Knowledge Bank

The correlation of normal frequencies in deuterated and non-deuterated molecules is aided by the Teller-Redlich product rule, and the order rule which places upper and lower bounds on each frequency of the deuterated molecules when arranged ordinally. Neither rule is of much utility in identifying modes which represent similar motions in the two molecules. To clarify this it is useful to consider a hypothetical continuous plot of the logarithm of each normal frequency versus the logarithm of mass of the substituted isotope (these continuous curves can be calculated in the harmonic approximation from estimated force constants.) The slope of each frequency curve is half the negative of the fraction of the kinetic energy of the normal mode contributed by the isotopic atoms, and lies between 0 and $-1/2$, which is the equivalent of the order $rule.1$ The \emph{sum} of the slopes for all frequencies of a given symmetry species is fixed by the product rule. In the harmonic approximation frequencies of the same symmetry species may occasionally be accidentally degenerate, but exact degeneracy or crossing within a species will be prohibited by Fermi resonance. The proposed rule states that near degeneracy of two frequencies of the same species, at an intermediate isotopic mass, will result in an interchange of the role of the isotopic atoms in the associated modes in the deuterated and non-deuterated molecules. Therefore, if these two frequencies are arranged by \emph{type} their order will be reversed. An example from $B5H9$ and $B5D9$ in which the small boron isotope splitting serves as an indicator will be $discussed.2,3$