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Theoretical Details To The Calculation Of Nuclear Spin-Rotation Constants

Interactions between the nuclear spin and the rotation of a molecule cause additional splittings in the rotational spectra. The corresponding molecular parameters are the so-called nuclear spin-rotation constants {\cal M}_n which are given as the second derivative of the molecular energy with respect to nuclear spin I_n of the n-th nucleus and rotational angular momentum \bf J

{\cal M}_n = - \frac{\partial^2 E}{\partial I_n \partial J}

The quantum chemical calculation of {\cal M}_n suffers from a slow basis-set convergence and an (unphysical) origin dependence concerning the electronic angular momentum (though an appropriate choice might be the corresponding nucleus). These problems can be overcome by using perturbation-dependent basis functions (often refered to as rotational London orbitals)

\chi_{\mu}({\bf r},{\bf J})  =  \exp(i ({\bf I}^{-1}{\bf J})\times{\bf R}_{\mu})\cdot{\bf r})\ \chi_{\mu}({\bf r})

In addition, it should be noted that a close relationship between the shielding tensor \sigma_n and the nuclear spin-rotaztion tensor {\cal M}_n. This relationship is usually stated as

{\cal M}_n = 2 \gamma_n \sigma_n^{para}({\bf R}_n) {\bf I}^{-1} + {\cal M}_n^{nuc}

with the paramagnetic part of the shielding computed with the corresponding nucleus as gauge origin, {\cal M}_n^{nuc} denoting the nuclear contribution to {\cal M}_n and \gamma_n as the gyromagnetic ratio. When using perturbation-dependent basis functions (rotational London orbitals), this expression has to be modified to

{\cal M}_n = 2 \gamma_n (\sigma_n^{GIAO} - \sigma_n^{dia}({\bf R}_n)) {\bf I}^{-1} + {\cal M}_n^{nuc}

with the diamagnetic contribution to \sigma_n calculated in the usual manner with \bf R_n as gauge origin.

Reference

J. Gauss, K. Ruud, and T. Helgaker, J. Chem. Phys. 105, 2804 (1996)

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CFOUR is partially supported by the U.S. National Science Foundation.