CCSD model amplitude

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

One of the special cases of coupled-cluster theory is the singles-and-doubles (CCSD) model [37]. The cluster operator Eq. (29) is restricted to contain only the singles and doubles excitation operators. The importance of this model can be seen from the fact that, for any coupled-cluster wave function, the singles and doubles amplitudes are the only ones that contribute directly to the coupled-cluster energy. In the explicitly correlated CCSD model the conventional cluster operator containing the T and T2 operators is supplemented with an additional term that takes care of the explicit correlation (written with red font)... 

In the case of the full CCSD(F12) model, to obtain the solution of the CC equations the Ti, T2 conventional amplitudes and coefficients have to be determined. Compared to the conventional CCSD scheme, the explicit electron correlation requires an additional equation for the qj coefficients [see Eqs. (41), (47)]. The CCSD(F12) amplitude equations, obtained by projecting Eq. (129) onto the excitation manifolds, have the following form... 

For the popular CCSD model recalling the definition of an exponential operator Eq. (3.71) and using the Mpller-Plesset partitioning of the Hamiltonian, Eq. (9.62), the expression for the energy and the amplitude equations then become... 

In the CCSD model, for example, the excited projection manifold comprises the fiill set of all singly and doubly excited determinants, giving rise to one equation (13.2.19) for each connected amplitude. For the full coupled-cluster wave function, the number of equations is equal to the number of determinants and the solution of the projected equations recovers the FCI wave function. The nonlinear equations (13.2.19) must be solved iteratively, substituting in eac iteration the coupled-cluster energy as calculated from (13.2.18). 

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. 

For high-precision work, the CCSD model is usually not sufficiently accurate and the CCSDT model is too expensive. Various hybrid methods, in which the contribution from the triples amplitudes is estimated by perturbation theory, have therefore been developed to provide descriptions intermediate between CCSD and CCSDT in cost and accuracy. The most pc ular is the CCSCHT) method, whose performance for the dissociation of the water molecule is displayed in Figure 15.18. 

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]. 

It is important to note, however, that there are fundamental differences between FSCC and SRCC with respect to the nature of their excitation operators. For a given truncation of the cluster operators beyond simple double excitations, the determinantal expansion space available in an FSCC calculation is smaller than those of SRCC calculations for the various model space determinants. A class of excitations called spectator triple excitations must be added to the FSCCSD method to achieve an expansion space that is in some sense equivalent to that of the SRCC. But even then, the FSCC amplitudes are restricted by the necessity to represent several ionized states simultaneously. Thus, we should not expect the FSCCSD to produce results identical to a single reference CCSD, nor should we expect triple excitation corrections to behave in the same way. The differences between FSCC and SRCC shown in Table I and others, below, should be interpreted as a manifestation of these differences. 

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... 

Recently, both valence bond (VB) and complete active space (CAS) SCF and CAS FCI wave functions were employed as a source of T3 and T< cluster amplitudes (11,13) (see also Ref. 12). The most satisfactory approach, however, that was developed very recently, relies on the MR CISD wave function, based on a relatively small model space. This approach is referred to as the reduced MR (RMR) CCSD method (15-17). 

It is interesting to compare the RMR CCSD and AL-RMR CCSD results obtained with two different model spaces, namely (2,2) and (4,4) ones. For example, when using the (2,2) model space, the difference between the RMR CCSD and AL-RMR CCSD energies is very small, often less than 50 /zhartree. Nonetheless, the largest difference, amounting to 1 mhartree, is found for a = 6.0 a.u. This indicates that in some cases, the and amplitudes from (2,2)-MR CISD are not very close to the final Ti and T2, and the AL version may cause some errors. When the large (4,4) model space is used, the difference between RMR CCSD and AL-RMR CCSD in fact disappears. This is easily understood since the S4 model is a four-electron system and the and from (4,4)-MR CISD should be very close to the real T and T2. 

In addition to the terms presented so far, the implementation of the CCSD(F12) model in Turbomole requires the inclusion of terms that are the analogues of the MP2-F12 terms responsible for the coupling between conventional and MP2-F12 amplitudes [47]. 

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