where we have not made any assumptions about the nature of the Gaussian exponents. It is worth noting that neither Eq. ( ) nor Eq. ( ) is bounded on the interval [0, 1], which is undesirable from a rigorous standpoint. This has likely gone unnoticed in practice because divergences occur only very close to R pq = 0 and in most real systems the steric repulsion will preclude access to these problematic geometries. Finally, we introduce two novel damping functions to be investigated alongside those discussed above. First, we write an exact Tang-Toennies expression in terms of the overlap, which puts the generalized formulas presented above on a more rigorous foundation: