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

Abstract

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 \rho atomic sites is considered. In the LRP approach vibrational frequencies \omega_k and normal modes $${; ; ; \left|{; ; ; \Psi_k}; ; ; \right\rangle}; ; ; $$ of the isotopomer B are expressed in terms of the vibrational frequencies \nu_i 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 \tau 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.