Seniority-based coupled cluster theory

Transcription

1 Senorty-bsed coupled cluster theory Thoms M. Henderson, Ireneusz W. Bulk, Tmr Sten, nd Gustvo E. Scuser Ctton: The Journl of Chemcl Physcs 141, (014); do: / Vew onlne: Vew Tble of Contents: Publshed by the AIP Publshng Artcles you my be nterested n Communcton: Convergence of nhrmonc nfrred ntenstes of hydrogen fluorde n trdtonl nd explctly correlted coupled cluster clcultons J. Chem. Phys. 135, (011); / A stte-specfc prtlly nternlly contrcted multreference coupled cluster pproch J. Chem. Phys. 134, (011); / Accurte clculton nd modelng of the dbtc connecton n densty functonl theory J. Chem. Phys. 13, (010); / A nturl lner sclng coupled-cluster method J. Chem. Phys. 11, (004); / Extended benchmrk studes of coupled cluster theory through trple excttons J. Chem. Phys. 115, 3484 (001); /

2 THE JOURNAL OF CHEMICAL PHYSICS 141, (014) Senorty-bsed coupled cluster theory Thoms M. Henderson, 1, Ireneusz W. Bulk, 1 Tmr Sten, 1 nd Gustvo E. Scuser 1, 1 Deprtment of Chemstry, Rce Unversty, Houston, Texs , USA Deprtment of Physcs nd Astronomy, Rce Unversty, Houston, Texs , USA (Receved 3 October 014; ccepted 4 December 014; publshed onlne December 014) Doubly occuped confgurton ntercton (DOCI) wth optmzed orbtls often ccurtely descrbes strong correltons whle workng n Hlbert spce much smller thn tht needed for full confgurton ntercton. However, the sclng of such clcultons remns combntorl wth system sze. Pr coupled cluster doubles (pccd) s very successful n reproducng DOCI energetclly, but cn do so wth low polynoml sclng (N 3, dsregrdng the two-electron ntegrl trnsformton from tomc to moleculr orbtls). We show here severl exmples llustrtng the success of pccd n reproducng both the DOCI energy nd wve functon nd show how ths success frequently comes bout. Wht DOCI nd pccd lck re n effectve tretment of dynmc correltons, whch we here dd by ncludng hgher-senorty cluster mpltudes whch re excluded from pccd. Ths frozen pr coupled cluster pproch s comprble n cost to trdtonl closed-shell coupled cluster methods wth results tht re compettve for wekly correlted systems nd often superor for the descrpton of strongly correlted systems. C 014 AIP Publshng LLC. [ I. INTRODUCTION The coupled cluster (CC) fmly of methods 1 3 offers powerful wve functon pproch to the descrpton of wekly correlted systems, to the pont tht the ccurte tretment of such systems s essentlly routne: provded tht the system s not too lrge, one cn smply pply coupled cluster wth sngle nd double excttons 4 (CCSD) or CCSD plus perturbtve trple excttons, 5 whch we refer to s CCSD(T). The sme, unfortuntely, cnnot be sd for the coupled cluster tretment of strongly correlted systems, for whch trdtonl snglereference methods such s CCSD or CCSD(T) my fl bdly. Much progress hs been mde n mult-reference coupled cluster theory, to be sure, but the technques re by no mens blck box or computtonlly nexpensve. Contnued developments of coupled cluster technques for strongly correlted systems re essentl. In 013, Ayers nd coworkers mde surprsng dscovery long these lnes: method whch they refer to s the ntsymmetrc product of 1-reference orbtl gemnls 6 9 (AP1roG) nd whch we wll refer to s pr coupled cluster doubles 10 (pccd) provdes remrkbly resonble descrpton of the strong correltons for wde vrety of systems. Wht mkes ths so surprsng s tht pccd looks lke coupled cluster doubles (CCD) restrcted to nclude only those excttons whch preserve electron prs, but pccd, unlke CCD, seems to be ble to descrbe strong correltons. Why should smplfcton of fundmentlly sngle-reference method be ble to descrbe mult-reference problems? In ths mnuscrpt, we seek to do three thngs. Frst, we wnt to provde self-contned descrpton of pccd, wth ll the equtons one needs to mplement the pproch. Second, we wsh to offer some perspectve on the method s successes. Thrd, we wsh to go beyond pccd nd nclude some of the dynmc correltons whch pccd does not provde. To ccomplsh ths, however, we frst must dscuss doubly occuped confgurton ntercton nd orbtl senorty. II. SENIORITY AND DOUBLY OCCUPIED CONFIGURATION INTERACTION Pr coupled cluster theory s bsed on the concept of the senorty of determnnt. The senorty s the number of unpred electrons. The de s smple: every spnorbtl φ p s pred wth one nd only one other spnorbtl, φ p, nd the senorty of determnnt s the number of spnorbtl prs whch between them contn only one