Monte Carlo Studies of the Fundamental Limits of the Intrinsic Hyperpolarizability

The off-resonant hyperpolarizability is calculated using the dipole-free sum-over-stats expression from a randomly chosen set of energies and transition dipole moments that are forced to be consistent with the sum rules. The process is repeated so that the distribution of hyperpolarizabilities can be determined. We find this distribution to be a cycloid-like function. In contrast to variational techniques that when applied to the potential energy function yield an intrinsic hyperpolarizability less than 0.71, our Monte Carlo method yields values that approach unity. While many transition dipole moments are large when the calculated hyperpolarizability is near the fundamental limit, only two excited states dominate the hyperpolarizability - consistent with the three-level ansatz.