, exept there s one extr denomntor from the delt-funton. If the funtonl dervtve hd een ten wth respet to the hghest or lowest vertex n the energy dgrm, the resultng dgrm orresponds to the produt Ω (n, n the nth-order of perturton theory. After pplyng the sme proedure to some ntermedte vertex, we hve n expresson whh ontns n- denomntors nd one extr denomntor from the delt-funton. To trnsform the resultng dgrm nto the produt of delt-funton nd two wve opertors, we need one extr lne, whh ross ll lnes t ts level. Suh lne n e otned, usng the rntz-mlls[8] ftorzton theorem. Sne ll ntl horzontl lnes ross ll lnes, whh go up from lower lyng vertexes, t s possle to represent the dgrms, whh pper fter pplyng the rntz-mlls[8] theorem n the form ( 39 Ω (+ Ω (n ( 39 Eh orresponds to some tme verson. When funtonl dervtves wll e ten from ll vertexes n ll dgrms, we wll hve ll possle ontrtons of the densty orreton opertor, whh hve lnes gong up or down, wth produts of wve opertors. So we hve prt of n =0 (Ω(+ Ω (n S. or exmple, n thrd order we wll hve: d d d = + ( 40 The dgrm on the left-hnd sde s the result of tng funtonl dervtves from typl energy dgrms of thrd order. Applyng the ftorzton theorem[8] to the rght-hnd sde we wll hve the sme result s on the left-hnd sde of ths relton. The 4

43 frst memer of the rght-hnd sde orresponds to Ω (3, whle the seond orresponds to Ω (+ Ω (. To fnlly estlsh the nteronneton etween the two pprohes, we need to show tht t s possle to trnsform dgrms whh ontns ret-type denomntors to stndrd dgrms. In prevous suseton ths proedure ws desred for the se where one horzontl lne rosses the ontour n the dgrm, whh orresponds to the energy n seond order MBPT. We need to prove, tht energy dgrms, whh ontn ontours rossed y m lnes, fter tng the funtonl dervtve whh respet to unouped (ouped ortls nd nsertng lnes of unouped (ouped ortls; n e trnsformed to lner omnton of m dgrms whh do not ontn ret-type denomntors. Consder the followng dgrm ( 4:. m ( 4 Numers on lnes show the ondtonl numer of lnes whh re entered for smplty of further mnpultons. The desrle sum of dgrms hs the form ( 4. l m +.. l m l m ( 4 Together wth dgrms whh orrespond to funtonl dervtves from denomntors, ths set of dgrms forms produts le Ω (+ Ω (l, where +l equls the order of MBPT. 43

44 Anlytl expressons whh orrespond to ths set of dgrms hve the form ( 43 Nom (ɛ + A (ɛ l + A...(ɛ + A m + Nom (ɛ l + A (ɛ + A (ɛ l + A...(ɛ + A m Nom (ɛ l + A...(ɛ l + A m (ɛ + A m ( 43 In these expressons Nom mens numertor,a mens the rest of the terms, whh re present n denomntors. After multplton nd dvson y (ɛ ɛ l we hve ( 44 Nom(ɛ ɛ l + A A (ɛ ɛ l (ɛ l + A (ɛ + A...(ɛ + A m + Nom(ɛ ɛ l + A A (ɛ ɛ l (ɛ l + A (ɛ l + A (ɛ + A...(ɛ + A m Nom(ɛ ɛ l + A m A m (ɛ ɛ l (ɛ l + A...(ɛ l + A m (ɛ + A m = = Nom (ɛ ɛ l (ɛ l + A (ɛ + A...(ɛ + A m Nom (ɛ ɛ l (ɛ + A (ɛ + A...(ɛ + A m + Nom (ɛ ɛ l (ɛ l + A (ɛ l + A (ɛ + A 3...(ɛ + A m Nom (ɛ ɛ l (ɛ l + A (ɛ + A...(ɛ + A m + + Nom (ɛ ɛ l (ɛ l + A (ɛ l + A (ɛ l + A 3 (ɛ + A 4...(ɛ + A m Nom (ɛ ɛ l (ɛ l + A (ɛ l + A (ɛ + A 3...(ɛ + A m Nom (ɛ ɛ l (ɛ l + A...(ɛ l + A m Nom (ɛ ɛ l (ɛ l + A...(ɛ + A m ( 44 After nellton of ll equvlent terms, we hve the expressons ( 45 Nom (ɛ ɛ l (ɛ + A...(ɛ + A m Nom (ɛ ɛ l (ɛ + A...(ɛ + A m ( 45 44

45 where Nom s the numertor, wth nterhnged ndexes nd l.these expressons n e represented y the followng two dgrms ( 46. l m +.. l m ( 46 Nmely these two dgrms wll pper fter tng funtonl dervtves usng dgrmmt rules for tng funtonl dervtves. Proof for the se where funtonl dervtves re ten wth respet to unouped ortls s the omplete nlog of ths one. Hene, we n formulte seond sttement. Sttement : When funtonl dervtves from some vertex of dgrm re ten wth respet to ouped (unouped ortls, nd lne whh orresponds to ouped (unouped ortl nserted, together wth dgrms whh rse from tng funtonl dervtves from denomntors; we lwys hve dgrmmt expressons, whh orresponds to prt of n =0 (Ω(+ Ω (n S. Ths sttement s dret orollry of the ove proved sttement out rettype denomntors nd sttement. Sne we hve orrespondene n the seond nd n-th order,usng the method of mthemtl nduton, t s possle to prove, tht the orrespondene ts ple n ll orders (fnte or nfnte..3.4 Interonneton n Infnte Order It should e noted tht summton of perturton orretons to the energy nd the wve funtons up to nfnte order provde orrespondene wth the full CI: n=0 E (n ( CI = E Ψ (n = Ψ ( CI ( 47 n=0 45

46 Usng the effetve opertor pproh t s possle to defne V x (n n ll possle orders. Ths proedure for n nfnte sum of orders gves V x. Insertng ths potentl nto the Kohn-Shm equtons, we n fnd ρ (,whh orresponds to the one-prtle densty n the full CI method.the sme densty ould e used n method le Zho-Morrson-Prr (ZMP[9] to extrt the orrespondng exhnge-orrelton potentl. Sne the full CI energy does not depend upon the hoe of ortl ss set, the OEP proedure[3, 5] nnot e used dretly for ths se. The nfnte sum of ll energy orretons does not depend upon hoe of ortl ss set, ut eh term of ths sum does depend upon the hoe of ortls. Ths ft enles us to onsder ll term of the nfnte sum of energy orretons s ortl-dependent funtonls. After the onstruton of the set V x (...V x ( nd summton up to nfnty, we wll hve the exhnge-orrelton potentl whh orresponds to the full CI. Usng the equvlene of the funtonl dervtve nd the effetve opertor pproh, t s possle to onlude tht fter summton of ths set of potentls, we gn wll hve the sme result s n the effetve opertor pproh. Redefned n suh wy, the OEP proedure[3, 5] for the full CI energy produes the sme densty s the ZMP method[9] would from full CI. 46