Vibrational contributions to the hyperpolarizabilities of homonuclear diatomic molecules

An expression for the off-resonance vibrational contribution to the hyperpolarizability of a homonuclear diatomic molecule is derived. This expression is written in terms of the Raman transition polarizability, and it explicitly gives the frequency dependence of all the independent tensor components of the hyperpolarizability of a randomly oriented, freely rotating molecule. The vibrational contributions to the hyperpolarizabilities of several molecules are evaluated and compared, both in the static limit, and also at optical frequencies. Several nonlinear optical processes are considered.