electron. Loosely spekng, senorty s relted to the number of broken electron prs. In ths work, s n our prevous work on the subject, 10 we restrct ourselves to snglet prng, n whch the orbtls tht re pred re the two spnorbtls correspondng to the sme sptl orbtl. In tht cse, the senorty opertor s just Ω = N D, (1) where N s the number opertor ( ) ( ) N = c p c p +c p c p = np + n p p nd D s double-occupncy opertor D = c p c p c p c p = n p n p. (3) p Throughout ths work, we wll use ndces, j, k, l for occuped sptl orbtls,, b, c, d for vrtul sptl orbtls, nd p, q, r, s for generl sptl orbtls. It s mportnt to notce tht senorty depends on whch orbtls we use to defne the double-occupncy opertor D, p p () /014/141(4)/44104/10/$ , AIP Publshng LLC

3 Henderson et l. J. Chem. Phys. 141, (014) becuse untry trnsformton whch mxes the orbtls leves N nvrnt but chnges the form of D. If we defne senorty wth respect to the moleculr orbtls of the restrcted Hrtree-Fock (RHF) determnnt RHF, then we see tht the RHF determnnt s senorty egenfuncton nd hs senorty zero. If we defne senorty wth respect to dfferent bss, ths need not be true. It s lso mportnt to note tht senorty s not symmetry of the moleculr Hmltonn [H, Ω] 0 whch mens tht the exct wve functon s not n egenfuncton of Ω. The utlty of the senorty concept comes from usng t s n lterntve to orgnze Hlbert spce. 11 Conventonlly, we descrbe determnnts n terms of ther exctton level, whch we cn extrct from the prtcle-hole number opertor ( ) ( ) N ph = n + n + n n. (4) As wth senorty, the exctton level s nether orbtlly nvrnt (becuse defnng prtcles nd holes wth respect to dfferent Ferm vcuum chnges the exctton level) nor symmetry of the Hmltonn, but t nevertheless provdes vluble frmework wthn whch we cn orgnze Hlbert spce nd solve the Schrödnger equton n subspce. The exct wve functon s generlly lner combnton of determnnts of ll possble exctton levels, nd smlrly, t s generlly lner combnton of determnnts of ll possble senortes. The success of sngle-reference coupled cluster theory for wekly correlted systems s grounded on the fct tht the coupled cluster expnson n terms of prtclehole excttons out of the Hrtree-Fock determnnt converges rpdly towrd full confgurton ntercton (FCI). The ground stte of wekly correlted systems, then, s chrcterzed by hvng low number of prtcle-holes. We post tht the ground stte of strongly correlted systems s chrcterzed by hvng low senorty number n sutble one-electron bss. One cn test ths by defnng confgurton ntercton (CI) restrcted to the zero senorty sector of Hlbert spce, whch we refer to s doubly occuped confgurton ntercton (DOCI) Becuse DOCI s not nvrnt to the orbtls wth respect to whch senorty s defned, we optmze ths choce energetclly. Ths s nlogous to optmzng the dentty of the reference determnnt n n exctton-truncted CI clculton, or to optmzng the orbtls n CAS-SCF, though DOCI s generlly sze consstent. As we nd others hve shown, DOCI wth orbtl optmzton provdes vluble tool for the descrpton of strong correltons. Ths cn be shown n Fg. 1, whch shows tht DOCI gves the correct lmt n the dssocton of the eqully spced H 8 chn nd gves most of the strong correlton n N s well. Note tht these plots re generted usng mnml ctve spce to remove, to the degree possble, dynmc correlton t dssocton. The chef drwbck of DOCI s tht of computtonl cost: the number of determnnts wth Ω = 0 s just the squre root of the number of ll determnnts wth gven prtcle number, so the cost of DOCI s the squre root of the cost of full CI. Worse yet, t s more dffcult to use symmetry to elmnte determnnts from DOCI thn t s to elmnte determnnts from FCI. For exmple, every DOCI determnnt s spn FIG. 1. Top pnel: Dssocton of the eqully spced H 8 chn. Bottom pnel: Dssocton of N. Both clcultons re done n the cc-pvdz bss set nd restrct the CI problem to mnml ctve spce. We emphsze tht curves re obtned wth n RHF wve functon. Results tken from Ref. 11. snglet wth our snglet prng scheme, so we cnnot use spn symmetry to reduce the number of determnnts to be ncluded. In prctce, DOCI clcultons on systems wth more thn few dozen electrons re prohbtvely expensve. Ths s where pccd enters the pcture: pccd generlly provdes results whch for the moleculr Hmltonn re nerly ndstngushble from those of DOCI, but wheres the computtonl cost of DOCI scles combntorlly wth system sze, the cost of pccd scles s O(N 3 ). III. PAIR COUPLED CLUSTER DOUBLES In pccd, we wrte the wve functon s Ψ = e T 0, (5) where 0 s closed-shell reference determnnt nd T = t P P (6) n terms of the pr opertors P nd P, where generclly, P q = c q c q (7) wth the snglet prng we re usng. As usul, one cn nsert ths nstz nto the Schrödnger equton to get E = 0 H 0, (8) 0 = 0 P P H 0, (8b)

