Theory of Hyper-Raman Effects (Nonlinear Inelastic Light Scattering): Selection Rules and Depolarization Ratios for the Second-Order Polarizability

The symmetry properties of the third-rank tensor β which comprises the set of coefficients of quadratic terms in the expansion of the induced-dipole moment in the electric field are investigated, and the various linear combinations belonging to the irreducible representations are tabulated for the important molecular symmetry groups. Depolarization ratios are calculated for a sample composed of randomly oriented molecules (liquid phase). Examples of the selection rules in both liquid and crystalline phases are discussed, and these selection rules are contrasted with the ones appropriate for infrared absorption and the ordinary Raman effect. Molecular modes allowed in the infrared are always allowed in the hyper-Raman effect and have a depolarization ratio (for linearly polarized incident radiation) ≤⅔, while allowed hyper-Raman transitions forbidden in the infrared have a depolarization ratio of ⅔.