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A Scale-Invarient Analysis of the Hyperpolarizability


Universal Plot

Scale Invariance

The nth 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 nonlinear-optical 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] off-resonant 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 nth 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 1-dimensional 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 one-dimensional molecule with a bridge of hetero-atomic 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 (C1-III to C5-III). 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.

Amylose Helix

Theoretical Questions

It is puzzling that many very different one-dimensional 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 one-dimensional system can be as large as unity.[13]


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.


[1] M. G. Kuzyk, “Fundamental limits on third-order 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 nonlinear-optical phenomena that are representable by a second-order susceptibility,“ J. Chem. Phys. 125, 154108. (2006).

[4] Juefei Zhou and Mark G. Kuzyk, “Intrinsic Hyperpolarizabilities as a Figure of Merit for Electro-optic Molecules,” J. Phys. Chem. C 112, 7978 (2008).

[5] Mark G. kuzyk, "Using fundamental principles to understand and optimize nonlinear-optical 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 Perez-Moreno, 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 Perez-Moreno, 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, 5084-5093 (2009).

[11] Javier Pérez-Moreno, Inge Asselberghs, Yuxia Zhao, Kai Song, Hachiro Nakanishi, Shuji Okada, Kyoko Nogi, Oh-Kil Kim, Jongtae Je, Janka Mátrai, Marc De Maeyer, and Mark G. Kuzyk  “Combined molecular and supramolecular bottom-up nano-engineering 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 pi-electron system electro-optic chromophores: Synthesis, solid-state and solution-phase 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).

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