4 Henderson et l. J. Chem. Phys. 141, (014) where the smlrty trnsformed Hmltonn H s gven by In AP1roG, one nsted wrtes but becuse H = e T H e T. (9) E = 0 H e T 0, E 0 P P e T 0 = 0 P P H e T 0, 0 e T = 0, 0 P P e T = 0 P P t 0 (10) (10b) (11) = 0 P P 0 P P e T 0 0, (11b) one cn see tht Eqs. (8) nd (10) re dentcl, nd consequently so too re Eqs. (8b) nd (10b). Explctly, the pccd energy nd mpltudes re gven by E = 0 H 0 + t v (1) 0 = v, + ( f f b j j v j j t j jb b ) v bb tb t ( v v v ) t t + v bb tb + v j j t j + v j j bb t j tb, (1b) where fq p s n element of the Fock opertor nd vr pq s = φ p φ q V ee φ r φ s s two-electron ntegrl n Drc notton. As promsed, these equtons cn be solved n O(N 3 ) computtonl cost wth the d of the ntermedte y j = bv j j bb tb. As wth trdtonl CC methods, we cn defne left-hnd egenvector L of H n CI-lke fshon, where Then, the expectton vlue of H s L = 0 (1+ Z), (13) Z = z P P. (14) E = 0 (1+ Z) H 0 = 0 (1+ Z) e T H e T 0. (15) The equtons for the mpltudes t re just 0 = E z nd gurntee by ther stsfcton tht (16) E = 0 H 0 (17) for ny vlue of Z; smlrly, we obtn the mpltudes z from 0 = E t. (18) We fnd tht the z equtons re 0 = v + ( f f v j j t j ) v bb tb z j b ( v v ) v t z ( v z j t j + ) b j j b z b tb + v bb z b + vj j z j + jb t b j ( v bb z j + v j j z b). (19) Agn, these cn be solved n O(N 3 ) tme. We should emphsze tht the pccd energy nd mpltude equtons for both T nd Z cn be extrcted from the usul RHF-bsed CCD 17,18 by smply retnng only the pr mpltudes t nd z whch we hve here wrtten s smply t nd z for compctness of notton nd to emphsze tht the pccd t nd z mpltudes re two-ndex qunttes. In prctce, one usully fnds tht Z T, s we mght expect. We note n pssng tht one cn redly dentfy the vrous chnnels 19,0 of the CCD mpltude equtons n Eq. (1b), where the ldder terms re found on the thrd lne, the rng nd crossed-rng terms pper on the second lne, nd wht we hve termed the Brueckner or mosc terms pper on the frst lne. For pccd, the vrous rng terms decouple, though our lmted numercl experence suggests tht pr rng CCD model s not useful. Lke DOCI, pccd s not nvrnt to the choce of whch orbtls re used to defne the pr opertors P p. Addtonlly, pccd depends on the choce of reference determnnt 0. In order to hve well-defned method, we must provde wy of fxng these choces. Ths cn be ccomplshed by orbtl optmzton 1, for whch purpose we ntroduce the one-body nthermtn opertor ( ) κ = κ pq c p c q c q c p p>q (0) whch, when exponentted, cretes untry orbtl rottons; here, ndexes spns (.e., =, ). Note tht n contrst to the typcl coupled-cluster orbtl optmzton whch requres only occuped-vrtul mxng, we must llow ll orbtls to mx. We hve tken κ to be rel. Gven the rotton opertor, we cn smply generlze the energy to E(κ) = 0 (1+ Z) e T e κ H e κ e T 0 (1) nd mke t sttonry wth respect to κ, whch gves us 0 = E(κ) κ pq κ=0 = 0 (1+ Z) e T [H,c p c q c q c p ] e T 0, () where we work t κ = 0 by trnsformng the bss n whch we express the Hmltonn (.e., by trnsformng the onend two-electron ntegrls). The commuttor cn be evluted redly, [H,c p c q ] = h r p c r c q hr q c p c r r r + v r s pt c r c s c t c q r st vr qt s c p c t c s c r, (3) r st where the Hmltonn s H = hq p c p c q + 1 vr pq s c p c q c s c r (4) pq pqr s

5 Henderson et l. J. Chem. Phys. 141, (014) n terms of one-electron ntegrls hq p nd the two-electron ntegrls vr pq s prevously defned. The energy grdent s then E(κ) ( = κ h r pq κ=0 p γr q hr q γp) r r ( ) + v r s pt Γr qt s vr qt s Γpt r s (p q), (5) r st where γq p nd Γr pq s re one-body nd two-body densty mtrces, gven by γq p = 0 (1+ Z) e T c q c p e T 0, (6) Γr pq s = 0 (1+ Z) e T c r c s c q c p e T 0. (6b) We use Newton-Rphson scheme to mnmze the norm of the orbtl grdent, whch fnds n orbtl sttonry pont. Hvng found such pont, we check the egenvlues of the coupled cluster orbtl Hessn nd, f there s negtve egenvlue, follow the nstblty untl we fnd locl energy mnmum or sddle pont (.e., we look for ponts wth zero grdent nd non-negtve Hessn). The nlytc formule for the densty mtrces nd the orbtl Hessn re presented n the Appendx. As hs been prevously ponted out, there re multple solutons to the orbtl optmzton equtons, nd becuse the optmzed orbtls re generlly locl n chrcter f the system s strongly correlted, 7,10,11 t proves convenent to strt from the RHF determnnt wth loclzed moleculr orbtls. We should lso pont out tht convergence of the pr mpltude nd response equtons s gretly ded by usng drect nverson n the tertve subspce. 