# The State-Universal Multi-Reference Coupled-Cluster Theory: An Overview of Some Recent Advances

^{*}

## Abstract

**:**

## 1 Introduction

^{1}A

_{1}−X

^{3}B

_{1}) [51,52] and singlet–singlet (2

^{1}A

_{1}− 1

^{1}A

_{1}; 1

^{1}A

_{1}≡ A

^{1}A

_{1}) [50] energy gaps in methylene. For example, the orthogonally spin-adapted (OSA) [102,103,104,105,106] two-reference SUMRCCSD (SUMRCC singles and doubles) approach [39,41,45,46,47,48,49,50,54] including all relevant direct and coupling terms [50], which contrary to the statements made in Ref. 107 represents the most complete formulation of this method to date, combined with the open-shell CCSD approach [108], gives 3133 cm

^{−1}for the singlet–triplet separation T

_{0}. This compares extremely well with the spectroscopically derived value of T

_{0}of 3147 ± 5 cm

^{−1}[109]. There are open problems in the SUMRCC theory (cf. the remarks below), but this and other examples show that the SUMRCC theory is a highly promising formalism, which needs to be developed further. Unfortunately, apart from the earlier advances in formulating, implementing, and testing the spin-adapted and spin-orbital SUMRCCSD methods [4,6,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56] and apart from the recent activity in our group (see, e.g., Refs. 57–59) and the group of Pal [110], who formulated the response SUMRCC theory, the development of the genuine SUMRCC method has practically stopped, which is a situation that this paper hopes to change (at least, to some extent).

^{(p)}is assigned to each reference configuration |Φ

_{p}〉 (p = 1, . . . , M). Aside from various mathematical difficulties that this assumption creates, the requirement of having a separate cluster operator T

^{(p)}for each reference configuration |Φ

_{p}〉 leads to an excessively large number of cluster amplitudes when the dimension of model space (M) is large and when we are only interested in a few low-lying states.

_{f}) analog [116].

## 2 The State-Universal Multi-Reference Coupled-Cluster Theory

_{0}, that of the wave operator U, and that of the effective Hamiltonian H

^{eff}[4,6]. Alternative formulations of genuine multi-reference methods, in which the effective Hamiltonians are obtained by similarity transformations of the Hamiltonian (this frees us from the necessity of solving for the wave operator) can be developed [93,125] (cf., also, Refs. 9, 10, 40), but we do not use them here, since the SUMRCC theory of Jeziorski and Monkhorst [38] is based on the exponential parameterization of the wave operator and solving the generalized Bloch equation for the wave operator represented by the Jeziorski-Monkhorst ansatz.

_{0},

_{p}〉, p = 1, . . . , M, that provide a reasonable zero-order description of the target space,

_{µ}〉, µ = 1, . . . , M, of the electronic Hamiltonian H. The wave operator U : $\mathcal{M}$

_{0}→ $\mathcal{M}$ is defined as a one–to–one mapping between $\mathcal{M}$

_{0}and $\mathcal{M}$. It is usually assumed that U satisfies the intermediate normalization condition,

_{0},

^{2}= U, so that the wave operator U, just like operators P and Q, is idempotent. However, unlike P and Q, the wave operator U is not Hermitian, U ≠ U

^{†}.

_{µ}of the electronic states |Ψ

_{µ}〉, µ = 1, . . . , M, are obtained by diagonalizing the effective Hamiltonian,

^{eff}≡ H

^{eff}(U) = PHU = PHUP,

_{0}. The corresponding wave functions |Ψ

_{µ}〉 are calculated using the formula,

^{eff},

^{(p)}is the cluster operator corresponding to reference configuration |Φ

_{p}〉. The intermediate normalization condition, Eq. (3), requires that each T

^{(p)}generates states belonging to , when acting on the corresponding |Φ

_{p}〉. The wave operator U, Eq. (12), reduces to the well-known wave operator of the single-reference CC theory, U

^{SRCC}= e

^{T}|Φ〉〈Φ|, when model space $\mathcal{M}$

_{0}is a one-dimensional space spanned by a single reference configuration |Φ〉.

_{0}and cluster operators T

^{(p)}. As in all multi-reference ab initio methods, in order to define the reference configurations |Φ

_{p}〉, we divide all molecular spin-orbitals into the three disjoint subsets of core, active, and virtual spin-orbitals. The core spin-orbitals (designated by

**i**,

**j**, . . .) are occupied and the virtual ones (designated by

**a**,

**b**, . . .) are unoccupied in all reference configurations |Φ

_{p}〉, p = 1, . . . , M. The reference configurations differ in the occupancies of active spin-orbitals (designated by

**I**,

**J**, . . . , for spin-orbitals occupied in a given reference determinant |Φ

_{p}〉, and by

**A**,

**B**, . . . , for spin-orbitals unoccupied in this |Φ

_{p}〉). All possible distributions of active electrons among active spin-orbitals result in a complete model (or active) space (CAS). The use of CAS is essential to obtain size extensive results [38], although it is possible to formulate the size extensive SUMRCC method employing incomplete model spaces by relaxing the intermediate normalization condition [126,127]. In this paper, we consider the CAS formulation of the SUMRCC theory.

