Scale Invariance
The n^{th} hyperpolarizability describes the effectiveness of a molecule in mediating the interaction of n+1 beams of light. Molecules rich in electrons, and with a low energy gap (i.e. the difference in energy between the ground and first excited state) are the most efficient. It is trivial to make better molecules simply by making them bigger (larger molecules have both more electrons and a lower energy gap). We call this simple scaling.
Molecules with a similar number of electrons and energy gap have a large range of hyperpolarizabilities. It is thus useful to take into account the effects of simple scaling to understand structural properties and other nuances that can effect the hyperpolarizability. This approach allows one to focus on the intrinsic properties that elucidate the fundamental origins of the nonlinearoptical response. Such an understanding would undoubtedly lead to new science as well as provide design guidelines for making better molecules.
The fundamental limits for the first[1] and second[2] offresonant hyperpolarizabilities were first reported in 2000. Later, the limits of the full dispersion were reported.[3] It has been shown that the ratio of the hyperpolarizability to the fundamental limit of the hyperpolarizability, called the intrinsic hyperpolarizability, removes the effects of simple scaling.[4,5] As such, the n^{th} intrinsic hyperpolarizability is a quantity that is less than unity, and can be used to compare a broad range of molecules, independent of their size.
The figure above shows a plot of the intrinsic first hyperpolarizability as a function of the wavelength of maximum absorption of a large number of molecules. Interestingly, three decades of research resulted in molecules of larger and larger hyperpolarizability; but, the intrinsic hyperpolarizability showed little improvement. All of the gains in hyperpolarizability were due to simple scaling. Indeed, the best molecules made prior to about 2007 fell about a factor of 30 short of the fundamental limit, that is, had an intrinsic hyperpolarizability of about 0.03.
The energies and eigen functions of the clipped harmonic oscillator(CHO) can be determined from the Schrodinger Equation analytically and without approximation.[6] The intrinsic first hyperpolarizability of a CHO, as shown as a light blue line in the plot above, is about 0.6. So, there appear to be no reasons why researchers can't make molecules that approach the fundamental limit.
New Paradigms
Using a universal scaling plot as shown above, recent research hints at potentially new paradigms to improve the nonlinear optical response. Here are some of the more interesting recent developments.
Modulation of Conjugation
It has been shown that when theoretically optimizing 1dimensional potentials, the intrinsic hyperpolarizability can be as large as 0.709,[7,8] as represented by the blue line in the figure. This work suggests the new paradigm of modulation of conjugation, that is, one way to optimize the hyperpolarizability is to make a onedimensional molecule with a bridge of heteroatomic bonds.[7,8] The series of molecules labeled LCW in the figure above led to molecules LCW6 and LCW7 with modulated conjugation that broke the longstanding apparent limit.[9,10] While these molecules are still far from the limit, modulation of conjugation may be a viable paradigm for getting there.
Amylose Helix
Placing a molecule in an amylose helix (see below) to make it planar may be another way of improving the intrinsic hyperpolarizability.[11] The series of molecules C1 to C5 were placed in an amylose helix (C1III to C5III). The results are summarized in the plot above.
Twisted Molecules
Twisted molecules have been reported to have the highest intrinsic hyperpolarizabilities ever measured (M2, MC2, and MC3 in the plot above).[4,12] If these results hold up, this may be a significant development in the search for new paradigms for molecular design.

Theoretical Questions
It is puzzling that many very different onedimensional potential energy functions all yield the same intrinsic hyperpolarizability of 0.709.[7,8]
The sum rules (SR), which directly follow from the Schrodinger Equation, relate the dipole matrix to the energy levels of a molecule. The SRs are the basis of calculating fundamental limits and intrinsic hyperpolarizabilities. Monte Carlo calculations, which randomly pick energies and dipole matrix elements that are forced to be consistent with the sum rules, show that the hyperpolarizability of a onedimensional system can be as large as unity.[13]
Conclusion
The intrinsic hyperpolarizability is a quantity that takes into account simple scaling. Universal plots, as shown above, can be used to compare lots of molecules of disparate shapes and sizes. This allows one to search for structural properties that correlate with a molecule's innate ability to mediate the interaction between beams of light.
References
[1] M. G. Kuzyk, “Fundamental limits on thirdorder molecular susceptibilities,” Optics Letters 25, 1183 (2000).
[2] M. G. Kuzyk, “Physical Limits on Electronic Nonlinear Molecular Susceptibilities,” Physical Review Letters 85, 1218 (2000).
[3] Mark G. Kuzyk, “Fundamental limits of all nonlinearoptical phenomena that are representable by a secondorder susceptibility,“ J. Chem. Phys. 125, 154108. (2006).
[4] Juefei Zhou and Mark G. Kuzyk, “Intrinsic Hyperpolarizabilities as a Figure of Merit for Electrooptic Molecules,” J. Phys. Chem. C 112, 7978 (2008).
[5] Mark G. kuzyk, "Using fundamental principles to understand and optimize nonlinearoptical materials," J. Mater. Chem (2009) DOI: 10.1039/b907364g.
[6] Kakoli Tripathy, Javier Perez Moreno, Mark G. Kuzyk, Benjamin J. Coe, Koen Clays, and Anne Myers Kelley, “Why Hyperpolarizabilities Fall Short of the Fundamental Quantum Limits,” J. Chem. Phys. 121, 7932 (2004)
[7] Juefei Zhou , Mark G. Kuzyk, and David S. Watkins, “Pushing the hyperpolarizability to the limit,” Optics Letters 31, 2891 (2006).
[8] Juefei Zhou, Urszula B. Szafruga, David S. Watkins, and Mark G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76, 053831 (2007).
[9] Javier PerezMoreno, Yuxia Zhao, Koen Clays, and Mark Kuzyk, “Modulated conjugation as a means for attaining a record high intrinsic hyperpolarizability,” Opt. Lett. 32, 59 (2007).
[10] Javier PerezMoreno, Y. X. Zhao, Koen Clays, Mark G. Kuzyk, Y. Q. Shen, Ling Qui, J. M. Hao, and K. P. Guo, “Modulated Conjugation as a Means of Improving the Intrinsic Hyperpolarizability,” J. Am. Chem.Soc. 131, 50845093 (2009).
[11] Javier PérezMoreno, Inge Asselberghs, Yuxia Zhao, Kai Song, Hachiro Nakanishi, Shuji Okada, Kyoko Nogi, OhKil Kim, Jongtae Je, Janka Mátrai, Marc De Maeyer, and Mark G. Kuzyk “Combined molecular and supramolecular bottomup nanoengineering for enhanced nonlinear optical response: Experiments, modeling and approaching the fundamental limit,” J. Chem. Phys. 126, 074705 (2007).
[12] Kang, H. and Facchetti, A. and Jiang, H. and Cariati, E. and Righetto, S. and Ugo, R. and Zuccaccia, C. and Macchioni, A. and Stern, C. L. and Liu, Z. F. and Ho, S. T. and Brown, E. C. and Ratner, M. A. and Marks, T. J., "Ultralarge hyperpolarizability twisted pielectron system electrooptic chromophores: Synthesis, solidstate and solutionphase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies," J. Am. Chem. Soc. 129, 3267 (2007).
[13] Mark C. Kuzyk and Mark G. Kuzyk, “Monte Carlo Studies of the Fundamental Limits of the Intrinsic Hyperpolarizability,” J. Opt. Soc. Am B 25, 103 (2008). 