A fundamental connection between symmetry and spatial localization properties of basis sets

The problem of the compatibility between symmetry and localization properties of basis sets is addressed here. It is shown that both concepts are closely related from a fundamental point of view through the notion of invariance extent. This quantity is a functional that depends on the symmetry group and the basis set choices, and it is shown that all basis sets adapted in a general way to symmetry, i.e. induced from irreducible bases of the subgroups, are stationary points of it. In particular, the usual irreducible bases of the full group display a maximal invariance extent, while those symmetry-adapted basis sets that display a minimal value of this quantity feature in most cases the same symmetry properties as localized functions obtained by means of the Boys scheme. The most relevant conclusions are illustrated by means of simple molecular and periodic examples.