^{(p)}is a sum of its many-body components . In the exact SUMRCC formalism,

^{(p)}is truncated at some (usually low) excitation level. Thus, if m

_{A}is the excitation level defining a given standard SUMRCC approximation, referred here and elsewhere in this work to as method A, the corresponding cluster operators T

^{(p)}have the following form:

_{A}is the same for all values of p. The SUMRCCSD method is obtained by setting m

_{A}= 2.

_{p}〉 = UHU|Φ

_{p}〉, where U is defined by Eq. (12), premultiplying this equation on the left by e

^{−T}(p), and projecting the resulting equation on the excited configurations relative to |Φ

_{p}〉 belonging to . The final equations for cluster operators defining the exact SUMRCC theory can be written as follows:

^{(p)}E

_{K}are the excitation operators, generating the excited configurations relative to |Φ

_{p}〉 belonging to , when acting on |Φ

_{p}〉. These operators are also used to represent cluster operators T

^{(p)},

^{(p)}t

_{K}are the corresponding cluster amplitudes. Because of this representation, the system of equations, Eq. (15), represents a system of coupled nonlinear algebraic equations for cluster amplitudes

^{(p)}t

_{K}. For the complete model space $\mathcal{M}$

_{0}, the excitation operators

^{(p)}E

_{K}and cluster amplitudes

^{(p)}t

_{K}carry at least one inactive (i.e., core or virtual) spin-orbital index [38].

_{µ}and the corresponding zero-order states |χ

_{µ}〉, µ = 1, . . . , M, by diagonalizing the effective Hamiltonian matrix in model space $\mathcal{M}$

_{0}, as described above [cf. Eq. (11)]. The SUMRCC wave functions |Ψ

_{µ}〉 are obtained by applying the wave operator U, Eq. (12), to zero-order states |χ

_{µ}〉. We obtain [cf. Eqs. (9) and (12)],

^{eff}in $\mathcal{M}$

_{0}, Eq. (11).

_{0}defined by Eq. (4), whereas operators Q

^{(p)}appearing in Eq. (23) are the projection operators onto the manifolds of excited configurations relative to |Φ

_{p}〉 belonging to that are generated by excitation operators

^{(p)}E

_{K}. If designates a projection operator onto the subspace of spanned by the n-tuply excited configurations relative to |Φ

_{p}〉 that belong to , we can write

^{(p)}, are identical, i.e., = for all p = 1, . . . , M. This is a consequence of the fact that

_{p}〉, the exact SUMRCC method is completely equivalent to an eigenvalue problem for eigenstates |Ψ

_{µ}〉, µ = 1, . . . , M [111] (see, also, Ref. 117). This is no longer the case, when cluster operators T

^{(p)}are truncated in a standard manner according to Eq. (14). Indeed, in the standard SUMRCC approximations, the system of equations (22) and (23) is replaced by a truncated system of equations for the many-body components of cluster operators , namely,

_{p}〉 (cf., e.g., the lists of excitations

^{(p)}E

_{K}, corresponding to different references |Φ

_{p}〉 of the OSA SUMRCCSD formalism of Refs. 39, 41, 45–50, 54, given in Ref. 47). As pointed out in Ref. 111, this asymmetric treatment of the manifolds of excitations corresponding to different reference configurations causes that the approximate SUMRCC schemes based on Eqs. (27) and (28) (including the existing SUMRCCSD methods) are not equivalent to any Hermitian eigenvalue problem. This significant distortion of the exact SUMRCC theory, resulting from the truncation of the many-body expansions of all operators T

^{(p)}at the same excitation level m

_{A},leads to a number of pathologies in approximate SUMRCC calculations based on Eqs. (27) and (28). These pathologies include the existence of an excessive number of real and complex solutions that lack physical interpretation and the appearance of the intruder solution problem [57]. The multi-reference extension of the MMCC theory, which we recently suggested in Ref. 117, and which we overview in the next section, offers a possibility of reducing the severity of problems encountered in the standard SUMRCC (e.g., SUMRCCSD) calculations by incorporating higher-order effects in the SUMRCC formalism and by reinforcing the symmetric treatment of the subspaces.

