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Alternative definition of excitation amplitudes in Multi-Reference state-specific Coupled Cluster Yann Garniron,1 Emmanuel Giner,2,3 Jean-Paul Malrieu,1 and Anthony Scemama1,a) 1)Laboratoire de Chimie et Physique Quantiques (CNRS 5626), IRSAMC, Universit´e P. Sabatier, Toulouse (France) 2)Dipartimento di Scienze Chimiche e Farmaceutiche, Universita di Ferrara, Via Fossato di Mortara 17, 7 I-44121 Ferrara (Italy) 1 3)Max Planck Institut for solid state research, Heisenbergstraße 1, 70569 Stuttgart, 0 Germany 2 A central difficulty of state-specific Multi-Reference Coupled Cluster (MR-CC) formalisms concerns the def- n inition of the amplitudes of the single and double excitation operators appearing in the exponential wave a operator. If the reference space is a complete active space (CAS) the number of these amplitudes is larger J thanthenumberofsinglyanddoublyexciteddeterminantsonwhichonemayprojecttheeigenequation,and 7 one must impose additional conditions. The present work first defines a state-specific reference-independent 1 operator Tˆ˜m which acting on the CAS component of the wave function Ψm maximizes the overlapbetween h] (1+Tˆ˜m)Ψm and the eigenvector of the CAS-SD CI matrix Ψm | 0.i This operator may be used to | 0 i | CAS−SDi p generate approximate coefficients of the Triples and Quadruples, and a dressing of the CAS-SD CI matrix, - according to the intermediate Hamiltonian formalism. The process may be iterated to convergence. As a m refinement towards a strict Coupled Cluster formalism, one may exploit reference-independent amplitudes he provided by (1 +Tˆ˜m)|Ψm0 i to define a reference-dependent operator Tˆm by fitting the eigenvector of the c (dressed) CAS-SD CI matrix. The two variants, which are internally uncontracted, give rather similar re- . sults. The new MR-CC version has been tested on the ground state potential energy curves of 6 molecules s c (up to triple-bond breaking) and a two excited states. The non-parallelism error with respect to the Full-CI si curves is of the order of 1 mEh. y h p I. INTRODUCTION TheCCequations,obtainedbyprojectingtheeigenequa- [ tion on each of the Singles and Doubles lead to coupled 1 The single-reference Coupled Cluster (CC) quadraticequations. Inpractice, guessvalues of the am- v formalism1–4 is the standard technique in the study of plitudes of the Tˆ0→i operators appearing in the Tˆ oper- 4 the ground state of closed-shell molecules, i.e. those ator may be taken as the coefficients of the Singles and 6 for which a mean-field treatment provides a reasonable Doubles i in the intermediate normalization of the SD 7 zero-order single-determinant wave-function Φ . This CI vector|.iThe solution of the CC equations may be ob- 4 0 method incorporates the leading contributions to the tained by treating the effect of the Triples and Quadru- 0 . correlationenergy in a given basis set, it is based on the ples as an iterative dressing of the SD CI matrix,6 ac- 1 linked-cluster theorem5 and is size-consistent since it is cordingto the IntermediateEffectiveHamiltonian(IEH) 0 free from unlinked contributions. The method generates theory.7,8Thefieldofapplicationofthismethod,whichis 7 1 anapproximatewavefunctionundertheactionofawave both intellectually satisfying and numericallyefficient, is : operator Ωˆ acting on the single-determinant reference however limited to the systems and the situations where v Φ , and assumes an exponential character to the wave a single-determinant zero-order description is relevant. i 0 X operator This is no longer the case when chemical bonds are bro- ken, creating open shells, as occurs in most of the chem- r a Ψ=ΩˆΦ =exp TˆΦ (1) ical reactions. The magnetic systems generally present 0 0 several open shells and the low spin-multiplicity states (cid:16) (cid:17) The most popular version only introduces single and are intrinsically of multiple-determinant character. Due double excitation operators in Tˆ, and is known as the to near degeneracies, most of the excited states are not Coupled Cluster Singles and Doubles (CCSD) approxi- only of multi-determinantal but of multi-configurational mation. It incorporates the fourth order correction of character. The conception of a multi-reference counter- the quadruply excited determinants. The lacking fourth part of the CCSD formalism is highly desirable, and has order contribution concerns the triply excited determi- been the subject of intense research. The most compre- nants, which may be added in a perturbative manner. hensive review has been given by R. Bartlett and his colleagues.9 Forformalreasonsandinparticularto treat correctly the breaking of bonds, the reference space, or modelspace,isusuallytakenasaCompleteActiveSpace a)Electronicmail: [email protected] (CAS), i.e. the Full-CI of a well-defined number of elec- 2 trons (the active electrons) in a well-defined set of or- reference-independentamplitudesoftheexcitations,may bitals (the active MOs). The other MOs are called inac- be exploited directly to generate approximate values of tive. Letus label I , J ,... the referencedeterminants. the coefficients of the triply and quadruply excited de- | i | i The determinants i , j ,... whichinteractwith the ref- terminants,accordingtotheexponentialstructureofthe | i | i erence space are obtained under purely inactive or semi- wave operator. From these coefficients one may dress active single and double excitations, they generate the the CAS-SD CI matrix, redefine amplitudes and iterate CAS-SD CI space, the diagonalizationof which provides the process to convergence. This solution, presented in a size-inconsistent energy Em and the correspond- section 2C, is not an MR-CC technique, one may call it CAS−SD ing eigenvector, an exponential dressing of the CAS-SD CI matrix. Sec- tion 2D redefines reference-dependent excitation ampli- Ψm = Ψm + Ψm | CAS−SDi | 0 i | SDi tudesfromthereference-independentamplitudesbyafit- = Cm+ I + cm i (2) ting of the previous amplitudes on the coefficients of the I | i i | i Singles and Doubles of the (dressed) CAS-SD CI eigen- I∈CAS i∈/CAS X X vector. This represents an alternative solution to multi- with Ψm Ψm =1. parentageproblemandopensthewaytoastrictMR-CC h CAS−SD| CAS−SDi One strategy, which is not very aesthetic since it formalism. Section 3 presents a series of numerical tests breaks the symmetry between degenerate reference de- on the bond breaking of single, double and triple bonds terminants, but which has given rather satisfactory re- in ground states of molecules as well as a few tests on sults, consists in selecting (eventually in an arbitrary excited states. The results are comparedto our previous manner) a specific single reference and in introducing in proposal and with full Configuration Interaction (FCI) the wave operator the multiple excitations which gener- results. ate the other references (the other determinants of the modelspace). The otherstrategiesconsideralltherefer- ences onanequalfooting, andarereallymulti-reference. II. FORMALISMS Let us call N the number of references, and n the num- ber of SD determinants. If the treatment pretended to In this section, all the presented formalisms are state- provide N eigenvectors simultaneously, one might define specific. To simplify the notations we will consider that the N n amplitudes sending from the references to the state superscriptm is implicit forthe wavefunctions × the outer-space determinants, in a unique manner but (Ψm Ψ) and for the excitation operators (Tˆm Tˆ). this state-universalapproachisnotpracticablewhenthe → → model space is a CAS. Most of the proposed formalisms are state-specific. In A. The multi-parentage problem in the this case one faces the famous multi-parentage prob- Jeziorski-Monkhorstapproach lem. This problem is recalled in section 2A. Sufficiency conditions have to be imposed.10 One solution was pro- Since one wants to produce a MR-CCSD method, one posedby Mukherjee andcoworkers,and has beenwidely may start from a preliminary CAS-SD CI calculation tested.11–13 Another one had been proposed earlier by whichwillhelptofixguessvaluesoftheamplitudesofthe one ofus (JPM) and coworkers.14 It consists,for a given excitationoperators. Letus call I , J ,... the determi- outer-spacedeterminant,inscalingtheamplitudesofthe | i | i nants of the CAS, i.e. the so-called reference vectors, various excitation operators Tˆ on the interaction be- I→i and i , j ,... the Singles andDoubles whichdo notbe- tween the outer-space determinant and its parents. A | i | i long to the CAS and interact with them. The resulting recent work has implemented this second solution of the approximate wave function of the targeted state Ψ is state-specific MR-CC problem and has tested its accu- | i written racyandrobustnessonaseriesofmolecularbenchmarks, comparingitsresultstotheFull-CI(FCI)energies.15The Ψ = C I + c i (4) CAS−SD I i presentworkproposesanalternativeprocesstodefinethe | i | i | i I i amplitudesoftheexcitationoperators. Thestate-specific X X MR-CC formalisms are usually based on the Jeziorski- Although this function is not size consistent one may Monkhorst16 splitting ofthewaveoperatorintoasumof note that the coefficients on the CAS determinants are operators acting individually on the various references no longer those of the CAS-CI : they incorporate the effect ofthe dynamicalcorrelationonthe compositionof Tˆm = Tˆm I I (3) the CAS component of the wave function. I | ih | In CC formalisms the wave operator Ωˆ is assumed to I X take an exponential form We shall leave in a first time this assumption and de- fine in section IIB a reference-independent operator Tˆ Ωˆ =exp(Tˆ) (5) which acting on the component of the desired state in the modelspace, Ψm , providesa vectoras closeaspos- and in our previous MR-CC formalism15 the Jeziorski- | 0 i sible to the CAS-SD eigenvector. This solution, defining Monkhorst structure of the wave operator was adopted, 3 introducing reference-specific wave operators acting B. Introduction of reference-independent amplitudes specifically on each reference vector (Eq.3). One may exploit the knowledge of the CAS-SD CI eigenvector to The present formalism will leave in the first step the