3 Our Newton-Rphson procedure typclly uses the dgonl Hessn nd turns on the full nlytc Hessn only ner convergence; ths vods gettng trpped n hgh energy locl mnm. It should be noted here tht the one-body densty mtrx γ s dgonl n the bss n whch we defne the prng. In other words, the moleculr orbtls defnng the pccd T nd Z opertors re lso the nturl orbtls of pccd. The twobody densty mtrx Γ s lso very sprse nd hs knd of sem-dgonl form where only Γp qq p, Γpq, nd Γpq q p re nonzero. These propertes re true both for pccd nd for DOCI (nd ndeed for ny zero-senorty wve functon method). Detled expressons for the densty mtrces cn be found n the Appendx. IV. PAIR COUPLED CLUSTER AND DOUBLY OCCUPIED CONFIGURATION INTERACTION Now tht we hve gven mple detl bout pccd nd hve ntroduced DOCI, t wll prove useful to compre results from the two methods for vrety of smll systems for whch the DOCI clcultons re fesble. We wll compre the energes from the two pproches nd lso look t overlps of the pccd nd DOCI wve functons; explctly, we wll compute E = E pccd E DOCI (7) to ssess the qulty of the pccd energy nd S = 0 (1+ Z) e T DOCI DOCI e T 0 (8) to ssess the qulty of the pccd wve functons. Note tht S 1 when pccd s close to DOCI; more explctly, we hve 0 (1+ Z) e T e T 0 = 1, (9) nd nsertng the projector DOCI DOCI should not substntlly chnge ths vlue when pccd nd DOCI roughly concde. Becuse pccd s borthogonl, we need not hve S < 1; ndeed, we wll frequently see tht S s slghtly lrger thn one. We emphsze here tht both pccd nd DOCI cn be symmetry dpted despte hvng ndvdul orbtls whch re not symmetry egenfunctons due to the orbtl optmzton; ndeed, for the exmples dscussed below, pccd wth optmzed orbtls ppers to respect pont-group symmetry, though we hve found model Hmltonns for whch ths s not the cse. We wll lwys compre DOCI nd pccd wth the sme orbtl set (usully orbtls optmzed for pccd). Spot checks show tht typclly orbtls optmzed for DOCI re vrtully ndstngushble from orbtls optmzed for pccd. All DOCI nd pccd clcultons n ths secton nd ndeed throughout the mnuscrpt use n-house progrms, s do the frozen-pr coupled cluster clcultons dscussed n Sec. V; other clcultons used the Gussn progrm pckge. 4 Throughout, we wll use Dunnng s cc-pvdz bss set, 5 becuse we need suffcently smll bss tht the DOCI s computtonlly trctble, though we wll use Crtesn rther thn sphercl d-functons. We strt by notng tht for H, s for ny two-electron snglet, pccd wth orbtl optmzton s exct (nd s equvlent to DOCI). Ths s just becuse one cn use occupedvrtul rottons to mke sngle excttons n CCSD vnsh (n other words, one cn do Brueckner coupled cluster doubles) nd then pck vrtul-vrtul rotton to elmnte the senorty two exctton mpltudes. One cn see ths by notng tht for two-electron snglet, we hve T = 1 t b 1,1 c c c b 1 c 1 ; (30) b the combnton of fermonc ntsymmetry nd spn symmetry mens tht t b 1,1 = tb 1,1, so we cn defne rel symmetrc mtrx M b = t b 1,1 whch cn be dgonlzed by vrtulvrtul rotton so tht T tkes the pccd form. Numerclly, we fnd tht wth optmzed orbtls, E pccd = E DOCI = E FCI nd S = 1, s we should. In Fg., we show results for the dssocton of LH. Becuse LH s qus-two electron problem, we would expect DOCI nd pccd to be very ccurte n ths cse. Indeed, Fg. shows tht pccd nd DOCI re energetclly ndstngushble nd both re essentlly supermposble wth FCI (errors re on the order of 0.4 me H throughout the dssocton). Moreover, the DOCI nd pccd wve functons hve ner unt overlp throughout the dssocton. Ths s exctly wht we would expect for such problem. We next turn our ttenton to the dssocton of eqully spced hydrogen chns. These serve s mportnt prototypes of strongly correlted systems nd mp n loose sense to the Hubbrd Hmltonn. 6 The top pnel of Fg. 3 shows the dfference between the DOCI nd pccd energes per electron pr, whle the bottom pnel shows the devton of the overlp S from unty, gn per electron pr. These results pper to

