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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.