## 3 A New Type of the Noniterative Corrections to Multi-Reference Coupled-Cluster Energies: The Method of Moments of the State-Universal Multi-Reference Coupled-Cluster Equations

_{f}) [116], and CCSDT(Q

_{f}) [116], so that they can correctly describe entire ground-state PESs in situations where the standard arguments based on MBPT, on which the CCSD(T), CCSD(TQ

_{f}), and similar approximations are based, completely fail [7,26,117,118,119,120,121,131,132,133] (cf., also, Ref. 134 for a rederivation of the renormalized CCSD(T) expressions, published one year earlier in Refs. 7, 118, 119, and for some additional tests). It has also been demonstrated that the EOMCC-based excited-state MMCC theory allows us to introduce a new hierarchy of simple noniterative EOMCC approximations that remove the pervasive failing of the EOMCCSD and perturbative EOMCCSDT approximations in describing excited-state PESs [122,123]. Clearly, the MMCC methodology provides us with new ways of systematically improving the CC or EOMCC results by adding simple noniterative corrections to the CC or EOMCC energies. Thus, it might be useful to investigate the possibility of extending the MMCC formalism to a multi-reference case.

_{µ}of the electronic states of interest. The main purpose of the approximate MM-SUMRCC calculations is to estimate corrections δ

_{µ}, so that the resulting energies + δ

_{µ}remain very close to the corresponding exact energies E

_{µ}.

_{µ}is a nontrivial functional of the corresponding exact electronic wave function |Ψ

_{µ}〉 and the generalized moments of the SUMRCC equations, i.e., the SUMRCC equations projected on the excited configurations whose excitation level exceeds that defining a given SUMRCC approximation. The precise mathematical definition of the generalized moments of the SUMRCC equations, which is consistent with Eq. (28) and which is used in the MM-SUMRCC formalism of Ref. 117, is as follows:

_{m}designates the m-body component of the corresponding operator expression. The generalized moments , Eq. (31), can be viewed as the most fundamental quantities of the Jeziorski-Monkhorst theory, as defined by Eqs. (27) and (28), since the system of equations for the many-body components of cluster operators , p = 1, . . . , M, Eq. (28), is immediately obtained by imposing a requirement that the lowest moments , with m = 1, . . . , m

_{A}, vanish, i.e.,

_{µ}, Eq. (30), are those with m > m

_{A}. As shown in Ref. 117, the explicit formula for corrections δ

_{µ}, in terms of moments , is as follows:

_{0},

_{µ}〉 and the SUMRCC wave function

_{0}, and |Ψ〉 is an N-electron wave function. For |Ψ〉 = |Ψ

_{µ}〉 and for |χ〉 equal to the corresponding eigenstate of the effective Hamiltonian , we immediately obtain

_{A}, vanish [cf. Eq. (32)] allow us to write

_{µ}〉 and |χ〉 = in Eq. (43) and using Eqs. (38) and (39), we obtain the desired Eq. (33).

_{A}using Eq. (31). Finally, we use cluster operators , generalized moments , and the right eigenvectors of , i.e., , to calculate corrections δ

_{µ}with the help of Eq. (33).

_{1}〉. In addition, when M = 1, the generalized moments of the SUMRCC equations, Eq. (31), reduce to the generalized moments of the single-reference CC equations defined in Refs. 7, 118, 119. We obtain [cf. Eq. (31)],

_{1}〉 with, respectively, the cluster operator and the reference configuration of the standard singlereference CC method.

_{µ}〉, µ = 1, . . . , M. However, we can calculate the approximate values of corrections δ

_{µ}which, when added to the SUMRCC energies , may give very good estimates of the exact energies E

_{µ}, if we use simple estimates of wave functions |Ψ

_{µ}〉, provided by one of the relatively inexpensive ab initio methods. Independent of the approximate form of |Ψ

_{µ}〉 chosen for such calculations, corrections δ

_{µ}can be calculated in a state-specific manner. Our belief that simple estimates of wave functions |Ψ

_{µ}〉 may be sufficient to obtain accurate δ

_{µ}values is based on the success of the single-reference MMCC methods and their renormalized CC analogs [7,26,117,118,119,120,121,122,123,131,132,133,134], in which simple perturbative or CI wave functions are used to construct the relevant energy corrections.

_{µ}〉 obtained in truncated multi-reference CI (MRCI) calculations (using, e.g., the popular MRCISD method or one of its approximate variants) and use the resulting corrections δ

_{µ}to improve the results of the SUMRCCSD calculations (the m

_{A}= 2 case). We can also think of using the CISDt or CISDtq [7,120,122,123] approaches, in which triply and quadruply excited configurations of the single-reference CI method are selected via active orbitals, to construct wave functions |Ψ

_{µ}〉 in Eq. (33). In either case, we should be able to significantly improve the quality of the SUMRCCSD results and reinforce a fully symmetric treatment of the manifolds of excitations corresponding to different reference configurations, which is broken by the SUMRCCSD and other SUMRCC approximations. Indeed, when m

_{A}< N, the subspaces spanned by the excited configurations relative to |Φ

_{p}〉 are usually different for different p values. As mentioned earlier, this asymmetric treatment of manifolds of excitations corresponding to different references |Φ

