  Because of the potentially large number of important applications of
nonlinear optics, researchers have expended a great deal of effort to optimize
the second-order molecular nonlinear-optical response, called the
hyperpolarizability. The focus of our present studies is the {\em intrinsic}
hyperpolarizability, which is a scale-invariant quantity that removes the
effects of simple scaling, thus being the relevant quantity for comparing
molecules of varying sizes. Past theoretical studies have focused on structural
properties that optimize the intrinsic hyperpolarizability, which have
characterized the structure of the quantum system based on the potential energy
function, placement of nuclei, geometry, and the effects of external electric
and magnetic fields. Those previous studies focused on single-electron models
under the influence of an average potential. In the present studies, we
generalize our calculations to two-electron systems and include electron
interactions. As with the single-electron studies, universal properties are
found that are common to all systems -- be they molecules, nanoparticles, or
quantum gases -- when the hyperpolarizability is near the fundamental limit.
