Sum Rules for the Vibration Frequencies of Isotopic Molecules

The squares of the frequencies of isotopically related molecules obey certain simple sum rules. Whenever there exists a chemical exchange reaction involving only isotopic variations of a single type of molecule, such that the reaction is balanced at each atomic position of the molecule, then a relation of identical form exists between the sums of the squares of the frequencies. Thus $${\mathrm{\Sigma \nu }}^2({\mathrm{H}}_2\mathrm{O}{\mathrm{)+\Sigma \nu }}^2({\mathrm{D}}_2\mathrm{O}{\text{)=2Σν}}^2(\mathrm{HDO}\mathrm{)\hspace{0.17em}\hspace{0.17em}}\mathrm{and}{\mathrm{\hspace{0.17em}\hspace{0.17em}\Sigma \nu }}^2({\mathrm{N}}^{14}{\mathrm{H}}_3{\mathrm{)+\Sigma \nu }}^2{\mathrm{(N}}^{15}{\mathrm{D}}_3{\mathrm{)=\Sigma \nu }}^2({\mathrm{N}}^{14}{\mathrm{D}}_3{\mathrm{)+\Sigma \nu }}^2({\mathrm{N}}^{15}{\mathrm{H}}_3\mathrm{).}$$ If, as in the latter example, the molecules all have the same symmetry, the sum rule may be applied separately to each factor of the secular equation. When forms of lower symmetry appear (as in the example involving the water molecule), symmetry factoring may or may not be possible. In such cases it is necessary to employ certain group‐theoretical considerations; the possible factoring is not in general identical with that appropriate for the product rule. Examples of the application of the sum rule are given, involving the acetylene, water, methane, and hydrocyanic acid molecules. The existence of higher order isotope rules, intermediate between the sum and product rules, is pointed out.