_{p}〉 causes that the conventional SUMRCC approaches based on Eqs. (27) and (28) are not equivalent to any Hermitian eigenvalue problem which, in turn, leads to various problems in SUMRCC calculations. However, if we do not truncate the summations over n and m in Eq. (33) in any arbitrary manner and if we simply let the projection onto a suitably chosen approximate wave function |Ψ

_{µ}〉 select terms in the summations over p, n, and m in the numerator of Eq. (33), we will obtain a fully symmetric treatment of the subspaces corresponding to different references |Φ

_{p}〉. In order for this scheme to work, we only have to assume that the CI expansions of wave functions |Ψ

_{µ}〉 contain some configurations whose excitation level relative to at least one of the M references |Φ

_{p}〉 exceeds m

_{A}. This is certainly true for the MRCISD wave functions and their CISDt and CISDtq analogs if we are interested in correcting the SUMRCCSD results. The projection onto |Ψ

_{µ}〉 in the numerator of Eq. (33) will select precisely those subsets of the generalized moments (usually, different subsets of Γ

^{(p)}’s for different values of p) that are needed to restore a symmetric treatment of the manifolds of excitations in the approximate SUMRCC (e.g., SUMRCCSD) calculations. Although this particular way of improving the SUMRCCSD results by using the MRCISD, CISDt, or CISDtq wave functions |Ψ

_{µ}〉 in Eq. (33) has not been tested yet in actual numerical calculations, we believe that we should be able to obtain significant improvements in the calculated SUMRCC energies, particularly in regions plagued by intruder states or intruder solutions, where there is an apparent need to incorporate higher–than–doubly excited clusters and have a more symmetric treatment of the subspaces in the SUMRCC calculations.

_{A},m

_{B}) approximations suggested in Refs. 7, 118, 119 (see, also, Refs. 120–123). The multi-reference MMCC(m

_{A},m

_{B}) approximations [referred to as the MMSUMRCC(m

_{A},m

_{B}) or MM-SUCC(m

_{A},m

_{B}) schemes] are obtained by truncating the summation over n in Eq. (33) at n = m

_{B}, where m

_{A}< m

_{B}< N. The multi-reference MMCC(m

_{A},m

_{B}) energy formula can be given the following form [117]:

_{A},m

_{B}) method uses moments with m = m

_{A}+ 1, . . . , m

_{B}. For typical applications of Eq. (46) (e.g., m

_{A}= 2 and m

_{B}= 3 or 4), the moments with m = m

_{A}+ 1, . . . , m

_{B}form a small subset of all Γ

^{(p)}’s.

_{A},m

_{B}) approximation is the MM-SUCC(2,3) scheme, in which we use Eq. (46) to correct the results of the SUMRCCSD calculations (the m

_{A}= 2 case). In this case, the only generalized moments of the SUMRCC equations that need to be considered are the moments. The MM-SUCC(2,3) energy expression is [117]

_{µ}〉 and the SUMRCCSD wave function [cf. Eq. (36)]

_{p}〉. If i, j, k, . . . (a, b, c, . . .) represent the spin-orbitals that are occupied (unoccupied) in the reference configuration |Φ

_{p}〉 and if are the excitation operators that generate the triply excited configurations relative to |Φ

_{p}〉 (X

_{a}and X

_{i}are the usual creation and annihilation operators, respectively), we can write

_{0}, considered here, at least one index among i, j, k, a, b, c in Eqs. (53) and (54) must be inactive.

_{µ}〉 in the MM-SUCC(2,3) energy expressions, Eqs. (48) and (49). The MRMBPT(2) wave function is the lowest-order MRMBPT wave function that contains information about the cluster components. The use of wave functions |Ψ

_{µ}〉 of this type in Eqs. (48) and (49) would lead to a multi-reference extension of the recently proposed completely renormalized CCSD(T) [CR-CCSD(T)] method [7,26,117,118,119,121,132,133]. The spectacular successes of the single-reference CR-CCSD(T) approach in calculations of groundstate PESs involving bond breaking, where the standard CCSD(T) approach completely fails, suggest that the multi-reference analog of the CR-CCSD(T) approach, obtained by inserting the MRMBPT(2)-like wave functions |Ψ

_{µ}〉 in Eqs. (48) and (49), may provide excellent results, particularly in difficult situations in which the use of the low-order MRMBPT theory alone to estimate the higher-order (e.g., ) effects is not entirely appropriate due to the presence of intruder states.

