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The off-resonant hyperpolarizability is calculated using the dipole-freesum-over-stats expression from a randomly chosen set of energies and transitiondipole moments that are forced to be consistent with the sum rules. The processis repeated so that the distribution of hyperpolarizabilities can bedetermined. We find this distribution to be a cycloid-like function. Incontrast to variational techniques that when applied to the potential energyfunction yield an intrinsic hyperpolarizability less than 0.71, our Monte Carlomethod yields values that approach unity. While many transition dipole momentsare large when the calculated hyperpolarizability is near the fundamentallimit, only two excited states dominate the hyperpolarizability - consistentwith the three-level ansatz.

Monte Carlo Studies of the Fundamental Limits of the Intrinsic Hyperpolarizability