Risser, SM; Beratan, DN, *What is the anharmonicity of a molecule's electronic wave function?*,
The Journal of Physical Chemistry, vol. 99 no. 7
(1995),
pp. 1935-1942 [pdf] .
**Abstract:**

*The dependence of a classical spring's hyperpolarizabilities on its anharmonicity constants is known, defining a simple structure-property relationship. Corresponding structure-property relationships for the electronic hyperpolarizabilities of quantum many-particle systems, molecules, remain elusive. The quasi-harmonic behavior of molecular electronic polarization is disguised by the Coulombic nature of the interaction potential. Yet, to the extent that a perturbation theory description of electronic hyperpolarizability is appropriate, the quantum electronic hyperpolarizability problem maps onto a single-particle effective anharmonic oscillator hamiltonian. Here, we derive this relationship, which provides a means of establishing the effective anharmonic oscillator constants for a molecule. We calculate these electronic anharmonicities analytically and show results for some simple hamiltonians. One barrier to the design of chemical chromophores with tailored nonlinear optical coefficients is an understanding of how molecular structure influences hyperpolarizability. Restating the problem in terms of anharmonicities demystifies the source and magnitude of the electronic hyperpolarizabilities. © 1995 American Chemical Society.*