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The Thomas Kuhn Reich sum rules and the sum-over-states (SOS) expression forthe hyperpolarizabilities are truncated when calculating the fundamental limitsof nonlinear susceptibilities. Truncation of the SOS expression can lead to anaccurate approximation of the first and second hyperpolarizabilities due toenergy 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. Theterms in the sum rule series, however, are weighted by the state energies, soconvergence of the series requires that the position matrix elements scale atmost in inverse proportion to the square root of the energy. Even if theconvergence condition is met, serious pathologies arise, including selfinconsistent sum rules and equations that contradict reality. As a result,using the truncated sum rules alone leads to pathologies that make any rigorouscalculations impossible, let alone yielding even good approximations. Thispaper discusses conditions under which pathologies can be swept under the rugand how the theory of limits, when properly culled and extrapolated usingheuristic arguments, can lead to a semi-rigorous theory that successfullypredicts the behavior of all known quantum systems, both when tested againstexact calculations or measurements of broad classes of molecules.

A Heuristic Approach for Treating Pathologies of Truncated Sum Rules in Limit Theory of Nonlinear Susceptibilities