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Vibrational isotope effect by the low rank perturbation method. Case study: out-of-plane vibrations of benzene (H, D)-isotopomers

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 ω k and normal modes |Ψk⟩ of the isotopomer B are expressed in terms of the vibrational frequencies ν i and normal modes |Φi⟩ of the parent molecule A. In those relations complete specification of the normal modes |Φi⟩ is not required. Only amplitudes ⟨τs|Φi⟩ 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 Δ≈0.48cm−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 (ΔLRP≈4.74cm−1) than scaled DFT frequencies (ΔDFT≈6.79cm−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.

low rank perturbation ; vibrational isotope effect ; harmonic approximation ; benzene isotopomers

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