Excitations as ground state variational parameters

Abstract

The ground state density matrix can be expressed in terms of transition amplitudes 〈r|a a + a b| g〉 by the following relation: 〈 g|a a +a ca b +a d|g〉= Σ r 〈 g|a a +a c|r〉〈r|a b +a d| g〉. A formalism is derived in which these transition amplitudes are used as ground state variational p parameters. The quadratic form with transition amplitudes as variational parameters keeps the above matrix non-negative definite. This represents one necessary N-representability condition. This condition plus symmetry and trace relations are used to bound the two-body density matrix. (It keeps also the one-body density matrix between 0 and 1.) the trial ground state is also restricted in such a way that already half of the particle-hole excited states exhausts the sum over r. The variational equations have the form of a secular equation similar to that in the random phase approximation. The single-particle energies are, however, introduced as Lagrange multipliers for the trace relations of the density matrix. The one-body density matrix is also determined using variation methods. Additional Lagrange multipliers are introduced to guarantee certain symmetry relations, which are disregarded in the RPA. The obtained ground state is then consistent with excited states having different quantum numbers. The calculation for a simple system with a schematic intershell pairing is compared with the exact solution, the RPA and some other methods. The ground state energies and transition amplitudes are good for the entire range of the interaction.