A Heuristic Approach for Treating Pathologies of Truncated Sum Rules in Limit Theory of Nonlinear Susceptibilities
Mark G. Kuzyk
The Thomas Kuhn Reich sum rules and the sum-over-states (SOS) expression for the hyperpolarizabilities are truncated when calculating the fundamental limits of nonlinear susceptibilities. Truncation of the SOS expression can lead to an accurate approximation of the first and second hyperpolarizabilities due to energy denominators, which can make the truncated series converge to within 10% of the full series after only a few excited states are included in the sum. The terms in the sum rule series, however, are weighted by the state energies, so convergence of the series requires that the position matrix elements scale at most in inverse proportion to the square root of the energy. Even if the convergence condition is met, serious pathologies arise, including self inconsistent sum rules and equations that contradict reality. As a result, using the truncated sum rules alone leads to pathologies that make any rigorous calculations impossible, let alone yielding even good approximations. This paper discusses conditions under which pathologies can be swept under the rug and how the theory of limits, when properly culled and extrapolated using heuristic arguments, can lead to a semi-rigorous theory that successfully predicts the behavior of all known quantum systems, both when tested against exact calculations or measurements of broad classes of molecules.