Formal expressions for the lowest order inelastic contribution to the degeneracy averaged total cross section for both distinguishable and indistinguishable rigid diatomic molecules are obtained for molecules interacting through a multiple potential. Explicit expressions for the inleastic cross section are obtained for molecules interacting through permanent dipole-dipole, dipole-quadrupole, or quadrupole-quadrupole potentials. These results require the summation over only a single quantum number; the relative angular momentum quantum number. Thus, they are useful for analytic or numerical calculations