  We have optimized the zero frequency first hyperpolarizability \beta of a
one-dimensional piecewise linear potential well containing a single electron by
adjusting the shape of that potential. With increasing numbers of parameters in
the potential, the maximized hyperpolarizability converges quickly to 0.708951
of the proven upper bound. The Hessian of \beta at the maximum has in each case
only two large eigenvalues; the other eigenvalues diminish seemingly
exponentially quickly, demonstrating a very wide range of nearby nearly optimal
potentials, and that there are only two important parameters for optimizing
\beta. The shape of the optimized wavefunctions converges with more parameters
while the associated potentials remain substantially different, suggesting that
the ground state wavefunction provides a superior physical description to the
potential for the conditions that optimize the hyperpolarizability. Prospects
for characterizing the two important parameters for near-optimum potentials are
discussed.
