We have optimized the zero frequency first hyperpolarizability β of a one-dimensional piecewise linear potential well containing a single electron by adjusting the shape of that potential. With increasing numbers of parameters in the potential, the maximized hyperpolarizability converges quickly to 0.708951 of the proven upper bound. The Hessian of β at the maximum has in each case only two large eigenvalues; the other eigenvalues diminish seemingly exponentially quickly, demonstrating a very wide range of nearby nearly optimal potentials, and that there are only two important parameters for optimizing β. The shape of the optimized wavefunctions converges with more parameters while the associated potentials remain substantially different, suggesting that the ground state wavefunction provides a superior physical description to the potential for the conditions that optimize the hyperpolarizability. Prospects for characterizing the two important parameters for near-optimum potentials are discussed.

The synthesis of several new truxenones and one tris(dicyanomethylene) derivative substituted by different amines at positions 4, 9, and 14 is reported. A complete characterization of the NLO properties of representative derivatives was carried out by hyper-Rayleigh scattering; the major electronic effects and the influence of the structural modifications on the NLO properties have been examined. Because of their C 3 symmetry and their large first hyperpolarizability, the chiral versions of the tris(dicyanomethylene) truxene derivatives are of interest for second-order nonlinear optics in uniaxially aligned chiral media.

It is shown that energy/length scaling complicates maximizing the first hyperpolarizability of a single electron as a function of the potential. A more transparent formula for this hyperpolarizability is given. Examining this formula demonstrates that Zhou et al.(1) have not proved that modulated conjugation results in large hyperpolarizability.

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