Coupled cluster amplitude

The construction of the coupled cluster amplitude equations is somewhat more complicated than the energy equation in that the latter requires only reference expectation values of the second-quantized operators. For the amplitude equations, we now require matrix elements between the reference, o, on the right and specific excited determinants on the left. We must therefore convert these into reference expectation value expressions by writing the excited determinants as excitation operator strings acting on Oq. For example, a doubly excited bra determinant may be written as... 

The first-order T2 amplitudes, which are required for Eq. [207], may be determined by left-projecting the first-order variant of Eq. [202] involving by a doubly excited determinant, as we did earlier in the construction of the coupled cluster amplitude equations,... 

Every term in the coupled cluster amplitude equations that is nonlinear in T may be factored into linear components. As a result, each step of the iterative solution of the CCSD equations scales at worst as ca. 0(X ) (where X is the number of molecular orbitals). The full CCSDT method in which all Tycon-taining terms are included requires an iterative 0(X ) algorithm, whereas the CCSD(T) method, which is designed to approximate CCSDT, requires a noniterative O(X ) algorithm. The inclusion of all T4 clusters in the CCSDTQ method scales as... 

The coupled cluster amplitude equations are often collectively called the coupled cluster vector function e with elements. 

In the case of the coupled cluster wavefunction the equations for the wavefunction parameters, i.e. for the coupled cluster amplitudes are simply the equations for the coupled cluster vector function in Eq. (9.81). The constraints are then = 0 and the coupled cluster Langrangian (Christiansen et ai, 1995a, 19986) is given as... 

Exercise 11.9 Derive the coupled cluster Jacobian in Eq. (11.75) as a derivative of the coupled cluster amplitude equations, i.e. prove Eq. (11.77). 

Projecting the similarity-transformed Schrodinger equation (13.2.20) against the same determinants as in (13.2.18) and (13.2.19), we arrive at the following set of equations for the coupled-cluster amplitudes and energy ... 

For the solution of the coupled-cluster amplitude equations, a number of techniques have been proposed. In the present section, some of these methods are discussed. We begin in Section 13.4.1... 

Consider the left-hand side of the coupled-cluster amplitude equations (13.2.23). Its elements constitute a vector function... 

Because of its relationship to Mpller-Plesset theory (as discussed in Section 14.2), we shall refer to (13.4.10) as the perturbation correction to the coupled-cluster amplitudes. The first correction to the wave function is obtained from (13.4.10) using zero cluster amplitudes, and higher corrections are generated by repeating this procedure. 

Thus, by solving these Nq equations, we ensure that the Nd spin equations (13.9.11) are satisfied. To determine the coupled-cluster amplitudes uniquely, we then solve the N projected... 

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. 

We shall include the effects of triples by combining the eoupled-cluster and Mpller-Plesset models. Let us begin by reviewing the form of the coupled-cluster equations for the coupled-cluster singles (CCS), CCSD and CCSDT wave functions. Introducing the following notation for the right-hand sides of the coupled-cluster amplitude equations... 

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