The dipole moment expansion for a tetrahedral molecule in the ground vibronic state

Abstract

The expansion of the electric dipole moment operator in terms of angular momentum operators is examined in detail for a tetrahedral molecule in the ground vibronic state. The components of this centrifugal distortion moment in the molecule fixed frame are formally expanded to arbitrary order, with the expansion coefficients being given in terms of rotation matrices. For terms of rank j less than 15, general expressions are given for the matrix elements in the symmetric top basis of the space-fixed components of the dipole moment. The dependence of these matrix elements on the K-quantum number is shown to factor in such a way that previous first order calculations can be extended to second order by replacing the first order dipole coupling constant μ 2 (2) with a function of J which involves μ 2 (2) and two second order constants, μ 2 (4) and μ 4 (4) . Different functions are required for Q- and R-branch matrix elements and explicit expressions are given for both. The third order terms are examined in detail in the Appendix. A recurrence relation is derived for j < 15 between the tetrahedral harmonics of types A 2 and F 2. The implications of this work for distortion moment spectroscopy in tetrahedral molecules are discussed.