determine guess operators Tˆ defined in such a manner I Jeziorski-Monkhorst formulation of the wave operator that andwillconsiderthe possibility to define a unique state- specific reference-independent operator Tˆ, written as a Ψ = C Tˆ I (6) | CAS−SDi I I| i sum of single and double excitation operators, I X The Tˆ operators are a sum of single and double exci- Tˆ = tmn→pqa†pa†qanam+ tm→pa†pam (12) I tations TˆI→i possible on I , multiplied by an amplitude mXnpq Xmp t | i = t Tˆ + t Tˆ (13) I→i mn→pq mn→pq m→p m→p mnpq mp Tˆ = t Tˆ (7) X X I I→i I→i where the indices p and q run on the virtual and active i X MOs and the indices m and n run on the inactive occu- In the single-reference CC the amplitudes of the excita- pied and active MOs, excluding the possible occurrence tion operators are obtained by projecting the eigenequa- of 4 active MOs. tion on the singly and doubly excited determinants, the This operator has the same form as the one intro- number of unknowns is equal to the number of equa- duced by the internally-contracted MR-CC (ic-MRCC) tions. This is no longer the case in the MR context : methodbyEvangelistaandGauss,17andbyHanauerand projecting the eigenequation on each on the singly or K¨ohn,18 but it differs by both its determination and by doubly excited vectors i is not sufficient to define the the way we use it, as will appear later. The ic-MRCC | i amplitudes tI→i since for many classes of excitation an method determines the amplitudes of the excitation by outer-spacedeterminantinteractswithseveralreferences, solving the projected Coupled Cluster equations, where i =TˆI→i I =TˆJ→i J . The condition the amplitudes appear as linear and quadratic terms. | i | i | i Hereafter we exploit the knowledge of the CAS-SD CI C = t C (8) eigenvector to determine guess values of the reference- i I→i I I independent amplitudes. These excitation amplitudes X will be used later on to estimate the coefficients of the is not sufficient to define the amplitudes, even if one re- Triples and Quadruples, and perform an iterative dress- strictstheexcitationoperatorstosingleanddoubleexci- ing of the CAS-SD CI matrix introducing the coupling tations. Additional constraints have to be introduced to between the Singles and Doubles with the Triples and fixtheamplitudes,andthisisthefamousmulti-parentage Quadruples. problem. The number of amplitudes is larger than the We propose a criterion to fix the amplitudes t = number of outer-space determinants so that one cannot t ,t . Given the factthat we haveat our dis- mn→pq m→p determine directly guess values of the amplitudes from { } posal the CAS-SD wave function, a natural way to solve Eq. 6. Different additional constraints have been pro- this overdeterminedproblem is to minimize the distance posed. Oneofthemconsistsinscalingtheamplitudeson between the CAS-SD vector and the vector obtained by the Hamiltonian interactions between the references and applying the (1+Tˆ) operatoronthe CAS wavefunction the outer space determinants, tI→i = hi|Hˆ|Ii. (9) argtmink(1+Tˆ)|Ψ0i−|ΨCAS−SDik (14) tJ→i iHˆ J =argmin Tˆ Ψ Ψ , h | | i k | 0i−| SDik t This constraint is expressed as Tˆ Ψ being normalized such that Tˆ Ψ = Ψ . t =λ iHˆ I (10) | 0i k | 0ik k| SDik I→i i To perform the minimization, we build the N N h | | i SD t transformationmatrixA = iTˆ Ψ w×hich i,mn→pq mn→pq 0 where h | | i maps from the outer space of determinants i to the {| i} λi = iHˆciΨ (11) sspeaarccehoffoerxtchiteedvewctaovreofufnacmtipolnitsu{dTeˆms nt→wpqh|iΨch0i}m,inainmdizwees 0 h | | i A.t c by solving the normal equations This solution has been recently implemented15 and k − k shown to provide excellent agreements with Full-CI re- (A†A)t=A†c (15) sults on a series of molecular problems. It only presents minor stability problems in comparisonwith the present Note that in the single-reference case, A is a permuta- suggestion when the term iHˆ Ψ is small. From now tion matrix and the CAS-SD wave function is exactly 0 h | | i on, we will refer to this method as the λ-MR-CCSD. recovered. 4 The matrix A is usually so large that the use of stan- TriplesandQuadruplesasobtainedbytheactionof 1Tˆ2. 2 dardsingularvaluedecomposition(SVD)routinestoob- Actually one may assume, in the spirit of the internally- tain the least squares solution is prohibitive. contracted MR-CC methods, that the wave operator Ωˆ Let us first consider the most numerous 2-hole-2- generating the correlated wave function Ψ from Ψ , 0 particle inactive double excitations Tˆ . Acting on jk→rs a determinant I the operator creates a determinant Ψ=ΩˆΨ0 (21) | i i =Tˆ I which can only be produced by this pro- |ceiss. Tjkh→errse|foire, the corresponding rows of A contain has an exponential structure, only one non-zero element located in the jk rs col- Ωˆ =exp(Tˆ) (22) → umn with value A =C . The condition fixing the i,jk→rs I amplitude tjk→rs is given by But this form will be simply used to estimate the coeffi- cients of the triply and quadruply excited determinants argmin Tˆ Ψ t Ψ (16) jk→rs 0 jk→rs SD α , leaving the internally-contracted structure of the k | i −| ik tjk→rs {o|utie}r-space. The coefficients of these determinants are which is obtained by minimizing estimated as 2 1 c = αTˆ2 Ψ . (23) min C t c iTˆ I (17) α 2h | | 0i I jk→rs i jk→rs tjk→rs − h | | i! I i X X In practice all the determinants α are generated. {| i} This condition turns out to be satisfied for For each α one finds the reference determinants I α | i {| i} which differ by at most 4 orbital substitutions from α C c tjk→rs = I CI2i (18) (itsgrand-parents). Onethenidentifiesthesetofcomp|lei- P I I mentaryexcitations (p,q) Tˆ Tˆ I = α as the prod- p q One may notice that this is thPe weighted average of the | i | i ratios between the coefficients of the doubly excited de- uαctscwonhtircihbugteensertaoteint|sαiasfsroo(cid:12)(cid:12)cmiat{e|Idi}cαo.effiTchieeongtecnealboygythoef terminants i andthecoefficientoftheiruniquereference | i α generator, | i quantity tptqCI. Knowing |αi, one also knows the Sin- gles and Doubles i with which it interacts through α {| i} 1 c the matrix elements iHˆ α , and in the eigenequation tjk→rs = ICI2 I CI2(cid:18)CiI(cid:19)! (19) relative to |ii h | | i X For all the remainPing active excitations, A remains iHˆ i E ci+ iHˆ J CJ+ h | | i− h | | i sparse since the maximum number of non-zero elements (cid:16) (cid:17) XJ (24) per column is equal to the number of reference determi- iHˆ j c + iHˆ α c =0. j α nants. Hence, we use Richardson’s iterative procedure19 h | | i h | | i j α X X Onemayreplacethelastsumbyadressingofthematrix t =A†c 0 (20) elements between the determinant i and the references (cid:26)tn+1 =A†c+ I−A†A tn which are grand-parents of α , | i | i which may be implemented v(cid:0)ery efficie(cid:1)ntly using sparse matrix products. There are cases where multiple amplitudes applied i∆I = iHˆ α  tptq (25) to different references lead to same determinant : h | | i h | | i Tˆ I = Tˆ J = i . If this determinant can Xα n(p,q) TˆpXTˆq|Ii=|αio  jk→rs| i lm→tv| i | i   be reached by no other process, there is an infinity of  (cid:12)  since (cid:12) solutions for the amplitudes. The solution of Eq.(15) is A+cwhereA+isthepseudo-inverseofA,sothesolution i∆I C = iHˆ α c . (26) obtained minimizes the norm of the amplitude vector.20 h | | i I h | | i α I α Inthisway,thearbitrarinessbroughtbythenullspaceof X X Ais minimized inthe amplitude vector,andone obtains The effect of the Triples and Quadruples is incor- the most sensible solution. porated as a change of the columns of the CAS-SD CI matrix concerning the interaction between the ref- erences and the Singles and Doubles. This type of C. Evaluation of the coefficients of Triples and dressing was already employed in our previous MR-CC Quadruples and iterative dressing of the CAS-SD CI matrix implementation.15 One will find in the same reference the practical procedure to make the dressedmatrix Her- The so-determined excitationoperator Tˆ may be used mitian without any loss of information. Of course the to generate approximate values of the coefficient of the whole process may be iterated. The diagonalization of 5 thedressedCAS-SDCImatrixprovidesnewvaluesofthe Astheoverlapbetween(1+Tˆ)Ψ and Ψ has 0 CAS−SD | i | i coefficients,notonlyoftheSinglesandDoubleswhichno beenmaximizedthecoefficientsc˜ andc areexpectedto i i longer suffer from the truncation, but also those of the be very close in particular if c is large, and the parame- i references : the method is fully non-contracted. From ter µ should be close to 1, at least for the determinants i the new wave function new amplitudes are obtained, a which contribute significantly to the wave function. In new dressing is defined and the process is repeated till practice we observe this tendency, but the smallest co- convergence, which is usually rapidly obtained (3-4 iter- efficients are sacrificed during the maximization of the ations). overlap and their µ can be very far from 1. This intro- i ThisformalismisnotastrictMR-CCmethodsincewe duces some instabilities in the iterations, so we chose to exploit the CAS-SD CI function, and since this function limit the values of µ in the [ 2,2] range. The effect on i − slightly differs from the vector resulting from the action the stabilityofthe iterationsissignificant,andthe effect ofTˆ onthe vector. Althoughthe distance betweenthese on the energy is not noticeable. two vectors has been minimized they are not identical, This version returns to the Jeziorski-Monkhorst for- (1+Tˆ)Ψ = Ψ˜ . malismas the waveoperatoragainis a sum ofreference- 0 CAS−SD | i6 | i Once the Tˆ operator has been obtained one might specific operators. The so-obtained amplitudes may be imagine a contracted exponential formalism calculating exploited to generate the coefficients of the Triples and Tˆ2 Ψ and the interaction between Tˆ Ψ and Tˆ2 Ψ , Quadruples, and one may follow the same strategy as in 0 0 0 but| thiis calculation requires to return| toithe dete|rmii- our previous formalism, with an iterative column dress- nants. This formalism would remain internally con- ing of the interactions between the Singles and Doubles tracted and would be less accurate than the procedure and the references. In a strict Coupled