C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology. September 1996

Transcription

1 Dervaon of he Confguraon Ineracon Sngles (CIS) Mehod for Varous Sngle Deermnan References and Exensons o Include Seleced Double Subsuons (XCIS) C. Davd Sherrll School of Chemsry and Bochemsry Georga Insue of Technology Sepember Inroducon The confguraon neracon expanson of an approxmae soluon o he elecronc Schrödnger equaon s ypcally wren Ψ = c 0 Φ 0 + a c a Φ a + <j,a<b c ab j Φ ab j + <j<k,a<b<c c abc jk Φ abc jk +... (1) Φ 0 s he so-called reference, ypcally obaned from a Harree-Fock self-conssen-feld (SCF) procedure as he bes sngle Slaer deermnan (or confguraon sae funcon, CSF) whch descrbes he elecronc sae of neres. Φ a s he deermnan formed by replacng spn-orbal n Φ 0 wh spn orbal a, ec. These noes follow he convenon ha, j, k denoe orbals occuped n he reference, a, b, c denoe orbals unoccuped n he reference, and p, q, r are general ndces. The wdely-employed CI sngles and doubles (CISD) wavefuncon ncludes only hose N-elecron bass funcons whch represen sngle or double subsuons relave o he reference sae and ypcally accouns for abou 95% of he correlaon energy for small molecules near her equlbrum geomeres. Head-Gordon, Pople, and ohers have advocaed he use of confguraon neracon wh only sngle subsuons (CIS) as he sarng pon for nvesgaons of exced elecronc saes. In her 199 paper, Foresman, Head-Gordon, Pople, and Frsch [1] ls he followng desrable properes of CIS: well defned (and dfferenable), applcable o large sysems, sze-conssen, varaonal, and provdng drecly comparable (.e. orhogonal) elecronc sae soluons. They go on o presen equaons for he CIS energy and graden when he reference s a sngle deermnan obaned from an SCF procedure. These noes presen a dervaon of he CIS energy for general 1

2 and several specfc ypes of sngle-deermnan references. Laer, we dscuss some exensons whch are necessary for relable reamens of open-shell sysems. CIS Energy Equaons Snce here are only wo ypes of deermnans accordng o excaon level (.e., he reference and sngle excaons), here are only wo relevan ypes of marx elemens, Φ 0 Ĥ Φa and Φ a Ĥ Φb j. Assumng ha he deermnans are made up of a common se of orhonormal spn orbals, hese marx elemens may be evaluaed usng Slaer s rules. The frs s gven by Φ 0 Ĥ Φa = h a + k Φ 0 k ak = F a, () where he Fock marx elemen F pq s defned as F pq = h pq + k Φ 0 pk qk. (3) The oher relevan marx elemens are of he form Φ a Ĥ Φb j. The sngly exced deermnans may dffer from each oher by wo spn orbals f j and a b. If so, he deermnans are already n maxmum concdence and he marx elemen s of he form or a j Ĥ b (4) j a Ĥ b (5) The marx elemens are aj b and ja b, respecvely, and hese ansymmerzed negrals are equal o each oher (and also o ja b and aj b ). For he case = j, a b, he marx elemens are Φ a Ĥ Φb = a Ĥ b = h ab + k Φ 0,k Lkewse, for he case j, a = b, he marx elemens are Φ a Ĥ Φa j = a Ĥ a j = h j Fnally, when = j and a = b, Φ a Ĥ Φa = k {Φ 0 +a} ak bk = F ab a b. (6) k jk = F j a ja. (7) h kk + 1 kl kl h + h aa k k + ka ka a a (8) k Φ 0 k,l Φ 0 k Φ 0 k Φ 0 = E 0 F + F aa a a,

