  Using sum rules, the dipolar terms can be eliminated from the commonly-used
sum-over-states (SOS) expression for nonlinear susceptibilities. This new
dipole-free expression is more compact, converges to the same results as the
common SOS equation, and is more appropriate for analyzing certain systems such
as octupolar molecules. The dipole-free theory can be used as a tool for
analyzing the uncertainties in quantum calculations of susceptibilities, can be
applied to a broader set of quantum systems in the three-level model where the
standard SOS expression fails, and more naturally leads to fundamental limits
of the nonlinear susceptibilities.
