The exponential ansatz of coupled-cluster theory

The product form (5.7.1) for the wave function is inconvenient for algebraic manipulations. We note that, since 

As in Cl theory, we now introduce a hierarchy of coupled-cluster wave functions by truncating the cluster operator (5.7.10) at different excitation levels. At the simplest level, we obtain the coupled-cluster singles-and-doubles (CCSD) model, omitting from the cluster operator all terms that involve higher than single and double excitations. At the next level, we also retain the triples and arrive at the CCSDT wave function, and so on. 

Unlike the wave functions considered so far (Hartree-Fock, MCSCF and Cl), the coupled-cluster wave function is not optimized according to the variation principle. Instead, the cluster amplitudes are determined by projecting the Schrodinger equation in the form 

The reasons for not invoking the variation principle in the optimization of the wave function are given in Chapter 13, which provides a detailed account of coupled-cluster theory. We here only note that the loss of the variational property characteristic of the exact wave function is unfortunate, but only mildly so. Thus, even though the coupled-cluster method does not provide an upper bound to the FCI energy, the energy is usually so accurate that the absence of an upper bound does not matter anyway. Also, because of the Lagrangian method of Section 4.2.8, the complications that arise in connection with the evaluation of molecular properties for the nonvariational coupled-cluster model are of little practical consequence. 

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In this section we examine some of the critical ideas that contribute to most wavefunction-based models of electron correlation, including coupled cluster, configuration interaction, and many-body perturbation theory. We begin with the concept of the cluster function which may be used to include the effects of electron correlation in the wavefunction. Using a formalism in which the cluster functions are constructed by cluster operators acting on a reference determinant, we justify the use of the exponential ansatz of coupled cluster theory. ... 

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