Phase dilemma in density matrix functional theory

For closed-shell systems, a particular parametrization of coefficients in a configuration interaction (CI) expansion provides a convenient formulation for the search over electronic wave functions constrained by a set of natural orbitals (NOs) and the corresponding occupation numbers that are invoked in every variational construction of the density matrix functional (DMF) V(ee)(Gamma) for the electron--electron repulsion energy. It produces an explicit expression for V(ee) in terms of the Coulomb and exchange integrals over NOs, and an idempotent matrix omega, diagonal elements of which equal the occupation numbers. At the same time, it reveals a very serious bottleneck affecting any rigorous approach to the DMF theory, namely the phase dilemma that stems from the necessity to carry out minimization over a large number of possible combinations of CI coefficient signs. While underscoring its lack of variational nature, a simple approximation for the phase factor products provides a strict derivation for the recently proposed Kollmar-Hess functional.