Péter G. Szalay Eötvös Loránd University, Budapest, Hungary

Transcription

1 cn0066 Configuration Interaction: Consistency Corrections for Size- Péter G. Szalay Eötvös Loránd University, Budapest, Hungary Abstract In this article, the size-consistency corrected MR-CI methods are reviewed. First, the origin of the size-consistency error of the CI method is analyzed and the possible corrections derived, in particular, the importance of the EPV (exclusion principle violating) terms is discussed. A complete list of methods using such a correction is presented. These methods belong to two groups depending on whether the correction modifies the original CI equations or just the CI energy is corrected a posteriori. The first group is referred to as CEPA-type methods, while the second one is known as Davidsontype methods. According to the approximation used for the correction, the methods can be classified whether they include EPV terms or not. Guidance for choosing the best among thse methods will be given on the basis of theoretical arguments, availability, and numerical experience. The latter analysis is mostly based on data from the literature but this article will also present some results to demonstrate the performance of the various methods. Importance of the possible calculation of analytical gradients is also discussed. Keywords configuration interaction; size-consistency; size-extensivity; Davidson correction; Pople correction; Meissner correction; Davidson Silver correction; multireference methods; coupled electron pair theory (CEPA); multireference averaged coupled pair functional (MR-ACPF); multireference averaged quadratic coupled cluster (MR-AQCC); self-consistent size-consistent configuration interaction ((SC) 2 -CI); multiconfiguration coupled electron pair theory (MC-CEPA); exclusion principle violating (EPV) 1

2 1 Introduction: Size-Consistency and the Configuration Interaction Method The Configuration Interaction (CI) method is perhaps the conceptually simplest model for the treatment of Electron Correlation. In this model, the wave function is chosen as a linear combination of a set of many electron functions Φ I (preferably Configuration State Function CSF): Ψ CI = I c I Φ I (1) where the coefficients c I need to be determined. If the set of CSFs {Φ I } used in the expansion spans the whole many-electron space, we have the so-called full CI wave function, which is exact (in the given one-electron basis). This is, however, achievable only for small systems and in practice truncation of the above expansion is necessary. The choice for the expansion has been discussed in detail in the article on Configuration Interaction. Although the truncation scheme can be very flexible, there is always a problem with the truncated CI wave function, namely, the result does not scale properly with the size of the system. It is therefore said that the truncated CI is not or. In this article, we will use the term size-consistent since it is conceptually simpler and can be quantified by comparing the sum of the calculated energy of the noninteracting systems and that of the supersystem. This error will, however, not disappear when the two monomers approach each other, we just will not be able to quantify it by performing two calculations. Therefore, the concept of sizeextensivity is useful; it has a precise mathematical meaning and originates in the linked cluster theorem. 1, 2 Whether a method is sizeextensive or not can be proven by inspecting the terms contributing to the energy and wave function. To understand the origin of the size-consistency problem of the truncated CI method, consider two subsystems at infinite separation. Let us use the CID wave function to describe this system, that is, only double excitations are allowed with respect to the HF 2

3 determinant. For the supersystem, we may write (using intermediate normalization): Φ CID = Φ HF + Φ D (2) where Φ HF is the Hartree Fock (HF) determinant of the supersystem and Φ D is the sum of all double excitations out of Φ HF, which are multiplied by the appropriate coefficient. For one of the subsystems we can write: A Φ CID = A Φ HF + A Φ D (3) and similarly for system B. The product of these two wave functions gives the other choice for the wave function of the supersystem: A+B Φ CID = A Φ CID B Φ CID (4) = A Φ HF B Φ HF + A Φ HF B Φ D + B Φ HF A Φ D + A Φ D B Φ D (5) = Φ HF + Φ D + A Φ D B Φ D (6) In the last line, the HF and double-excited functions of the supersystem have been recognized. Comparison of equation (2) and equation (6) shows that the two wave functions are not identical; the latter includes an additional term that is in fact a quadruple excitation with respect to the HF determinant. It is therefore clear that the two ansätze will not give the same energy, that is, CID is not sizeconsistent. This simple model enables us to identify the origin of the sizeconsistency error of the truncated CI: simultaneous excitations of maximum excitation rank (double excitation in our example) on the subsystems are missing from the CI wave function of the supersystem. One way to avoid this problem is to use an exponential wave function as is done in Coupled-cluster Theory. The accuracy of the single reference coupled-cluster (CC) methods is so much better than that of the single reference CI that the former actually completely replaced the latter in actual applications. Therefore, the discussion of the sizeconsistency correction is obsolete as well. The situation is, however, somewhat different for the multireference case (see Configuration Interaction): despite the tremendous effort, at the time of writing 3

4 this article there is no generally accepted MR-CC method that could replace MR-CI. The aim of this article is therefore to discuss the size-consistency corrections for MR-CI. To make the discussion easier to understand, the single-reference case will be discussed first and generalization to MR cases will be given subsequently. 2 Theoretical Background 2.1 The Single-Reference Case Let us construct the intermediately normalized full CI wave function starting from a single Slater-determinant φ 0 : Ψ FCI = φ 0 + i,a c a i φ a i + i>j,a>b c ab ij φ ab ij + i>j>k,a>b>c c abc ijkφ abc ijk +... (7) Here, and later in this article i, j, k... refer to occupied orbitals and a, b, c... to virtual orbitals in the reference function (φ 0 ). Thus, φ a i represents single excited determinant, φ ab ij double-excited determinant, and so on, and c a i, cab ij,... are the corresponding coefficients. The coefficients are determined by a variational procedure. For example, for the double-excited coefficients we have: φ ab ij H N φ 0 + k,c c c k φ ab ij H N φ c k (8) k>l,c>d k>l>m,c>d>e c cd kl φ ab ij H N φ cd kl k>l>m>n,c>d>e>f = c ab ij E c cde klm φ ab ij H N φ cde klm c cdef klmn φab ij H N φ cdef klmn with H N = H E 0, E 0 = φ 0 H φ 0, H is the Hamiltonian, while E = E E 0 is the correlation energy. Provided that φ 0 is a Hartree Fock determinant, the correlation energy can be exactly calculated solely using the double-excited coefficients: E = k>l,c>d c cd kl φ 0 H N φ cd kl (9) 4

