We have optimized the zero frequency first hyperpolarizability β 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 β 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 β. 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.

The dimensionless zero-frequency intrinsic second hyperpolarizability γint = γ/4E −5 10 m −2 (e ) 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 γint 0.60; potentials optimizing gamma were arbitrarily close to the lower bound and were within ∼ 0.5% of the upper bound. All optimal potentials posses parity symmetry. Analysis of the Hessian of γint 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.

We optimize the first and second intrinsic hyperpolarizabilities for a 1D piecewise linear potential dressed with Dirac delta functions for N non-interacting electrons. The optimized values fall rapidly for N > 1, but approach constant values of βint = 0.40, γ + int = 0.16 and γ − int = −0.061 above N 8. These apparent bounds are achieved with only 2 parameters with more general potentials achieving no better value. In contrast to previous studies, analysis of the hessian matrices of βint and γint taken with respect to these parameters shows that the eigenvectors are well aligned with the basis vectors of the parameter space, indicating that the parametrization was well-chosen. The physical significance of the important parameters is also discussed.

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