A new dipole-free sum-over-states expression for the second hyperpolarizability

TL;DR

This paper introduces a dipole-free sum-over-states expression for the second hyperpolarizability, simplifying calculations for non-dipolar systems and enabling more efficient modeling and convergence testing.

Contribution

The authors derive a new SOS expression that eliminates dipolar dependence, improving analysis of non-dipolar systems and enabling simplified three-state models.

Findings

The new expression is equivalent to the traditional one in fundamental limits.

Both expressions show the same frequency dependence in tested models.

The new formula can serve as a convergence test for molecular calculations.

Abstract

The generalized Thomas-Kuhn sum rules are used to eliminate the explicit dependence on dipolar terms in the traditional sum-over-states (SOS) expression for the second hyperpolarizability to derive a new, yet equivalent, SOS expression. This new dipole-free expression may be better suited to study the second hyperpolarizability of non-dipolar systems such as quadrupolar, octupolar, and dodecapolar structures. The two expressions lead to the same fundamental limits of the off-resonance second hyperpolarizability; and when applied to a particle in a box and a clipped harmonic oscillator, have the same frequency-dependence. We propose that the new dipole-free equation, when used in conjunction with the standard SOS expression, can be used to develop a three-state model of the dispersion of the third-order susceptibility that can be applied to molecules in cases where normally many more states would have been required. Furthermore, a comparison between the two expressions can be used as a convergence test of molecular orbital calculations when applied to the second hyperpolarizability.