5 Note that equation (8) yields the required coefficients but these depend on the single excited, triple excited, and even quadruple excited coefficients. If we knew these, equation (8) and equation (9) could be used to calculate the exact energy. Calculation of the triple and quadruple excited coefficients are, however, rather expensive and approximations are necessary. For example, in the CISD method, just the single and doubly excited terms are retained in the wave function. Equation (8) can be simplified using Slater rule: φ ab ij H N φ cdef klmn = δ inδ jm δ ae δ bf φ 0 H N φ cd kl (10) Also, the quadruple excited coefficients can be approximated by using the cluster condition (known from Coupled-cluster Theory) as: c abcd ijkl c ab ij c cd kl (11) Note that terms like c ab ik ccd jl, and so on, that is, when the indices are mixed, are not considered, although these play an important role in CC theory. Completely neglected are the connected quadruple excitations, that is, the real fourfold excitations and therefore the methods discussed below should not be expected to closely approximate full CI. However, this approximation is solely aimed at giving a correction to the size-consistency problem through the simultaneous excitation of electron pairs i,j and k,l. For the discussion below, it is also very important to keep in mind that repeated indices cannot appear on the left-hand side since these would correspond to excitations from or to the same orbital twice. Therefore, indices of the second coefficient (kl,cd) cannot coincide with (ij,ab). The corresponding restriction on the summation indices will be denoted as ij, ab below. Similar approximation can be introduced for triple excited coefficients; to simplify the discussion, these will not be considered here and in fact we will leave out the triple excited terms from the presentation below. Inserting equation (11) and equation (10) into equation (8), expanding the correlation energy according to equation (9) (and neglecting 5

6 triple excitations), one arrives at: φ ab ij H N φ 0 + k,c + + = c ab ij k>l,c>d ij,ab k>l,c>d k>l,c>d c cd kl φ ab ij H N φ cd kl c ab ij c cd kl φ 0 H N φ cd kl c cd kl φ 0 H N φ cd kl c c k φ ab ij H N φ c k (12) Inspecting this equation we can recognize the similarity of the right hand side term and the last term on the left-hand side. In CISD, the latter is neglected while the former one is retained. It is clear from this analysis that this is not the best approximation since these two terms almost match. The goal with the corrected CISD methods is therefore to include the term: ij,ab c ab ij Kij ab = c ab ij c cd kl φ 0 H N φ cd kl (13) k>l,c>d that is, the corrected equation will have the form: φ ab ij H N φ 0 + k,c + k>l,c>d c c k φ ab ij H N φ c k (14) c cd kl φ ab ij H N φ cd kl + c ab ij Kij ab = c ab ij E Notice that the sum on the right-hand side of equation (13) is almost equal to the correlation energy. After subtracting the right-hand side, we get: φ ab ij H N φ 0 + k,c + k>l,c>d + c ab ij R ab ij = 0 c c k φ ab ij H N φ c k (15) c cd kl φ ab ij H N φ cd kl 6

7 with ij,ab c ab ij Rij ab = c ab ij E c ab ij c cd kl φ 0 H N φ cd kl (16) k>l,c>d = c ab ij =ij,ab k>l,c>d c cd kl φ 0 H N φ cd kl (17) The two terms in equation (16) do not cancel completely due to the restricted summation in the second term. The surviving term consists of contributions where occupied and/or virtual indices coincide (denoted by = ij, ab ). These terms are the so-called EPV (exclusion principle violating) terms. 3 This is a rather strange concept since it suggests that these terms are not physical and therefore should not appear in the equations. The contrary is true: as has been discussed above the restriction on the indices in the last term of the left-hand side of equation (12) is due to exclusion principle, that is, we cannot excite the same electron twice (cf. equation (11)). Consequently, neglecting the term given by equation (17) from equation (15) is equivalent to the inclusion of the terms that violate Pauli principle. Therefore, the inclusion of the term given in equation (17), contrary to its name is necessary to prevent exclusion principle violating contributions in the equations. The other important fact to be recognized is that the inclusion of the c ab ij Kab ij term in the equations is equivalent to the approximate inclusion of certain higher excitations in the CISD equation. More specifically, according to equation (11), the higher excitations are approximated by products of double excitations, terms that have been found to be responsible for the size-consistency error (see the Introduction). It follows from the above discussion that the size-consistency correction to CI requires the appropriate inclusion of Rij ab in equation (15) or Kij ab in equation (14). What follows here will summarize the possible approximations. 7

8 2.2 Approximations to the EPV Terms For some of the approximations to Rij ab, for example, the series of CEPA (coupled electron pair approximation) methods, it is useful to introduce the concept of pair energies: ɛ ij = ab c ab ij φ 0 ĤN φ ab ij (18) In this expression, the energy contribution of those configurations that have electrons excited from the same occupied orbitals are summed over virtual orbitals. The sum of pair energies gives the correlation energy (cf. equation (9)): ɛ ij = E (19) ij With pair energies, the EPV terms can be approximated as =ij c ab ij Rij ab c ab ij ɛ kl (20) Since an unrestricted summation over virtual indices is included in ɛ kl, EPV terms due to coincidence of virtual indices cannot be considered. Most of the methods discussed below will use this approximation. The consequence of using pair energies is that the method will not be invariant under transformation of occupied orbitals CEPA(0) Approximation Clearly, the simplest step in correcting the size-consistency error of CISD is to use a full cancellation in equation (15). This means that EPV terms are completely neglected: or, equivalently, for the K ab ij kl R ab ij = 0 (21) quantity of equation (14): K ab ij = E (22) This approximation has been introduced in different contents and therefore it is known under different names: CEPA(0), 4 L-CPMET (linearized coupled pair many electron theory), 5 LCCD (linearized 8

