Analysis of nearly degenerate bands of spherical top molecules: Matrix elements of the vibrational-rotational Hamiltonian

We formulate a general method for calculating energy eigenvalues and eigenfunctions in a simultaneous analysis of nearly degenerate vibrational-rotational bands of spherical top molecules. The basis functions are products of rigid rotor and N-dimensional harmonic oscillator eigenfunctions. General explicit expressions are derived for the matrix elements of vibrational operators in the basis of N-dimensional harmonic oscillator eigenfunctions. Using these general expressions, the matrix elements of vibrational operators in the basis of five-dimensional harmonic oscillator eigenfunctions applicable, for example, in the analysis of the nearly degenerate ν2 and ν4 fundamental vibrational-rotational bands of tetrahedral XY4 molecules like CH4, are calculated explicitly.