3 where E 0 = Φ 0 Ĥ Φ 0 ; f Φ 0 s obaned by an SCF procedure, hen hs s he SCF energy. Usng he permuaonal symmeres of he ansymmerzed wo-elecron negrals, he woelecron erms for he precedng four cases can be combned (rearrangng he negral a ja requres he assumpon ha he orbals are real). Ths yelds he fnal, compac resul Φ a Ĥ Φb j = E 0 δ j δ ab + F ab δ j F j δ ab + aj b. (9) Ths s equaon (11) of Maurce and Head-Gordon [], who exended he CIS mehod o he case of resrced open-shell (ROHF) and unresrced (UHF) reference deermnans. Noe ha E 0 occurs along he dagonal of he enre marx H; hs means ha we can subrac E 0 before dagonalzng and add back laer o each of he egenvalues. If all marx elemens F a = 0, as hey ofen are, hen he reference deermnan does no mx wh any of he exced deermnans and Φ 0 s already an egenfuncon of he CIS Hamlonan wh egenvalue E 0 ; furhermore, he egenvalues of he CIS Hamlonan less he E 0 dagonal erms represen excaon energes. From hs pon onward, E 0 wll be subraced from he Hamlonan. Gven he above marx elemens, remans o wre down he CIS energy expresson. Recall ha he CIS wavefuncon s expanded as Ψ = c 0 Φ 0 + a c a Φ a. (10) Assumng real CI coeffcens, he energy s gven by E CIS = E 0 + a c 0 c a F a + c a c b F ab c a c a j F j + c a aj b. (11) ab ja jab For a closed-shell SCF reference Φ 0, off-dagonal erms of he Fock marx vansh, and he expresson becomes E CIS = E SCF + (c a ) (ɛ ɛ j ) + c a aj b, (1) a jab whch maches equaon (.15) of Foresman e al. [1] once he wo-elecron negral s rearranged. Of course, hs equaon s only useful once he CI coeffcens are known. In general, he lowes several egenvecors are of neres n a CIS sudy; hese can be obaned by eravely dagonalzng he CIS Hamlonan usng Davdson s mehod [3] or he Davdson-Lu Smulaneous Expanson Mehod [4]. Ierave soluon calls for dagonalzaon of he Hamlonan n a small subspace of ral vecors, wh he se of vecors beng expanded every eraon unl convergence. Ths requres calculaon of he followng quanes, usually called he σ vecors: σ I = J H IJ c J. (13) 3

4 One σ vecor mus be compued for each c n he se of ral vecors. For CIS, σ can be wren σ 0 = Φ 0 H Φ b j (14) σ a = Φ a H Φ 0 c 0 + Φ a H Φ b j, (15) where he bar over H s a remnder ha E 0 has been subraced from he Hamlonan. These expressons can be expanded o σ 0 = F (16) σ a = c 0 F a + [F ab δ j F j δ ab + aj b ]. (17) The above equaons make clear ha σ can be compued drecly from he one- and wo-elecron negrals whou he need o explcly compue or sore he one- and wo-elecron couplng coeffcens γpq IJ and Γ IJ pqrs as separae quanes; hs makes he CIS mehod a drec CI procedure. As noed by Foresman e al. [1], he CIS eraons can acually be performed n a double-drec fashon;.e., he negrals can also be compued on-he-fly as needed. As shown by Maurce and Head-Gordon [], he conrbuon of he wo-elecron negrals o σ a can be wren as a Fock-lke marx, F a = = µν aj b (18) CµaC ν µλ νσ Cλjc b jc σb, λσ (19) where C µ are he coeffcens defnng he ransformaon from aomc orbals (AOs) o molecular orbals (MOs). Evaluaon of Fa can be carred ou as a seres of marx mulples. The pseudodensy marx, P λσ = Cλjc b jc σb, (0) can be mulpled by he wo-elecron negrals as hey are formed n he aomc orbal bass o yeld he AO Fock-lke marx, F µν = µλ νσ P λσ, (1) λσ whch s ransformed back no he MO bass by F a = µν C µa F µν C ν. () 4

