The local group K(4) in the algebraic approach to vibrational spectra of tetrahedral molecules: Application to silane
In a previous paper, Michelot and Moret-Bailly ( J. Phys., 48, 51 (1987)) proposed an algebraic treatment of vibrational stretching modes in polyatomic molecules. They used the properties of the group chain U( p + 1) ⊃ U( p) ⊃ S( p) ⊃ G for the study of p identical oscillators. The molecule, with p equivalent bonds described as a system of p oscillators, has a symmetry group G. We develop in this paper an application to p = 4 equivalent oscillators. We show that, for a tetrahedral molecule, the group chain U(5) ⊃ U(4) ⊃ S(4) ≈ Td can be completed, in a local point of view, with a particular group K(4): U(5) ⊃ U(4) ⊃ K(4) ⊃ S(4) ≈ T d This group provides us with available labels which clearly identify the different local states. A set of K(4) invariants allows the straightforward computation of the energy of these states. We apply our model to the silane molecule.