9 coupled-cluster doubles) 6 method or doubles many-body perturbation theory infinite order (D-MBPT( )). 7 Since, in this approximation, the EPV terms are completely neglected, it usually overestimates the effect of higher excitations Other CEPA Approximations Equation (20) offers a way of approximating the effect of EPV terms using the orbital energies. It was Kelly 8, 9 who first suggested the following approximation: ( ) c ab ij Rij ab = c ab ij ɛ ij + k (ɛ ik + ɛ kj ) (23) Later, Meyer 10, 11 suggested several variants, the so-called CEPA(2) approximation uses c ab ij R ab ij = c ab ij ɛ ij (24) that is, only the diagonal EPV term is considered. This formulation ensures that the method is, in contrast to CEPA(0), exact for separated electron pairs if localized orbitals are used. The version now known as CEPA(1) takes the form: ( ) c ab ij Rij ab = c ab ij ɛ ij + 1 (ɛ ik + ɛ kj ) 2 k (25) which is exact for the separated electron pair problem even with nonlocalized orbitals. This approximation differs from Kelly s approximation in just the factor of half, which is required to get the desired orbital-invariance property. Though the invariance is satisfied only for the model system and not for molecules, it was assumed that this will not cause serious problems in applications. These and other CEPA variants have been systematically compared by Koch and Kutzelnigg Averaged CEPA Approximations The solution to the orbital-invariance problem is possible if the EPV terms are considered only in an averaged way. It was suggested by Gdanitz and Ahlrichs 13 (note that the same idea also appears in the paper by Meissner, 14 see below also) to partition the system 9

10 into n e 2 equivalent noninteracting electron pairs (n e is the number of electrons in the system). For such a system, the total correlation energy is equally distributed among the pairs, therefore, the pair energy can be written as: ɛ = E n e 2 (26) Since, in this approximation, all pairs give EPV term only with itself, we can write: c ab ij Rij ab = c ab ij ɛ = c ab E ij n e /2 (27) This approximation has been used in the ACPF (averaged coupled pair functional) 13 method and the simplified form of the Pople correction 15 can be explained with the same model. Since the method considers the EPV for one pair only, it can be viewed as an averaged version of the CEPA(2) approximation. By construction, this choice is exact for noninteracting electron pairs. Meissner 14 has also suggested a slightly different approximation, namely, to consider ) all possible electron pairs, the number of which ( ne is now. Again, the correlation energy is distributed equally 2 among all pairs, that is, the averaged pair energy becomes: ɛ = ( E ) (28) ne 2 and EPV terms are calculated by summing for those electron pairs that have indices in common: (( ) ( )) c ab ij Rij ab = c ab ne ne 2 E ij ( ) (29) 2 2 ne 2 = c ab ij ( 1 (n ) e 2)(n e 3) E (30) n e (n e 1) Unlike in CEPA(2), in this approximation, not only are the diagonal EPVs considered but the overcounting of certain terms are also 10

11 avoided by the averaging. Therefore, the approximation by equation (30) can be viewed as an averaged CEPA in between Kelly s CEPA and CEPA(1). This approximation is used in the Meissner correction 14 and in the AQCC (averaged quadratic CC) 16 method. One important note on the n e constant appearing in the above formula: as has been already pointed out by Gdanitz and Ahlrichs, 13 the best choice for n e is the number of valence electrons. Exclusion of the core electrons from the correction can be justified by the fact that their contribution to the correlation energy is smaller and therefore they would bias the averaging procedure The Exact CEPA Since equation (20) is already an approximation and its use does not allow the calculation of the EPV terms arising from coincidence of the virtual orbitals, it might be necessary to return to equation (17) and calculate all terms. This procedure is sometimes termed as exact CEPA. One realization is the self-consistent, size-consistent CI ((SC) 2 -CI) method by Daudey et al. 17 or the state-specific MRCEPA (SS-MRCEPA(I) version) by Mukherjee et al The AQCC-v 19 Approximation It is also possible to use the averaging idea in connection with exact CEPA. As proposed by Füsti-Molnár and Szalay, 19 the total correlation energy can be distributed among individual configurations: ( ) ( since ) (neglecting spin and spatial symmetry) there are ne Nv configurations (where N 2 2 v is the number of virtual orbitals), their averaged energy contribution is: φ 0 ĤN φ cd kl ( ne 2 E ) ( ) (31) Nv 2 Considering ( ) ( a quartet ) (i, j, a, b) of labels, there are ne 2 Nv 2 quartet (k, l, c, d) of labels where none 2 2 of the labels coincide. These are the ones that do not contribute to 11

12 EPV terms. Since all determinants are assumed to have the same contribution (equation (31)), we have: ( ) ( ) ne 2 Nv 2 c ab ij Rij ab = c ab ij ( ) ( ) ne Nv E (32) 2 2 This approximation has been used in the MR-AQCC-v method (v stands for virtual ) Multireference Case As usual in the theory of multireference approaches, we divide the full CI space into three subspaces: 20 P space includes the reference functions (φ P p ), Q space includes all single and double excitations (φ Q q ) out of the reference functions, and finally R space includes all other functions (φ R r ). The full CI wave function may be written as: Ψ Full CI = p P c P p φ P p + q Q c Q q φ Q q + r R c R r φ R r (33) while the MR-CISD wave function can be written as Ψ MR CISD = p P c P p φ P p + q Q c Q q φ Q q (34) The indices p, q, and r refer to configurations in the P, Q, and R spaces respectively. The indices s and t may refer to any configurations. The correlation energy naturally does not depend on the coefficients of the R space explicitly: E = c t Φ 0 H N φ t (35) t {P,Q} where H N = H Φ 0 H Φ 0 and Φ 0 is an arbitrarily chosen function in the P space (and the coefficients are intermediately normalized with respect to Φ 0 ). For any φ s P, Q, R we have the following equation: c P p φ s H N φ P p + c Q q φ s H N φ Q q (36) p P q Q + r R c R r φ s H N φ R r = c s E 12

