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.