Restoring rotational invariance of Löwdin populations

Abstract

A decade ago it has been realized that Löwdin populations (atomic populations calculated in a Löwdin-orthogonalized basis) are not rotationally invariant if one uses Cartesian d- (f-) basis orbitals, as is the case for the standard 6-31G* or 6-31G** basis sets. It was shown that the reason for this behavior is that invariance is conserved only if the rotation induces a unitary transformation of the basis orbitals on each atom. Davidson pre-orthogonalizes the basis on every atom separately; then the rotational invariance is restored, but the numbers change wildly. Here the “best compromise” is proposed, in which one pre-orthogonalizes only those basis orbitals that transform between each other during the rotation. In this manner, the rotational invariance is restored and the numbers remain close to the range obtained by the conventional Löwdin-orthogonalization. It is also demonstrated that the situation with Wiberg indices (bond orders in the Löwdin-orthogonalized basis) is the same as for the populations: the condition of the invariance is the unitary character of the transformations induced by the rotations. In their case, the partial pre-orthogonalization proposed here is adequate, too.