  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.