Cluster formal- we propose. Actually in this version the deviations of ism, one should redefine the amplitudes of the double the approximate reference-independent amplitudes from excitations by subtracting the products of the comple- optimalones,thosewhichwouldgeneratetheexactcoef- mentary single excitations they contain. For the sake of ficients of the Singles and Doubles, only affects the eval- simplicity, as we did in our previous work, we did not uation of the coefficients of the Triples and Quadruples, proceed to this revision, the difference concerning only and these deviations representa minor source oferrorin fifth-order perturbative corrections. the correction restoring the size extensivity. This relia- Inwhatfollows,wewillrefertothis method asµ-MR- bility will be illustrated in the numerical tests. CCSD as it involves the µi (Eq.28). D. State-specific MR-CC variant III. NUMERICAL TESTS In order to return to a MR-CC formalism, one may In this section, we compare the here-proposeddressed simply exploit the reference-independent amplitudes as CAS-SD and MR-CCSD to the MR-CCSD presented aninitialguesstodefinereference-dependentamplitudes. in ref15 on standard benchmark systems.11,13,17,18,21–30 Currently the determinant i belonging to the Singles To differentiate those two variants, we will label λ-MR- and Doubles has a coefficien|tic˜ in Tˆ Ψ CCSD the variant of ref15 (Eq. 11) and the MR-CCSD i 0 | i of this work will be labeled µ-MR-CCSD (Eq. 28). c˜ = iTˆ Ψ = t C (27) The basis set used is Dunning’s cc-pVDZ,31 and the i 0 l I h | | i molecular orbitals were obtained using the CAS-SCF n(I,l)XTˆl|Ii=|iio code present in GAMESS.32 All the following calcula- tionsweremadeusingtheQuantumPackage,33anopen- (cid:12) which differs from the coefficien(cid:12)t ci in ΨSD . One can source library developed in our group. Full-CI energies | i define a parameter µi, specific of the determinant i , were obtained using the CIPSI algorithm,34–36 and the | i accuracyofthetotalenergiesisestimatedtobeoftheor- c µi = i (28) der of 10−5Eh. In all the calculations (Full-CI, CAS-SD c˜ i and MR-CC), only the valence electrons are correlated (frozen core approximation). which multiplying c˜ will produce the exactcoefficient c i i of i in the (dressed) CAS-SD CI eigenvector. So the | i previous reference-independentamplitudes havenow be- come reference-dependent. The excitation Tˆ which ex- A. Bond breaking l cites I to i (i =Tˆ I )receivesareference-dependent l ampli|tuide | i | i | i For all the applications we compare the dressed CAS- SD and µ-MR-CCSD with the λ-MR-CCSD and the t =t =µ t . (29) CAS-SD values. Results are also given using the I→i l,I i l reference-independent dressing of the CAS-SD CI ma- The same excitation will receive a somewhat different trix. All the applications are presented as energy differ- amplitude when it acts on another reference t =t . ences with respect to the Full-CI energy estimated by a l,J l,I 6 6 CIPSI calculation with a second-order perturbative cor- whichislargerthantheNPEof1.3mE obtainedbythe h rection. Figure 1 shows the difference of energy with λ-MR-CCSD. This is due to only one point of the curve, respectto the Full-CI alongthe reactioncoordinate. Ta- the maximum which is higher by 0.4 mE , all the other h ble I summarizes the non-parallelism errors (NPE) and points being very close by less than 0.1 mE . Here, the h themaximumoftheerrorobtainedalongthecurve. The dressed CAS-SD and the µ-MR-CCSD are equivalent. MR-CC treatment reduces the average and maximum error of the CAS-SD with respect to Full-CI by a fac- tor close to 4. The correction is larger when the sys- Two bond breaking tem involves an important number of inactive electrons (F2, C2H6) than when this number is small (BeH2, N2). For breaking two bonds we have used CAS(4,4) wave Oneactuallyknowsthatthe size-consistencyerrorofthe functions as the reference space. The first example is CAS-SDtreatmentincreaseswiththenumberofinactive the simultaneous breakingofthe twoO—Hbonds ofthe electrons, this error disappears in the MRCC treatment, watermoleculebystretching. Here,theCAS-SDexhibits which essentially misses some fourth-order connected ef- a NPE of 1.8 mE which is significantly improved to h fects of the Triples. 0.2 mE with the dressed CAS-SD. The µ-MR-CCSD, h with an NPE of 0.5 mE , is slightly more parallelto the h Full-CI curve than the λ-MR-CCSD which has an NPE Single-bond breaking of 0.7 mE . h The second example is the double-bond breaking of We present here the single bond breaking of the σ ethylene by stretching. One should first clarify that the bonds of C H and F molecules and of the π bond of energydifferencesinthefiguredonotmatchthoseofthe 2 6 2 ethylene. The active spaces were chosen with two elec- torsion along the bond because in the former example trons in two MOs, the minimum wavefunctions to de- the reference was a CAS(2,2), and here it is a CAS(4,4). scribe properly the dissociation of the molecules. In the Dressing the CAS-SD reduces the NPE from 2.8 mEh to case of ethane, the NPE of the CAS-SD is 5.1 mEh, and 1.7 mEh. One can remark a discontinuity in the curve is reduced to 3.5 mE with the µ-MR-CCSD. The curve at large distances. The µ-MR-CCSD and λ-MR-CCSD h of the dressed CAS-SD has the lowest NPE (1.3 mEh). slightlyimprovetheNPEtoavalueof1.6mEh,andboth The curves obtained by both MR-CCSD methods give variants of the MR-CCSD are equivalent with smooth equivalent results, with NPEs of 3.5 and 3.6 mE . curves. h In the case of F the NPE of the dressed CAS-SD is 2 0.9 mE and the NPE of the µ-MR-CCSD is 1.6 mE , h h Triple-bond breaking bothbetterthanthe NPEoftheλ-MR-CCSDwhichhas anNPEof3.1mE . Also,onecanremarkheresomenu- h merical instabilities in the λ-MR-CCSD where the curve N2 isthetypicalbenchmarkforbreakingatriplebond. is not perfectly smooth. Here, we have used a CAS(6,6) reference wave function. In the next example, the π bond of ethylene is bro- At the CAS-SD level, the NPE is 1.7 mEh, and the ken by the rotation of the CH fragments. The CAS-SD dressedCAS-SDdoesn’treducetheNPE.Here,itisnec- 2 has an NPE of 1.5 mE , and using the dressed CAS-SD essary to use reference-dependant amplitudes to recover h reduces the NPE to 0.7 mEh. The µ-MR-CCSD gives alowNPE:1.0mEhwiththeλ-MR-CCSD,and0.7mEh an NPEs of 0.5 mE , and the NPE obtained with the with the µ-MR-CCSD. h λ-MR-CCSD is slightly better with an NPE of 0.3 mE . h B. Excited states Insertion of Be in H 2 Triplet state of F 2 We present the results obtained by the insertion of a We report here calculations on the triplet state 3Σ+ beryllium atominto the H2 molecule, which is a popular u of F . The reference wave function was prepared in two benchmark for MR-CC methods. The reference is still a 2 different ways, both using restricted open-shell Hartree- CAS(2,2)forcomparisonwiththeliterature,eventhough this choice of reference is not the most appropriate for a Fockmolecularorbitals. Thefirstreferencewavefunction labeled m = 1 is a single open-shell determinant, and correct description of the reaction. The geometries are s given by the relation the second wave function is the triplet ms = 0, made of two determinants 1/√2(αβ βα). − z =2.54 0.46x (a.u.) (30) To ensure that the CAS-SD is a strict eigenfunction − of the Sˆ2 operator, we have included in Ψ all the de- SD where the beryllium atom is at the origin and the hy- terminants with the same space part as the Singles and drogen atoms are at the coordinates (x,0, z). In this DoubleswithrespecttotheCAS.Thesedeterminantsare ± particularcase,theµ-MR-CCSDgivesaNPEof1.8mE treated in the same way as Singles and Doubles and are h 7 TABLEI. Non-parallelism errors (NPE) and maximum errors with respect to the Full-CI potential energy surface (mE ) h CAS-SD λ-MR-CCSD Dressed CAS-SD µ-MR-CCSD NPE Max Error NPE Max Error NPE Max Error NPE Max Error C H 5.1 35.5 3.6 8.4 1.3 8.3 3.5 8.3 2 6 F 3.8 19.8 3.1 4.0 0.9 4.2 1.6 3.9 2 C H twist 1.5 27.7 0.3 6.7 0.7 7.2 0.5 6.7 2 4 BeH 2.9 4.1 1.3 1.8 2.0 2.4 1.8 2.2 2 H O 1.8 4.6 0.7 1.2 0.2 1.3 0.5 1.2 2 C H stretch 2.8 22.1 1.6 5.3 1.7 6.2 1.6 5.2 2 4 N 1.8 9.0 1.0 2.2 1.7 3.9 0.7 2.8 2 F2 3Σ+u(ms=1) 2.5 18.6 1.3 3.3 1.3 3.5 1.2 3.3 F2 3Σ+u(ms=0) 2.5 18.6 1.2 1.8 1.3 3.5 1.1 3.3 HF(ground state) 2.6 14.6 1.8 3.4 2.1 4.4 1.8 4.1 HF(excited state) 3.3 20.9 8.8 8.5 10.5 10.1 7.1 8.3 F (local) 3.8 19.8 1.2 3.2 1.5 3.1 1.0 4.6 2 N (local) 1.8 9.0 3.8 5.0 1.1 3.5 1.1 2.8 2 treated variationally in the diagonalizations. Of course, Avoided crossing in HF and LiF those which are Triples or Quadruples with respect to Ψ are excluded from the set of the α and have no ref { } We have calculated the potential energy surfaces of effect in the dressing. the two lowest 1Σ+ states of HF, using as reference wave function the CAS(2,2) with state-averaged CAS- To reduce the computational cost, the Triples and SCF molecular orbitals in the aug-cc-pVDZ basis set. Quadruples were not augmented with all the determi- Figure 3 shows the NPEs of the ground and excited nants with the same space part. The absence of some states. Inthegroundstate,the NPEis1.8mE forboth h determinantsgivesrisetoaslightdeviation(<10−6a.u.) MR-CCSD variants, but the λ-MR-CCSD shows some of Sˆ2 fromthe desiredeigenvalue,anditis expected to numerical instabilities, as opposed to the µ-MR-CCSD h i have some impact on the iterative dressing. It is worth which gives a very smooth curve. checking the effect of this deviation from the exact spin In the excited state, the situation is different : sur- multiplicity. The first test concerns the comparison of prisingly the best NPE is obtained by the CAS-SD, but the m =0 and m =1 components of a triplet state. s s this may be due to the fact the molecule is particularly favorable to the CAS-SD : both H and H+ have no cor- In all the cases, the NPE of the CAS-SD (2.6 mE ) is relation energy, so the wave functions of the dissociated h improvedtoavalueof1.1–1.5mE . Asexpectedthetwo moleculecanbeexpressedasaproduct. Thetwovariants h variantsoftheMR-CCSDarestrictlyequivalentform = of the MR-CCSD agree at short and long distances, but s 1. Indeed, for both variants the usual single-reference they differ significantly between 2 and 3.0 ˚A, after the amplitudes