CCSD approximation

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

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

Using the connected cluster form of H defined above, as well as the techniques of Wick s theorem and normal ordering, we may derive a programmable form of the energy expression in the CCSD approximation. In accord with Eq. [50] and the normal-ordered Hamiltonian, the energy is given by... 

This equation is not restricted to the CCSD approximation, however. Since higher excitation cluster operators such as T3 and T4 cannot produce fully contracted terms with the Hamiltonian, their contribution to the coupled cluster energy expression is zero. Therefore, Eq. [134] also holds for more complicated methods such as CCSDT and CCSDTQ. Higher excitation cluster operators can contribute to the energy indirectly, however, through the equations used to determine the amplitudes, and t-h, which are needed in the energy equation above. 

As discussed earlier, the cluster amplitudes that parameterize the coupled cluster wavefunction may be determined from the projective Schrodinger equation given in Eq. [51]. In the CCSD approximation, the single-excitation amplitudes, t- , may be determined from... 

At the ab-initio level, the most obvious possibility is offered by CAS SCF or CAS FCI (i.e., Cl within the CAS or, equivalently, CAS SCF without the orbital reoptimization based on RHF orbitals, cf. [33, 34]) wave functions based on the smallest possible active-space that warrants the correct description of the dissociation channel at hand. This option was also suggested by Stolarczyk [29], although we are not aware of any concrete implementation. Our testing proved to be very encouraging [33, 34], particularly for open shell systems, in which case we employed the spin-adapted CCSD based on the unitary group approach (UGA) [16, 36]. Even in the case of triple bond breaking, the applicability of the CCSD approximation can be significantly extended, as will be shown in Sect. 4. Most recently, we have explored the MR CISD wave function as an external source, as described in the next section. 

The explicit expressions for h and hpq relate these quantities to one- and two-electron molecular integrals / and vpq and, in case of the CCSD approximation, to cluster amplitudes tla and t Jb [34] (cf., however, Ref. 83 see, also Appendix A). We use these expressions in this paper as well, since some many-body components of Hn can easily be related to and hpq (provided that we use the XCCSD rather than CCSD amplitudes t a and... 

The explicit equations of the EOMECC theory proposed above are almost identical to equations presented in Appendices A and B [provided, of course, that we use the CCSD approximation, T = T + T2, E = Ei + E2,... 

In this coupled-cluster singles and doubles (CCSD) approximation, none of the coefficients in the FCI expansion is ignored. Instead, they are approximated by a much smaller set of singles and doubles amplitudes. 

To factor out basis set effects, we first compare in Table 24 full configuration interaction (FCI) excitation energies and dipole strengths (the values in parentheses) to those obtained with the TDA, RPA, ACISD, and EOM-CCSD approximations. Because FCI is a very expensive method, which has a factorial dependence on the number of electrons, we are restricted to a small system and present comparisons only for the beryllium atom and the CH" molecule." ... 

In this work, in addition to the CCSD approximation, we examine two different ways of correcting the CCSD energy for the effects of the connected triply excited clusters, namely, the CCSD(T) method and its completely renormalized CR-CC(2,3) extension. Since the CCSD(T) approach can be obtained as a natural approximation to CR-CC(2,3) [24, 25], we begin our brief description of both methods with the key equations of CR-CC(2,3). 

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]. 

An important question which can be raised is how the r-electron CCSD approximation in (hyper)polarizability calculations compare with the results of other methods, particularly with the ab initio and semiempirical all-valence methods. It is also important to compare obtained theoretical results with available experimental data. 

For a more accurate treatment of electron correlation, coupled-cluster (CC) approaches [23] enter into play. While the full CC singles, doubles, triples (CCSDT) model [24] and augmented CCSDT approaches that feature corrections for quadruple and even higher excitations [25, 26] are currently too expensive, CC singles and doubles (CCSD) approximation [27] and CCSD augmented by a perturbative treatment of triple excitations, CCSD(T) [28], are more feasible and rather widely adopted. 

General experience, however, is that fourth-order MBPT overestimates the triples correction. A somewhat more reliable approach is obtained when the second-order MBPT amplitudes in equation (44) are replaced by the converged CCSD amplitudes. This leads to the so-called CCSD-I-T(CCSD) approximation. Its performance is reasonable, but it still tends to exaggerate the triples contribution in more difficult cases. ... 

Kobayashi et al. implemented in 1994 the approach outlined above for the calculation of frequency-dependent polarizabilities within the CCSD approximation. Hattig et al. have recently extended it further to frequency-dependent hyperpolarizabilities. Results reported in Table 9 for CH illustrate the accuracy achieved in CC calculations of dynamic polarizabilities. Comparison with SCF indicates the importance of electron correlation in this challenging case, while the good agreement with FCI demonstrates the excellent performance of the CCSD approximation. 

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