Symmetry Equiincidence of Natural Orbitals

The symmetry equiincidence principle quantifies the apportionment of the natural orbitals (NOs), ordered according to their nonascending occupation numbers, among the irreducible representations (irreps) of the point group pertaining to the underlying on-top two-electron density. This principle, which is rigorously proven for the resolvable C_s, C_2v, C_3v, C_4v, C_6v, D_2h, D_3h, D_4h, D_6h, and O_h point groups, states that the symmetry incidences, i.e., the asymptotic probabilities with which the NOs belonging to different irreps occur, are proportional to the squares of irreps' dimensions. Since its proof hinges upon a sufficient number of planes of symmetry among the elements of a given point group, it yields only linear combinations of the symmetry incidences for the quasiresolvable groups with too few such planes and fails for the unresolvable C₁, C_i, C_n, D_n, S_2n, T, O, and I groups whose nontrivial elements comprise only symmetry axes and/or the center of inversion.