The projected coupled-cluster equations

For many purposes, it is convenient to express the projected coupled-cluster equations in a slightly different form. First, we multiply the Schrodinger equation (13.2.16) from the left by the... 

To solve the projected coupled-cluster equations (13.2.23) for the CCSD wave function, we need... 

Let us now consider the evaluation of the projected coupled-cluster equations [25,26]. We begin with the singles projection here, discussing the doubles projection in Section 13.7.8. The equation for projection against the singles space is given by ... 

The modifications necessary in the orbital representation become apparent when we consider the projected coupled-cluster equations. For the singles and doubles, we shall still assume that (Till and (/[ are related to ju,i) and lju,2) in the biorthogonal fashion (13.7.54) and (13.7.55)... 

We now wish to establish to what order Nvr in the fluctuation potential the coupled-cluster wave function and the coupled-cluster energy of order Ncc are correct. We begin by recalling that the rank-n operator T is of order n — 1 and greater in the fluctuation potential. From this observation, we proceed to determine the perturbation order of the projected coupled-cluster equations (14.6.2). We introduce the notation... 

Next, let us examine the relationship between the projected coupled-cluster equations of different orders. Inspection of (14.6.5)-( 14.6.9) reveals the following recursive relationships ... 

The resulting projected coupled-cluster equations may be written as... 

Having examined the coupled-cluster energy and seen that it is no higher than quadratic in the cluster amplitudes, let us now turn our attention to the structure of the linked projected coupled-cluster equations (13.2.23) ... 

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). 

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 field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

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 ... 

Like the variational coupled-cluster conditions (13.1.20), the projected equations (13.1.22) are nonlinear in the amplitudes. However, unlike the variational conditions, the expansion of the wave function in (13.1.22) and (13.1.23) terminates after a few terms since the Hamiltonian operator couples determinants that dilfer by no higher than double excitations, making the solution of the projected equations and the calculation of the energy tractable. Of course, the calculated coupled-cluster energy no longer represents an upper bound to the FCI energy. In practice, the deviation from the variational energy turns out to be small and of little practical consequence. 

Truncated coupled-cluster wave functions cannot satisfy this equation exactly and, as discussed in Section 13.1.4, we use projection to determine the wave function. The optimized coupled-cluster wave function then satisfies the Schrtidinger equation (13.2.16) projected onto the Hartree-Fock state and onto the excited projection manifold... 

In the CCSD model, for example, the excited projection manifold comprises the fiill set of all singly and doubly excited determinants, giving rise to one equation (13.2.19) for each connected amplitude. For the full coupled-cluster wave function, the number of equations is equal to the number of determinants and the solution of the projected equations recovers the FCI wave function. The nonlinear equations (13.2.19) must be solved iteratively, substituting in eac iteration the coupled-cluster energy as calculated from (13.2.18). 

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. 

Whereas the singles occur to fourth order, the doubles appear only quadratically in these exjxessions. Moreover, from (13.2.39), it is easily verified that, except for the singles and the doubles, the amplitudes of the highest excitation level always occur oidy linearly in the coupled-cluster equations. These addifional simplifications in the coupled-cluster equations - beyond what is dictated by the termination of the BCH expansion (13.2.37) - occur because of the restrictions on the excitation levels in the projection space (/r,j. If instead we had calculated the energy fhrm the variation principle, the expansion of the exponentials would not terminate (except, of coinse, for the fact that we have only a finite number of electrons to excite from the occupied spin orbitals). The evaluation... 

Clearly, the fulfilment of conditions (13.3.26) and (13.3.27) for the noninteracting systems A and B implies the fulfilment of conditions (13.3.28)-(13.3.30) for the compound system - as expected for a size-extensive computational model. Note that, in our demonstration of size-extensivity in the unlinked formulation, no assumptions were made about the projection space. Thus, although the linked and unlinked coupled-cluster equations may give different solutions when the projection space is not closed under de-excitations, both solutions are size-extensive. 

However, since the bra states are used only for projection of the coupled-cluster equations (which is zero for the optimized wave function), normalization is unimportant. In fact, use of the biorthogonal basis (13.7.59) rather than the biorthonormal basis (13.7.60) simplifies our algebraic manipulations considerably. 

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,... 

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 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,... 

The projections against all possible states that might be generated by the excitation operators present in the cluster operator Eq. (29) according to Eq. (32) give the recipe how to compute the coupled-cluster state from the following nonlinear equations... 

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