13 However, it would be better to avoid the solution of the equations for coefficients with s R and consider equation (36) for φ s {Q, P } only. It is possible if c r can be eliminated from the equations. As in single-reference theory, we would like to replace the matrix elements φ s H N φ R r by matrix elements with functions in the P and Q spaces, and approximate c R r. The former can be achieved by defining the operators E s as: E s Φ 0 = φ s (37) for all s {P, Q}. Considering equation (36) with s Q, the function φ R r appearing in the third term can be written as: φ R r = E t E Q s Φ 0 (38) with E t corresponding to a function φ t, which belongs to the P or Q spaces, since terms with φ R r being higher than double excited with respect to φ s will not give a contribution. The last integral in equation (36) can thus be written as: φ s H N φ R r = E Q s Φ 0 H N E t E Q s Φ 0 Analogously to the Slater-rule, these matrix elements can be approximated as: E Q s Φ 0 H N E t E Q s Φ 0 Φ 0 H N E t Φ 0 (39) The validity of this approximation has been analyzed by Ruttink et al. 21 As in the single-reference case, we introduce a cluster condition, as well: c R r = c t c s (40) (Note, that this is not a unique choice for the cluster condition.) Finally, changing the summation index r to t: c P p Es Q Φ 0 H N Ep P Φ 0 (41) p P + c Q q Es Q Φ 0 H N Eq Q Φ 0 q Q + c s K s = c s c t Φ 0 H N E t Φ 0 t {P,Q} 13

14 with K s = restr t Q c Q t Φ 0 H N E t Φ 0 (42) The restriction on the summation over t (denoted by restr ) is due to the fact that by changing the summation index from r to t, only those t {P, Q} are allowed that fulfill equation (38). There are two distinct cases here: (a) if E s and E t act on the same orbital, their product will necessarily vanish, as in the single-reference case. The terms surviving the cancellation due to this restriction will be referred to as the EPV terms; (b) not all E t E s Φ 0 functions belong to R, since some products may belong to the P or Q spaces. These clearly should not be included since these are already accounted for in the ansatz. The corresponding terms surviving the cancellation will be referred to as the redundancy terms. 22 Note that equation (41) is similar to equation (14) of the single-reference theory, the quantity K being a restricted sum of the energy contributions. After the cancellation, the equation can be put in a form similar to the SR case (cf. equation (15)): c P p Es Q Φ 0 H N Ep P Φ 0 (43) p P + q Q c Q q E Q s Φ 0 H N E Q q Φ 0 + c s R s = 0 The quantity R s is defined by comparing equation (43) and equation (41). The approximations for the EPV terms have been discussed in the previous section; these can be applied here, as well. The methods treating redundancy terms will be introduced in the next subsection. 2.4 Approximations for the Redundancy Terms Redundancy Terms According to Ruttink et al. 21 One way of approximating redundancy terms was introduced by Ruttink et al. 21 So-called excitation classes, 21 which are characterized by doublet of numbers (k,l) are defined where k is the number of holes in the inactive orbitals (double occupied in all functions of P) and l is the number of particles on the virtual orbitals (empty in all functions of P). The reference functions belong to class (0,0). The higher than double excitations have k>2 and/or l>2. 14

15 The functions E t E s Φ 0 with t s will belong to the class (k s + k t, l s + l t ) where E s (k s, l s ) and E t (k t, l t ). The value of (k s +k t, l s +l t ) can be used to decide whether the function generated by the product of excitations is redundant or not. For example, if k s + k t >2 or l s + l t >2 the corresponding operator produces higher than double excitation, that is, there is no redundancy term associated with it. The energy contribution from excitation class (k,l) is given by ɛ(k, l) = c j Φ 0 H N φ j (44) j (k,l) which allows the redundancy terms to be written as c s R s = c s M[(k s, l s ), (k t, l t )]ɛ(k t, l t ) ; (45) (k t,l t ) φ s (k s, l s ) where M[(k s, l s ), (k t, l t )] is one for redundant terms and zero otherwise. The definition of classes and, consequently, the subset of E s operators considered in the redundancy terms depend on the representation of the space spanned by φ i s. Ruttink et al. 21 used a determinental basis, while configurations have been used by Füsti- Molnár and Szalay. 19 As noted by Ruttink et al., 21 this has only a small effect on the results (see also Ref. 19) Exact Account for Redundancy Terms In the multireference version of the (SC) 2 CI method, Malrieu et al. 22 proposed an exact way of calculating the redundancy terms. Unlike the presentation above, Malrieu et al. 22 generate the functions in Q space by acting on the individual functions of the P space. In this way, some of the Q space functions are generated more than once. Since this causes additional redundancies, Malrieu et al. 22 chose a weighted inclusion of the corresponding terms (neglecting EPV 15

16 ones): c i R i = Ec i + c i a ĤN φ j /c a (46) q Q φ a P φ a ĤN Êjφ a /c a ρ ia q Q Ê j φ a Q with the genealogical weight given by ρ ia = H ia c a b P H ibc b (47) The formula proposed by Tanaka et al. 23, 24 is similar, but due to the use of effective Hamiltonian, the weight factor is absent. Similarly, in the recent method by Mukherjee and coworkers, 18 the redundancy is accounted for by appropriate sufficiency conditions maintaining extensivity and other desirable properties. For more detail, see the original paper 18 and Ref Choice of the Reference Function There are several ways of choosing the reference function in the multireference calculations. In this subsection, three distinct ways will be discussed which are used in one or in the other methods. This choice usually does not influence the quality of the size-consistency correction but methods based on the different choices are expected to perform differently in extreme situations such as near degeneracies. (In my opinion this is a rather technical aspect; thus, the reader who is not interested in these details can skip this subsection.) Internally Contracted Reference Space The coefficients in the reference space (c P p ) can be obtained by solving the eigenvalue equation in the P space: c p φ P p = E 0 c p φ P p with ˆP = φ P p φ P p (48) ˆP Ĥ ˆP p P p P In this case, the relaxation of the reference space due to the inclusion of the Q space will completely be neglected. p P 16

