Analytic connection between configuration–interaction and coupled-cluster solutions

Abstract

The coupled-cluster (CC) equations in the work of Coester, Kümmel, Čižek, Paldus, and others are inhomogeneous, nonlinear and algebraic in the cluster operators to be determined. If taken to all orders, they are equivalent to complete configuration–interaction (CI) equations, except for states orthogonal to the reference state Φ. However, if taken only to nth order, they are not equivalent to the nth order CI equations, and due to their nonlinear form, the existence and the number of the solutions is not guaranteed. Also, the reality of the associated energy values is not certain since these values do not arise as eigenvalues of a Hermitian operator. We show that the equations can be cast in the form of perturbed CI equations, with the ’’perturbations’’ being non-Hermitian and nonlinear in the CI-like coefficients to be calculated. In the case of a finite number of single-particle states, we construct the solutions to the CC equations by analytic continuation from the CI solutions. Singularities peculiar to the method are identified and studied, and conditions for reality and the maximum multiplicity of solutions are given. In general, the energy will be real, and the number of solutions equals that of the associated CI problem. Singularities or instabilities in the coupled-cluster equations can be traced to unphysical assumptions in the basis set Hamiltonian, or a poor description to highly excited states.