Coupled-cluster amplitude equations

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. 

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). 

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... 

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... 

This is the natural truncation of the coupled cluster energy equation an analogous phenomenon occurs for the amplitude equation (Eq. [45]). This truncation depends only on the form of f and not on that of T or on the number of electrons. Equation [49] is correct even if T is truncated to a particular excitation level. 

The coupled cluster Schrodinger equation, which leads to the energy and amplitude expressions given in Eqs. [50] and [51], may be written as... 

Equation (1.26) shows that the T amplitude equations correspond to the projection of the coupled-cluster Schrddinger equation for the molecular solute... 

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... 

The Jacobian matrix A can be shown [see Exercise 11.9) to be the first derivative of the time-independent coupled cluster amphtude equations, i.e. the coupled cluster vector function e, Eq. (9.81), with respect to the time-independent amplitudes... 

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 ... 

The projected coupled-cluster Schrodinger equation (13.2.32) therefore yields at most quartic equations in the cluster amplitudes - even for the full cluster expansion. The BCH expansion terminates because of the special structure of the cluster operators, which are linear combinations of commuting excitation operators of the form (13.2.6) and (13.2.7). 

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 the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

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