6 Henderson et l. J. Chem. Phys. 141, (014) FIG.. Dssocton of LH. Top pnel: Dssocton energes from FCI, DOCI, nd pccd. Bottom pnel: Dfference between DOCI nd pccd energes ( E, defned n Eq. (7) nd mesured on the left xs) nd n the overlp (1 S, mesured on the rght xs wth S defned n Eq. (8)). FIG. 3. Dssocton of eqully spced hydrogen chns. Top pnel: Dfferences between DOCI nd pccd energes ( E, defned n Eq. (7)) per electron pr. Bottom pnel: Devtons n the overlp (1 S, wth S defned n Eq. (8)) per electron pr. be sturtng, though unfortuntely, the DOCI clcultons on H 10 re mprctcbly expensve wth our code. We should note tht whle the equvlence between DOCI nd pccd hs been estblshed for energetclly optmzed orbtls, we see the sme generl behvor when DOCI nd pccd pr cnoncl RHF orbtls nsted, though not to the sme degree. Tht s, even prng cnoncl RHF orbtls rther thn optmzed orbtls, pccd nd DOCI gve energes tht gree to wthn few mllhrtree, wth the greement predctbly degrdng s the systems become more strongly correlted. We cn see ths n hydrogen chns n Fg. 4. Strngely, the greement between DOCI nd pccd ppers to mprove s we move from H 4 to H 6 to H 8 when usng cnoncl RHF orbtls, whle n the optmzed orbtl cse, we see the opposte behvor. We should emphsze tht the devtons n the energy nd overlp n Fg. 4 re not shown per electron pr. Our next exmple s the symmetrc double dssocton of H O, s shown n Fg. 5. Agn, pccd nd DOCI provde nerly dentcl energes throughout the dssocton process, nd the overlps of the pccd nd DOCI wve functons re lrge. The concdence of DOCI nd pccd, n other words, s true not just for one pr of strongly correlted electrons, but for two prs s well. At dssocton, DOCI nd pccd gve essentlly the unrestrcted Hrtree-Fock (UHF) result, despte beng closed-shell wve functons, though s we shll see lter, ths s somewht fortutous. These methods mss sgnfcnt mount of the correlton compred to UHF-bsed CCSD nd CCSD(T); the dynmc correlton, then, s clerly not well descrbed. Smlr conclusons cn be reched from exmnng the dssocton of N. As Fg. 1 revels, DOCI does not gve ll the strong correlton needed to dssocte the trple bond n N correctly but does offer substntl mprovements over RHF. We see smlr results n Fg. 6. In these clcultons, we froze the ntrogen 1s core orbtls fter the orbtl optmzton nd compre the frozen-core DOCI to the frozen-core pccd. We lso note tht our procedure of repetedly followng nstbltes n the pccd orbtl Hessn led to n unphyscl reference determnnt for whch the pccd broke down; we hve thus used sttonry pont rther thn mnmum of the pccd energy functonl to defne the reference. Our results reterte tht pccd nd DOCI get most but not ll of the strong correlton n N nd fl to ccount for the dynmc correlton effectvely. Nonetheless, even for ths trple bond, we see tht DOCI nd pccd hve close greement. One cn see tht DOCI nd pccd do not descrbe dynmc correlton prtculrly well by consderng the neon tom, s seen n Tble I. Whle DOCI nd pccd re n excellent greement wth one nother, they only retreve bout 36% of the correlton energy even fter orbtl optmzton, wth optmzed orbtls very close to the cnoncl RHF moleculr orbtls. The bulk of the correltons must then nvolve determnnts of hgher senorty. In order to remedy

7 Henderson et l. J. Chem. Phys. 141, (014) FIG. 4. Dssocton of eqully spced hydrogen chns n the cnoncl RHF bss rther thn the pccd-optmzed bss used elsewhere. Top pnel: Dfferences between DOCI nd pccd energes ( E, defned n Eq. (7)). Bottom pnel: Devtons n the overlp (1 S, wth S defned n Eq. (8)). ths defcency, we turn to wht we cll frozen-pr coupled cluster, 10 s we wll descrbe shortly. Frst, however, t my be nstructve to tke closer look t the T-mpltudes of pccd nd the CI coeffcents of DOCI to understnd why the two methods concde so netly. Often, wht we fnd, s n the exmples bove, s tht the pccd T-mpltudes re such tht ech occuped orbtl s strongly correlted wth t most one vrtul orbtl, so tht ech row of the mtrx t hs t most one lrge entry, whle most of the mpltudes re smll. The DOCI vector follows ths sme bsc structure, whch s unsurprsng snce the DOCI nd pccd wve functons re essentlly the sme. In these cses, pccd nd DOCI re smlr to knd of perfect prng wve functon. 