17 2.5.2 Inclusion of the Orthogonal Complement of the Reference Space in Q In conventional MR-CI calculations, all c P p and cq q coefficients are optimized and, in fact, the two types of coefficients are equally handled. The generalization of this scheme for the methods that include size-consistency corrections can be done formally by using Φ 0 = p cp p φ P p as the only function in P space and treat its orthogonal complement space as part of the Q space. In this case, to avoid most of the redundancy effect, the equations belonging to the latter are not corrected Effective Hamiltonian Formalism The third possibility is to define the reference function Φ 0 in a selfconsistent way, that is, to obtain the coefficients c P p in the presence of the Q space. Consequently, Φ 0 must also be updated when the c P p coefficients change. The corresponding equations can be derived by projecting the Schrödinger equation also onto φ P p functions, which leads to the following equation in the reference space: ( ˆP Ĥ ˆP + ˆP Ĥ ˆQ( ˆQ(E x Ĥ ˆQ) 1 ˆQ) Ĥ ˆP )Φ 0 = EΦ 0 (49) with E x being a constant depending on the approximation used for EPV and redundancy terms. This approach has been used in the QDVPT (Quasi-degenerate Variational Perturbation Theory) method of Cave and Davidson 26 and by Tanaka in the MRCPA methods. 23, 24, 27, 28 If E x = E, it can be viewed as the partitioned form of CI. 29 Note that CI based on this and the previous ansatz are equivalent. 2.6 Solution of the Equations Equation (14) and equation (41) represent a convenient common form of the equations for corrected CI methods. Irrespective of the approximation used for K s, the term c s K s can be added to the term c s φ s H φ s, that is, the correction can be applied as diagonal shift to the Hamiltonian. The equation to be solved becomes: c t φ s H N + K φ t = c s E (50) t {P,Q} 17

18 with the operator K defined by φ s K φ t = δ ts K s (51) The diagonal-shift form of the equations had been first recognized by Heully and Malrieu. 30 Equation (50) is in the form of a matrix eigenvalue problem but one should remember that this equation is not a real eigenvalue problem for most methods since K s can depend on the coefficients or the correlation energy. Still, the usual CI techniques with small modifications can be used. Meller et al. 31 proposed a two-level iterative scheme: for given values of c s the shift is calculated and the eigenvalue problem of the resulting modified operator is solved. Then a new shift is calculated from the new c coefficients and this procedure is repeated until convergence is reached. Ruttink et al. 21 in connection with their MRCEPA 21 method recalculated K s in every iteration of the diagonalization procedure. A general procedure using the Davidson iterative diagonalization algorithm has been suggested in Ref. 32, 33. This formulation has several advantages, the most important being that this way all of the methods can be more or less easily implemented into existing CI codes. In addition, excited states can be calculated as higher roots of the matrix using of course the appropriate (state dependent) shift. For more detail see Ref Methods After summarizing the possible approximations to size-consistency correction of CISD, one needs to address the question on how these can be incorporated into the calculations. There are essentially two ways. From the presentation above, it appears natural to include c ab ij Kab ij or c sk s terms into the equations. As a simpler choice, the converged CISD coefficients could be corrected accordingly, which would lead to a simple correction to the energy. There are advantages and disadvantages with both the procedures; these will be discussed in section 4. In what follows, we review the methods belonging to both classes, starting with the simpler a posteriori corrections. 18

19 3.1 A Posteriori or Davidson Corrections The a posteriori corrections are often called Davidson corrections 35, 36 because the first such correction was introduced by Davidson. As before, we first summarize the single-reference corrections; their generalization to MR case will be discussed at the end of this subsection. Corrected CI coefficients due to the term c ab can be given as c ab ij = c ab ij ij Kab ij in equation (14) c ab ij Kab ij φ ab ij H N φ ab ij (52) where c ab ij represents a CISD coefficient and c ab ij represents the corrected one. The corresponding corrected correlation energy (up to second order) can be written as E = E + (c cd kl) 2 Kkl cd (53) k>l,c>d The simplest approximation considered in the previous section is the CEPA(0) approximation (equation (22)), for which the corrected correlation energy becomes: E = E + (c cd kl) 2 E (54) k>l,c>d Using the normalization condition (assuming CID): (c 0 ) 2 + (c cd kl) 2 = 1 (55) k>l,c>d the correction of the CI energy will be: E DC = (1 (c 0 ) 2 ) E (56) This is the well-known form of the first Davidson correction, which can be evaluated at essentially no cost. A little-known, but more rigorous formula was in fact given much earlier by Bruckner: 38 E BC = (1 (c 0) 2 ) (c 0 ) 2 E (57) 19

