A rigorous upper bound follows from the Cauchy-Schwarz (CS) inequality for the scalar product between two vectors: ⟨u, v⟩ ≤ |u||v|. Let µ(r) and ν(r) be atomic orbitals centered on different atoms and O(r) = x m x y m y z m z r -k 1e -ar 2 q be the polarization operator with the cutoff function included. The polarization integral can be expressed as a scalar product between the product µ(r)ν(r) and O(r).