Second-Order Sum Rule for the Vibrations of Isotopic Molecules and the Second Rule of the Mean

Abstract

A second‐order sum rule for the vibrational frequencies of isotopic molecules is derived. The rule relates the second power of the λ's, Σλ_i² and Σ_i≠j λ_iλ_j, for isotopic molecules of the type X′Y_z—rY_r′ and XY_z—rY_r′ with the second power of the λ's for the symmetrical molecules X′Y_z′, XY_z′, X′Y_z, and XY_z. Special consideration is given to the case where X′ and X are in fact Y′ and Y in positions structurally equivalent to the z Y atoms in XY_z. In the latter case, if z is greater than 2, the rule may involve isotopic isomers. Just as in the case of the first‐order sum rule, the second‐order rule can be applied separately to the individual symmetry classes common to all of the molecules. It is shown that the second‐order sum rules for a homologous series of isotopic molecules can be found easily by a simple tabulation of the interactions between the equivalent isotopic atoms. Interaction tables are given for the deutero‐ethylenes and the deutero‐benzenes. The second‐order sum rule is used to derive the second rule of the mean for isotopic exchange equilibria. Examples of the latter are given involving the carbon dioxide, ethylene, and benzene molecules.