Implications of the unitary invariance and symmetry restrictions on the development of proper approximate one-body reduced-density-matrix functionals

In many of the approximate functionals in one-body reduced-density-matrix (1RDM) functional theory, the approximate two-body reduced density matrix (2RDM) in the natural orbital representation only depends on the natural occupation numbers. In Phys. Rev. A 92, 012520 (2015)PLRAAN1050-294710.1103/PhysRevA.92.012520, Wang and Knowles initialized a discussion of to what extent this simplification is valid by introducing two different H4 geometries with identical natural occupation numbers but different 2RDMs. Gritsenko has argued that this feature is due symmetry [Phys. Rev. A 97, 026501 (2018)2469-992610.1103/PhysRevA.97.026501]. This work aims to contribute to the discussion on the following points: (1) one should rather speak of symmetry-restricted variants of the universal functional than saying that the universal functional is symmetry dependent; (2) the unitary invariance of degenerate NOs can lead to large deviations in the 2RDM elements, especially the phase of the NOs; (3) symmetry-restricted functionals are constructed for the H4 geometries considered by Wang and Knowles, whose structure could serve as guide in the construction of approximate 1RDM functionals.