  The generalized Thomas-Kuhn sum rules are used to eliminate the explicit
dependence on dipolar terms in the traditional sum-over-states (SOS) expression
for the second hyperpolarizability to derive a new, yet equivalent, SOS
expression. This new dipole-free expression may be better suited to study the
second hyperpolarizability of non-dipolar systems such as quadrupolar,
octupolar, and dodecapolar structures. The two expressions lead to the same
fundamental limits of the off-resonance second hyperpolarizability; and when
applied to a particle in a box and a clipped harmonic oscillator, have the same
frequency-dependence. We propose that the new dipole-free equation, when used
in conjunction with the standard SOS expression, can be used to develop a
three-state model of the dispersion of the third-order susceptibility that can
be applied to molecules in cases where normally many more states would have
been required. Furthermore, a comparison between the two expressions can be
used as a convergence test of molecular orbital calculations when applied to
the second hyperpolarizability.