c /c are recovered. The amplitudes of the region of the avoided crossing. To understand these dif- i 0 λ-MR-CCSDlowerthe curveby 1mE whengoingfrom ferences, we have plotted the two eigenvalues of the two h m = 1 to m = 0. The dressed CAS-SD also gives state-specific Hamiltonians, one dressed for the ground s s a lower energy, but only by 0.5 mE . This is due to state and one dressed for the excited state. It appears h the increased number of degrees of freedom in the fit of that between 2 and 3.0 ˚A, the lowest eigenvalue of the the amplitudes as no additional constraint is imposed Hamiltonian dressed for the excited state is very badly to enforce the m -invariance. But when the reference- described. The reason is that the fitting procedure for s dependence is introduced via the µ , it is imposed to theamplitudesisaleast-squaresfitontheCAS-SDwave i recovertheCAS-SDwavefunctionwhichism -invariant, function of the state of interest, so the quality of the s andthisstepcompensatestheadditionalfreedomgained dressing for the determinants which have small coeffi- in the fitting, and the m = 1 and m = 0 MR-CCSD cients on the state of interest but large coefficients on s s curves differ by less than 0.1 mE . the other state will be very low. The λ-MR-CCSD has h amplitudes which depend less on the wave function, so Ifoneconsiderstheerroronthesinglet-tripletgapwith thequalityisequivalentonbothstates,andthechoiceof respect to the Full-CI reference, it appears clearly that these amplitudes is better suited for calculating excited the µ-MR-CCSD gives the most accurate results, with states within the same symmetry. errors lying between 0.1 mE and 1.3 mE along the Infigure5wehaverepresentedthe avoidedcrossingof h h curve. LiF,alsocalculatedwiththeaug-cc-pVDZbasisset. The 8 CH 2 6 F CI) (a.u.) 0000....000033333456 000...000122802 2 E - E(FCI) (a.u.) 0000000.......000000033300010127890 E - E(FCI) (a.u.) 000000......000000001111680246 E - E(F 000...000000456 000...000000024 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1 1.5 2 2.5 3 3.5 4 R (Angs.) R (Angs.) CH twist 2 4 BeH u.) 0.0280 0.0045 2 CI) (a. 00..00227705 0.0040 F 0.0035 E - E(CI) (a.u.) 00000.....00000220006677805050 E - E(FCI) (a.u.) 00000.....00000000001122305050 F E( 0.0065 0.0005 E - 0.0060 0.0000 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 4 Angle (degrees) z (a.u.) CH HO 2 4 000...000000445050 2 FCI) (a.u.) 0000....000022221234 E - E(FCI) (a.u.) 000000......000000000000112233050505 E - E(FCI) (a.u.) 000000......000000120000905678 00..00000005 E - E( 00..000034 1 1.5 2 2.5 3 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 R (Angs.) R (Angs.) N 2 u.) 0.010 CI) (a. 0.009 F 0.008 E( E - 0.007 0.004 CI) (a.u.) 0.003 Dresµλs--eMMd RRCC--AACCSSCC--SSSSDDDD E(F 0.002 E - 0.001 1.00 1.50 2.00 2.50 3.00 3.50 4.00 R (Angs.) FIG.1. Dissociation curves. Differencewith respect totheFull-CI energyusingtheMR-CCSDmethod presentedin ref15 and with theMR-CCSD method proposed in thiswork, as well as theCAS-SDand thedressed CAS-SD. physicalsituationis similarto HF, butthe energydiffer- position of the avoided crossing is very well reproduced ence between the ground and the excited states is much bythethreemethods: theCAS-SDcrossesat6.3˚A,the smaller. A striking result is that the λ-MR-CCSD, al- Full-CI crossesat6.8 ˚Aandthe dressedCAS-SD andthe thoughbeingstate-specific,isabletoreproduceverywell twoMR-CCSDvariantscrossat6.9˚A.Theµ-MR-CCSD the whole potential energy surfaces of both states. The andλ-MR-CCSDcoincideintheshort-range( 5˚A)and ≤ 9 F2 (3Σu) HF 0.0190 0.0185 Excited state CI) (a.u.) 000...000111778050 00..002202 E(F 0.0165 0.018 E - 0.0160 0.0155 u.) 0.016 0.0150 a. 0.014 0.0040 CAS-SD CI) ( 0.012 E(FCI) (a.u.) 00000.....00000000001223350505 DDrreessµµλλss----eeMMMMdd RRRRCC----AACCCCSSCCCC--SSSSSSDDDDDD mmmmmmssssss======010101 E - E(F 000...000001680 E - 0.0010 0.004 0.0005 0.002 0.0000 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 R (Angs.) Ground state 0.0030 CAS-SD 0.022 ∆E - E(FCI) (a.u.) 00000.....00000000000112250505 DDrreessµµλλss----eeMMMMdd RRRRCC----AACCCCSSCCCC--SSSSSSDDDDDD mmmmmmssssss======010101 E - E(FCI) (a.u.) 00000000........000000000011111268024680 Dresµλs--eMMd RRCC--AACCSSCC--SSSSDDDD ∆ 0.0000 0.004 0.002 -0.0005 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 R (Angs.) -0.0010 -0.0015 1.00 1.50 2.00 2.50 3.00 3.50 4.00 FIG. 3. Difference with respect to theFull-CI energy for the R (Angs.) two lowest 1Σ+ states of HF. FIG.2. F 3Σ+. DifferencewithrespecttotheFull-CIenergy 2 u -99.5 forthems =0andms =1wavefunctions(top),anderroron thesinglet-tripletgap∆E =E(3Σ+)−E(1Σ+)(bottom). On -99.6 u g bothgraphics,thetwocurvesofthedressedCAS-SDcoincide. -99.7 u.) -99.8 a. gy ( -99.9 cinomtheevloernygcrlaonsgeein(≥en7.e2rg˚Ay),inbutthewhreegniotnheotfwtohestcartoesssbineg- otal Ener --110000..10 the dressedCAS-SD and the µ-MR-CCSD are unable to T -100.2 give sensible values. This disappointing result motivates -100.3 Ground-state amplitudes a future work on a multi-state µ-MR-CCSD. Excited-state amplitudes -100.4 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 R (Angs.) Sensitivity to the choice Molecular Orbitals FIG.4. Potentialenergysurfacesofthetwolowest1Σ+states The µ-MR-CCSD algorithm we propose is in the ofHFwiththeµ-MR-CCSDmethod. Theenergyofthestate Jeziorski-Monkhorst framework, so it is not invariant corresponding to the dressing is plotted in plain curves, and with respect to the choice of molecular orbitals. In this theenergy of theother state is plotted in dashed curves. section,wecheckeditssensitivitytothechoiceoftheMO setbycomparingresults obtainedwithpseudo-canonical CAS-SCForbitalsandwith localizedMOsinthe F and 0.7 mE to 1.1 mE . On the other hand, the dressed 2 h h N molecules (figure 6). CAS-SD gives a better NPE with local orbitals, going 2 In the F molecule, using localized MOs is a better from 1.7 mE to 1.1 mE . 2 h h choice than the pseudo-canonical MOs. The best NPE The fact that the µ-MR-CCSD is less sensitive to the is obtained by the µ-MR-CCSD method with a value MO set than the λ-MR-CCSD can be understood. By of 1.1 mE . In the case of N , the situation is differ- changing the MO set, a single excitation rotates into h 2 ent: the NPEofthe λ-MR-CCSD goesfrom0.9mE to a combination of single and double excitations. In the h 3.7mE ,andtheNPEoftheµ-MR-CCSDincreasesfrom λ-MR-CCSD method, the amplitudes are calculated by h 10 IV. CONCLUSIONS LiF -106.96 We have proposed a method to determine reference- independentamplitudesbyfittingtheCAS-SDCIvector. -106.98 Theseamplitudesmaybeusedtoperformastate-specific u.) iterativedressingoftheCAS-SDHamiltonianinorderto a. E ( -107.00 takeintoaccounttheeffectoftheTriplesandQuadruples CAS-SD in the spirit of the Coupled Cluster formalism. Alterna- Full-CI -107.02 Dressed CAS-SD tively, these amplitudes may be rescaled to reproduce µ-MR-CCSD the exact coefficients of the singles and doubles to in- λ-MR-CCSD -107.04 troduce a reference-dependent character. In that case, 4 4.5 5 5.5 6 6.5 7 7.5 8 the CAS-SD CI vector is recoveredby the application of R (Angs.) (1+Tˆ) on the reference wave function, so we reach here the Jeziorski-MonkhorstCoupled Cluster formalism. FIG.5. Potentialenergysurfacesofthetwolowest1Σ+states The CAS-SD dressed with reference-independent am- of LiF. plitudes gives excellent results for single-bond breaking (F andethane)andthesimultaneousbreakingofthetwo 2 O—Hbonds ofwater,with a non-parallelismerrorlower F2 than the milli-Hartree. When the active space becomes 0.0045 larger, it is necessary to go to the reference-dependent Dressed CAS-SD 0.0040 µ-MR-CCSD MR-CCSD introducing the µ factors in Eq. 28. In the 0.0035 λ-MR-CCSD case of ethylene and N2, this keeps the NPE to a value u.) close to the milli-Hartree. a. 0.0030 CI) ( 0.0025 We have shown numerically that the here-proposed E(F 0.0020 amplitudes are not very sensitive to the value of ms for E - open-shell systems, and to the choice of the molecular 0.0015 orbitals. This is clearly an improvement compared the 0.0010 amplitudes proposed earlier15. But we have also shown 0.0005 thattheformeramplitudesareabetterchoicewhencom- 1.00 1.50 2.00 2.50 3.00 3.50 4.00 puting excited states of the same symmetry because the R (Angs.) here-proposedamplitudeshaveamuchmorepronounced N2 state-specific character which may be disadvantageous if 0.0050 the states are too close in energy. This problem can be Dressed CAS-SD 0.0045 µ-MR-CCSD curedby leavingthe state-specific formalismfor a multi- λ-MR-CCSD 0.0040 state formalism37, and this will be the object of a future a.u.) 0.0035 work. CI) ( 0.0030 Acknowledgments. This work has been made F E( 0.0025 through generous computational support from CALMIP E - (Toulouse) under the allocation 2015-0510, and GENCI 0.0020 under the allocation x2015081738. 0.0015 0.0010 1.00 1.50 2.00 2.50 3.00 1F.Coester,Nucl.Phys.7,421(1958). R (Angs.) 2F.CoesterandH.Ku¨mmel,Nucl.Phys.17,477(1960). 3J.Cˇi´zˇek,J.Chem.Phys.45,4256(1966). 4R. J. Bartlett, J. Watts, S. Kucharski, and J. Noga, FIG. 6. Comparison between pseudo-canonical (dashed Chem.Phys.Lett.165,513(1990). curves) and localized (plain curves) MOs in F and N . Dif- 5J.Goldstone, Proc.Roy.Soc.A239,267(1957). 2 2 ference with respect to theFull-CI energy. 6I.Nebot-Gil,J.San´chez-Marin´,J.P.Malrieu,J.L.Heully, and D.Maynau,J.Chem.Phys.103,2576(1995). 7J. P. Malrieu, P. Durand, and J. P. Daudey, J.Phys.A:Math.Gen.18,809(1985). 8B.Kirtman,J.Chem.Phys.75,798(1981). taking into account the matrix elements of the Hamil- 9D. I. Lyakh, M. Musial , V. F. Lotrich, and R. J. Bartlett, tonian, which are of different nature depending on the Chem.Rev.112,182(2012). degree of excitation, so the amplitudes are expected to 10U. S. Mahapatra, B. Datta, and D. Mukherjee, changesignificantly. Intheµ-MR-CCSDvariant,theam- Mol.Phys.94,157(1998). plitudesareadjustedinsuchawaythattheyfittheCAS- 11S. Das, D. Mukherjee, and M. K´allay, J.Chem.Phys.132,074103(2010). SD wave function, which is invariant by rotation of the 12A.Szabados, J.Chem.Phys.134,174113(2011). MOs. Therefore, it is expected to be more robust with 13U. S. Mahapatra, B. Datta, and D. Mukherjee, respect to the MO set. J.Chem.Phys.110,6171(1999).

