Coupled-cluster theory perturbative corrections

A number of types of calculations begin with a HF calculation and then correct for correlation. Some of these methods are Moller-Plesset perturbation theory (MPn, where n is the order of correction), the generalized valence bond (GVB) method, multi-conhgurational self-consistent held (MCSCF), conhgu-ration interaction (Cl), and coupled cluster theory (CC). As a group, these methods are referred to as correlated calculations. 

In this section we examine the fundamental relationship between many-body perturbation theory (MBPT) and coupled cluster theory. As originally pointed out by Bartlett, this connection allows one to construct finite-order perturbation theory energies and wavefunctions via iterations of the coupled cluster equations. The essential aspects of MBPT have been discussed in Volume 5 of Reviews in Computational Chemistry,as well as in numerous other texts. We therefore only summarize the main points of MBPT and focus on its intimate link to coupled cluster theory, as well as how MBPT can be used to construct energy corrections for higher order cluster operators such as the popular (T) correction for connected triple excitations. 

Recognition of this relationship between coupled cluster theory and MBPT has inspired research efforts to construct perturbation-based corrections to the CCSD energy to account for higher excitation contributions. Undoubtedly, the most successful and popular of these is the (T) correction first described for closed-shell molecular systems by Raghavachari et al. " In the next section, we will describe the structure of this correction using diagrammatic techniques. 

M.. O. Deegan and P.. Knowles, Chem. Phys. Lett., 117, 321 (1994). Perturbative Corrections to Account for Triple Excitations in Closed- and Open-Shell Coupled-Cluster Theories. 

Perturbation and coupled cluster theories (e.g., MP2 or CCSD) provide correlation corrections. Density functional theory (DPT) appears to offer the best of all worlds, correlation-quality results at single-determinant prices. However, there is always a limit somewhere. The choice today seems to be between correlated methods with large basis sets, such as CCSDT, which systematically approach the "correct" answer at appreciable cost, and DFT, with its relatively economical efficiency, but which cannot be systematically improved (5). 

A posteriori corrections can be developed for calculations performed by using the Brillouin-Wigner perturbation expansion. These a posteriori corrections can be obtained for the Brillouin-Wigner perturbation theory itself and, more importantly, for methods, such as limited configuration interaction or multi-reference coupled cluster theory, which can be formulated within the framework of a Brillouin-Wigner perturbation expansion. 

Couple cluster methods differ from perturbation theory in that they include specific corrections to the wavefunction for a particular type to an infinite order. Couple cluster theory therefore must be truncated. The exponential series of functions that operate on the wavefunction can be written in terms of single, double and triple excited states in the determinantl " . The lowest level of truncation is usually at double excitations since the single excitations do not extend the HF solution. The addition of singles along with doubles improves the solution (CCSD). Expansion out to the quadruple excitations has been performed but only for very small systems. Couple cluster theory can improve the accuracy for thermochemical calculations to within 1 kcal/mol. They scale, however, with increases in the number of basis functions (or electrons) as N . This makes calculations on anything over 10 atoms or transition-metal clusters prohibitive. 

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. 

More recent investigations with the RPH employ standard correlation-corrected methods such as M0ller-Plesset (MP) perturbation theory (see M0ller-Plesset Perturbation Theory) at second or fourth order (MP2, MP4) or coupled cluster (CC) methods (see Coupled-cluster Theory) in connection with DZP or TZP basis sets. The repertoire of methods has recently been extended by applying density functional theory (DFT) (see Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field) and some convincing results have been published (see Section 3). 

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