Vibrational isotope effect of planar molecules by the low rank perturbation method: Deuterated ethylenes

Planar molecules A and B, which are identical except for isotopic substitutions at various atomic sites tau, are considered. Out-of-plane frequencies omega(k) and normal modes Psi(k) of the perturbed isotopic molecule B are expressed in terms of out-of-plane frequencies nu(i) and out-of-plane normal modes Phi(i) of unperturbed molecule A. Complete specification of unperturbed normal modes Phi(i) is not required. All that is needed are amplitudes [tau\Phi(i)] of normal modes Phi(i) at sites tau of isotopic substitution. The interlacing rule that interlaces frequencies omega(k) with frequencies nu(k) is derived. The method is applied to all deuterated ethylenes. Out-of-plane frequencies of deuterated ethylenes are in excellent agreement with the interlacing rule. There is only one serious discrepancy, which might be due to an experimental error. In the ethylene case, out-of-plane amplitudes [tau\Phi(i)] are determined by symmetry alone. Hence, out-of-plane frequencies and normal modes of all deuterated ethylenes depend only on the three experimental out-of-plane frequencies of ethylene. Standard error of the calculated out-of-plane frequencies (expressed in cm(-1)) is only 2.92 cm(-1).