Coupled-cluster method in Fock space. I. General formalism

The problem of finding the spectrum of the Fock-space Hamiltonian Ĥ \H{} for a system of many fermions is analyzed. The quasiparticle formalism is employed, with ``holes'' and ``particles'' defined with respect to some N-particle determinantal wave function (the model vacuum). The basic idea behind the proposed approach is to perform a similarity transformation of Hamiltonian Ĥ \H{}, such that the resulting effective Hamiltonian Ĝ \G{} is, unlike Ĥ \H{}, a quasiparticle-number--conserving operator. It is shown that eigenvalues of Ĝ \G{}, corresponding to small numbers of quasiparticles (0,1,2) can be easily calculated. This is equivalent to finding eigenvalues of Ĥ \H{} for certain states of N, N±{}1, and N±{}2 particles. The construction of the operator transforming Ĥ \H{} into Ĝ \G{} (the wave operator) stems from an analysis of the structure of the algebra of linear operators acting in a (finite-dimensional) Fock space. The exponential Ansatz for the wave operator is used, resulting in a generalization of the coupled-cluster (CC) method of Coester [Nucl. Phys. 7, 421 (1958)]. The generalized CC equations determining the wave operator, and equations determining the effective Hamiltonian Ĝ \G{}, are presented in a diagrammatic form. An effort has been made to obtain a concise notation for expressing these equations in an algebraic form. Approximation schemes, necessary for practical applications of the proposed method, are also studied.