5 .1 Resrced Harree-Fock References Now consder he case of a closed-shell RHF reference deermnan. In hs case, F pq = δ pq ɛ p, so ha σ 0 becomes zero. σ a smplfes o σ a = c a (ɛ a ɛ ) + aj b. (3) Conservng M s requres ha he spns of j and b are equal. Therefore, σ a = c a (ɛ a ɛ ) + aj b + aj b. (4) Afer negrang over spn, hs becomes n chemss noaon σ a = c a (ɛ a ɛ ) + [(a ) (ab j)] + (a ). (5) Tme-reversal symmery mposes ceran condons on he CI coeffcens. srng noaon [5], for M s = 0 cases, In alpha and bea c(i α, I β ) = ( 1) S c(i β, I α ). (6) An analogous equaon also holds for σ. Ths means ha for a closed shell RHF reference, c a = c a (σ a = σ a) for sngles, and ca = c a (σ a = σ a ) for rples; hus only half of he CI coeffcens mus be compued explcly. These sgn rules are also evden from he observaon ha Φ a and Φ a are no spn egenfuncons, bu ha he oal CI wavefuncon wll be a spn egenfuncon (f he requred deermnans are presen n he CI space). Usng he deermnan sgn convenon of Szabo and Oslund [6], spn egenfuncons (or CSFs) assocaed wh he above deermnans are Usng hese relaonshps beween CI coeffcens, we oban 1 Φ a = 1 ( Φ a + Φ a ) (7) 3 Φ a = 1 ( Φ a Φ a ). (8) 1 σ a (RCIS) = c a (ɛ a ɛ ) + 3 σ a (RCIS) = c a (ɛ a ɛ ) [(a ) (ab j)] (9) (ab j). (30) Furhermore, 1 σ a (RCIS) =1 σ a (RCIS) and 3 σ a (RCIS) = 3 σ a (RCIS). 5

6 Consder how equaons (0)-() change when spn s explcly accouned for. There wll be wo pseudodensy marces, P α λσ = P β λσ = C λjc σb (31) C λj C σb. (3) Due o equaon (6), P λσ α = P β λσ for sngles, and P λσ α = P β λσ for rples. Thus s necessary o α form only one of he Fock-lke marces, F µν or F µν. β The former s consruced as 1 F α µν = λσ [(µν λσ) (µσ λν)] P α λσ (33) 3 α F µν = (µσ λν) P λσ α (34) λσ Fnally F α a s consruced accordng o eq (). In erms of hese quanes, he σ vecor can be wren 1 σ a (RCIS) = 1 σ a (RCIS) = ca (ɛ a ɛ ) + 1 F α a (35) 3 σ a (RCIS) = 3 σ a (RCIS) = ca (ɛ a ɛ ) + 3 F α a, (36). Unresrced Harree-Fock References Now consder he case when Φ 0 s obaned by he UHF procedure. Once agan, he Fock marx s dagonal: F pq = δ pq ɛ p. However, s cusomary o spl he Fock marx no wo marces, one for α and one for β spn orbals (mxed erms such as F pq and F pq are zero). As for he RHF case, σ 0 = 0 and σ a can be wren σ a = c a (ɛ a ɛ ) + aj b + aj b. (37) Unforunaely, for a UHF reference equaons (6)-(8) no longer hold, so ha he above equaon can be smplfed only o σ a = c a (ɛ a ɛ ) + [(a ) (ab j)] + (a ). (38) Even hough spn has been negraed ou, he overbars mus be kep n he above equaon because he spaal pars of spn orbals and are no necessarly equal. An analogous equaon holds for σ a, σ a = c a (ɛ a ɛ ) + [(a ) (ab j)] + (a ), (39) 6

