Symmetry-restricted functionals in one-body reduced density matrix functional theory

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) Wang and Knowles initialised the discussion to which extend this simplification is valid, by introducing two different H$_4$ geometries with identical natural occupation numbers, but different 2RDMs. Gritsenko has argued that this feature is due symmetry dependence of the exact functional [Phys. Rev. A 97, 026501 (2018)]. This work aims to contribute to the discussion on the following points: 1) the feature that the exact functional can yield different 2RDMs for the same set of natural occupations can be explained without symmetry, so is more general; 2) the exact functional is not symmetry dependent, though it is possible to define symmetry-restricted variants of the exact functionals; 3) symmetry-restricted functionals are derived for the H$_4$ geometries considered by Wang and Knowles, which can serve as guide in the construction of approximate 1RDM functionals.