7 9 For exmple, for the stretched H O cse, the pccd nd DOCI wve functons re qulttvely Ψ O 1s O4 lp OH α OH, (31) where α pproches 1 t dssocton nd where O 1s, O lp, OH, nd OH, respectvely, denote the oxygen 1s orbtl, oxygen lone-pr orbtls, OH bondng orbtls, nd OH ntbondng orbtls. In the cse of stretched H O, t s the smll devtons from ths perfect prng structure whch cuse the energy to be close to the UHF lmt. Tht s, the only wve functon mpltudes lrger thn 0.05 correspond to excttons from n OH bondng orbtl nto ts ntbondng orbtl, but correltng the bondng orbtls lone yelds n energy somewht bove the sum of restrcted open-shell FIG. 5. Symmetrc double dssocton of H O. Top pnel: Dssocton energes from DOCI nd pccd, s well s from UHF nd CCSD nd CCSD(T) bsed thereon. Bottom pnel: Errors n the energy ( E, defned n Eq. (7) nd mesured on the left xs) nd n the overlp (1 S, mesured on the rght xs wth S defned n Eq. (8)). Hrtree-Fock tomc energes. Thus, we mght not expect pccd to descrbe strong correltons beyond those ccessble wth the perfect prng structure, even though we must emphsze tht the pccd wve functon s not nherently lmted to ths form. Indeed, t s mportnt to note tht we hve found cses n the repulsve Hubbrd Hmltonn 6 for whch nether pccd nor DOCI dopt perfect prng structure, yet the two methods stll gree closely. We lso note tht for the ttrctve prng Hmltonn 30 or the ttrctve Hubbrd Hmltonn (results not shown), one cn fnd nstnces n whch pccd does not resemble DOCI. In these cses, the DOCI coeffcents nd the pccd mpltudes re dense nd nether DOCI nor pccd dsplys perfect prng structure. Whle pccd nd DOCI nclude perfect prng wve functon s specl cse, they re more generl methods. The fct tht pccd closely resembles DOCI seems key feture of fermonc repulsve Hmltonns lke the moleculr one. V. FROZEN PAIR COUPLED CLUSTER The bsc de of frozen pr coupled cluster s very smple. One could mgne decomposng the T doubleexctton opertor nto pr prt T (0) nd non-pr prt T ; one would then solve the pccd equtons for the pr

8 Henderson et l. J. Chem. Phys. 141, (014) FIG. 7. Frozen pr symmetrc double dssocton of H O. FIG. 6. Dssocton of N. Top pnel: Dssocton energes from DOCI nd pccd, s well s from RHF, UHF, nd RHF- nd UHF-bsed CCSD. Bottom pnel: Errors n the energy ( E, defned n Eq. (7) nd mesured on the left xs) nd n the overlp (1 S, mesured on the rght xs wth S defned n Eq. (8)). mpltudes nd then solve the usul CCD equtons wthout llowng the pr mpltudes to chnge. Note tht the non-pr opertor T cretes senorty non-zero determnnts, whch we rely upon to provde the dynmc correlton whch pccd lcks; T on senorty zero determnnt returns lner combnton of determnnts wth senortes two nd four. Note lso tht the Fock opertor for orbtl-optmzed pccd s n generl nether dgonl nor n the semcnoncl form whch dgonlzes the occuped-occuped nd vrtul-vrtul blocks, so the full non-cnoncl form of the mpltude equtons must be used. Ths s not concern for pccd, where only the dgonl elements of the (generlly non-dgonl) Fock opertor contrbute to the mpltude equtons. Wht we hve descrbed bove we would cll frozen pr CCD (fpccd). One could of course extend ths bsc de to nclude sngle excttons nd trple or hgher excttons n the cluster opertor. Wht we wsh to do here s to brefly consder frozen pr coupled cluster wth sngle-, double-, nd trple-exctton mpltudes (fpccsdt). 31,3 In Fg. 7, we show the symmetrc double dssocton of H O, ths tme wth the frozen pr pproxmton. The effect of sngle excttons s n ths cse smll (fpccd nd fpccsd gve smlr results) nd fpccsd gves results frly smlr to the UHF-bsed CCSD nd CCSD(T) curves. Addng full trple excttons n fpccsdt gves lrger correlton t dssocton nd probbly overcorreltes somewht. For comprson purposes, we show results from FCI nd RHF-bsed CCSD nd CCSDT n Fg. 8. These clcultons fx the H-O-H bond ngle t 110 rther thn t the used n our other clcultons, nd use sphercl d functons; the CCSD, CCSDT, nd FCI dt re tken from Ref. 33. We see tht s one stretches the bond, CCSD nd CCSDT go through mxmum nd turn over; for lrger bond lengths, we would expect CCSD nd CCSDT to overcorrelte more. In contrst, fpccd s concdentlly very close to FCI, nd whle fpccsd nd fpccsdt overcorrelte somewht more, they provde sensbly shped dssocton curves wthout requrng symmetry brekng. Tble II shows fpccd nd fpccsd results for the neon tom. Whle pccd undercorreltes sgnfcntly compred to TABLE I. Energes nd overlps n the neon tom. Here, E Ref denotes the energy of the reference determnnt. We show results for both the optmzed determnnt for pccd nd for the cnoncl RHF determnnt s reference. Optmzed Cnoncl E Ref E DOCI E pccd E CCSD S FIG. 8. Symmetrc double dssocton of H O t 110 bond ngle wth frozen pr coupled cluster nd trdtonl coupled cluster methods. FCI nd CCSD dt tken from Ref. 33. All results use closed-shell (restrcted) wve functons.

