CC theory

The amplitudes C- iT. f, etc, which play the role of the Cj coefficients in CC theory, are detemiined tln-ough the set of equations generated by projecting the Scln-ddinger equation in the fomi... 

These quartic equations are solved in an iterative maimer and, as such, are susceptible to convergence difficulties. In any such iterative process, it is important to start with an approximation reasonably close to the final result. In CC theory, this is often achieved by neglecting all of tlie temis tliat are nonlinear in the t amplitudes (because the ts are assumed to be less than unity in magnitude) and ignoring factors that couple different doubly-excited CSFs (i.e. the sum over i, f, m and n ). This gives t amplitudes that are equal to the... 

The t , t j ", ete. amplitudes, whieh play the role of the Cl eoeffieients in CC theory, are determined through the set of equations generated by projeeting the Sehrodinger equation in the form... 

For our purposes, CC theory and its finite order MBPT approximations offer a convenient, compact description of the correlation problem and give rapid convergence to the basis set (i.e. full Cl) limit with different categories of correlation operators (see Fig. 15.1). The coupled-cluster wave function is... 

Let us recall that in the single-reference CC theory, the ground-state wave function, ) of an A/-electron system, described by the Hamiltonian H, is given by the following expression ... 

Harmonic and cubic force fields of Sis were calculated using coupled-cluster (CC) theory (25) and a correlation-consistent basis set. Specifically, the CC singles and doubles (CCSD) method (24) was used in conjunction with the cc-pVTZ basis set (25) developed by Dunning and co-workers. The force... 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

Figures 7 and 8 plot deviations of total energies from FCI results for the various methods. It is clear that the CASSCF/L-CTD theory performs best out of all the methods smdied. (We recall that although the canonical transformation operator exp A does not explicitly include single excitations, the main effects are already included via the orbital relaxation in the CASSCF reference.) The absolute error of the CASSCF/L-CTD theory at equilibrium—1.57 mS (6-31G), 2.26 m j (cc-pVDZ)—is slightly better than that of CCSD theory—1.66m j (6-31G), 3.84 m j (cc-pVDZ) but unlike for the CCSD and CCSDT theories, the CASSCF/L-CTD error stays quite constant as the molecule is pulled apart while the CC theories exhibit a nonphysical turnover and a qualitatively incorrect dissociation curve. The largest error for the CASSCF/L-CTD method occurs at the intermediate bond distance of 1.8/ with an error of —2.34m (6-3IG), —2.42 mE j (cc-pVDZ). Although the MRMP curve is qualitatively correct, it is not quantitatively correct especially in the equilibrium region, with an error of 6.79 mEfi (6-3IG), 14.78 mEk (cc-pVDZ). One measure of the quality of a dissociation curve is the nonparallelity error (NPE), the absolute difference between the maximum and minimum deviations from the FCI energy. For MRMP the NPE is 4mE (6-3IG), 9mE, (cc-pVDZ), whereas for CASSCF/ L-CTD the NPE is 5 mE , (6-3IG), 6 mE , (cc-pVDZ), showing that the CASSCF/L-CTD provides a quantitative description of the bond breaking with a nonparallelity error competitive with that of MRMP.

Rapid convergence of molecular properties with respect to the excitation rank is obtained with the systematic approximations of coupled-cluster (CC) theory... 

The ansatz of CC-R12 was introduced by Noga et al. [31,32] as a generalization of the MP2-R12 ansatz to CC theory. Taking CC-R12 with connected singles and doubles (CCSD-R12) as an example, its wave function can be parameterized as... 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

The only, yet essential, limitation that is imposed on the SR version of the CC theory is the nondegeneracy of the reference configuration [ )... 

Unfortunately, in contrast to the Cl method, an extension of the SR CC theory to the MR case is far from being straightforward, since there is no unique way in which to generalize the SR exponential Ansatz for the exact N-electron wave function jT), i.e.,... 

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]. 

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