Eighth rank isotropic tensors and rotational averages

Isotropic tensors play an important role in the theory of many physical processes, which take place in gases and liquids. In such systems it is usually necessary to perform a rotational average on products of direction cosines relating the space-fixed and molecular coordinate frames. The average is generally expressible in terms of isotropic tensor. The authors discuss the isotropic tensors of eighth rank, and relations between them are demonstrated. The rotational average of eighth rank is then evaluated in both reducible and irreducible form; the results are applicable to a number of processes, for example optical seventh harmonic generation and four-photon absorption.