Lِwdin population analysis is not rotationally invariant

It is discussed that the Lِwdin atomic populations (atomic populations computed in a Lِwdin-orthogonalized basis) are invariant only under unitary transformations of the basis orbitals centered on the same atom, but not under the general rotational-hybridizational transformations as are the Mulliken populations. As a consequence, if basis sets containing 6 d-orbitals (or 10 f-orbitals, etc.) are used, then the Lِwdin populations do not possess correct rotational invariance and equivalent atoms may be assigned different populations. Therefore, usual Lِwdin populations are not appropriate tools of analysis if such basis sets are used. No such difficulty arises in the version proposed by Davidson, in which the basis orbitals belonging to the individual atoms are pre-orthogonalized; it, however, gives quite different results than the conventional scheme.