The problem considered is that of the diagonal N -representability of a p th-order reduced density matrix, p ≥ 2 p \geq 2 , for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I. M = N + p M = N + p ; Case II. M > N + p M > N + p ; Case III. M > N + p M > N + p . Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.
The
N-representability problem for the first- and second-order density matrices is considered when
N is an odd integer. Attention is restricted to those density matrices derivable from functions whose natural spin orbitals have certain paired properties. Neither pairing of spatial orbitals nor special spin properties are required in general, but they are discussed as special cases. New sufficient conditions for
N-representability of the first-order density matrix are given. The second-order density matrix is studied in detail. Special cases in which the
N-representability problem for the second-order density matrix can be solved are considered.