CCSD theory projected equations

Let us now compare the coupled-cluster equations in the linked and unlinked forms. We b n by reiterating that these two forms of the coupled-cluster equations are equivalent for the standard models in the sense that they have the same solutions. Moreover, applied at the important CCSD level of theory, neither form is superior to the other, requiring about the same number of floating-point operations. The energy-dependent unlinked form (13.2.19) exhibits mcMe closely the relationship with Cl theory, where the projected equations may be written in a similar form (13.1.18). On the other hand, the linked form (13.2.23) has some important advantages over the unlinked one (13.2.19), making it the preferred form in most situations. 

In this section, we study the relationship between coupled-cluster and Mpller-Plesset theories in greater detail. We begin by carrying out a perturbation analysis of the coupled-cluster wave functions and energies in Section 14.6.1. We then go on to consider two sets of hybrid methods, where the coupled-cluster approximations are improved upon by means of perturbation theory. In Section 14.6.2, we consider a set of hybrid coupled-cluster wave fiinctions, obtained by simplifying the projected coupled-cluster amplitude equations by means of perturbation theory. In Section 14.6.3, we examine the CCSE)(T) approximation, in which the CCSD energy is improved upon by adding triples corrections in a perturbative fashion. Finally, in Section 14.6.4, we compare numerically the different hybrid and nonhybrid methods developed in the present chapter and in Chapter 13. 

The general nature of the MMCC theory, on which all renormalized and completely renormalized CC methods described here are based, allows us to proposed many other potentially useful approximations. We can, for example, introduce the MMCC(2,6) method, in which the CCSD results are corrected by considering all nonzero moments of the CCSD equations, including those corresponding to projections on pentuply and hextuply excited configurations. We can also introduce the active-space variants of the renormalized and completely renormalized CC approaches, in which we consider small subsets of the generalized moments of CC equations defined... 

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