9 Henderson et l. J. Chem. Phys. 141, (014) TABLE II. Energes n the neon tom. Here, E Ref denotes the energy of the reference determnnt. Method Energy E Ref E pccd E fpccd E CCD E fpccsd E CCSD E fpccsdt E CCSDT CCSD, mkng the frozen pr pproxmton yelds results tht dffer from those wthout freezng T (0) by bout 4 mllhrtree. As wth the double dssocton of H O, frozen pr coupled cluster overcorreltes slghtly. As fnl exmple, we consder fpccsd for the dssocton of N, s seen n Fg. 9. As should by now be fmlr, fpccsd gves resonble ccountng for dynmc correlton but overcorreltes somewht. Both fpccsd nd RHF-bsed CCSD brek down for lrge bond lengths, nd hve n rtfcl bump n the dssocton curve; whle fpccsd does not elmnte ths unphyscl effect, t t lest mtgtes t somewht. Our results show tht frozen pr coupled cluster should be understood s n esy wy to ncorporte the resonble pccd descrpton of strong correlton whle retnng much of the blty of trdtonl coupled cluster to lso descrbe dynmc correlton. However, whle esy to mplement nd conceptully smple, t s lso mportnt to note tht frozen pr full coupled cluster pproch would gve the wrong nswer. In other words, n the exct theory, one must clerly llow the zero-senorty T mpltudes to relx from ther pccd vlues. In prctce, fpccsd should llow for resonble descrpton of both strongly nd wekly correlted systems t essentlly the cost of CCSD clculton, wthout brekng spn symmetry, lthough fpccsd would be expected to brek down somewht for cses where pccd s unble to cpture ll the strong correltons, s s the cse wth N. For two-electron snglets, fpccd nd fpccsd re both exct, becuse s we hve prevously noted, pccd s lredy FCI, whch mples tht T 1 nd the non-zero senorty prts of T vnsh. VI. CONCLUSIONS Whle trdtonl coupled cluster theory s hghly successful for the descrpton of wekly correlted systems, t generlly fls to descrbe strong correlton. Prdoxclly, by smply elmntng the vst bulk of the cluster opertor, one cn form pr coupled cluster doubles, whch ccurtely reproduces DOCI, nd to the extent tht DOCI cn descrbe strong correltons, so too cn pccd. Moreover, pccd ccomplshes ths tsk wth men-feld computtonl sclng for the coupled cluster prt. Not only does pccd reproduce the DOCI energy but t lso reproduces the DOCI wve functon. The DOCI wve functon, n other words, s essentlly fctorzble nto the pccd form. Loosely, ths cn be ccomplshed becuse, upon orbtl optmzton, the pccd nd DOCI wve functons studed n ths work dopt perfect prng-lke structure. Whle pccd cn descrbe strong correltons, t s much less successful t modelng dynmc correlton, whch pprently requres the brekng of electron prs to obtn hgher senorty determnnts when we defne prs n terms of the sptl orbtls n prtcle-hole representton. Usng pccd to obtn the zero-senorty prt of the cluster opertor nd then solvng the trdtonl coupled cluster equtons for the rest of the mpltudes yelds frozen-pr coupled cluster, whch seems to be ble to descrbe both wekly nd strongly correlted systems wth resonble ccurcy nd wth computtonl cost not much dfferent from tht of stndrd coupled cluster methods. Of course, pccd s not pnce nd there re occsons when pccd fls to ccount for the strong correlton present n the DOCI wve functon, lthough we hve not seen such cse for the moleculr Hmltonn. Lkewse, t s possble tht the DOCI form s too restrcted to llow for complete descrpton of the strong correltons present, s ppers to hppen n the dssocton of N, for exmple. In such cses, the frozen-pr coupled-cluster pproch would be of less utlty. We speculte tht t my be possble to nclude these strong correltons by generlzng the prng structure