Variational coupled cluster doubles

C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, /. Chem. Phys., 109, 4171 (1998). Energies and Analytic Gradients for a Coupled-Cluster Doubles Model Using Variational Brueckner Orbitals. Application to Symmetry Breaking in O4. 

Like the variational coupled-cluster conditions (13.1.20), the projected equations (13.1.22) are nonlinear in the amplitudes. However, unlike the variational conditions, the expansion of the wave function in (13.1.22) and (13.1.23) terminates after a few terms since the Hamiltonian operator couples determinants that dilfer by no higher than double excitations, making the solution of the projected equations and the calculation of the energy tractable. Of course, the calculated coupled-cluster energy no longer represents an upper bound to the FCI energy. In practice, the deviation from the variational energy turns out to be small and of little practical consequence. 

Quadratic Cl (QCI) and coupled cluster (CC) exemplify more complex methods that are not strictly variational in character, but include physical corrections similar to those of higher-order perturbation theory. Keywords for these methods also include a specification of substitutions from the reference FIF configuration, such as QCISD or CCSD, respectively, for QCI or CC methods with all single and double substitutions. More complete descriptions of these methods are beyond the scope of this appendix. 

Figure 1. The shape of the potential curve for nitrogen in a correlation-consistent polarized double-zeta basis set is presented for the variational 2-RDM method as well as (a) single-reference coupled cluster, (b) multireference second-order perturbation theory (MRPT) and single-double configuration interaction (MRCl), and full configuration interaction (FCl) wavefunction methods. The symbol 2-RDM indicates that the potential curve was shifted by the difference between the 2-RDM and CCSD(T) energies at equilibrium.

Coupled Cluster Singles and Doubles A non-variational method of solving the Schrddinger equation with the wave function in the form of an exponential operator (with the explicit presence of the single and double excitations, their contribution to be determined in the method) acting on the Hartree-Fock wave function. 

Alternatively, there are perturbation methods to estimate Ecorreiation- Briefly, in these methods, you take the HF wavefunction and add a correction—a perturbation—that better mimics a multi-body problem. Moller-Plesset theory is a common perturbative approach. It is called MP2 when perturbations up to second order are considered, MP3 for third order, MP4, etc. MP2 calculations are commonly used. Like CISD, MP2 allows single and double excitations, but the effects of their inclusion are evaluated using second-order perturbation theory rather than variationally as in CISD. An even more accurate type of perturbation theory is called coupled-cluster theory. CCSD (coupled-cluster theory, singles and doubles) includes single and double excitations, but their effects are evaluated at a much higher level of perturbation theory than in an MP2 calculation. 

Within the variation (supramolecular) approach, definitely the method of choice for interaction energies would be the coupled cluster CCSD(T) method (in which the single and double excitations are evaluated iteratively while the triple excitations are included in a non-iterative way). The CCSD(T) method yields a significant portion of the correlation energy. The MP2 method, including the double electron excitations at the second order of perturbation theory, overestimates the correlation interaction energy for stacking, as noted above. 

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