20 This correction is often called the renormalized Davidson correction and has also been obtained by Luken 39 as a limiting case of a more rigorous derivation (see below). The factor of 1 comes from the c 2 0 fact that in equation (53) intermediately normalized coefficients are required, while equation (55) assumes normalization to unity. A slightly different form of the correction can be derived if one iteratively replaces c ab ij by c ab ij in equation (52). This correction has been suggested independently by Davidson and Silver 40 and by Siegbahn 41 and it takes the form: E DS = (1 (c 0) 2 ) E 2(c 0 ) 2 (58) 1 All these formulas use the CEPA(0) approximation, that is, a complete cancellation in equation (16). To improve upon this approximation, EPV terms need to be considered. This was first suggested by Luken, 39 but the first useful formula has been given by Pople: 15 n 2 E PC = e + 2n e tan 2 (2θ) n e E (59) 2(sec(2θ) 1) with cos(θ) = c 0. To show that this formula includes EPV terms, we rewrite it (assuming c 0 1) as: 14 E PC = (1 (c 0) 2 ) ) E (1 2ne (c 0 ) 2 (60) This formula can be easily derived by considering the averaged CEPA approximation given by equation (27). As it was discussed already, the advantage of this approximation is that it vanishes for two electrons, which is necessary since CISD is exact for two electrons. The correction proposed by Meissner 14 uses equation (30) instead: E MC = (1 (c 0) 2 ( ) ) (ne 2)(n e 3) E (c 0 ) 2 (61) n e (n e 1) This correction vanishes for three electrons, as well, as has been pointed out by Meissner; 14 this is also necessary since for three electrons there should be no contribution from quadruple excited configurations. 20

21 The corrections listed above are the ones most often used in the literature. For the sake of completeness, we refer to some other suggestions, too: starting from the Pople correction, Duch and Diercksen 42 derived a slightly modified version of the Davidson Silver correction. Meissner 43 suggested to approximate the couple-cluster energy using the CI coefficients. Note, however, that such a correction falls already beyond the goal of the presented treatment, since it uses a more accurate approximation than the one given by equation (11). The calculation of this correction is more involved than the usual corrections since besides the coefficients, some Hamiltonian-matrix elements are also necessary, 43 which might not be so easily available in a direct CI program (see Configuration Interaction). As mentioned already, Davidson-type corrections are obsolete for single reference CI wave functions, since one can easily use coupledcluster methods instead. The generalization of the above formulas for MR case is, however, of interest. Redundancy effects are usually not considered in the generalized corrections at all. Bruna, Peyerimhoff, and Buenker 44 suggested the use of the Davidson correction (E DC ) to correct the MR-CISD energy: E BPB = (1 p P c 2 p)(e MR CISD E 0 ) (62) that is, c 0 has been replaced by the sum of the square of the reference coefficients and the correlation energy by the difference of the MR- CI energy and that in the reference space (Multi-configurational SCF (MC SCF) Method MCSCF). Theoretical justification for this formula along with the similar generalization of the renormalized form (E BC ) has been given by Meissner 14 who has also pointed out that the use of the Meissner correction (E MC ) for the MR case is the most appropriate theoretically. Shepard 45 (in particular, see pg 417) discusses alternative ways for the MR generalization: he suggests using the overlap of the MCSCF and MR-CI wave functions as c 0 and/or using the expectation value of the Hamiltonian in the reference space using the MR-CI instead of the MCSCF coefficients as reference energy. His suggestions have never been tested numerically. Some more involved corrections using MR-CC argumentation in the derivation have been introduced by Duch and Diercksen, 42 by Meissner and coworkers, 46, 47 and by Hubač et al

22 3.2 MR-CEPA-Type Methods In this subsection, those methods that include the corrections in the equations are summarized. Owing to the historical development and the underlying theory, these methods are often referred to as CEPA-type methods. The particular procedures worked out for the single reference case will not be discussed here since their present importance is strongly weakened by the powerful CC methods. One should remember, however, that these methods had been very important in the 1970s and the 1980s; therefore, I call the reader s attention to the detailed review by Koch and Kutzelnigg. 12 In what follows, multireference CEPA approximations will be compared. These methods are strongly related although the ansatz used to derive the approximation might be very different. For this reason, some of the methods are equivalent, even though they have different names. The main aspect of the comparison will be whether the methods include EPV terms as well as redundancy contributions in the equations. We will also mention the choice for handling the reference space and discuss whether the calculation of the energy gradient is possible via an energy functional. The details of latter analysis cannot be given in this article due to space limitations. Let me mention only that the existence of the functional makes the calculations of energy derivatives conceptually easy and computationally cheap. If such a functional does not exist, additional equations, which increase the necessary computer time by a factor of 2 compared to the energy calculations, must be solved. The details can be found in Ref. 32. Although the motivation in introducing these methods was to account for the size-consistency error, most of the methods given here are not size-consistent in the general sense; rather, they fulfill size-consistency for special model systems. The remaining sizeconsistency error does not bias the applications. 49 Table 1 gives the summary of this comparison. For a more detailed analysis, we list R s quantity used in the particular methods (see equation (43)) and the related diagonal shift (K s ) used in equation (41). 22

23 3.2.1 MR-LCCM 50, 51 and MR-CEPA-0 52 The first multireference method with this type of approximation was proposed by Bartlett and coworkers in their multireference linearized coupled-cluster method (MR-LCCM). 50, 51 The derivation is based on linearizion of the MR-CC equations as proposed by Paldus. 53 The method uses the CEPA(0) approximation for EPV terms and neglects redundancy effects: R s = 0 or K s = E (63) For the wave function, this method uses configurations and the orthogonal complement of the reference space is excluded. Although the inclusion thereof was also considered in one of the MR-LCCM papers, 51 the exclusion was preferred for two reasons: (a) it was the logical choice in CC-type derivation and (b) improved the convergence behavior of the equations. The simplest version of MR-CEPA-n series by Fulde and Stoll 52 referred to as MR-CEPA-0 is equivalent to MR-LCCM as acknowledged already by Fulde and Stoll. 52 Functional can be associated with this method; therefore, the analytic gradients are available MR-CEPA(0), 13 UCEPA, 54 and VPT 55 Methods The story of these four methods shows nicely the parallel effort of different groups. Exactly the same method was proposed by three groups within a year using completely different derivations: the method called MR-CEPA(0) by Gdanitz and Ahlrichs 13 was just a by-product of MR-ACPF (see later), the Unitary CEPA (UCEPA) of Hoffmann and Simons 54 was based on unitary CC ansatz, 4 while the variational perturbation theory (VPT) method of Cave and Davidson 55 used perturbation theory. Moreover, these methods differ from MR-LCCM method just slightly: they all use the CEPA(0) approximation but include the orthogonal complement of the reference space in the wave function. The difference does not affect R s, K s values, that is, equation (63) does also apply here and calculations of gradients via energy functional is possible. 23

