There is standard methodology available to facilitate electronic structure computations on a space that is invariant under a symmetry group. Here, we focus on additional consequences that arise if the basis itself is invariant under the symmetry group (i.e., in the case that application of symmetry operations to each basis vector yields, up to proportionality, a single basis vector). In illustration of the formal development, examples are considered where the symmetries are point-group symmetries and the basis vectors are Slater determinants over singly occupied atomic orbitals, as for an open-shell valence bond (VB) model. Several other types of examples are mentioned, e.g., a basis of chemically motivated resonance structures, as for a VB model, or an orbital basis of atomic orbitals for a one-electron Hckel-type model.
