Maximizing the hyperpolarizability of 1D potentials with multiple electrons

We optimize the first and second intrinsic hyperpolarizabilities for a 1D piecewise linear potential dressed with Dirac delta functions for $N$ noninteracting electrons. The optimized values fall rapidly for $N>1$, but approach constant values of ${\beta }_{\mathrm{int}}=0.40$, ${\gamma }_{\mathrm{int}}^{+}=0.16$, and ${\gamma }_{\mathrm{int}}^{-}=-0.061$ above $N\gtrsim 8$. These apparent bounds are achieved with only two parameters with more general potentials achieving no better value. In contrast to previous studies, analysis of the Hessian matrices of ${\beta }_{\mathrm{int}}$ and ${\gamma }_{\mathrm{int}}$ taken with respect to these parameters shows that the eigenvectors are well aligned with the basis vectors of the parameter space, indicating that the parameterization was well-chosen. The physical significance of the important parameters is also discussed.