_{µ}〉 in Eq. (33) and considering the truncated MM-SUCC(m

^{A},m

_{B}) schemes will cause the resulting energies to be no longer strictly size extensive (in a sense of introducing the unlinked terms into the MM-SUMRCC energies). However, our experience with the CI-based singlereference MMCC methods [7,120,122,123] and the MBPT-based renormalized CC approaches, such as CR-CCSD(T) [7,26,117,118,119,121,132,133], demonstrates that the presence of unlinked terms in the MMCC approximations does not have an effect on the excellent performance of the approximate MMCC schemes. In fact, we have recently performed a number of calculations showing that the CR-CCSD(T) approach provides approximately size extensive results, as long as we remain within the range of general applicability of this approach, which is a single bond breaking (cf., e.g., Refs. 133, 135). Moreover, a number of studies by the Paldus Waterloo group involving the so-called reduced MRCCSD (RMRCCSD) approach indicate that using the relatively inexpensive MRCI wave functions to estimate the higher-order contributions of the CC theory [6,136,137,138,139,140,141], at the risk of introducing unlinked terms into the calculations, tremendously benefits the CC results. The direct use of the final energy expressions in CC calculations, as is done in our MMCC theory [7,118,119,120,121,122,123] and its multi-reference extension discussed here and in Ref. 117, which may result in the introduction of unlinked terms, is also exploited in the Brillouin-Wigner MRCC method [32,33,34,35,36,37]. As in the approximate MMCC case, the Brillouin-Wigner MRCC approach is not size extensive. However, the Brillouin-Wigner MRCC results are excellent (even for molecular systems containing heavier atoms), which is again suggesting to us that the presence of unlinked terms in the approximate MM-SUMRCC energy expressions may not have a detrimental effect on the final results. We should also keep in mind that all approximate MMCC methods (just like the RMRCCSD approach of Paldus and Li) introduce unlinked terms in very high orders, so that it is quite likely that the results of approximate MM-SUMRCC calculations will be very good, in spite of the presence of unlinked terms in the approximate MM-SUMRCC energy expressions.

## 4 The State-Universal Multi-Reference Coupled-Cluster Method with Perturbative Description of Core-Virtual Excitations: The SUMRCCSD(1) Approach

^{(p)}is assigned to each reference configuration |Φ

_{p}〉 (p = 1, . . . , M). This requirement leads to an excessively large number of cluster amplitudes when the number of reference configurations is large. In the Jeziorski-Monkhorst formalism, we are forced to solve for all M T

^{(p)}cluster operators, even if we are interested in calculating a few low-lying states. In particular, each individual core–virtual excitation is in the SUMRCC theory represented by as many independent cluster amplitudes as the number of references. This is somewhat counterintuitive, since ideally we should only be required to determine as many cluster amplitudes for a given core–virtual excitation as is the number of electronic states under consideration. For example, in the popular MRCISD approach, we are required to determine as many CI coefficient vectors as is the number of calculated states. The latter number is usually much smaller than the number of reference configurations used in such calculations. As a matter of fact, at least in the first-order MRMBPT, the values of the core–virtual amplitudes representing the doubly excited clusters do not depend on the reference label p [38]. It is, therefore, quite reasonable to introduce an approximation, in which all core–virtual amplitudes of the SUMRCC theory are approximated by their first-order MRMBPT estimates. This new approximation, referred to as the SUMRCCSD(1) method [59], is discussed in this section.

_{p}〉. In terms of these operators, the singly and doubly excited clusters of the SUMRCCSD approach take the usual form,

_{0}is complete, the operators and and the corresponding cluster amplitudes and must carry at least one inactive (i.e., core or virtual) spin-orbital label.

_{1}〉 is the ground-state Hartree-Fock determinant. Let us construct the remaining reference configurations |Φ

_{p}〉, p = 2, . . . , M, in a usual way by choosing some occupied and some unoccupied spin-orbitals in |Φ

_{1}〉 as active spin-orbitals and by promoting active electrons (electrons occupying active spin-orbitals in |Φ

_{1}〉) to active spin-orbitals that are unoccupied in |Φ

_{1}〉. Let us also introduce the Fock matrix elements

_{p}〉. In addition, the (1) amplitudes vanish for p = 1 and are, in general, nonzero for p > 1. The order-by-order analysis of the SUMRCCSD method demonstrates that differences between cluster amplitudes associated with different references |Φ

_{p}〉 are even larger in higher orders of MRMBPT [38].

_{p}〉, which are not treated symmetrically in the standard SUMRCCSD scheme, has been discussed in the previous sections.

^{1}A

_{1}states of methylene, as described by the double-zeta plus polarization (DZP) basis set of Refs. 142, 143, is 1341 (720 for |Φ

_{1}〉 and 621 for |Φ

_{2}); see Table 1). The number of core–virtual amplitudes for each reference is in this case 318, so that the total number of core–virtual amplitudes considered in the two-reference OSA SUMRCCSD calculations for the DZP methylene molecule is 636. These 636 amplitudes are estimated in the SUMRCCSD(1) method by the first-order MRMBPT expression, Eq. (64), and by Eq. (65), and only the remaining 705 cluster amplitudes that carry at least one active orbital index are determined iteratively by solving the relevant subset of all SUMRCCSD equations. A similar ∼ 50 % reduction in the number of amplitudes that have to be determined in the SUMRCCSD(1) iterative procedure characterizes the calculations for methylene employing a larger [5s4p3d2f1g/3s2p1d] basis set of Ref. 144 (cf., also, Ref. 52). In this case, the total number of core–virtual amplitudes considered in the two-reference OSA SUMRCCSD calculations is 10432, while the number of all singly and doubly excited cluster amplitudes is 22611 (see Table 1). The 10432 core–virtual amplitudes are estimated in the SUMRCCSD(1) calculations via Eqs. (64) and (65), whereas the remaining 12179 amplitudes are determined iteratively. This significant reduction in the number of cluster amplitudes that have to be determined in the SUMRCCSD(1) iterative procedure would be even larger for larger many-electron systems, since the number of core–virtual amplitudes increases with the number of core electrons.

