Theory of Nuclear Hyperfine Interactions in Spherical-Top Molecules

Abstract

The problem of nuclear hyperfine interactions in spherical-top molecules is treated by the methods of group theory. The tetrahedral molecules C${\mathrm{H}}_4$ and C${\mathrm{D}}_4$ are considered as specific examples. For these systems in their ground electronic and vibrational states, the complete effective rotational and hyperfine Hamiltonian $W$ is discussed and put into a form which is manifestly irreducible and invariant under the operations of the full three-dimensional rotation-reflection group $R\left(3\right)\mathrm{}$ and the molecular point group $T_d$. The spin-rotation, direct spin-spin, electron-coupled spin-spin, nuclear quadrupole, and shielding interactions are treated in detail, along with the centrifugal-distortion Hamiltonian $W_{\mathrm{dist}}$. The physical mechanism giving rise to each of these terms is discussed briefly. A representation $\Gamma$ is found which is suitable for the calculation of the eigenvalues of the total Hamiltonian. The total wave functions required for carrying out this calculation to first order in high-field perturbation theory are derived for arbitrary rotational angular momentum $J$. In particular, both the spin functions and the rotational functions which transform by the $T_1$ and $T_2$ representations of the tetrahedral point group $T_d$ are developed. The rotational functions are constructed so as to diagonalize $W_{\mathrm{dist}}$ exactly. Their explicit form depends on coefficients which have been calculated numerically. The methods used in generating the total wave functions are sufficiently general that they can be easily extended to more complex spherical molecules. In terms of the theory presented here, it will subsequently be possible to analyze a series of molecular-beam magnetic-resonance experiments which has been recently completed. Moreover, the treatment of nuclear spin relaxation for these molecules should be considerably simplified.