We investigate the effects of an external electric and magnetic field on the first hyperpolarizability tensor of a quantum system, such as a molecule or nanoparticle, whose nonlinear response is well below the fundamental limit. We find that the intrinsic hyperpolarizability is optimized when the applied electric and magnetic fields are comparable to the internal molecular fields. Indeed, the nonlinear response is just as large for an electron in the presence of the external field without the nuclei as it is for an electron bound to a molecule and in the presence of the applied field. We find that all combinations of fields and molecular structures that optimize the largest diagonal component of the intrinsic hyperpolarizability share the same universal properties: The three-level ansatz is obeyed, the normalized transition moment to the dominant state is about 0.76, the ratio of the two dominant excited state energies is about 0.48, the electron density tends toward being one-dimensional, and the intrinsic hyperpolarizability is less than 0.71. Thus, strategies for optimizing the hyperpolarizability should focus on ways to achieve these universal properties. On the other hand, when beta(xxy) is optimized, the three level ansatz appears to hold for a pair of degenerate states. In this case, the energy ratio between the pairs of degenerate states is 0.42 and the normalized transition moment to the pair of dominant states is 0.87. Most importantly, the intrinsic hyperpolarizability is 0.9, the largest ever calculated for a system described by a potential energy function.