24 3.2.3 QDVPT 26 and MRCPA(2) 23, 27 Methods Shortly after VPT, 55 Cave and Davidson proposed a method called quasi-degenerate variational perturbation theory (QDVPT) method. 26 It still uses CEPA(0) approximation, but in order to avoid the convergence problem of MR-CEPA(0) (or VPT) and account for relaxation in the reference space, the effective Hamiltonian equation (49) with E x = E 0 is used to define Φ 0. Since (a) neglecting the relaxation in the reference space is normally less important than the effect of CEPA(0) approximation and (b) for CI wave function, effective Hamiltonian approach and the full diagonalization are equivalent, it is expected that QDVPT will give very similar results to MR-CEPA(0). This idea is indeed supported by calculations. 26 Different behavior can be expected, however, for near-degenerate cases or for cases where the reference space is not a good zeroth-order space. In this latter instance, however, the CEPA(0) approximation is not valid and neither of these methods should be used. R s, and K s values of QDVPT are the same as in MR-LCCM, that is, given by equation (63). Owing to the effective Hamiltonian form, however, the functional form does not exist and gradients are more difficult to obtain. The MRCPA(2) method of Tanaka et al. 23, 27 is equivalent to QD- VPT MRCEPA Method 21 Ruttink et al. 21 in the derivation of their MRCEPA method considered redundancy effects the first time. Their averaged procedure is discussed in Section 2.3 in detail. On the other hand, EPV terms were completely neglected, that is, MRCEPA is still a CEPA(0) type method. The constants are given by R s = M[(k s, l s ), (k t, l t )]ɛ(k t, l t ) (64) (k t,l t ) K s = (k s,l 2 ) (1 M[(k s, l s ), (k t, l t )]) ɛ(k t, l t ); φ s (k s, l s ) For the definition of the quantities involved here, see equation (44) and equation (45). 24

25 The coefficients of the reference functions are relaxed in this procedure. The original derivation used determinental formulation, but as it was pointed out in the original paper, 21 application of the formulas on spin-adapted configuration instead should not have a large effect on the results as this was also demonstrated numerically. 19 Since R s now depends explicitly on c t coefficients, no functional can be defined and the analytic gradient is not available for this method. 23, 24, 27, MRCPA Methods Several variants of the so-called multireference coupled pair approximation (MRCPA) have been worked out by Tanaka and coworkers. 23, 24, 27, 28 These methods use the effective Hamiltonian formalism like QDVPT. There are two levels of approximations: MRCPA(2) (formerly know as MRCPA(0) 27, 28 ) uses a CEPA(0) approximation 23, 24 and as such, it is equivalent to QDVPT. 24 MRCPA(4) 23 (which is a slight modification of the variant formerly know as MRCPA(2) 27, 28 ) includes redundancy effects but no EPV terms (see equation (68) in Ref. 24). In the limit of a singlereference function, this correction simplifies to a CEPA(0). This view is supported by the results obtained by MRCPA(4), which shows that the method, like all other methods using CEPA(0) for the EPV terms, overestimate the effect of higher excitation considerably (see Table 3 and Figure 2). The method is claimed to fulfill the size-consistency criteria for noninteracting electron pairs. 23 The effective Hamiltonian formalism does not allow the construction of a functional, but, at the same time, the method is ready for excited states MR-CEPA-n (n = 0, 1, 2) Methods 52 Fulde and Stoll 52 derived a series of methods that are analogous to single reference CEPA(n) (n = 0, 1, 2) methods. In the derivation, a different mathematical technique (cumulants) was used. In all variants, the reference function is from MCSCF calculation and not relaxed later on. The correction in MR-CEPA-2 is given by: R ab ij = ɛ ij + cd c cd ij Φ 0 φ ab ij (65) 25

26 The first term is responsible for EPV terms, exactly as in singlereference CEPA(2) and second term is due to the nonorthogonality of configurations. In case of MR-CEPA-1 method: Rij ab = ɛ ij + 1 (ɛ ik + ɛ kj ) + c cd ij Φ 0 φ ab ij (66) 2 k cd + 1 (c cd 2 ik Φ 0 φ cd ik + c cd ki Φ 0 φ cd ki ) kcd which is related to CEPA(1). Redundancy effects are not considered in the equations but through the energy expression. Fulde and Stoll 52 do not prove sizeconsistency explicitly, they argue that the derivation through cumulants assures this property. Implementation into CI programs would be easy in a diagonal-shift form and no substantial computational effort compared to CI would be needed. Still, there is no report on the implementation of any variants. Calculation of gradient would, however, be difficult since functional of the energy cannot be associated with the equations and the method also lacks invariance with respect to transformation of occupied orbitals. Note that these disadvantages are already present in the original, single reference CEPA(1), and CEPA(2) methods MR-ACPF 13, 56 and MR-AQCC 16, 19, 49 Methods As has been discussed in Subsection 2.2, Gdanitz and Ahlrichs 13 proposed an averaged formula in their MR-ACPF method to account for EPV terms. This pioneering idea allows the energy to be written in a functional form, that is, relation to CI is apparent, which makes implementation easy and allows for calculation of analytic gradients just like for CI. The approximation used for the EPV terms was discussed in Subsection 2.2: R s = E n e /2, K s = n e 2 n e E (67) The method is formulated with configurations and orthogonal complement of reference space is included in the wave function. Although the method is not strictly size-consistent, this choice of R s 26

