Theory of Vibrational Isotope Effects in Polyatomic Molecules

Following Dennison's general, noncentral force treatment of the normal modes of vibration of symmetrical triatomic and tetratomic molecules, we derive expressions for the effects of isotopy on the normal frequencies. The isotope effect of any particular normal frequency depends on and may serve to evaluate the force constants of the molecule, but sums of certain isotope effects are independent of the constants. For triatomic molecules, the isotope effect of the vibration perpendicular to the symmetry axis depends on the value of the apex angle of the molecule. The sum of the isotope effects of the parallel vibrations is calculable from the masses alone. Definite criteria of collinearity of the molecule from isotope effects are given. For tetratomic molecules, the sum of isotope effects of vibrations parallel to the altitude of the molecular pyramid is calculable from the masses alone, the sum of isotope effects perpendicular to the altitude from the masses and from the ratio of the altitude to the length of side of the triangular base. Whereas the molecule $Y{X}_3$ has four distant normal frequencies of vibration, it is shown that replacement of one of the $X$ atoms of mass $m$ by an atom of mass $m+\Delta m$ removes the degeneracy of the motion, the resulting molecule having six normal frequencies of vibration. Special relations between the various isotope effects serve as criteria for co-planar molecules.