  The hyperpolarizability has been extensively studied to identify universal
properties when it is near the fundamental limit. Here, we employ the Monte
Carlo method to study the fundamental limit of the second hyperpolarizability.
As was found for the hyperpolarizability, the largest values of the second
hyperpolarizability approaches the calculated fundamental limit. The character
of transition moments and energies of the energy eigenstates are investigated
near the second hyperpolarizability's upper bounds using the missing state
analysis, which assesses the role of each pair of states in their contribution.
In agreement with the three-level ansatz, our results indicate that only three
states (ground and two excited states) dominate when the second
hyperpolarizability is near the limit.
