Polynomial potentials determined from the energy spectrum and transition dipole moments that give the largest hyperpolarizabilities

TL;DR

This paper develops a method to determine polynomial potentials from energy spectra and transition dipole moments, aiming to identify potentials that maximize hyperpolarizability, and finds that the maximum approaches known limits without exceeding them.

Contribution

It introduces a polynomial inverse approach to find potentials from spectral data and explores simple potentials that nearly reach the hyperpolarizability limit.

Findings

Largest hyperpolarizabilities approach the known limit.

No real potentials exceed the apparent hyperpolarizability limit.

Simple one-parameter potentials can scan nearly the full hyperpolarizability range.

Abstract

We attempt to get a polynomial solution to the inverse problem, that is, to determine the form of the mechanical Hamiltonian when given the energy spectrum and transition dipole moment matrix. Our approach is to determine the potential in the form of a polynomial by finding an approximate solution to the inverse problem, then to determine the hyperpolarizability for that system's Hamiltonian. We find that the largest hyperpolarizabilities approach the apparent limit of previous potential optimization studies, but we do not find real potentials for the parameter values necessary to exceed this apparent limit. We also explore half potentials with positive exponent, which cannot be expressed as a polynomial except for integer powers. This yields a simple closed potential with only one parameter that scans nearly the full range of the intrinsic hyperpolarizability. The limiting case of vanishing exponent yields the largest intrinsic hyperpolarizability.