Knowledge Bank

It is shown that the accurate rotation-vibrational Hamiltonian evaluated in the finite basis set \begin{equation}|J_{g} R_{\tau} k_{R}M >=(-1)^{J-1-K_{R}} (2r + 1)^{1} \sum_{K,M} \left(\begin{array}{ccc}J & l & R\K & m & K_{R}\end{array}\right)|JKM>|1M\end{equation} \begin{equation}|1 m> = \sum_{\alpha}^{(1)}G_{F_{1a}}^{m}[F_{1} \alpha v>T=,ufor i=1,2,\end{equation} where the vibrational wave functions $|F1\alpha v>$ of $F1$ symmetry with respect to the feasible permutation $(Oh)$ or permutation-inversion $(Td)$ group are eigenfunctions of the accurate purely vibrational Hamiltonian, can be represented in the tensor from: \begin{equation}B_{eff} -B_{v}J^{2} + 2/3B_{v}\zeta_{v}T^{110}+Z_{vs}/5 T^{220} +Z_{v1}(120) T^{224}\end{equation} We can thus introduce the assigned'' group SO(3) composed of body-fixed rotations followed by appropriate transformations of the vibrational wave functions $|1_{1}m>$. Note that elements of the group introduced in such a way are not transformations of coordinates and hence cannot be used for deriving selection rules. It can be independently proved that matrix elements of a space-fixed component of the electric dipole moment for transitions from states (1a) to a totally symmetric vibrational state of $T_{d}$ or $o_{h}$ molecule (or for the $\nu_{6}(F_{2})-\nu_{5} (f_{2g})$ band of $SF_{6}$) behave as if this component were a spherical invariant under action of elements of the assigned'' SO(3) group. For tetrahedral molecules it is also useful to introduce the assigned'' inversion -- the operation which multiplies functions (1a) by the factor (-1). The products of feasible permutation-inversions and the assigned'' inversion generate transformations which belong to the SO(3) group in question and lead to Moret-Bailly's labeling scheme. It is important that levels in clusters have the same assigned symmetry in the ground and first excited states of triply degenerate normal modes.