A deficiency of the cumulant expansion of the anharmonic temperature factor

It is shown that the cumulant expansion of the anharmonic temperature factor is a function whose inverse Fourier transform either does not exist or has negative regions. Since the probability density function for an atom should always be non-negative, the inverse Fourier transform of the cumulant expansion may be a poor approximation to the true probability density function. Correspondingly, the cumulant expansion may be an inadequate tool for describing anharmonic motions. Five examples from the literature are quoted where the cumulant expansion gave worse results than other anharmonic expansions.