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Alternative definition of excitation amplitudes in Multi-Reference state-specific Coupled Cluster PDF

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Preview Alternative definition of excitation amplitudes in Multi-Reference state-specific Coupled Cluster

Alternative definition of excitation amplitudes in Multi-Reference state-specific Coupled Cluster Yann Garniron,1 Emmanuel Giner,2,3 Jean-Paul Malrieu,1 and Anthony Scemama1,a) 1)Laboratoire de Chimie et Physique Quantiques (CNRS 5626), IRSAMC, Universit´e P. Sabatier, Toulouse (France) 2)Dipartimento di Scienze Chimiche e Farmaceutiche, Universita di Ferrara, Via Fossato di Mortara 17, 7 I-44121 Ferrara (Italy) 1 3)Max Planck Institut for solid state research, Heisenbergstraße 1, 70569 Stuttgart, 0 Germany 2 A central difficulty of state-specific Multi-Reference Coupled Cluster (MR-CC) formalisms concerns the def- n inition of the amplitudes of the single and double excitation operators appearing in the exponential wave a operator. If the reference space is a complete active space (CAS) the number of these amplitudes is larger J thanthenumberofsinglyanddoublyexciteddeterminantsonwhichonemayprojecttheeigenequation,and 7 one must impose additional conditions. The present work first defines a state-specific reference-independent 1 operator Tˆ˜m which acting on the CAS component of the wave function Ψm maximizes the overlapbetween h] (1+Tˆ˜m)Ψm and the eigenvector of the CAS-SD CI matrix Ψm | 0.i This operator may be used to | 0 i | CAS−SDi p generate approximate coefficients of the Triples and Quadruples, and a dressing of the CAS-SD CI matrix, - according to the intermediate Hamiltonian formalism. The process may be iterated to convergence. As a m refinement towards a strict Coupled Cluster formalism, one may exploit reference-independent amplitudes he provided by (1 +Tˆ˜m)|Ψm0 i to define a reference-dependent operator Tˆm by fitting the eigenvector of the c (dressed) CAS-SD CI matrix. The two variants, which are internally uncontracted, give rather similar re- . sults. The new MR-CC version has been tested on the ground state potential energy curves of 6 molecules s c (up to triple-bond breaking) and a two excited states. The non-parallelism error with respect to the Full-CI si curves is of the order of 1 mEh. y h p I. INTRODUCTION TheCCequations,obtainedbyprojectingtheeigenequa- [ tion on each of the Singles and Doubles lead to coupled 1 The single-reference Coupled Cluster (CC) quadraticequations. Inpractice, guessvalues of the am- v formalism1–4 is the standard technique in the study of plitudes of the Tˆ0→i operators appearing in the Tˆ oper- 4 the ground state of closed-shell molecules, i.e. those ator may be taken as the coefficients of the Singles and 6 for which a mean-field treatment provides a reasonable Doubles i in the intermediate normalization of the SD 7 zero-order single-determinant wave-function Φ . This CI vector|.iThe solution of the CC equations may be ob- 4 0 method incorporates the leading contributions to the tained by treating the effect of the Triples and Quadru- 0 . correlationenergy in a given basis set, it is based on the ples as an iterative dressing of the SD CI matrix,6 ac- 1 linked-cluster theorem5 and is size-consistent since it is cordingto the IntermediateEffectiveHamiltonian(IEH) 0 free from unlinked contributions. The method generates theory.7,8Thefieldofapplicationofthismethod,whichis 7 1 anapproximatewavefunctionundertheactionofawave both intellectually satisfying and numericallyefficient, is : operator Ωˆ acting on the single-determinant reference however limited to the systems and the situations where v Φ , and assumes an exponential character to the wave a single-determinant zero-order description is relevant. i 0 X operator This is no longer the case when chemical bonds are bro- ken, creating open shells, as occurs in most of the chem- r a Ψ=ΩˆΦ =exp TˆΦ (1) ical reactions. The magnetic systems generally present 0 0 several open shells and the low spin-multiplicity states (cid:16) (cid:17) The most popular version only introduces single and are intrinsically of multiple-determinant character. Due double excitation operators in Tˆ, and is known as the to near degeneracies, most of the excited states are not Coupled Cluster Singles and Doubles (CCSD) approxi- only of multi-determinantal but of multi-configurational mation. It incorporates the fourth order correction of character. The conception of a multi-reference counter- the quadruply excited determinants. The lacking fourth part of the CCSD formalism is highly desirable, and has order contribution concerns the triply excited determi- been the subject of intense research. The most compre- nants, which may be added in a perturbative manner. hensive review has been given by R. Bartlett and his colleagues.9 Forformalreasonsandinparticularto treat correctly the breaking of bonds, the reference space, or modelspace,isusuallytakenasaCompleteActiveSpace a)Electronicmail: [email protected] (CAS), i.e. the Full-CI of a well-defined number of elec- 2 trons (the active electrons) in a well-defined set of or- reference-independentamplitudesoftheexcitations,may bitals (the active MOs). The other MOs are called inac- be exploited directly to generate approximate values of tive. Letus label I , J ,... the referencedeterminants. the coefficients of the triply and quadruply excited de- | i | i The determinants i , j ,... whichinteractwith the ref- terminants,accordingtotheexponentialstructureofthe | i | i erence space are obtained under purely inactive or semi- wave operator. From these coefficients one may dress active single and double excitations, they generate the the CAS-SD CI matrix, redefine amplitudes and iterate CAS-SD CI space, the diagonalizationof which provides the process to convergence. This solution, presented in a size-inconsistent energy Em and the correspond- section 2C, is not an MR-CC technique, one may call it CAS−SD ing eigenvector, an exponential dressing of the CAS-SD CI matrix. Sec- tion 2D redefines reference-dependent excitation ampli- Ψm = Ψm + Ψm | CAS−SDi | 0 i | SDi tudesfromthereference-independentamplitudesbyafit- = Cm+ I + cm i (2) ting of the previous amplitudes on the coefficients of the I | i i | i Singles and Doubles of the (dressed) CAS-SD CI eigen- I∈CAS i∈/CAS X X vector. This represents an alternative solution to multi- with Ψm Ψm =1. parentageproblemandopensthewaytoastrictMR-CC h CAS−SD| CAS−SDi One strategy, which is not very aesthetic since it formalism. Section 3 presents a series of numerical tests breaks the symmetry between degenerate reference de- on the bond breaking of single, double and triple bonds terminants, but which has given rather satisfactory re- in ground states of molecules as well as a few tests on sults, consists in selecting (eventually in an arbitrary excited states. The results are comparedto our previous manner) a specific single reference and in introducing in proposal and with full Configuration Interaction (FCI) the wave operator the multiple excitations which gener- results. ate the other references (the other determinants of the modelspace). The otherstrategiesconsideralltherefer- ences onanequalfooting, andarereallymulti-reference. II. FORMALISMS Let us call N the number of references, and n the num- ber of SD determinants. If the treatment pretended to In this section, all the presented formalisms are state- provide N eigenvectors simultaneously, one might define specific. To simplify the notations we will consider that the N n amplitudes sending from the references to the state superscriptm is implicit forthe wavefunctions × the outer-space determinants, in a unique manner but (Ψm Ψ) and for the excitation operators (Tˆm Tˆ). this state-universalapproachisnotpracticablewhenthe → → model space is a CAS. Most of the proposed formalisms are state-specific. In A. The multi-parentage problem in the this case one faces the famous multi-parentage prob- Jeziorski-Monkhorstapproach lem. This problem is recalled in section 2A. Sufficiency conditions have to be imposed.10 One solution was pro- Since one wants to produce a MR-CCSD method, one posedby Mukherjee andcoworkers,and has beenwidely may start from a preliminary CAS-SD CI calculation tested.11–13 Another one had been proposed earlier by whichwillhelptofixguessvaluesoftheamplitudesofthe one ofus (JPM) and coworkers.14 It consists,for a given excitationoperators. Letus call I , J ,... the determi- outer-spacedeterminant,inscalingtheamplitudesofthe | i | i nants of the CAS, i.e. the so-called reference vectors, various excitation operators Tˆ on the interaction be- I→i and i , j ,... the Singles andDoubles whichdo notbe- tween the outer-space determinant and its parents. A | i | i long to the CAS and interact with them. The resulting recent work has implemented this second solution of the approximate wave function of the targeted state Ψ is state-specific MR-CC problem and has tested its accu- | i written racyandrobustnessonaseriesofmolecularbenchmarks, comparingitsresultstotheFull-CI(FCI)energies.15The Ψ = C I + c i (4) CAS−SD I i presentworkproposesanalternativeprocesstodefinethe | i | i | i I i amplitudesoftheexcitationoperators. Thestate-specific X X MR-CC formalisms are usually based on the Jeziorski- Although this function is not size consistent one may Monkhorst16 splitting ofthewaveoperatorintoasumof note that the coefficients on the CAS determinants are operators acting individually on the various references no longer those of the CAS-CI : they incorporate the effect ofthe dynamicalcorrelationonthe compositionof Tˆm = Tˆm I I (3) the CAS component of the wave function. I | ih | In CC formalisms the wave operator Ωˆ is assumed to I X take an exponential form We shall leave in a first time this assumption and de- fine in section IIB a reference-independent operator Tˆ Ωˆ =exp(Tˆ) (5) which acting on the component of the desired state in the modelspace, Ψm , providesa vectoras closeaspos- and in our previous MR-CC formalism15 the Jeziorski- | 0 i sible to the CAS-SD eigenvector. This solution, defining Monkhorst structure of the wave operator was adopted, 3 introducing reference-specific wave operators acting B. Introduction of reference-independent amplitudes specifically on each reference vector (Eq.3). One may exploit the knowledge of the CAS-SD CI eigenvector to The present formalism will leave in the first step the determine guess operators Tˆ defined in such a manner I Jeziorski-Monkhorst formulation of the wave operator that andwillconsiderthe possibility to define a unique state- specific reference-independent operator Tˆ, written as a Ψ = C Tˆ I (6) | CAS−SDi I I| i sum of single and double excitation operators, I X The Tˆ operators are a sum of single and double exci- Tˆ = tmn→pqa†pa†qanam+ tm→pa†pam (12) I tations TˆI→i possible on I , multiplied by an amplitude mXnpq Xmp t | i = t Tˆ + t Tˆ (13) I→i mn→pq mn→pq m→p m→p mnpq mp Tˆ = t Tˆ (7) X X I I→i I→i where the indices p and q run on the virtual and active i X MOs and the indices m and n run on the inactive occu- In the single-reference CC the amplitudes of the excita- pied and active MOs, excluding the possible occurrence tion operators are obtained by projecting the eigenequa- of 4 active MOs. tion on the singly and doubly excited determinants, the This operator has the same form as the one intro- number of unknowns is equal to the number of equa- duced by the internally-contracted MR-CC (ic-MRCC) tions. This is no longer the case in the MR context : methodbyEvangelistaandGauss,17andbyHanauerand projecting the eigenequation on each on the singly or K¨ohn,18 but it differs by both its determination and by doubly excited vectors i is not sufficient to define the the way we use it, as will appear later. The ic-MRCC | i amplitudes tI→i since for many classes of excitation an method determines the amplitudes of the excitation by outer-spacedeterminantinteractswithseveralreferences, solving the projected Coupled Cluster equations, where i =TˆI→i I =TˆJ→i J . The condition the amplitudes appear as linear and quadratic terms. | i | i | i Hereafter we exploit the knowledge of the CAS-SD CI C = t C (8) eigenvector to determine guess values of the reference- i I→i I I independent amplitudes. These excitation amplitudes X will be used later on to estimate the coefficients of the is not sufficient to define the amplitudes, even if one re- Triples and Quadruples, and perform an iterative dress- strictstheexcitationoperatorstosingleanddoubleexci- ing of the CAS-SD CI matrix introducing the coupling tations. Additional constraints have to be introduced to between the Singles and Doubles with the Triples and fixtheamplitudes,andthisisthefamousmulti-parentage Quadruples. problem. The number of amplitudes is larger than the We propose a criterion to fix the amplitudes t = number of outer-space determinants so that one cannot t ,t . Given the factthat we haveat our dis- mn→pq m→p determine directly guess values of the amplitudes from { } posal the CAS-SD wave function, a natural way to solve Eq. 6. Different additional constraints have been pro- this overdeterminedproblem is to minimize the distance posed. Oneofthemconsistsinscalingtheamplitudeson between the CAS-SD vector and the vector obtained by the Hamiltonian interactions between the references and applying the (1+Tˆ) operatoronthe CAS wavefunction the outer space determinants, tI→i = hi|Hˆ|Ii. (9) argtmink(1+Tˆ)|Ψ0i−|ΨCAS−SDik (14) tJ→i iHˆ J =argmin Tˆ Ψ Ψ , h | | i k | 0i−| SDik t This constraint is expressed as Tˆ Ψ being normalized such that Tˆ Ψ = Ψ . t =λ iHˆ I (10) | 0i k | 0ik k| SDik I→i i To perform the minimization, we build the N N h | | i SD t transformationmatrixA = iTˆ Ψ w×hich i,mn→pq mn→pq 0 where h | | i maps from the outer space of determinants i to the {| i} λi = iHˆciΨ (11) sspeaarccehoffoerxtchiteedvewctaovreofufnacmtipolnitsu{dTeˆms nt→wpqh|iΨch0i}m,inainmdizwees 0 h | | i A.t c by solving the normal equations This solution has been recently implemented15 and k − k shown to provide excellent agreements with Full-CI re- (A†A)t=A†c (15) sults on a series of molecular problems. It only presents minor stability problems in comparisonwith the present Note that in the single-reference case, A is a permuta- suggestion when the term iHˆ Ψ is small. From now tion matrix and the CAS-SD wave function is exactly 0 h | | i on, we will refer to this method as the λ-MR-CCSD. recovered. 4 The matrix A is usually so large that the use of stan- TriplesandQuadruplesasobtainedbytheactionof 1Tˆ2. 2 dardsingularvaluedecomposition(SVD)routinestoob- Actually one may assume, in the spirit of the internally- tain the least squares solution is prohibitive. contracted MR-CC methods, that the wave operator Ωˆ Let us first consider the most numerous 2-hole-2- generating the correlated wave function Ψ from Ψ , 0 particle inactive double excitations Tˆ . Acting on jk→rs a determinant I the operator creates a determinant Ψ=ΩˆΨ0 (21) | i i =Tˆ I which can only be produced by this pro- |ceiss. Tjkh→errse|foire, the corresponding rows of A contain has an exponential structure, only one non-zero element located in the jk rs col- Ωˆ =exp(Tˆ) (22) → umn with value A =C . The condition fixing the i,jk→rs I amplitude tjk→rs is given by But this form will be simply used to estimate the coeffi- cients of the triply and quadruply excited determinants argmin Tˆ Ψ t Ψ (16) jk→rs 0 jk→rs SD α , leaving the internally-contracted structure of the k | i −| ik tjk→rs {o|utie}r-space. The coefficients of these determinants are which is obtained by minimizing estimated as 2 1 c = αTˆ2 Ψ . (23) min C t c iTˆ I (17) α 2h | | 0i I jk→rs i jk→rs tjk→rs − h | | i! I i X X In practice all the determinants α are generated. {| i} This condition turns out to be satisfied for For each α one finds the reference determinants I α | i {| i} which differ by at most 4 orbital substitutions from α C c tjk→rs = I CI2i (18) (itsgrand-parents). Onethenidentifiesthesetofcomp|lei- P I I mentaryexcitations (p,q) Tˆ Tˆ I = α as the prod- p q One may notice that this is thPe weighted average of the | i | i ratios between the coefficients of the doubly excited de- uαctscwonhtircihbugteensertaoteint|sαiasfsroo(cid:12)(cid:12)cmiat{e|Idi}cαo.effiTchieeongtecnealboygythoef terminants i andthecoefficientoftheiruniquereference | i α generator, | i quantity tptqCI. Knowing |αi, one also knows the Sin- gles and Doubles i with which it interacts through α {| i} 1 c the matrix elements iHˆ α , and in the eigenequation tjk→rs = ICI2 I CI2(cid:18)CiI(cid:19)! (19) relative to |ii h | | i X For all the remainPing active excitations, A remains iHˆ i E ci+ iHˆ J CJ+ h | | i− h | | i sparse since the maximum number of non-zero elements (cid:16) (cid:17) XJ (24) per column is equal to the number of reference determi- iHˆ j c + iHˆ α c =0. j α nants. Hence, we use Richardson’s iterative procedure19 h | | i h | | i j α X X Onemayreplacethelastsumbyadressingofthematrix t =A†c 0 (20) elements between the determinant i and the references (cid:26)tn+1 =A†c+ I−A†A tn which are grand-parents of α , | i | i which may be implemented v(cid:0)ery efficie(cid:1)ntly using sparse matrix products. There are cases where multiple amplitudes applied i∆I = iHˆ α  tptq (25) to different references lead to same determinant : h | | i h | | i Tˆ I = Tˆ J = i . If this determinant can Xα n(p,q) TˆpXTˆq|Ii=|αio  jk→rs| i lm→tv| i | i   be reached by no other process, there is an infinity of  (cid:12)  since (cid:12) solutions for the amplitudes. The solution of Eq.(15) is A+cwhereA+isthepseudo-inverseofA,sothesolution i∆I C = iHˆ α c . (26) obtained minimizes the norm of the amplitude vector.20 h | | i I h | | i α I α Inthisway,thearbitrarinessbroughtbythenullspaceof X X Ais minimized inthe amplitude vector,andone obtains The effect of the Triples and Quadruples is incor- the most sensible solution. porated as a change of the columns of the CAS-SD CI matrix concerning the interaction between the ref- erences and the Singles and Doubles. This type of C. Evaluation of the coefficients of Triples and dressing was already employed in our previous MR-CC Quadruples and iterative dressing of the CAS-SD CI matrix implementation.15 One will find in the same reference the practical procedure to make the dressedmatrix Her- The so-determined excitationoperator Tˆ may be used mitian without any loss of information. Of course the to generate approximate values of the coefficient of the whole process may be iterated. The diagonalization of 5 thedressedCAS-SDCImatrixprovidesnewvaluesofthe Astheoverlapbetween(1+Tˆ)Ψ and Ψ has 0 CAS−SD | i | i coefficients,notonlyoftheSinglesandDoubleswhichno beenmaximizedthecoefficientsc˜ andc areexpectedto i i longer suffer from the truncation, but also those of the be very close in particular if c is large, and the parame- i references : the method is fully non-contracted. From ter µ should be close to 1, at least for the determinants i the new wave function new amplitudes are obtained, a which contribute significantly to the wave function. In new dressing is defined and the process is repeated till practice we observe this tendency, but the smallest co- convergence, which is usually rapidly obtained (3-4 iter- efficients are sacrificed during the maximization of the ations). overlap and their µ can be very far from 1. This intro- i ThisformalismisnotastrictMR-CCmethodsincewe duces some instabilities in the iterations, so we chose to exploit the CAS-SD CI function, and since this function limit the values of µ in the [ 2,2] range. The effect on i − slightly differs from the vector resulting from the action the stabilityofthe iterationsissignificant,andthe effect ofTˆ onthe vector. Althoughthe distance betweenthese on the energy is not noticeable. two vectors has been minimized they are not identical, This version returns to the Jeziorski-Monkhorst for- (1+Tˆ)Ψ = Ψ˜ . malismas the waveoperatoragainis a sum ofreference- 0 CAS−SD | i6 | i Once the Tˆ operator has been obtained one might specific operators. The so-obtained amplitudes may be imagine a contracted exponential formalism calculating exploited to generate the coefficients of the Triples and Tˆ2 Ψ and the interaction between Tˆ Ψ and Tˆ2 Ψ , Quadruples, and one may follow the same strategy as in 0 0 0 but| thiis calculation requires to return| toithe dete|rmii- our previous formalism, with an iterative column dress- nants. This formalism would remain internally con- ing of the interactions between the Singles and