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We attempt to get a polynomial solution to the inverse problem, that is, todetermine the form of the mechanical Hamiltonian when given the energy spectrumand transition dipole moment matrix. Our approach is to determine the potentialin the form of a polynomial by finding an approximate solution to the inverseproblem, then to determine the hyperpolarizability for that system'sHamiltonian. We find that the largest hyperpolarizabilities approach theapparent limit of previous potential optimization studies, but we do not findreal potentials for the parameter values necessary to exceed this apparentlimit. We also explore half potentials with positive exponent, which cannot beexpressed as a polynomial except for integer powers. This yields a simpleclosed potential with only one parameter that scans nearly the full range ofthe intrinsic hyperpolarizability. The limiting case of vanishing exponentyields the largest intrinsic hyperpolarizability.

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