The 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... 

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... 

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 ... 

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. 

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. 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

Using the truncated Hausdorff expansion, we may obtain analytic expressions for the commutators in Eq. [52] and insert these into the coupled cluster energy and amplitude equations (Eqs. [50] and [51], respectively). However, this is only the first step in obtaining expressions that may be efficiently implemented on the computer. We must next choose a truncation of T and then derive expressions containing only one- and two-electron integrals and cluster amplitudes. This is a formidable task to which we will return in later sections. 

The projective techniques described above for solving the coupled cluster equations represent a particularly convenient way of obtaining the amplitudes that define the coupled cluster wavefunction, e o However, the asymmetric energy formula shown in Eq. [50] does not conform to any variational conditions in which the energy is determined from an expectation value equation. As a result, the computed energy will not be an upper bound to the exact energy in the event that the cluster operator, T, is truncated. But the exponential ansatz does not require that we solve the coupled cluster equations in this manner. We could, instead, construct a variational solution by requiring that the amplitudes minimize the expression ... 

This equation is not restricted to the CCSD approximation, however. Since higher excitation cluster operators such as T3 and T4 cannot produce fully contracted terms with the Hamiltonian, their contribution to the coupled cluster energy expression is zero. Therefore, Eq. [134] also holds for more complicated methods such as CCSDT and CCSDTQ. Higher excitation cluster operators can contribute to the energy indirectly, however, through the equations used to determine the amplitudes, and t-h, which are needed in the energy equation above. 

As discussed earlier, the cluster amplitudes that parameterize the coupled cluster wavefunction may be determined from the projective Schrodinger equation given in Eq. [51]. In the CCSD approximation, the single-excitation amplitudes, t- , may be determined from... 

The coupled cluster energy, on the other hand, does not suffer from this lack of size extensivity for two reasons (1) the amplitude equations in Eq. [50] are independent of the coupled cluster energy and (2) the Hausdorff expansion of the similarity-transformed Hamiltonian in Eq. [106], for example, guarantees that the only nonzero terms are those in which the Hamiltonian is con-... 

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