Problems in the comparison of theoretical and experimental hyperpolarizabilities

Frequently it is useful to compare experimental values of the hyperpolarizabilities β and γ with calculated values. It is also often helpful to compare experimental values of β obtained from dc-electric field induced second harmonic generation (dc-SHG) experiments, e.g., with values obtained using the solvatochromism method. In order to do this the hyperpolarizabilities must be defined using consistent conventions. In this paper, four commonly used conventions are discussed and simple factors for converting between them presented. In addition, the sum-over-states expression for the calculation of β and γ is described and its correct use in comparing with hyperpolarizabilities obtained using other experimental and theoretical techniques discussed. As an illustration of the consistent use of conventions, ab initio and semiempirical calculations on para-nitroaniline are compared with experimental dc-SHG values. This comparison highlights the difference between theoretical values of the hyperpolarizability with the molecule in a gas phase environment and experimental values obtained in polar solvents−a difference that has in the past been obscured by inconsistent choice of conventions.