SR CISD

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]. 

Turning, next, our attention to the CC methods, we first consider the SR approaches. These results are summarized in Table III. We first note a much better performance of SR CCSD over the SR CISD (2nd column in Tables III and II, respectively) The CCSD deviations from FCI are much smaller than the CISD ones, as are the corresponding NPEs. This seems to be a general feature of the SR CCSD method, which often performs amazingly well even in severely quasidegenerate situations. 

Another approach for treating the quasi-degeneracy is adopted by the various MR-based CEPA methods, which have appeared parallely along with the MRCC and MRPT methods. The earlier developed state-specific MRCEPA methods [37,65-70] avoided the redundancy problem using non-redundant cluster operators to compute the dynamical correlation on the zeroth order MR wave function. The MR version of (SC) CI method, termed as MR-(SC) CI [37], can be viewed as the size-extensive dressing of the MR-CISD method just as the (SC) CI [71] is considered to be the size-extensive dressing of the SR-CISD method. Similar to the SR-case, they include all EPV terms in an exact manner. 

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. 

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]. 

Table II. The SR, 2R, 4R and (2/2)R CISD ground state energies together with various Davidson-type corrections relative to the FCI energy (in mH) for the DZP H4 model with 0 < a < 0.5. The nonparallelism errors (NPE) are givenin the last two rows. See the text for details. 

Relying on the above discussed complementarity of the SR CCSD and MR CISD ansatze, it seems particularly attractive to employ the latter as an external source of 7) and T4 corrections. In order to explicitly illustrate this complementarity and the scope of the formalism involved, let us consider a minimal 2-reference case, i.e. let us assume that a given SCF reference becomes quasidegenerate with another configuration. For a CS system this case arises when the one-electron active-space involves only two MOs, each belonging to a different... 

Table 2. Comparison of various limited SR Cl, SR CCSD, and selected RMR CCSD and corresponding MR CISD energies with the exact FCI result for the N2 molecule, obtained with a DZ basis for three internuclear separations R, R = Re = 2.068 bohr, R = 1.5Re, and R = 2Re [23], Except for the FCI total energy, which is reported as — (E + 108) (in hartree), the energy differences (in millihartree) relative to the FCI result are given in all cases. The nonparallelism error (NPE) for the intervals R e [Re, 1.5i e] and R e [Re,2Re] (in millihartree) is given for easier comparison (see the text for details) ...

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... 

Ab initio pseudopotential calcul ons at the Hartree-Fodr (HF), MP2 and CISD levels have been conductedi on the equilibrium geom es of q>2M (M=Ca, Sr or Ba). At the HF level, linear sandwkh-type equilibrium structures were indicated. Large-scale MP2... 

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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