MR CCSD

At the same time, Meissner, Kucharski and others [56,57] developed the quadratic MR CCSD method in a spin-orbital form which does not exploit the BCH formula. The unknown cluster amplitudes are calculated from the so-called generalized Bloch equation [45-47,49,64,65] (or in our language the Bloch equation in the Rayleigh-Schrodinger form)... 

Linear MR CCSD approximation. Symbol NC indicates that no convergence is achieved when the ground and first biexcited states are calculated simultaneously. 

MR CCSD-3 approximation taken from [55]. Missing values were not present. 

Quadratic MR CCSD approximation [58]. Since not all geometries were present, we recomputed the whole curve again. 

Table IV. Various SU, SS and BW MR-CCSD ground state energies relative to the FCI energy (in inH) for the DZP H4 model with 0 < a < 0.5. The nonparallelism errors (NPE) are given in the last two rows. See the text and Table II for details. 

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. 

Coupled Cluster (cc) expansions Approximations based on CC expansions [3, 91,92] include ccd, ccsd, ccsdt, etc., and their multireference variants mr-CCD, MR-ccsD, MR-ccsDT, etc. The CCSD approximation is also known as the coupled pair approximation (cpa) [59] or the coupled pair many electron theory (CPMET) [20,21,93]. 

Figure 2. Potential energy curves for twisting of the tetramethyleneethane diradical (TME), computed by the CCSD, two-reference Hilbert space MR-CCSD (TDCCSD), and MR-BWCCSD methods.

TABLE 3. The Bergman reaction a comparison of heats of reaction and activation enthalpies supported by CCSD and MR-CCSD methods. 

The next step in the development and implementation of the MR ccsd method is to include the quadratic terms and, in general, non-linear terms. Here, we should mention the orthogonally spin-adapted MR ccsD-1, mr ccsd-2 and MR ccsd-3 approximations developed by Paldus et al. [105] and tested for the H4 model system. The first two approximations were designed just for testing purposes in order to better assess the importance of various non-linear terms. All three approximations are extensive. They differ by the presence of quadratic and bi-linear terms in the direct component, as well as in the coupling terms in the equation for cluster amplitudes (4.87). To be more precise, in addition to absolute and linear terms, the MR ccsd- 1 method contains the quadratic term involved in the direct term the MR ccsd-2 method contains the quadratic term involved in both the direct component, as well as in the coupling terms, and, finally, the MR ccsd-3 method represents a fully quadratic MR ccsd approximation which considers all bi-linear terms. The main conclusions to be drawn from these studies are that the inclusion of quadratic terms eliminates the singular behaviour of the linear mr ccsd approximation, mr l-ccsd, (even at the mr ccsd- 1 level) and that the inclusion of bi-linear components usually further improves the results. 

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