In prior work, we have studied the pairing interaction by means of the equations of motion satisfied by the matrix elements of single-fermion operators (coefficients of fractional parentage). The previous numerical calculations incorporated approximations which we seek to avoid here by developing a self-consistent theory fully consonant with the intrinsic structure of the equations of motion and associated conditions. Unlike other methods, where a distinction is usually made between the ground state and other low-lying excited states, we here consider the subspace defined by all these states on an equal footing. An elaborate self-consistency procedure is designed with the following salient features: (i) The Hamiltonian is rendered diagonal in a subspace of states of both an even and a neighboring odd nucleus. (ii) The Pauli principle, as represented by sum rules imposed on the coefficients of fractional parentage, is satisfied as accurately as possible in a least-squares sense. (iii) The number of particles is conserved exactly in all states of the system. Results of calculation for a simple model are compared with exact solutions and found to be vastly improved over previous results. The exact results were obtained by means of a new self-consistent version of exact shell-model diagonalization, which is described.

• Received 1 April 1968

DOI:https://doi.org/10.1103/PhysRev.172.1022