7 and here s no general relaonshp beween σ a and σ a. The pseudodensy marces P α λσ and P β λσ are defned as n eq. (31) and (3), bu hey are no longer smply relaed. Thus s necessary o compue wo Fock-lke marces, accordng o F α µν = λσ F β µν = λσ [(µν λσ) (µσ λν)] P α λσ + (µν λσ) P β λσ (40) [(µν λσ) (µσ λν)] P β λσ + (µν λσ) P α λσ. (41) In conras o RCIS, here are no longer separae sngle and rple cases, snce he UCIS egenfuncons are no CSFs. Afer ransformng o he MO bass by he expressons for σ become F α a = µν F β a = µν C µa F α µνc ν (4) C µa F β µνc ν, (43) σ a (UCIS) = c a (ɛ a ɛ ) + F α a (44) σ a (UCIS) = ca (ɛ a ɛ ) + F β a. (45).3 Resrced Open-Shell Harree-Fock References A sngle-deermnan resrced open-shell Harree-Fock (ROHF) wavefuncon descrbng a hghspn open-shell sysem wll be an egenfuncon of Ŝ (.e., a CSF). Ths s easy o verfy by drec applcaon of he Ŝ operaor, whch s where Ŝ = Ŝ Ŝ = N N ŝ() ŝ(j) j = Ŝ Ŝ+ + Ŝz + Ŝ z, (46) Ŝ z = Ŝ z = Ŝ ± = N ŝ z () (47) N ŝ z() (48) N ŝ ± () (49) 7

8 and ŝ z ()a = 1 a ŝ + ()a = 0 ŝ ()a = a ŝ z ()a = 1 a ŝ ()a = 0 ŝ +()a = a. (50) The hgh-spn ROHF wavefuncon can be wren as Φ ROHF = docc a a where and u wll denoe open-shells. Applyng Ŝ, hs becomes Ths s easy o evaluae: Ŝ Φ ROHF = [ Ŝ Ŝ + + Ŝz + Ŝ z ] docc a, (51) a a a. (5) docc Ŝ z docc Ŝz a a a a a = a = [ 1 (N α N β )] docc [ 1 (N α N β )] docc a a a a a (53) a. (54) The rasng operaor Ŝ+ yelds zero when acng on Φ ROHF : rasng operaors appled o α elecrons always yeld zero, and rasng operaors appled o he β elecrons yeld α spn orbals whch are already occuped (so he deermnan vanshes by he Paul prncple). Hence he fnal resul s [ 1 Ŝ Φ ROHF = 4 (N α N β ) + 1 (N α N β )] Φ ROHF [ ( 1 1 = N s N s + 1)] Φ ROHF, (55) where N s = N α N β. Now s worhwhle o consder how o form CSFs for he sngle excaons. Deermnans whch promoe elecrons from he sngly-occuped space o he vrual space, as well as deermnans whch promoe β elecrons n he doubly-occuped orbals o he sngly-occuped orbals, are already spn-adaped (he proof s compleely analogous o ha above for he ROHF reference deermnan). The only oher relevan sngle excaons are hose whch move an elecron from a doubly-occuped orbal o a vrual orbal. These deermnans are no spn-adaped, as we wll proceed o demonsrae. Consder he acon of Ŝ on he deermnan Φ a : ] docc [Ŝ Ŝ + + Ŝz + Ŝ z a a a 8 j a ja j a. (56)

9 The resul of Ŝz + Ŝ z s easly deermned o be [Ŝz + Ŝ z ] Φ a = [ Ns ( Ns + 1 )] Φ a. (57) Now all ha remans s he rasng and lowerng operaors. These are somewha more nvolved and requre ha aenon be pad o he sgn. docc Ŝ + Φ a = Ŝ+a aa j a ja j a. (58) By argumens smlar o hose presened above, all rasng operaors yeld zero excep for ŝ + (). Usng he ancommuaon relaons for creaon and annhlaon operaors, ŝ + ()a aa Φ 0 = ŝ + () ( a a + a a ) a a a Φ 0 = ŝ + ()a a a aa Φ 0 = a a a aa Φ 0 = a aa Φ 0. (59) The Ŝ operaor can now affec elecrons n any of he followng orbals: a,, and any of he open-shell orbals. Thus overall, [ Ŝ a a a Φ 0 ] = a aa Φ 0 + a aa Φ 0 + a aa a a Φ 0 Ŝ Φ a = [ Ns The analogous equaon for Φ a s Ŝ Φ a = [ Ns socc = Φ a Φ a + ( ) ] Ns Clearly a spn egenfuncon can be consruced as Then he operaon of Ŝ s socc Φ a. (60) socc Φ a Φ a + ( ) ] socc Ns Φ a Φa Φ a. (61) Φ a. (6) (Ns+1) Φ a = 1 ( Φ a + Φ a ). (63) Ŝ (Ns+1) Φ a = [ Ns ( Ns + 1 )] (Ns+1) Φ a. (64) 9

