  Any rigorous approach to first-order reduced density (1RDM) matrix 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 the 1RDM,
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[1RDM] 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.
