Knowledge Bank

The rotational wave functions v(JKM) and the direction cosines (or D(1 K M)) can be seen as tensors of two 0(3) groups: the first, (m)0(3),describes transformations in the molecule fixed frame and the second, (F)0(3) in the space fixed frame. For a spherical top molecule, the groops of degeneracy of rotational and rovibrational energies are $(m)0(3)\times (F)0(3)$ reapectiely $(G=Td$ for $XY4$, $G=0h$ for $XY6$). Energy levels are generally studied with the help of the reduced chain $G[FIGURE]\subset (m)0(3)$, but $G\times (F)0(2)[FIGURE]\subset (m)0(3)\times (F)0(3)$ seems more appropriate for the calculation of transition amplitudes In this scheme, the states are labelled by |J$\ell$Rp; JM where J,$\ell$,R stand for the total, vibrational and rotational angular moments respectively; p and M are cubic and magnetic quantum numbers. Any transition Operator can be written in the form: $$^{(F)}A^{(X,L)}{A{1}M}= (D^{(L,L)} X^{(m)} A^{(\ell,0)} ){}^{(X,L)}{A{1}M}$$ where $D(L,L)$ are products of direction cosines. Its matrix element is given by Wigner-Eckart theorem in : \begin{eqnarray*} &&(J^{\prime\prime} \ell^{\prime\prime} R^{\prime\prime} P^{\prime\prime} ; J^{\prime\prime} M^{\prime\prime} | ^{(F)} A^{(X , L)}{A{1}M } | J^{\prime} \ell^{\prime} R^{\prime} P^{\prime} ; J^{\prime} M^{\prime} ) =\ &&\quad (- 1)^{R^{\prime\prime}} F^{(X R^{\prime\prime} R^{\prime\prime})}{A{1}P^{\prime\prime} P^{\prime}} (- 1)^{J^{\prime\prime} -M^{\prime\prime}} ({}^ {J^{\prime\prime} J^{\prime} L}_{-M^{\prime\prime} M^{\prime} M}) (J^{\prime\prime} \ell^{\prime} R^{\prime\prime} ; J^{\prime\prime} | {}^{(F)}A^{(X,L)}| J^{\prime} \ell^{\prime} R^{\prime} ; J^{\prime} )\end{eqnarray*} using cubic and standard 3j-symbols. Applications are given for Infrared, microwave, and Raman spectra.