10 Now ha he relevan CSFs have been obaned, hey can be used o defne he ROHF convergence crera: he fnal ROHF wavefuncon wll no mx wh any of he sngly subsued CSFs. Thus Φ ROHF Ĥ Φa = 0 socc vr Φ ROHF Ĥ Φ = 0 docc socc Φ ROHF Ĥ (Ns+1) Φ a = 0 docc vr, (65) whch mples he followng condons on he Fock marx elemens F a = 0 (66) F = 0 (67) F a = F a. (68) Usng hese resuls, we can wre down expressons for he σ vecors. Snce deermnans Φ a mus ener wh he same coeffcens as Φ a, σ 0 = 0 once agan. Furhermore, snce he ROHF reference canno mx wh any oher sngly exced confguraons, he c 0 conrbuon o σ a and σ a may be safely gnored. We wll herefore consder four cases: σ a, σ a, σa, and σ, where once agan, u represen sngly occuped orbals. The equaon for σ a s readly seen o be σ a = [F ab δ j F j δ ab + aj b ] + aj b + b c b [ F δ ab + a b ] + j c j aj. (69) Separang he Fock operaor erms from he wo-elecron negrals, and negrang ou spn, hs yelds σ a = [F ab δ j F j δ ab ] b c b F δ ab + [(a ) (ab j)] + b c b [(a b) (ab )] + j c j (a j), (70) where we have used he relaon = c b. The analogous equaon for β spns s j σ a = + b [ ] Fab δ j F j δ ab + c j F aδ j + j c b (a b) + j c j [(a ) (ab j)] [(a j) (a j)], (71) 10

11 If he equaly = c b s o be mananed, we mus have j σa convenen o form hese quanes as = σ a. I s hen compuaonally σ a = σ a = 1 (σa + σ a ). (7) Thus σ a = σ a = b c b [F ab + F ab ] j c a j [ ] Fj + F j c a F + c F a + [(a ) (ab j)] + [(a ) (ab j)] + b c b [(a b) (ab )] + j c j [(a j) (a j)]. (73) Now s clear ha he wo-elecron negrals can be reaed all ogeher. To begn condensng he noaon once agan, le us defne he followng quanes: P DV λσ = P DV λσ = P SV λσ = b P DS λσ = j C λjc σb (74) C λj cb j C σb (75) C λc b C σb (76) C λj c j C σ (77) P + λσ = 1 [ P DV λσ + P DV λσ + P SV λσ + ] DS P λσ (78) where P λσ + he same as ha defned n eq. (17) of Maurce and Head-Gordon []. Then σa can be evaluaed as σ a = σ a = 1 c b [F ab + F ab ] [ ] c a j Fj + F j c a F + c F a + F a, + (79) b j where F + a = µν C µ F µν C νa (80) F + µν = λσ [(µν λσ) (µσ λν)] P + λσ (81) 11