to non-snglet prng, so tht the prs ncluded n pccd nd DOCI re not just the two electrons n the sme sptl orbtl. Regrdless, we hope tht pccd nd ts frozen pr extensons wll be useful tools for the descrpton of both wekly nd strongly correlted systems wthout the need for symmetry brekng or hgher exctton opertors. FIG. 9. Dssocton of N wth vrous coupled cluster methods. ACKNOWLEDGMENTS Ths work ws supported s prt of the Center for the Computtonl Desgn of Functonl Lyered Mterls, n Energy Fronter Reserch Center funded by the U.S. Deprtment of Energy, Offce of Scence, Bsc Energy Scences under Awrd #DE-SC GES s Welch Foundton chr (C-0036). T.S. s n wrdee of the Wezmnn Insttute

10 Henderson et l. J. Chem. Phys. 141, (014) of Scence Ntonl Postdoctorl Awrd Progrm for Advncng Women n Scence. We would lke to thnk Crlos Jménez-Hoyos for helpful dscusson. APPENDIX: DENSITY MATRICES AND ORBITAL HESSIAN For completeness, we nclude here expressons for the pccd densty mtrces nd orbtl rotton Hessn; together wth the orbtl rotton grdent of Eq. (5), these provde everythng needed for the Newton-Rphson lgorthm we use for orbtl optmzton. Recll tht the energy s wrtten s wth E(κ) = 0 (1+ Z) e T e κ H e κ e T 0 ( ) κ = κ pq c p c q c q c p, p>q (A1) (A) where the orbtl rotton s gven by the untry trnsformton exp(κ). At every step of the Newton-Rphson scheme, we solve for κ, buld exp(κ) whch rottes to new orbtl bss, trnsform the ntegrls, nd begn new terton. We hve lredy seen tht the grdent s smply E(κ) = P pq [H, c κ pq κ=0 p c q ], (A3) where P pq s permutton opertor P pq = 1 (p q) nd the notton for the expectton vlue mens Smlrly, the Hessn s O = 0 (1+ Z) e T O e T 0. H pq,r s = E(κ) κ pq κ r s κ=0 = 1 P pq P r s [[H, c p c q ], c r η c sη ],η (A4) + 1 P pq P r s [[H, c r η c sη ], c p c q ], (A5),η where η s nother spn ndex. We obtn H pq,r s = P pq P r s 1 ( δqr h u p γu s + hu s γ u p u + 1 ( δqr v uv pt Γuv st + vuv st Γpt uv uv tuv ( + v uv pr Γuv qs + vuv qs Γpr uv ) tu ) ( + δps h u r γu q + hu q γr u h s p γr q + hr q γp) s ) + δps v qt uv Γ uv r t + v uv r t Γuv qt ( v st pu Γ qu r t + v pu t s Γ qu tr + v qu r t Γ st pu + v qu tr pu). (A6) Γt s The one-prtcle densty mtrx we hve defned s γq p = 0 (1+ Z) e T c q c p e T 0. (A7) Becuse T nd Z both preserve the senorty of the wve functon, nd the reference 0 hs senorty zero, t s mmedtely cler tht the one-prtcle densty mtrx s dgonl n the bss n whch we hve defned the prng; the optmzed orbtl bss for pccd, n other words, s lso ts nturl orbtl bss. We then hve γ j = ( 1 x j ) δ j, (A8) γ b = x b δ b, γ = γ = 0, (A8b) (A8c) where δ pq s the Kronecker delt nd where we hve defned x j = t z, j (A9) x b = t b z. (A9b) Recll tht nd re, respectvely, occuped nd vrtul orbtl ndces. Smlr consdertons show tht the two-prtcle densty mtrx s lso sprse n the nturl orbtl bss. The non-zero elements of the two-prtcle densty mtrx re = x j + δ j 1 x, Γ j j Γ = t + x t Γ = z, Γ bb = x b, x + x t z, (A10) (A10b) (A10c) (A10d) Γ j j = 4 ( 1 x x j j) + δ j 3 x 1, (A10e) Γ = Γ = 4 x t z, (A10f) Γ b b = δ b x, Γpq q p = 1 qp Γpq pq. We hve defned the ddtonl ntermedte x = t b t j z j b. jb (A10g) (A10h) (A11) Note tht the sprsty of the one- nd two-prtcle densty mtrces llows one to consderbly reduce the cost of evlutng the Hessn. 1 J. Pldus nd X. L, Adv. Chem. Phys. 110, 1 (1999). R. J. Brtlett nd M. Musł, Rev. Mod. Phys. 79, 91 (007). 3 I. Shvtt nd R. J. Brtlett, n Mny-Body Methods n Chemstry nd Physcs (Cmbrdge Unversty Press, New York, 009). 4 G. D. Purvs nd R. J. Brtlett, J. Chem. Phys. 76, 1910 (198).

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