Algebraic approach to vibrational spectra of tetrahedral molecules: Application to methane

Abstract

We present an algebraic approach to describe the vibrational excitations of polyatomic molecules. The model is based on the coupling of anharmonic oscillators and is characterized by combining the strengths of the Lie-algebraic methods with those of point symmetry techniques. We illustrate the algebraic method for tetrahedral molecules and apply it to the construction of the complete vibrational spectra of methane up to four quanta.

A Symmetry Adapted Algebraic Approach to Molecular Spectroscopy

“…(13). The states with a higher number of quanta | V φ Γ γ can be constructed using the Clebsch-Gordan coefficients of T d [14,24]. Since all operators are expressed in terms of powers of the U i (2) generators, their matrix elements can be easily evaluated in closed form.…”

“…The five interaction terms in Eq. (16) correspond to linear combinations of the Casimir operators of [14]. However, for a good description of the vibrational energies of Be 4 it is necessary to include interactions which are related to the vibrational angular momenta associated with the degenerate modes E and F 2 .…”

“…In the present approach each relevant interatomic interaction is associated with a U i (2) algebra. The operators in the model are expressed in terms of the generators of these algebras, and the symmetry requirements of the tetrahedral group T d can be readily imposed [14,24]. The local operators {Ĝ i } acting on bond i can be projected to any of the fundamental irreps Γ = A 1 , E and F 2 .…”

“…The Hamiltonian operator can be constructed by repeated couplings of these tensors to a total symmetry A 1 , since it must commute with all operations in T d [14]. All calculations are carried out in a symmetry-adapted basis, which is projected from the local basis…”

“…In 1984 Van Roosmalen et al proposed a U (2) based model to describe the stretching vibrational modes in ABA molecules [11] which was later extended to describe the stretching vibrations of polyatomic molecules such as octahedral and benzene like molecules [12]. Recently the bending modes have also been included in the framework, which was subsequently applied to describe C 2v -triatomic molecules [13] and the lower excitations of tetrahedral molecules [14], using a scheme which combines Lie-algebraic and point group methods. In a different approach, it has also been suggested [15] to use a U (k + 1) model for the k = 3n − 3 rotational and vibrational degrees of freedom of a n-atomic molecule.…”