Externally corrected CCSD amplitude correcting

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. 

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... 

As already pointed out in Ref. 13, the externally corrected CCSD is equivalent to (truncated) CCSDTQ with zero-iteration on and T4 amplitudes that are in turn obtained from some external sources. Depending on the source of these amplitudes, we usually deal with only a proper subset of all possible T3 and T4 amplitudes. This subset is fixed in the externally corrected CCSD calculations. The RMR CCSD is then a special case of the general externally corrected CCSD in which the MR CISD wave function is used as the external source. The RMR CCSD method represents in fact a multireference approach in the sense that it is uniquely defined by the choice of the reference space and the fact that the RMR CCSD wave function involves the same number of connected cluster amplitudes as the corresponding genuine MR CCSD, such as the state-universal CCSD employing the same reference space. 

The split-amplitude strategy represents the total amplitudes as the sum of an a priori known approximate value, obtained from some external source, and an unknown correcting term. Assuming, further, that the known amplitudes represent a good approximation to the true ones, the unknown corrections can be obtained to a high degree of accuracy from a set of linear equations. The results of this article show that when a proper reference space is used, the connected clusters obtained from the MR CISD wave function represent indeed a very good approximation, and the almost linear versions of the RMR CCSD method performs very well. 

In order to overcome the shortcommings of standard post-Hartree-Fock approaches in their handling of the dynamic and nondynamic correlations, we investigate the possibility of mutual enhancement between variational and perturbative approaches, as represented by various Cl and CC methods, respectively. This is achieved either via the amplitude-corrections to the one- and two-body CCSD cluster amplitudes based on some external source, in particular a modest size MR CISD wave function, in the so-called reduced multireference (RMR) CCSD method, or via the energy-corrections to the standard CCSD based on the same MR CISD wave function. The latter corrections are based on the asymmetric energy formula and may be interpreted either as the MR CISD corrections to CCSD or RMR CCSD, or as the CCSD corrections to MR CISD. This reciprocity is pointed out and a new perturbative correction within the MR CISD is also formulated. The earlier results are briefly summarized and compared with those introduced here for the first time using the exactly solvable double-zeta model of the HF and N2 molecules. 

---