RELATIONS AMONG VIBRATIONAL FREQUENCIES OF ISOTOPICALLY SUBSTITUTED MOLECULES

Author Institution: Department of Chemistry, University of CaliforniaA simple derivation is given for the product, sum, and complete isotope rules. Additional sum rules and their implications are presented. In particular it is shown that if equivalent atoms of a molecule are isotopically substituted then one relationship (the product rule) exists between two isotopic species; two more relationships (the first order sum rules) exist among three isotopic species; two more relationships exist among four isotopic species; and for five or more isotopic species all frequencies are related. If the isotopic substitutions retain the full symmetry of the molecule, the above-mentioned relationships exist for each symmetry block. Even if isotopic substitution reduces the symmetry, the number of relationships is not diminished. Furthermore if equivalent atoms are isotopically substituted, then for the in-plane vibrations of planar molecules, all the vibrational frequencies are related among four or more isotopic species; for the out-of-plane vibrations of planar molecules and for linear molecules, all the vibrations are related among three or more isotopic species (the complete isotope rule)