Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the low rank perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at ρ atomic sites is considered. Relations which express vibrational frequencies ωk and normal modes Ψk of the perturbed isotopic molecule B in terms of the vibrational frequencies νi and normal modes Φi of the unperturbed molecule A are derived. In these relations complete specification of the unperturbed normal modes Φi is not required. Only amplitudes 〈τ|Φi〉 of normal modes Φi at sites τ affected by the isotopic substitution are needed.