Vibrational isotope effect by the low rank perturbation method

Abstract

Listen

Translate article

Mathematical formalism of the Low Rank Perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation with a standard assumption that force field does not change under isotopic substitutions. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at ρ atomic sites is considered. In the LRP approach vibrational frequencies ωk and normal modes \({\left|{\Psi_k}\right\rangle}\) of the isotopomer B are expressed in terms of the vibrational frequencies νi and normal modes \({\left|{\Phi_i}\right\rangle}\) of the parent molecule A. In those relations complete specification of the normal modes \({\left|{\Phi_i}\right\rangle}\) is not required. Only amplitudes \({\left\langle{\tau s}|{\Phi_i}\right\rangle}\) at sites τ affected by the isotopic substitutions and in the coordinate direction s (s = x, y, z) are needed. Out-of-plane vibrations of the (H,D)-benzene isotopomers are considered. Standard error of the LRP frequencies with respect to the DFT frequencies is on average \({\Delta \approx 0.48\,{\rm cm}^{-1}}\) . This error is due to the uncertainty of the input data (± 0.5 cm−1) and in the absence of those uncertainties and in the harmonic approximation it should disappear. In comparing with experiment, one finds that LRP frequencies reproduces experimental frequencies of (H,D)-benzene isotopomers better (\({\Delta_{LRP}\approx 4.74\,{\rm cm}^{-1}}\)) than scaled DFT frequencies (\({\Delta_{DFT}\approx 6.79\,{\rm cm}^{-1}}\)) which are designed to minimize (by frequency scaling technique) this error. In addition, LRP is conceptually and numerically simple and it also provides a new insight in the vibrational isotope effect in the harmonic approximation.