The coupled-cluster wave function

With the coupled cluster wave function (4.46) the Schrddinger equation becomes... 

For our purposes, CC theory and its finite order MBPT approximations offer a convenient, compact description of the correlation problem and give rapid convergence to the basis set (i.e. full Cl) limit with different categories of correlation operators (see Fig. 15.1). The coupled-cluster wave function is... 

In section c above, two different ways of writing the wave-function have been described by (4.12) and (4.13), which in principle can be extended to the exact wave-function. There is a third alternative which has many advantages and is connected with one of the presently dominating ways of solving the Schiodinger equation, and this is the coupled cluster wave-function. To define this wave-function it is convenient to define certain excitation operators. A general n-tuple excitation operator T is defined as... 

Using a cluster operator, T, and an exponential ansatz [60,61], the coupled cluster wave function is written as... 

For the optimization of the coupled cluster wave function in the presence of the classical subsystem we write the CC/MM Lagrangian as [24]... 

The second general approach to correlation theory, also based on perturbation theory, is the coupled-cluster method, which can be thought of as an infinite-order perturbation method. The coupled-cluster wave function T cc is expressed as a power series,... 

MR-MBPT methods, however, is not straightforward. The main problem here is similar With the coupled cluster wave function (4.46) the Schrodinger equation becomes... 

This defines an independent electron pair approximation in terms of extremal pairs, which can be regarded as a generalization of the independent electron pair approximation (IEPA) [4, 8] in terms of pairs (ij) constructed form (preferably) localized orbitals. As in the discussion in Paper I for MP2 [5], one can show that the extremal pairs defined in this section are related to approximate natural geminals corresponding to the coupled-cluster wave function. 

An operator annihilates an electron in spin orbital i, while creates an electron in spin orbital a. The creation and annihilation operators satisfy the usual algebra [a, aj]+ = 6pg. Introducing the generic notation for excitation operators and noting that [r, iv] = 0, we may express the coupled-cluster wave function in the standard exponential form [35]... 

The coupled-cluster electronic state is uniquely defined by the set of the cluster amplitudes and these amplitudes are used to obtain the coupled-cluster energy from Eq. (33). Due to the fact that the Ansatz of the coupled-cluster wave function has the exponential parametrization [Eq. (28)] the energy is size-extensive. This is an obvious advantage of the coupled-cluster formalism compared to some other techniques (e.g. configuration interaction). For a general discussion of coupled-cluster theory and the coupled-cluster equations see Refs. [5, 36]. 

As the reference function in the EOM-CC method, we take the coupled-cluster wave function for the ground state ... 

For actual applications of the cluster ansatz even in its truncated form it is necessary that no infinite sums as in Eq. (8.238) occur. Fortunately, we can benefit from the properties of the creation and annihilation operators. In order to understand how, we write the electronic Schrodinger equation for the coupled-cluster wave function. 

If the reference is a closed-shell determinant, it is symmetric under time reversal. We require that the coupled-cluster wave function is also symmetric under time reversal, because the wave function is nondegenerate and has an even number of electrons. Then... 

Before proceeding to open-shell theory, it is worth noting that CCSD properly treats the nondynamical effects that are missing in a single-determinant reference function, which were discussed in section 12.1. This is because the coupled-cluster wave function is an infinite-order expansion to the given excitation level the coefficients of the determinants that complete the reference expansion and all the excitations from these are included and optimized in the coupled-cluster wave function. Also, the presence of single excitations accounts for the orbital relaxation that would correct the distortion of the reference determinant. 

The coupled-cluster wave function is defined by the exponential ansatz [15-18]... 

The variational criteria at the zero and first order are satisfied when the Coupled-cluster wave-functions are solution of the PCM coupled-cluster Eqs. (1.24) and (1.28). 

Given the product ansatz for the coupled-cluster wave function (13.1.7), let us consider its optimization. We recall that, in Cl theory, the wave function (13.1.8) is determined by minimizing the expectation value of the Hamiltonian with respect to the linear expansion coefficients ... 

In Section 13.1, the coupled-cluster wave function was expressed as a product of correlating operators working on the Hartree-Fock state (13.1.7). This expression for the coupled-cluster wave function is useful for displaying the relation to Cl theory and for exhibiting the fundamental role played by the excitation operators in coupled-cluster theory. In general, however, the coupled-cluster wave function is expressed as the exponential of an operator applied to the Hartree-Fock determinant. In the present section, we introduce and explore this exponential ansatz for the coupled-cluster wave function. 

We assume that we have calculated the coupled-cluster wave functions for the individual systems A and B... 

The unsymmetric Jacobian matrix has previously appeared in the optimization of the coupled-cluster wave function (13.4.4) and in the calculation of the coupled-cluster Lagrange multipliers (13.5.6). 

Let us consider the simultaneous solution of the Schrodinger and spin equations for high-spin open-shell states. The coupled-cluster wave function may be expressed in terms of a singledeterminant reference function with the excitation operators chosen to yield excitations from this reference state. The full coupled-cluster wave function, being equivalent to the FCI wave function in the same orbital basis, is a simultaneous eigenfunction of the Schrodinger equation and the spin equations ... 

At this stage, it is interesting to compare the Mpller-Plesset corrections (14.2.21) and (14.2.40) with the coupled-cluster wave function as analysed in Section 13.2.2 - see, in particular (13.2.12)-(13.2.15). The MPl correction contains only the connected first-order doubles - no disconnected terms appear at this level. To second order, the MP2 correction contains connected contributions from the second-order singles, doubles and triples amplitudes as well as a disconnected contribution from the first-order doubles - there are no eontributions from the connected quadruples to the MP2 wave 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 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... 

In order to include geminal functions into the coupled-cluster wave-function, the pair function Q 2f f 2) (Pk9i) (formally) projected onto the complete space (due to Qu the complete virtual space suffices). 

If we include single excitations, other mechanisms become possible as well. With each excitation mechanism, we associate a characteristic probability amplitude. The final electronic state (5.7.1) is thus a linear combination of excited configurations, each with a total weight equal to the combined probabilities of all mechanisms leading to this particular configuration. In this respect, the coupled-cluster wave function differs fundamentally from the Cl wave fimction, where we parametrize the total weight of each excited configuration individually. 

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