12 Nex consder he erm σ a : σ a = [ aj b F j δ ab ] + aj b + [ Fab δ u F u δ ab + aj u c b u] + aj u c u j ub ju = [ F j δ ab + (a ) (ab j)] + (a ) + ub c b u [(a ub) (ab u) + F ab δ u F u δ ab ] + ju c u j (a ju) = F a j c a j F j + b c b F ab u c a uf u (8) I s compuaonally more effcen o evaluae σ a a he same me as σ a. The value of σ a compued n hs manner mus of course be correced for he dfference n he formulas for σ a and σ a, bu hs correcon scales as only O(N ) (see Maurce and Head-Gordon []). The wo-elecron par s hus compued by F a = F + a 1 c b u(ab u) + 1 ub ju Smlar consderaons apply o σ, whch s σ = j b + [ Fb δ j + j b ] + c b u u b + ub ju = b c b F b + u c u F u j c j F j + ( ) + c u j (a ju) (83) c u j [( ) (b j)] [ Fu δ j F j δ u + j u ] + ub c b u( ub) + ju c u j [( ju) (u j)] = b c b F b + u c u F u j c j F j + F (84) where F s acually evaluaed accordng o F = F c b u( ub) 1 ub ju c u j (u j) (85) 3 Exensons of CIS o Include Ceran Double Subsuons (XCIS) The ROCIS mehod appears o be superor o UCIS for open-shell molecules []. Neverheless, ROCIS s no as relable for open-shell cases as RCIS s for closed-shell cases. As explaned by 1

13 Maurce and Head-Gordon [7], a careful analyss of he falures of ROCIS ndcaed ha ceran double subsuons whch are negleced n UCIS and ROCIS can of crucal mporance n openshell sysems. The spn-adaped confguraons are of he form: Φ a (1) = 1 ( Φ a 6 Φa ) + Φ a. (86) 6 Alhough he hrd deermnan s a double subsuon as far as spn-orbals are concerned, s only a sngle subsuon when spaal orbal occupaons are consdered. Hence, s very reasonable o assume ha hs CSF may be of comparable mporance o he sngles ncluded n ROCIS. The exended CIS mehod (XCIS) s a spn-adaped CI mehod ncludng hese exended sngle subsuons. I s helpful o frs verfy ha eq. (86) s ndeed an egenfuncon of Ŝ. Usng prevous resuls for ROCIS, s rval o see ha Ŝ ( Φ a Φa ) = [ Ns ( ) ] Ns ( Φ + 1 a + Φa ) I remans o be seen wha s he effec of Ŝ acng on Φ a. Ŝ Φ a = Ŝ a aa docc a a a a The facor Ŝ+ acs on unpared β spns; hus, = Ŝ Ŝ+ Φ a + [ Ns u a u ( Ns + 1 )] Ŝ + a aa a a Φ 0 = ŝ + ()a aa a a Φ 0 = a a aa a Φ 0 socc Φ a. (87) Φ a. (88) = a aa Φ 0. (89) The resul of Ŝ a aa Φ 0 has already been worked ou n eq. (60). Thus overall, and s easy o see ha [ ( )] Ŝ Φ a = Ns Ns + 1 Φ a + Φa Φ a + u Φ au u, (90) Ŝ ( [ ( )] Φ a Φa + Φ a Ns Ns ( Φ ) = + 1 a Φa + Φ a ). (91) Hence Φ a (1) (eq. 86) s a CSF. 13

14 References [1] J. B. Foresman, M. Head-Gordon, J. A. Pople, and M. J. Frsch, J. Phys. Chem. 96, 135 (199). [] D. Maurce and M. Head-Gordon, In. J. Quanum Chem. Symp. 9, 361 (1995). [3] E. R. Davdson, J. Comp. Phys. 17, 87 (1975). [4] B. Lu, The smulaneous expanson mehod for he erave soluon of several of he lowes egenvalues and correspondng egenvecors of large real-symmerc marces, Techncal Repor LBL-8158, Lawrence Berkeley Laboraory, Unversy of Calforna, Berkeley, [5] N. C. Handy, Chem. Phys. Le. 74, 80 (1980). [6] A. Szabo and N. S. Oslund, Modern Quanum Chemsry: Inroducon o Advanced Elecronc Srucure Theory. McGraw-Hll, New York, [7] D. Maurce and M. Head-Gordon, J. Phys. Chem. 100, 6131 (1996). 14