_{0}is spanned by two closed-shell configurations |Φ

_{1}〉 and |Φ

_{2}〉 involving two active electrons and two active orbitals that belong to different symmetry species of the spatial symmetry group of the system. The M = 2 model space, $\mathcal{M}$

_{0}= span{|Φ

_{1}〉, |Φ

_{2}〉}, is complete if we are only interested in the totally symmetric singlet eigenstates of the Hamiltonian [39,41,45,46,47,48,49,50,54]. The explicit equations of the two-reference OSA SUMRCCSD(1) theory, in terms of the OSA mono- and biexcited cluster amplitudes, are a straightforward modification of the two-reference OSA SUMRCCSD equations presented elsewhere [39,41,45,50].

^{1}A

_{1}states of the DZP H4 model [146,147] (see Table 2 and Fig. 1), the lowest two states of the Li

_{2}molecule, as described by the double zeta (DZ) basis set of Refs. 148, 149 (see Table 3), and the lowest two

^{1}A

_{1}states of methylene, as described by the DZP basis set [142,143] and the [5s4p3d/3s2p] [52] and [5s4p3d2f1g/3s2p1d] atomic natural orbital [144] basis sets (see Table 4). The results of the SUMRCCSD(1) calculations for H4 and CH

_{2}were reported earlier [59]. The results for Li

_{2}are new.

^{1}A

_{1}states of the DZP model of methylene can be found in Refs. 50, 58 (recall that the lowest

_{1}A

_{1}state represents in this case the first-excited state; the ground state of CH

_{2}is

^{3}B

_{1}; cf. the Introduction). The results of the two-reference SUMRCCSD calculations for methylene for the [5s4p3d/3s2p] and [5s4p3d2f1g/3s2p1d] basis sets were reported in Refs. 52, 58.

_{2}molecules arranged in an isosceles trapezoidal configuration, with all nearest-neighbor H–H separations fixed at 2.0 bohr. The geometry of the H4 model is determined by a single parameter . The α = 0 and limits correspond to square and linear conformations, respectively [146].

_{2}) to the lowestenergy unoccupied orbital (2a

_{1}). Indeed, in the α = 0 limit, the absolute values of the coefficients at configurations |Φ

_{1}〉 and |Φ

_{2}), which dominate the full CI expansions of the lowest two

^{1}A

_{1}states in this region, are identical. For α = 0.5, the ground-state wave function becomes essentially nondegenerate, with |Φ

_{1}〉 representing the leading configuration in the corresponding full CI expansion.

^{1}A

_{1}states of the DZP H4 model (the 1

^{1}A

_{1}and 2

^{1}A

_{1}states) are shown in Table 2 and Fig. 1. In the SUMRCCSD(1), SUMRCCSD, and MRMBPT(2) calculations, we chose orbitals 1b

_{2}and 2a

_{1}as active. This choice of active orbitals is justified by the dominant role of the |Φ

_{1}〉 and |Φ

_{2}〉 configurations, Eqs. (66) and (67), respectively, in the full CI expansions of the lowest two

^{1}A

_{1}states of the H4 system.

^{1}A

_{1}states of are shifted by ∼ 1 millihartree or less relative to the corresponding SUMRCCSD curves. The errors in the SUMRCCSD(1) results, relative to full CI, range between 0.5 (α ≈ 0) and 2.7 (α ≈ 0.5) millihartree for the ground state and between 4.1 (α ≈ 0) and 3.0 (α ≈ 0.5) millihartree for the first-excited

^{1}A

_{1}state. The errors obtained with the SUMRCCSD method are virtually identical to those obtained with the simpler SUMRCCSD(1) approximation. The SUMRCCSD(1) and SUMRCCSD vertical excitation energies ∆ corresponding to the 1

^{1}A

_{1}→ 2

^{1}A

_{1}transition agree to within 0.5–1.5 millihartree for all values of α. For comparison, the MRMBPT(2) approach completely fails for α > 0.15 due to intruder states. The errors in the MRMBPT(2) results for the 1

^{1}A

_{1}→ 2

^{1}A

_{1}excitation energies, relative to full CI, are as large as 172.8 millihartree in the α ≈ 0.5 region. Even in the region of small α values, where the MRMBPT(2) approach works best, the errors in the MRMBPT(2) results for the 1

^{1}A

_{1}→ 2

^{1}A

_{1}excitation energies are much larger than those obtained with the SUMRCCSD(1) method (cf., e.g., the 9.4 millihartree error in the MRMBPT(2) value of ∆ at α = 0 with the 3.6 millihartree error obtained with the SUMRCCSD(1) approach for the same value of α).

_{2}. We considered three geometries in this case: the equilibrium geometry, R = R

_{e}= 5.051 bohr (R is the internuclear separation), and two stretches of the Li–Li bond, R = 1.5R

_{e}and R = 2R

_{e}. For larger distances R, the ground and the first-excited states of the symmetry are dominated by the RHF configuration

_{g}and 2σ

_{u}, respectively, in our SUMRCCSD(1), SUMRCCSD, and MRMBPT(2) calculations. The results of the two-reference SUMRCCSD(1), SUMRCCSD, and MRMBPT(2) calculations for Li

_{2}, using configurations |Φ

_{1}〉, Eq. (68), and |Φ

_{2}〉, Eq. (69), as references, along with the corresponding full CI results, are shown in Table 3.

_{2}that are in excellent agreement with the high quality curves obtained with the SUMRCCSD method. The SUMRCCSD(1) energies for the lowest two states of Li

_{2}are only slightly above the corresponding SUMRCCSD energies [the difference between the SUMRCCSD(1) and SUMRCCSD energies is ∼ 2.5 millihartree, independently of the value of R]. In consequence, the errors in the SUMRCCSD(1) results, relative to full CI, are very small (2.7–2.8 millihartree for the ground state and 3.1–3.2 millihartree for the first-excited state). The errors in the 1 → 2 vertical excitation energies (designated by ∆) obtained with the SUMRCCSD(1) method, relative to full CI, are 0.4–0.5 millihartree, independently of the value of R. The SUMRCCSD results for the 1 → 2 excitation energies are virtually identical to those obtained with the simpler SUMRCCSD(1) approach. The two-reference MRMBPT(2) method works reasonably well in the quasi-degenerate region corresponding to larger R values, where the lowest two states are dominated by configurations (68) and (69), but the SUMRCCSD(1) results are much more accurate [cf., e.g., the 12.8 millihartree error in the MRMBPT(2) result for the vertical excitation energy ∆ at R = 2R

_{e}with the 0.3–0.4 millihartree errors obtained at the same value of R with the SUMRCCSD and SUMRCCSD(1) approaches]. In the R ≈ R

_{e}region, the ground state is dominated by the RHF configuration |Φ

_{1}〉, Eq. (68), but the first-excited state of the symmetry has significant contributions from configurations belonging to . As a result, the error in the MRMBPT(2) value for the vertical excitation energy ∆ is as large as 54.0 millihartree at R = R

_{e}. Remarkably enough, the errors in the SUMRCCSD and SUMRCCSD(1) results for the 1 → 2 excitation energy are as little as 0.5 millihartree at R = R

_{e}. This clearly shows that we can tremendously benefit from incorporating the MRMBPT ideas into the SUMRCCSD scheme.

^{3}B

_{1}, and the lowest excited state of the

^{1}A

_{1}symmetry, A

^{1}A

_{1}≡ 1

^{1}A

_{1}[51,52]. The OSA SUMRCCSD method is also capable of providing the excellent description of the singlet– singlet (2

^{1}A

_{1}− 1

^{1}A

_{1}) energy separation [50]. For example, the full CI value of the singlet–singlet energy gap in the DZP methylene molecule is 168.907 millihartree [143]. The two-reference SUMRCCSD method gives 169.885 millihartree [50], in excellent agreement with full CI.

^{1}A

_{1}− 1

^{1}A

_{1}energy gap in methylene is largely (but not entirely) related to the quasi-degenerate character of the lowest two

^{1}A

_{1}states, which are both dominated by two closed-shell configurations,

_{1}and 1b

_{1}. Thus, by choosing orbitals 3a

_{1}and 1b

_{1}as active orbitals in the SUMRCCSD calculations, we should (and we do) obtain excellent results for the singlet– singlet energy separation. It is interesting to note though that the two-reference MRMBPT(2) method does not provide very good results in this case, in spite of the apparently two-reference character of the lowest two

^{1}A

_{1}states. For example, the error in the 1

^{1}A

_{1}→ 2

^{1}A

_{1}excitation energy obtained with the MRMBPT(2) approach for the DZP methylene model, relative to full CI, is 11.344 millihartree. This should be compared to a much smaller, 0.978, millihartree error obtained with the two-reference SUMRCCSD method (see Table 4).

