The relationship between the models of anharmonically and harmonically coupled anharmonic oscillators

The relationship between the model of anharmonically coupled anharmonic (Morse) oscillators (ACAO) and the model of harmonically coupled anharmonic oscillators (HCAO) used to interpret X-H stretch overtone spectra in terms of local modes is examined by means of a non-linear co-ordinate transformation described recently by Cooper (1987, Chem. Phys., 112, 67). The transformation allows the ACAO model hamiltonian to be rewritten in an harmonic oscillator basis. Van Vleck perturbation theory may then be applied to further transform the hamiltonian to block diagonal form. With the hamiltonian treated to second order in the perturbation parameter β [=(-2xm /ω m )1/2, where ωm is the harmonic wavenumber of the bond oscillation and xm is the Morse anharmonicity], the eigenvalues of the original ACAO hamiltonian can be reproduced to within a few tenths of a cm-1 at low vibrational levels and a few cm-1 at high vibrational levels, as indicated by calculations on model H2O. The perturbation treatment reveals important effects arising from the quadratic interbond coupling hamiltonian, in addition to effects from anharmonic coupling. The neglect of some correction terms from the perturbation treatment yields the HCAO model, and the models are compared by reference to results for H2O.