Vibrational Frequencies of PH₃ and NF₃: Lie Algebraic Method

Introduction

In the last part of the 19th century, Marius Sophus Lie proposed Lie algebras. Nevertheless, these methods have been useful in the study of the problems in the beginning portion of the 20th century, after the evaluation of quantum mechanics1, 2. Iachello (1981) presented Lie algebraic method (vibron model) for the study of vibrational spectra of molecules. This method is based on the second quantization of the Schrodinger wave equation with a 3- Dimensional Morse potential function and is described as ro-vibration spectra of diatomic molecules. Considerable attention has been given to determining vibration spectra of molecules. Therefore the study of molecular spectroscopy is fundamental to understanding both experimental and theoretical approaches. There are two general methods which are used to describe molecular vibrations. In the classical approach, the molecular Hamiltonian is parameterized in terms of internal coordinates and the potential energy function is represented in terms of force field constants by spectroscopic data. Although diatomic molecules have very accurate information on force fields, but not in the case of polyatomic molecules. The molecular rotation-vibration spectrum is provided by the Dunham expansion. It is a series of expansion of energy levels of rotational (J) and vibrational (v) quantum numbers. The constant coefficients yij are called Dunham parameters. For diatomic molecules, the expansion is

This approach is useful when the spectra can be expressed by a small number of parameters. The major drawbacks of this approach are that there is no Hamiltonian operator and it requires more number of parameters for large polyatomic molecules. The second approach is the potential approach. This approach is based on the Schrodinger wave equation. In this approach, the energy levels are computed by fitting the Schrodinger wave equation in terms of interatomic potential. The potential is expanded in terms of interatomic variable. For diatomic molecules, the possible expansion is

 

The potential coefficients, a’s are evaluated by fitting the experimental data on energy levels. The basic drawback of this approach is that it is very difficult to fit potential coefficients if the numbers of atoms are more in the molecule6. To overcome the difficulties raised in analysing the vibrational spectra, we consider the third approach (Lie algebraic method).

Lie Algebraic Method

The Hamiltonian in the case of stretching vibrations for the polyatomic is of the form 5

Here i vary from 1 to n for n stretching bonds and Ai , Aij and λij are algebraic parameters, which are determined by spectroscopic data. Where Ci is an invariant operator of the uncoupled bond with eigenvalue

 and the operator C_ij for coupled bonds are diagonal with matrix elements

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