Dispersion formulas for the second hyperpolarizability components γ||, γ⊥ and γK

The frequency-dependence of the isotropic parallel average γ|| of the second hyperpolarizability can be expanded in a compact way using dispersion coefficients, which are independent of the optical process, if three effective frequencies are introduced. We propose similar expansions for the second hyperpolarizability components γ⊥ and γK and give explicit expressions for the hyperpolarizability dispersion coefficients in terms of the derivatives of the hyperpolarizabilities with respect to their frequency arguments. The relation of the hyperpolarizability dispersion coefficients to the Cauchy moments of linear response theory is discussed and comparison is made with previously proposed dispersion formulas.