The Rayleigh–Ritz Variation Method: An Illustrative Application to Anharmonicity of the Hydrogen Chloride Molecule Vibrations

In this paper a new approach to presenting the Rayleigh–Ritz method to chemistry students is described. The emphasis is on the methodological aspects that students should concentrate on, rather than on computational issues (e.g., calculations of integrals). The method is illustrated by the example of its application to the problem of anharmonicity of the HCl molecule vibrations. An Excel workbook to perform calculations without distracting students from the essence of the method was prepared. The relevant integrals, including integrals for the anharmonic terms up the sixth order, were programmed using the well-known formulas. The Jacobi diagonalization procedure was implemented in Visual Basic as an Excel macro. This procedure can be easily used provided the Hamiltonian matrix is available. This matrix can be readily assembled from the precomputed integrals and adjusted according to the users’ needs, e.g., by straightforward inclusion of the desired anharmonic terms to the potential. Issues to be discussed with students based on the results obtained for the fourth-order anharmonic potential are also suggested. It is also highly recommended that students carry out similar calculations, e.g., by including higher order terms to see the changes in the oscillator energy levels. The workbook can be easily adapted for calculations on other diatomic molecules at any computational level. In addition, a Wolfram Mathematica notebook for performing similar calculations was also provided for the users’ convenience.