Maximizing the hyperpolarizability poorly determines the potential

TL;DR

This study optimizes the first hyperpolarizability of a one-dimensional potential well, revealing that only two parameters significantly influence the maximum, and that the wavefunction shape is more indicative of optimality than the potential shape.

Contribution

It demonstrates that hyperpolarizability optimization is governed by two key parameters and highlights the importance of wavefunction shape over potential shape for optimal nonlinear optical properties.

Findings

Maximized hyperpolarizability converges to 0.708951 of the upper bound.

Only two eigenvalues dominate the Hessian at the maximum, indicating two key parameters.

Wavefunction shape converges with more parameters, unlike the potential shape.

Abstract

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.