Phase dilemma in natural orbital functional theory from the N-representability perspective

Any rigorous approach to first-order reduced density matrix (Γ) functional theory faces the phase dilemma, that is, having to deal with a large number of possible combinations of signs in terms of the electron–electron interaction energy. This problem was discovered by reducing a ground-state energy generated from an approximate N-particle wavefunction into a functional of Γ, known as the top-down method. Here, we show that the phase dilemma also appears in the bottom-up method, in which the functional E[Γ] is generated by progressive inclusion of N-representability conditions on the reconstructed two-particle reduced density matrix. It is shown that an adequate choice of signs is essential to accurately describe model systems with strong non-dynamic (static) electron correlation, specifically, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, the Piris natural orbital functional 7 (PNOF7), with phases equal to −1 for the inter-pair energy terms containing the exchange-time-inversion integrals, agrees with exact diagonalization results.