Necessary conditions for the N-representability of the second-order reduced density matrix: Upper bounds on the P and Q matrices

Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the “P” and “Q” matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians H=−P and H=−Q) can be implemented in practical calculations. This necessary condition for N-representability should help ensure that the second-order reduced density matrix is not “overcorrelated.”