The CCSD Energy Equation

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 

For all other terms, we may use the advantage of normal-ordering of the operators to determine all the fully contracted terms of the operator product. For example, for the second term on the right-hand side of Eq. [122], insertion of the definition of the normal-ordered Hamiltonian gives 

The two-electron component, on the other hand, contributes nothing to the energy expression because no fully contracted terms can be generated from it  

Therefore, the energy contribution from the linear Tj operator is 

Next consider the contribution to the energy from the linear T2 rm in 

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As an example, consider the CCSD energy equation derived earlier in Eq. [134] using Wick s theorem. Each term of the general expression... 

The CCSD energy is given by the general CC equation (4.53), and amplitude equations are derived by multiplying (4.50) with a singly excited determinant and integrating (analogously to eq. (4.54)). 

Here, Ti and T2 are the singly and doubly excited clusters obtained by solving the CCSD equations, is the CCSD energy, and... 

Let us now examine the contents of the QMMCC theory, in a somewhat greater detail, by discussing the QMMCC equations for the special case, where the QMMCC corrections are added to the CCSD energy (7 = Ti + T2) and Z is... 

Once the system of CCSD equations, Eqs. (5) and (6), is solved for the cluster amplitudes and the CCSD energy is calculated using the expression... 

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). 

Figure 3 Diagrammatic representation of the CCSD energy and equations together with the corresponding translation into algebraic expressions...

The starting point for our discussion is equation (13.2.30) for the CCSD energy and equations (13.2.40) and (13.2.41) for the CCSD amplitudes. Omitting from these equations all terms containing commutators that are quadratic or higher in f and from (13.2.41) also the term containing [[H, f i], til we arrive at the equations that define the ( CISD model ... 

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 CC2 wave function and energy are correct to the same orders in the fluctuation potential as the MP2 wave function and energy. However, we would expect the CC2 wave function to be slightly more accurate and more robust than the MPl wave function since the CC2 wave function has the additional merit of fulfilling the CCSD amplitudes equations to first order. The improvements are expected to be small, however. A direct comparison is given after the discussion of the CC3 approximation. 

However, if all we wish to accomplish by our post-CCSD treatment of the electronic system is to improve the accuracy of the energy from third to fourth order in the fluctuation potential, then we are overdoing it with the CC3 approximation. In a perturbational sense, we may achieve the same improvement in the energy by not iterating the CC3 equations. Even simpler, we may obtain the necessary corrections to the CCSD energy directly from perturbation theory by examining the lowest-order terms that contain connected triples. In particular, let us identify all terms in CCPT that involve connected triples to fourth and fifth orders. 

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