Some rules forS(-2k) dipole sums?

Abstract

Stieltjes conditions and the ratio test provide necessary but not sufficient conditions onS(-2k) dipole sums. If the dipole sums are accurate the associated [n, n −1] Pade approximant provides a better representation of α (Ω), the frequency-dependent dipole polarizability, than a truncated series expression and, in addition, should bound α(Ω) below. It is shown how constraints on the dipole sums effect the form of the [2,1] Pade approximant and an additional constraint is derived that ensures the analyticity of the approximant on 0≤ Ω < Ω1. There then follows a discussion of the reliability of available literature dipole sum values for small molecules containing H, C, N and O.