^{1}A

_{1}states of methylene by the two-reference SUMRCCSD(1) approach is almost as good as that provided by its parent SUMRCCSD analog. Although the differences between the SUMRCCSD(1) and SUMRCCSD individual energies are somewhat larger than in the case of the H4 and Li

_{2}systems, the vertical excitation energies corresponding to the 1

^{1}A

_{1}→ 2

^{1}A

_{1}transition in methylene (the ∆ values in Table 4) obtained in the SUMRCCSD(1) and SUMRCCSD calculations are essentially identical, independent of the basis set employed. Indeed, the differences between the ∆ values resulting from the SUMRCCSD(1) and SUMRCCSD calculations are 0.958 millihartree for the DZP basis set, 1.389 millihartree for the [5s4p3d/3s2p] basis set, and 1.503 millihartree for the largest [5s4p3d2f1g/3s2p1d] basis set. The difference between the ∆ values obtained in the SUMRCCSD(1) and full CI calculations with the DZP basis set is 1.936 millihartree, which should be compared to a 0.978 millihartree difference between the SUMRCCSD and full CI values of ∆.

## 5 Summary

## Acknowledgements

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**Figure 1.**The SUMRCCSD, SUMRCCSD(1), MRMBPT(2), and full CI potential energy curves for the lowest two

^{1}A

_{1}states of the DZP H4 model.

**Table 1.**Numbers of the spin- and symmetry-adapted singly and doubly excited cluster amplitudes used in the two-reference OSA SUMRCCSD calculations for molecular systems discussed in this work.

^{a}

^{a}N

_{S}and N

_{D}designate, respectively, the numbers of all singly and doubly excited cluster amplitudes used in the two-reference OSA SUMRCCSD calculations. N

_{S}

_{(CV)}and N

_{D}

_{(CV)}designate, respectively, the numbers of singly and doubly excited cluster amplitudes corresponding to core–virtual excitations.

^{b}The H4 model of Ref. 146.

^{c}The [2s1p] basis set taken from Ref. 147.

^{d}The [4s2p] contraction of the Dunning-Hay basis set of Ref. 148 (see Ref. 149).

^{e}The lowest-energy molecular orbital (1a

_{1}) is kept frozen and the Cartesian components of the d and (if present in a basis set) f and g orbitals are employed.

^{f}The DZP basis set taken from Refs. 142, 143.

^{g}Basis set taken from Ref. 52.

^{h}Basis set taken from Refs. 52, 144.

**Table 2.**The SUSD(1) ≡ SUMRCCSD(1), SUSD ≡ SUMRCCSD, and full CI energies (in hartree) for the lowest two

^{1}A

_{1}states of the DZP H4 model and the SUMRCCSD(1), SUMRCCSD, MRMBPT(2), and full CI values of the 1

^{1}A

_{1}→ 2

^{1}A

_{1}excitation energies (∆, in millihartree) as functions of the parameter α [59].

**Table 3.**The SUSD(1) ≡ SUMRCCSD(1), SUSD ≡ SUMRCCSD, and full CI energies (in hartree) for the lowest two states of the DZ model of Li

_{2}and the SUMRCCSD(1), SUMRCCSD, MRMBPT(2), and full CI values of the 1 → 2 excitation energies (∆, in millihartree) for the equilibrium valueof the Li–Li separation R (R = R

_{e}= 5.051 bohr) and two stretches of the Li–Li bond (R = 1.5R

_{e}and R = 2R

_{e}).

**Table 4.**The SUSD(1) ≡ SUMRCCSD(1), SUSD ≡ SUMRCCSD, and full CI energies (in hartree) for the lowest two

_{1}A

_{1}states of methylene and the SUMRCCSD(1), SUMRCCSD, MRMBPT(2), and full CI values of the 1

_{1}A

_{1}→ 2

_{1}A

_{1}excitation energy (∆, in millihartree).

_{a}

^{a}Based on the results reported in Ref. 59. The lowest-energy molecular orbital was kept frozen and the Cartesian components of the d and (if present in a basis set) f and g orbitals were employed.

^{b}Basis set, geometries, and full CI results taken from Refs. [142,143].

^{c}Basis set and geometries taken from Ref. [52].

^{d}Basis set and geometries taken from Refs. [52,144].

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Piecuch, P.; Kowalski, K.
The State-Universal Multi-Reference Coupled-Cluster Theory: An Overview of Some Recent Advances. *Int. J. Mol. Sci.* **2002**, *3*, 676-709.
https://doi.org/10.3390/i3060676

**AMA Style**

Piecuch P, Kowalski K.
The State-Universal Multi-Reference Coupled-Cluster Theory: An Overview of Some Recent Advances. *International Journal of Molecular Sciences*. 2002; 3(6):676-709.
https://doi.org/10.3390/i3060676

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Piecuch, Piotr, and Karol Kowalski.
2002. "The State-Universal Multi-Reference Coupled-Cluster Theory: An Overview of Some Recent Advances" *International Journal of Molecular Sciences* 3, no. 6: 676-709.
https://doi.org/10.3390/i3060676