Doubles tracted and would be less accurate than the procedure and the references. In a strict Coupled Cluster formal- we propose. Actually in this version the deviations of ism, one should redefine the amplitudes of the double the approximate reference-independent amplitudes from excitations by subtracting the products of the comple- optimalones,thosewhichwouldgeneratetheexactcoef- mentary single excitations they contain. For the sake of ficients of the Singles and Doubles, only affects the eval- simplicity, as we did in our previous work, we did not uation of the coefficients of the Triples and Quadruples, proceed to this revision, the difference concerning only and these deviations representa minor source oferrorin fifth-order perturbative corrections. the correction restoring the size extensivity. This relia- Inwhatfollows,wewillrefertothis method asµ-MR- bility will be illustrated in the numerical tests. CCSD as it involves the µi (Eq.28). D. State-specific MR-CC variant III. NUMERICAL TESTS In order to return to a MR-CC formalism, one may In this section, we compare the here-proposeddressed simply exploit the reference-independent amplitudes as CAS-SD and MR-CCSD to the MR-CCSD presented aninitialguesstodefinereference-dependentamplitudes. in ref15 on standard benchmark systems.11,13,17,18,21–30 Currently the determinant i belonging to the Singles To differentiate those two variants, we will label λ-MR- and Doubles has a coefficien|tic˜ in Tˆ Ψ CCSD the variant of ref15 (Eq. 11) and the MR-CCSD i 0 | i of this work will be labeled µ-MR-CCSD (Eq. 28). c˜ = iTˆ Ψ = t C (27) The basis set used is Dunning’s cc-pVDZ,31 and the i 0 l I h | | i molecular orbitals were obtained using the CAS-SCF n(I,l)XTˆl|Ii=|iio code present in GAMESS.32 All the following calcula- tionsweremadeusingtheQuantumPackage,33anopen- (cid:12) which differs from the coefficien(cid:12)t ci in ΨSD . One can source library developed in our group. Full-CI energies | i define a parameter µi, specific of the determinant i , were obtained using the CIPSI algorithm,34–36 and the | i accuracyofthetotalenergiesisestimatedtobeoftheor- c µi = i (28) der of 10−5Eh. In all the calculations (Full-CI, CAS-SD c˜ i and MR-CC), only the valence electrons are correlated (frozen core approximation). which multiplying c˜ will produce the exactcoefficient c i i of i in the (dressed) CAS-SD CI eigenvector. So the | i previous reference-independentamplitudes havenow be- come reference-dependent. The excitation Tˆ which ex- A. Bond breaking l cites I to i (i =Tˆ I )receivesareference-dependent l ampli|tuide | i | i | i For all the applications we compare the dressed CAS- SD and µ-MR-CCSD with the λ-MR-CCSD and the t =t =µ t . (29) CAS-SD values. Results are also given using the I→i l,I i l reference-independent dressing of the CAS-SD CI ma- The same excitation will receive a somewhat different trix. All the applications are presented as energy differ- amplitude when it acts on another reference t =t . ences with respect to the Full-CI energy estimated by a l,J l,I 6 6 CIPSI calculation with a second-order perturbative cor- whichislargerthantheNPEof1.3mE obtainedbythe h rection. Figure 1 shows the difference of energy with λ-MR-CCSD. This is due to only one point of the curve, respectto the Full-CI alongthe reactioncoordinate. Ta- the maximum which is higher by 0.4 mE , all the other h ble I summarizes the non-parallelism errors (NPE) and points being very close by less than 0.1 mE . Here, the h themaximumoftheerrorobtainedalongthecurve. The dressed CAS-SD and the µ-MR-CCSD are equivalent. MR-CC treatment reduces the average and maximum error of the CAS-SD with respect to Full-CI by a fac- tor close to 4. The correction is larger when the sys- Two bond breaking tem involves an important number of inactive electrons (F2, C2H6) than when this number is small (BeH2, N2). For breaking two bonds we have used CAS(4,4) wave Oneactuallyknowsthatthe size-consistencyerrorofthe functions as the reference space. The first example is CAS-SDtreatmentincreaseswiththenumberofinactive the simultaneous breakingofthe twoO—Hbonds ofthe electrons, this error disappears in the MRCC treatment, watermoleculebystretching. Here,theCAS-SDexhibits which essentially misses some fourth-order connected ef- a NPE of 1.8 mE which is significantly improved to h fects of the Triples. 0.2 mE with the dressed CAS-SD. The µ-MR-CCSD, h with an NPE of 0.5 mE , is slightly more parallelto the h Full-CI curve than the λ-MR-CCSD which has an NPE Single-bond breaking of 0.7 mE . h The second example is the double-bond breaking of We present here the single bond breaking of the σ ethylene by stretching. One should first clarify that the bonds of C H and F molecules and of the π bond of energydifferencesinthefiguredonotmatchthoseofthe 2 6 2 ethylene. The active spaces were chosen with two elec- torsion along the bond because in the former example trons in two MOs, the minimum wavefunctions to de- the reference was a CAS(2,2), and here it is a CAS(4,4). scribe properly the dissociation of the molecules. In the Dressing the CAS-SD reduces the NPE from 2.8 mEh to case of ethane, the NPE of the CAS-SD is 5.1 mEh, and 1.7 mEh. One can remark a discontinuity in the curve is reduced to 3.5 mE with the µ-MR-CCSD. The curve at large distances. The µ-MR-CCSD and λ-MR-CCSD h of the dressed CAS-SD has the lowest NPE (1.3 mEh). slightlyimprovetheNPEtoavalueof1.6mEh,andboth The curves obtained by both MR-CCSD methods give variants of the MR-CCSD are equivalent with smooth equivalent results, with NPEs of 3.5 and 3.6 mE . curves. h In the case of F the NPE of the dressed CAS-SD is 2 0.9 mE and the NPE of the µ-MR-CCSD is 1.6 mE , h h Triple-bond breaking bothbetterthanthe NPEoftheλ-MR-CCSDwhichhas anNPEof3.1mE . Also,onecanremarkheresomenu- h merical instabilities in the λ-MR-CCSD where the curve N2 isthetypicalbenchmarkforbreakingatriplebond. is not perfectly smooth. Here, we have used a CAS(6,6) reference wave function. In the next example, the π bond of ethylene is bro- At the CAS-SD level, the NPE is 1.7 mEh, and the ken by the rotation of the CH fragments. The CAS-SD dressedCAS-SDdoesn’treducetheNPE.Here,itisnec- 2 has an NPE of 1.5 mE , and using the dressed CAS-SD essary to use reference-dependant amplitudes to recover h reduces the NPE to 0.7 mEh. The µ-MR-CCSD gives alowNPE:1.0mEhwiththeλ-MR-CCSD,and0.7mEh an NPEs of 0.5 mE , and the NPE obtained with the with the µ-MR-CCSD. h λ-MR-CCSD is slightly better with an NPE of 0.3 mE . h B. Excited states Insertion of Be in H 2 Triplet state of F 2 We present the results obtained by the insertion of a We report here calculations on the triplet state 3Σ+ beryllium atominto the H2 molecule, which is a popular u of F . The reference wave function was prepared in two benchmark for MR-CC methods. The reference is still a 2 different ways, both using restricted open-shell Hartree- CAS(2,2)forcomparisonwiththeliterature,eventhough this choice of reference is not the most appropriate for a Fockmolecularorbitals. Thefirstreferencewavefunction labeled m = 1 is a single open-shell determinant, and correct description of the reaction. The geometries are s given by the relation the second wave function is the triplet ms = 0, made of two determinants 1/√2(αβ βα). − z =2.54 0.46x (a.u.) (30) To ensure that the CAS-SD is a strict eigenfunction − of the Sˆ2 operator, we have included in Ψ all the de- SD where the beryllium atom is at the origin and the hy- terminants with the same space part as the Singles and drogen atoms are at the coordinates (x,0, z). In this DoubleswithrespecttotheCAS.Thesedeterminantsare ± particularcase,theµ-MR-CCSDgivesaNPEof1.8mE treated in the same way as Singles and Doubles and are h 7 TABLEI. Non-parallelism errors (NPE) and maximum errors with respect to the Full-CI potential energy surface (mE ) h CAS-SD λ-MR-CCSD Dressed CAS-SD µ-MR-CCSD NPE Max Error NPE Max Error NPE Max Error NPE Max Error C H 5.1 35.5 3.6 8.4 1.3 8.3 3.5 8.3 2 6 F 3.8 19.8 3.1 4.0 0.9 4.2 1.6 3.9 2 C H twist 1.5 27.7 0.3 6.7 0.7 7.2 0.5 6.7 2 4 BeH 2.9 4.1 1.3 1.8 2.0 2.4 1.8 2.2 2 H O 1.8 4.6 0.7 1.2 0.2 1.3 0.5 1.2 2 C H stretch 2.8 22.1 1.6 5.3 1.7 6.2 1.6 5.2 2 4 N 1.8 9.0 1.0 2.2 1.7 3.9 0.7 2.8 2 F2 3Σ+u(ms=1) 2.5 18.6 1.3 3.3 1.3 3.5 1.2 3.3 F2 3Σ+u(ms=0) 2.5 18.6 1.2 1.8 1.3 3.5 1.1 3.3 HF(ground state) 2.6 14.6 1.8 3.4 2.1 4.4 1.8 4.1 HF(excited state) 3.3 20.9 8.8 8.5 10.5 10.1 7.1 8.3 F (local) 3.8 19.8 1.2 3.2 1.5 3.1 1.0 4.6 2 N (local) 1.8 9.0 3.8 5.0 1.1 3.5 1.1 2.8 2 treated variationally in the diagonalizations. Of course, Avoided crossing in HF and LiF those which are Triples or Quadruples with respect to Ψ are excluded from the set of the α and have no ref { } We have calculated the potential energy surfaces of effect in the dressing. the two lowest 1Σ+ states of HF, using as reference wave function the CAS(2,2) with state-averaged CAS- To reduce the computational cost, the Triples and SCF molecular orbitals in the aug-cc-pVDZ basis set. Quadruples were not augmented with all the determi- Figure 3 shows the NPEs of the ground and excited nants with the same space part. The absence of some states. Inthegroundstate,the NPEis1.8mE forboth h determinantsgivesrisetoaslightdeviation(<10−6a.u.) MR-CCSD variants, but the λ-MR-CCSD shows some of Sˆ2 fromthe desiredeigenvalue,anditis expected to numerical instabilities, as opposed to the µ-MR-CCSD h i have some impact on the iterative dressing. It is worth which gives a very smooth curve. checking the effect of this deviation from the exact spin In the excited state, the situation is different : sur- multiplicity. The first test concerns the comparison of prisingly the best NPE is obtained by the CAS-SD, but the m =0 and m =1 components of a triplet state. s s this may be due to the fact the molecule is particularly favorable to the CAS-SD : both H and H+ have no cor- In all the cases, the NPE of the CAS-SD (2.6 mE ) is relation energy, so the wave functions of the dissociated h improvedtoavalueof1.1–1.5mE . Asexpectedthetwo moleculecanbeexpressedasaproduct. Thetwovariants h variantsoftheMR-CCSDarestrictlyequivalentform = of the MR-CCSD agree at short and long distances, but s 1. Indeed, for both variants the usual single-reference they differ significantly between 2 and 3.0 ˚A, after the amplitudes c /c are recovered. The amplitudes of the region of the avoided crossing. To understand these dif- i 0 λ-MR-CCSDlowerthe curveby 1mE whengoingfrom ferences, we have plotted the two eigenvalues of the two h m = 1 to m = 0. The dressed CAS-SD also gives state-specific Hamiltonians, one dressed for the ground s s a lower energy, but only by 0.5 mE . This is due to state and one dressed for the excited state. It appears h the increased number of degrees of freedom in the fit of that between 2 and 3.0 ˚A, the lowest eigenvalue of the the amplitudes as no additional constraint is imposed Hamiltonian dressed for the excited state is very badly to enforce the m -invariance. But when the reference- described. The reason is that the fitting procedure for s dependence is introduced via the µ , it is imposed to theamplitudesisaleast-squaresfitontheCAS-SDwave i recovertheCAS-SDwavefunctionwhichism -invariant, function of the state of interest, so the quality of the s andthisstepcompensatestheadditionalfreedomgained dressing for the determinants which have small coeffi- in the fitting, and the m = 1 and m = 0 MR-CCSD cients on the state of interest but large coefficients on s s curves differ by less than 0.1 mE . the other state will be very low. The λ-MR-CCSD has h amplitudes which depend less on the wave function, so Ifoneconsiderstheerroronthesinglet-tripletgapwith thequalityisequivalentonbothstates,andthechoiceof respect to the Full-CI reference, it appears clearly that these amplitudes is better suited for calculating excited the µ-MR-CCSD gives the most accurate results, with states within the same symmetry. errors lying between 0.1 mE and 1.3 mE along the Infigure5wehaverepresentedthe avoidedcrossingof h h curve. LiF,alsocalculatedwiththeaug-cc-pVDZbasisset. The 8 CH 2 6 F CI) (a.u.) 0000....000033333456 000...000122802 2 E - E(FCI) (a.u.) 0000000.......000000033300010127890 E - E(FCI) (a.u.) 000000......000000001111680246 E - E(F 000...000000456 000...000000024 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1 1.5 2 2.5 3 3.5 4 R (Angs.) R (Angs.) CH twist 2 4 BeH u.) 0.0280 0.0045 2 CI) (a. 00..00227705 0.0040 F 0.0035 E - E(CI) (a.u.) 00000.....00000220006677805050 E - E(FCI) (a.u.) 00000.....00000000001122305050 F E( 0.0065 0.0005 E - 0.0060 0.0000 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 4 Angle (degrees) z (a.u.) CH HO 2 4 000...000000445050 2 FCI) (a.u.) 0000....000022221234 E - E(FCI) (a.u.) 000000......000000000000112233050505 E - E(FCI) (a.u.) 000000......000000120000905678 00..00000005 E - E( 00..000034 1 1.5 2 2.5 3 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 R (Angs.) R (Angs.) N 2 u.) 0.010 CI) (a. 0.009 F 0.008 E( E - 0.007 0.004 CI) (a.u.) 0.003 Dresµλs--eMMd RRCC--AACCSSCC--SSSSDDDD E(F 0.002 E - 0.001 1.00 1.50 2.00 2.50 3.00 3.50 4.00 R (Angs.) FIG.1. Dissociation curves. Differencewith respect totheFull-CI energyusingtheMR-CCSDmethod presentedin ref15 and with theMR-CCSD method proposed in thiswork, as well as theCAS-SDand thedressed CAS-SD. physicalsituationis similarto HF, butthe energydiffer- position of the avoided crossing is very well reproduced ence between the ground and the excited states is much bythethreemethods: theCAS-SDcrossesat6.3˚A,the smaller. A striking result is that the λ-MR-CCSD, al- Full-CI crossesat6.8 ˚Aandthe dressedCAS-SD andthe thoughbeingstate-specific,isabletoreproduceverywell twoMR-CCSDvariantscrossat6.9˚A.Theµ-MR-CCSD the whole potential energy surfaces of both states. The andλ-MR-CCSDcoincideintheshort-range( 5˚A)and ≤ 9 F2 (3Σu) HF 0.0190 0.0185 Excited state CI) (a.u.) 000...000111778050 00..002202 E(F 0.0165 0.018 E - 0.0160 0.0155 u.) 0.016 0.0150 a. 0.014 0.0040 CAS-SD CI) ( 0.012 E(FCI) (a.u.) 00000.....00000000001223350505 DDrreessµµλλss----eeMMMMdd RRRRCC----AACCCCSSCCCC--SSSSSSDDDDDD mmmmmmssssss======010101 E - E(F 000...000001680 E - 0.0010 0.004 0.0005 0.002 0.0000 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 R (Angs.) Ground state 0.0030 CAS-SD 0.022 ∆E - E(FCI) (a.u.) 00000.....00000000000112250505 DDrreessµµλλss----eeMMMMdd RRRRCC----AACCCCSSCCCC--SSSSSSDDDDDD mmmmmmssssss======010101 E - E(FCI) (a.u.) 00000000........000000000011111268024680 Dresµλs--eMMd RRCC--AACCSSCC--SSSSDDDD ∆ 0.0000 0.004 0.002 -0.0005 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 R (Angs.) -0.0010 -0.0015 1.00 1.50 2.00 2.50 3.00 3.50 4.00 FIG. 3. Difference with respect to theFull-CI energy for the R (Angs.) two lowest 1Σ+ states of HF. FIG.2. F 3Σ+. DifferencewithrespecttotheFull-CIenergy 2 u -99.5 forthems =0andms =1wavefunctions(top),anderroron thesinglet-tripletgap∆E =E(3Σ+)−E(1Σ+)(bottom). On -99.6 u g bothgraphics,thetwocurvesofthedressedCAS-SDcoincide. -99.7 u.) -99.8 a. gy ( -99.9 cinomtheevloernygcrlaonsgeein(≥en7.e2rg˚Ay),inbutthewhreegniotnheotfwtohestcartoesssbineg- otal Ener --110000..10 the dressedCAS-SD and the µ-MR-CCSD are unable to T -100.2 give sensible values. This disappointing result motivates -100.3 Ground-state amplitudes a future work on a multi-state µ-MR-CCSD. Excited-state amplitudes -100.4 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 R (Angs.) Sensitivity to the choice Molecular Orbitals FIG.4. Potentialenergysurfacesofthetwolowest1Σ+states The µ-MR-CCSD algorithm we propose is in the ofHFwiththeµ-MR-CCSDmethod. Theenergyofthestate Jeziorski-Monkhorst framework, so it is not invariant corresponding to the dressing is plotted in plain curves, and with respect to the choice of molecular orbitals. In this theenergy of theother state is plotted in dashed curves. section,wecheckeditssensitivitytothechoiceoftheMO setbycomparingresults obtainedwithpseudo-canonical CAS-SCForbitalsandwith localizedMOsinthe F and 0.7 mE to 1.1 mE . On the other hand, the dressed 2 h h N molecules (figure 6). CAS-SD gives a better NPE with local orbitals, going 2 In the F molecule, using localized MOs is a better from 1.7 mE to 1.1 mE . 2 h h choice than the pseudo-canonical MOs. The best NPE The fact that the µ-MR-CCSD is less sensitive to the is obtained by the µ-MR-CCSD method with a value MO set than the λ-MR-CCSD can be understood. By of 1.1 mE . In the case of N , the situation is differ- changing the MO set, a single excitation rotates into h 2 ent: the NPEofthe λ-MR-CCSD goesfrom0.9mE to a combination of single and double excitations. In the h 3.7mE ,andtheNPEoftheµ-MR-CCSDincreasesfrom λ-MR-CCSD method, the amplitudes are calculated by h 10 IV. CONCLUSIONS LiF -106.96 We have proposed a method to determine reference- independentamplitudesbyfittingtheCAS-SDCIvector. -106.98 Theseamplitudesmaybeusedtoperformastate-specific u.) iterativedressingoftheCAS-SDHamiltonianinorderto a. E ( -107.00 takeintoaccounttheeffectoftheTriplesandQuadruples CAS-SD in the spirit of the Coupled Cluster formalism. Alterna- Full-CI -107.02 Dressed CAS-SD tively, these amplitudes may be rescaled to reproduce µ-MR-CCSD the exact coefficients of the singles and doubles to in- λ-MR-CCSD -107.04 troduce a reference-dependent character. In that case, 4 4.5 5 5.5 6 6.5 7 7.5 8 the CAS-SD CI vector is recoveredby the application of R (Angs.) (1+Tˆ) on the reference wave function, so we reach here the Jeziorski-MonkhorstCoupled Cluster formalism. FIG.5. Potentialenergysurfacesofthetwolowest1Σ+states The CAS-SD dressed with reference-independent am- of LiF. plitudes gives excellent results for single-bond breaking (F andethane)andthesimultaneousbreakingofthetwo 2 O—Hbonds ofwater,with a non-parallelismerrorlower F2 than the milli-Hartree. When the active space becomes 0.0045 larger, it is necessary to go to the reference-dependent Dressed CAS-SD 0.0040 µ-MR-CCSD MR-CCSD introducing the µ factors in Eq. 28. In the 0.0035 λ-MR-CCSD case of ethylene and N2, this keeps the NPE to a value u.) close to the milli-Hartree. a. 0.0030 CI) ( 0.0025 We have shown numerically that the here-proposed E(F 0.0020 amplitudes are not very sensitive to the value of ms for E - open-shell systems, and to the choice of the molecular 0.0015 orbitals. This is clearly an improvement compared the 0.0010 amplitudes proposed earlier15. But we have also shown 0.0005 thattheformeramplitudesareabetterchoicewhencom- 1.00 1.50 2.00 2.50 3.00 3.50 4.00 puting excited states of the same symmetry because the R (Angs.) here-proposedamplitudeshaveamuchmorepronounced N2 state-specific character which may be disadvantageous if 0.0050 the states are too close in energy. This problem can be Dressed CAS-SD 0.0045 µ-MR-CCSD curedby leavingthe state-specific formalismfor a multi- λ-MR-CCSD 0.0040 state formalism37, and this will be the object of a future a.u.) 0.0035 work. CI) ( 0.0030 Acknowledgments. This work has been made F E( 0.0025 through generous computational support from CALMIP E - (Toulouse) under the allocation 2015-0510, and GENCI 0.0020 under the allocation x2015081738. 0.0015 0.0010 1.00 1.50 2.00 2.50 3.00 1F.Coester,Nucl.Phys.7,421(1958). R (Angs.) 2F.CoesterandH.Ku¨mmel,Nucl.Phys.17,477(1960). 3J.Cˇi´zˇek,J.Chem.Phys.45,4256(1966). 4R. J. Bartlett, J. Watts, S. Kucharski, and J. Noga, FIG. 6. Comparison between pseudo-canonical (dashed Chem.Phys.Lett.165,513(1990). curves) and localized (plain curves) MOs in F and N . Dif- 5J.Goldstone, Proc.Roy.Soc.A239,267(1957). 2 2 ference with respect to theFull-CI energy. 6I.Nebot-Gil,J.San´chez-Marin´,J.P.Malrieu,J.L.Heully, and D.Maynau,J.Chem.Phys.103,2576(1995). 7J. P. Malrieu, P. Durand, and J. P. Daudey, J.Phys.A:Math.Gen.18,809(1985). 8B.Kirtman,J.Chem.Phys.75,798(1981). taking into account the matrix elements of the Hamil- 9D. I. Lyakh, M. Musial , V. F. Lotrich, and R. J. Bartlett, tonian, which are of different nature depending on the Chem.Rev.112,182(2012). degree of excitation, so the amplitudes are expected to 10U. S. Mahapatra, B. Datta, and D. Mukherjee, changesignificantly. Intheµ-MR-CCSDvariant,theam- Mol.Phys.94,157(1998). plitudesareadjustedinsuchawaythattheyfittheCAS- 11S. Das, D. Mukherjee, and M. K´allay, J.Chem.Phys.132,074103(2010). SD wave function, which is invariant by rotation of the 12A.Szabados, J.Chem.Phys.134,174113(2011). MOs. Therefore, it is expected to be more robust with 13U. S. Mahapatra, B. Datta, and D. Mukherjee, respect to the MO set. J.Chem.Phys.110,6171(1999).

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