  The dimensionless zero-frequency intrinsic second hyperpolarizability
\gamma_{int}=\gamma/4E_{10}^{-5}m^{-2}(e\hbar)^{4} was optimized for a single
electron in a 1D well by adjusting the shape of the potential. Optimized
potentials were found to have hyperpolarizabilities in the range
-0.15\lessapprox\gamma_{int}\lessapprox0.60 ; potentials optimizing gamma were
arbitrarily close to the lower bound and were within \sim0.5% of the upper
bound. All optimal potentials posses parity symmetry. Analysis of the Hessian
of \gamma_{int} around the maximum reveals that effectively only a single
parameter, one of those chosen in the piecewise linear representation adopted,
is important to obtaining an extremum. Prospects for designing new chromophores
based on the design principle here elucidated are discussed.
