Higher-than-pair clusters

We have carried out several applications showing the promise of this procedure [63,64], as well as addressed the question of the size-consistency and size-extensivity [65-67], to which we wish to turn our attention again in this paper. Finally, we have also extended the idea of externally corrected (ec) SR CCSD methods [68-70] (see also Refs. [21,24]) to the MR case, introducing the (N, M)-CCSB method [71], which exploits an Preference (A R) CISD wave functions as a source of higher-than-pair clusters in an M-reference SU CCSD method. Both the CMS and (N, M)-CCSD allow us to avoid the undesirable intruder states, while providing very encouraging results. 

Let us recall, finally, that ec CCSD approaches exploit the complementarity of the Cl and CC methods in their handling of the dynamic and nondynamic correlations. While we use the Cl as an external source of higher than pair clusters, Meissner et al. [10,72-74] exploit the CC method to correct the Cl results (thus designing the CC-based Davidson-type corrections). This aspect will also be addressed below. 

Another way to exploit the complementarity of Cl and CC approaches was explored earlier by Meissner et al. [10]. Instead of using Cl as a source of higher-than-pair clusters and correcting CCSD, it exploits the CC theory to correct the MR CISD results. In the spirit of an earlier work on Davidson-type corrections for SR CISD [10], Meissner et al. formulated a CCSD-based corrections for both SR [72] and MR [74] CISD. The latter was later extended to higher lying excited states [73]. 

The three- and four-body clusters play an even more important role in MR CC theories. In contrast to the SR formalism, where the energy is fully determined by one- and two-body clusters, the higher-than-pair clusters enter already the effective Hamiltonian. Consequently, even with the exact one- and two-body amplitudes, we can no longer recover the exact energies [71]. Here we must also keep in mind that the excitation order of various configurations from is not uniquely defined, since a given configuration... 

The above presented data clearly demonstrate the usefulness of the ec CC approaches at both the SR and MR levels. While in the SR case the energy is fully determined by the one- and two-body clusters, and the truncation of the CC chain of equations at the CCSD level can be made exact by accounting for the three- and four-body clusters, the MR case is much more demanding, since the higher-than-pair clusters appear already in the effective Hamiltonian. An introduction of the external corrections is thus... 

Numerous earlier studies of the ec GG methods clearly indicate that the modest size MR GISD wave functions represent the most suitable and easily available source of higher-than-pair clusters for this purpose (see Ref. [21] for an overview). Indeed, these wave functions can be easily transformed to a SR form, whose cluster analysis is straightforward. Moreover, the resulting three- and four-body amplitudes represent only a very small subset of all such amplitudes, namely those which are most important, and which at the same time implicitly account for all higher-order cluster components that are present in the MR GISD wave functions. 

Our recently developed reduced multireference (RMR) CCSD method [16, 21, 22, 23, 24, 25] represents such a combined approach. In essence, this is a version of the so-called externally corrected CCSD method [26, 27, 28, 29, 30, 31, 32, 33, 34] that uses a low dimensional MR CISD as an external source. Thus, rather than neglecting higher-than-pair cluster amplitudes, as is done in standard CCSD, it uses approximate values for triply and quadruply excited cluster amplitudes that are extracted by the cluster analysis from the MR CISD wave function. The latter is based on a small active space, yet large enough to allow proper dissociation, and thus a proper account of dynamic correlation. It is the objective of this paper to review this approach in more detail and to illustrate its performance on a few examples. 

Although the existing applications of RMR CCSD, and of other versions of the ecCCSD method, have shown considerable promise, much work remains to be done in order to establish optimal sources of higher than pair clusters that would be both reliable and computationally affordable, as well as to determine the limits of applicability of this type of approaches. Here we shall only present a few typical examples that illustrate the potential of this technique, drawing on both the existing applications and recently generated new results. 

A rapid increase in the importance of higher-than-pair clusters is clearly illustrated by the sequence of SR Cl results. Even the SR CISDTQ NPEs amount to 12.6 and 38 mhartree. This clearly indicates the role played by higher-than-4-body (both connected and disconnected) clusters as R —> oo. 

For these and other reasons, much attention was given to the so-called state-selective or state-specific (SS) MR CC approaches. These are basically of two types (i) essentially SR CCSD methods that employ MR CC Ansatz to select a subset of important higher-than-pair clusters that are then incorporated either in a standard way [163,164], or implicitly [109-117], or via the so-called externally corrected (ec) approaches of either the amplimde [214-219] or energy [220,221] type, and (ii) those actually exploiting Bloch equations, but focusing on one state at a time [222]. The energy-correcting ec CC approaches [220,221] are in fact very closely related to the renormalized CCSD(T) method of Kowalski and Piecuch mentioned earlier [146,147]. 

With the GMS-based SU CCSD method, we were able to carry out a series of test calculations for model systems that allow a comparison with full Cl results, considering GMSs of as high a dimension as 14. These results are most promissing. Moreover, we have formulated a generalization of the RMR CCSD method, resulting in the so-called (M, N)-CCSD approach [226] that employs an M-reference MR CISD wave functions as a source of higher-than-pair clusters in an Al-reference MR SU CCSD (clearly, we require that M S N). In this way, the effect of intruders can be taken care of via external corrections, which are even more essential at the MR level than in the SR theory, because, in contrast to... 

The basic idea of the externally corrected CCSD methods relies on the fact that the electronic Hamiltonian, defining standard ah initio models, involves at most two body terms, so that the correlation energy is fully determined by one (Ti) and two (T2) body cluster amplitudes, while the subset of CC equations determining these amplitudes involves at most three (T3) and four (T4) body connected clusters. In order to decouple this subset of singly and doubly projected CC equations from the rest of the CC chain, one simply neglects all higher than pair cluster amplitudes by setting... 

The effect of higher than pair clusters (primarily of triples) is often estimated perturbatively. This is particularly efficient as long as the state considered remains nondegenerate, as is the case for nearly equilibrium geometries. In such cases, the often-employed CCSD(T) method 11) yields invariably excellent results (5,72). Unfortunately, with the increasing quasidegeneracy, the perturbative treatment of triples breaks down, even in cases... 

In our earlier work we have examined various ways of accounting for the nondynamic correlation in the SR-type CC approaches. Our main aim was directed towards the improvement of one- and two-body cluster amplitudes by relying on the ecCCSD approach, particularly on RMR CCSD which exploit the MR CISD wave function of a modest size as the source of higher than pair clusters. The capability of this approach to generate highly accurate spectroscopic data was briefly summarized in Sec. II. 

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