27 ensures that (a) the method is exact for noninteracting electron pairs and (b) it is size-consistent for identical subsystems using a single function in the reference function. MR-ACPF is a successful method but it is known to overestimate the effect of the higher excitations. 16 For this reason, Szalay and Bartlett 16 suggested a modified version under the name multireference averaged quadratic CC (MR-AQCC) method. 16, 49 Here, the EPV terms are approximated by equation (30) instead of equation (27), that is, the correlation energy is distributed among all possible electron pairs instead of the noninteracting ones in MR-ACPF. The R s and K s values are given by: R s = ( 1 (n e 2)(n e 3) n e (n e 1) K s = (n e 2)(n e 3) E n e (n e 1) ) E, (68) Since these differ from the MR-ACPF values only in the numerical factor multiplying E, all the technical advantages of MR-ACPF has been retained. By construction, MR-AQCC is not exactly sizeconsistent and, unlike MR-ACPF, it is not exact for noninteracting electron pairs, but the correction vanishes for both two and three electrons 14 in accord with the fact that no EPV terms exist in these cases. We note that in practical applications, size-consistency error of MR-AQCC and MR-ACPF has been shown to be negligible. 49 Recently, Gdanitz 56 suggested a modified version of the MR-ACPF method, called MR-ACPF-2, which is essentially a combination of the original MR-ACPF and the MR-AQCC parametrization: while for the double excitation space equation (67) is used, equation (68) is applied for the single excitation block of the Hamiltonian. In this way, the notorious overestimation of higher excitation effect of MR- ACPF has been damped and similar results to MR-AQCC have been achieved. 56 MR-ACPF 13 and MR-AQCC 16 neglect redundancy effects. Füsti- Molnár and Szalay 19 proposed a modified version called MR- AQCC-mc (where mc refers to multi configuration ), which accounts for the redundancy terms in a procedure similar to the one in MR-CEPA. 21 Equations are given in Ref. 19 and Ref. 32. Note that they also proposed a corresponding modification to MR-ACPF 27

28 (called MR-ACPF-mc). In the test calculations, 19 these methods performed excellently, but unfortunately the existence of the energy functional had to be sacrificed for the inclusion of the redundancy effects and thus analytic gradients are not available for these variants MR-AQCC-v Method 19 This version of MR-AQCC includes EPV terms of virtual orbitals ( v refers to virtual) and can be described by: ( ) ( ) ne 2 Nv 2 R s = ( ) ( ) ne Nv E, (69) 2 2 ( ) ( ) ne 2 Nv 2 K s = 2 2 ( ) ( ) ne Nv E 2 2 where N v is the number of virtual orbitals. The method differs from MR-AQCC only in the constant multiplying of E. For infinite basis, the MR-AQCC-v is equivalent to MR-AQCC. These versions 19, 32 do not appear to be very successful QDVPT-APC Method 57 Murray et al. 57 extended the QDVPT method 26 to include EPV terms. They showed that using E x = E n e E in equation (49), an ACPF version of QDVPT can be defined. This method is called QDVPT-APC (QDVPT with averaged pair correction) and it can also be viewed as a modification of MR- ACPF via the effective Hamiltonian equation (49). Since MR- ACPF does not show convergence problems as MR-CEPA(0) does, this other relaxation scheme does not seem to have an advantage over the original one. On the contrary, no functional exists for QDVPT-APC, which is a disadvantage. The QDVPT-APC method is, however, expected to perform perhaps better than MR-ACPF, if 28

29 quasi-degeneracy is present and it is always better than QDVPT, since the latter neglects EPV terms. QDVPT type generalization of MR-AQCC can trivially be made by using: 32 ( E x = E (n ) e 3)(n e 2) E (70) n e (n e 1) MC-CEPA Method 58 The Multiconfigurational reference CEPA method of Fink and Staemmler 58 can be described by: R s = ( 1 1 ) 1 (71) δ t st ( E s φ t 2 + E t φ i 2 )c t Φ 0 H N φ t where E s is defined by equation (37). This formula reduces to Kelly s formula 8, 9 in the single-reference case. 58 Nonorthogonality of functions produced by products of operators are considered by the norms. Redundancies are neglected. Implementation in the form of diagonal shift is easy, the additional computational effort is due to the calculation of the norms. Since no averaging of the pair energies are used, the method might be superior to MR-ACPF and MR-AQCC, if the averaged pair approximation is not valid. The form of R i does not allow the construction of the functional; thus, gradient calculations can be challenging MR-(SC) 2 CI Method 22 The Multireference size-consistent self-consistent CI method of Malrieu et al. 22 includes both EPV and redundancy terms exactly as described at the end of Subsection 2.1 and Subsection 2.3. The 29

30 formula of R i and ii is given by: c s R s = Ec s + c s p H φ q (72) q Q φ q Q E q φ p Q p P φ a H E q φ q φ a H E q φ p q Q E q φ p =0 c p ρ sp K s = p H φ q p P q Q φ q Q φ p H E q φ p q Q E q φ p =0 E q φ p Q c p ρ sp φ a H E j φ a (73) The method is proved to be size-consistent, 31 which is formally clear since both EPV and redundancy terms are considered exactly. Diagonal-shift implementation is possible but the calculation of K s is the most expensive among the methods discussed. It requires additional storage of a vector of length n ref times the number of Q space functions, while the cost scales to the fourth power of the number of orbitals. 31 Therefore, even in this case, the calculation of the shift is much cheaper than the calculation of the matrix-vector product of the CI method. 22, 31 Determinental formulation was presented in the original papers. The only disadvantage with this method is that no rigorous functional can relate to this method and therefore the calculation of gradients would be much more expensive for MR-(SC) 2 -CI than for MR-AQCC or MR-ACPF. Note that recently a functional form of the MR(SC) 2 -CI method has been defined under the name MR-FCPF (multireference full coupled pair functional). 59 However, since the MR-(SC) 2 -CI equations do not make this functional stationary, this 30