RMR CCSD

It turns out that MR CISD represents again the most suitable source of the required higher-order clusters. Carefully chosen small reference space MR CISD involves a very small, yet representative, subset of such cluster amplitudes. Moreover, in this way we can also overcome the eventual intruder state problems by including such states in MR CISD, while excluding them from CMS SU CCSD. In other words, while we may have to exclude some references from Ado in order to avoid intruders, we can safely include them in the MR CISD model space Adi. In fact, we can even choose the CMS for Adi. Thus, designating the dimensions of Ado and Adi spaces by M and N, respectively, we refer to the ec SU CCSD method employing an NR-CISD as the external source by the acronym N, M)-CCSD. Thus, with this notation, we have that (N, 1)-CCSD = NR-RMR CCSD and (0, M)-CCSD = MR SU CCSD. Also, (0,1)-CCSD = SR CCSD. For details of this procedure and its applications we refer the reader to Refs. [63,64,71]. 

SR NR-RMR CCSD CCSD- CCSD- 2R-RMR ACCSD ACCSD CCSD CCSDQ CCSDQ ... 

RMR CCSD methodologies, their performance is illustrated by a few examples, and their potential and relationship with other approaches is discussed. 

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. 

We first outline the basic idea and origins of the externally corrected CCSD methods in Sect. 2, followed by the formulation and discussion of its special version, the RMR CCSD method, in Sect. 3. In Sect. 4 we present a few illustrative examples and summarize the general conclusions in Sect. 5. 

Assuming now that the r3(0) and 74 01 amplitudes so obtained represent a reasonable approximation of actual T3 and T4 clusters, we can employ the RMR CCSD ansatz... 

To summarize, the RMR CCSD method involves the following three steps (i) We choose a suitable reference space and compute the corresponding MR CISD. Next, (ii) we compute r3(0) and r4(0) clusters by cluster analyzing the MR CISD wave function of step (i), and finally (iii) we use these amplitudes to generate and solve ecCCSD equations. The details of the actual implementation of the RMR CCSD method for various types of reference spaces can be found in our earlier papers [21, 22, 23, 24, 25]. 

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. 

Table 1. Comparison of the SR CCSD, RMR CCSD, and ASTQ CCSD energies with the exact FCI result for the X2If state of OH at three internudear separations R, R = Re = 1.832 bohr, R = l.5Re> and R = 2Re. Except for the SCF and FCI total energies, which are reported as — (E + 75) (in hartree), the energy differences (in millihartree) relative to the FCI result are given in all cases. The nonparallelism error (NPE) for the interval R e [Re, 2Re] (in millihartree) is also given for easier comparison (see the text for details) ...

It is immediately apparent from the results in Table 1 that the RMR CCSD strategy is yet more effective for reasons given in Sect. 3. Already the 2R version... 

JWe note here that since RMR CCSD is based on a special version of MR CISD, our codes are not yet spin-adapted so that the corresponding dimensions are given in terms of the number of Slater determinants rather than CSFs. Note also that, on average, the number of CSFs is about one-third of the number of determinants or less. 

In addition to CAS(n, n) RMR CCSD results, we also present those based on incomplete active-spaces. These model spaces (MSz s) are obtained by imposing certain restrictions on the orbital occupancies of active orbitals (see [23]). Only three typical spaces of the latter type, namely MSz for i = 1,8 and 9, involving 8, 16 and 48 determinants, respectively, are included. 

The RMR CCSD approach certainly represents the optimal ecCCSD method relying on the Cl-type wave function. In particular, we must emphasize its conceptual simplicity and the fact that the method is unambiguously defined by the choice of the reference space, whose dimension can be very small indeed. Already a 2-dimensional 2R-RMR CCSD often provides excellent results. This is important, since the computational cost, in contrast to ASTQ approaches, is proportional to the number of references employed. On the other hand, in view of the fact that in computing a subset of r3<0) and T4<0) amplitudes we... 

The other possibility is to focus on the MR CISD wave function and exploit the Tj(0) and r20) clusters it provides to account for the dynamic correlation due to disconnected triples and quadruples that are absent in the MR CISD wave function. This approach, recently proposed and tested by Meissner and Gra-bowski [42], may thus be characterized as a CC-ansatz-based Davidson-type correction to MR CISD. The duplication of contributions from higher-than-doubly excited configurations that arise in MR CISD as well as through the CC exponential ansatz is avoided by a suitable projection onto the orthogonal complement to the MR CISD N-electron space. The results are very encouraging, particularly in view of their affordability, though somewhat inferior to RMR CCSD. 

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 objective of this study is to explore the possibilities offered by the AL version of RMR CCSD, since in the RMR-type approaches one obtains the three and four body clusters from the corresponding MR CISD wave function. 

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