Eur. Phys. J. B (2018) 91: 109
https://doi.org/10.1140/epjb/e2018-90078-8

Regular Article

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

¹ Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), P.K. 1072, 20080 Donostia, Euskadi, Spain
² Donostia International Physics Center (DIPC), 20018 Donostia, Euskadi, Spain
³ Institut de Química Computacional i Catalisi, 17003 Girona, Catalonia, Spain
⁴ KERBASQUE, Basque Foundation for Science, 48013 Bilbao, Euskadi, Spain

^a e-mail: mario.piris@ehu.eus

Received: 20 February 2018
Received in final form: 13 April 2018
Published online: 11 June 2018

Abstract

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.