Coupled cluster formalisms effective operators

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. ... 

Recent developments include exact [12-14, 44, 90, 91] and approximate [14, 90, 92-94] iterative schemes to determine Hg, the intermediate Hamiltonian method [21, 24, 95], the use of incomplete model spaces [43, 44] and some multireference open-shell coupled-cluster (CC) formalisms [16-20, 96, 97]. Only some eigenvalues of the intermediate Hamiltonian H, are also eigenvalues of H. The corresponding model eigenvectors of H, are related to their true counterparts as in Bloch s theory. Provided effective operators a are restricted to act solely between these model eigenvectors, the possible a definitions from Bloch s formalism (see Section VI.A) can be used. 

Solutions to the eigenvalue equation (16) can be obtained by any of the standard quantum chemical methods, such as Hartree-Fock SCF, multiconfiguration SCF (MCSCF), Mpller-Plesset perturbation, coupled cluster, or density functional theories. The matrix elements of Hr, a one-electron operator, are readily computed, thus formally the inclusion of solvent effects in the quantum mechanical description of the solute molecule appears quite simple. Moreover, gradients of the eigenvalue E are readily computed. 

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