Anharmonically-coupled local mode to normal mode Hamiltonian transformations: Beyond the x, K-relations

Attention is focused on the quantitative relationships between the parameters of the normal mode and the local mode effective Hamiltonian models of anharmonic stretching vibrations. R. G. Della Valle (1988, Molec. Phys., 63, 611) has demonstrated the general relationship between the harmonically coupled anharmonic (Morse) oscillators (HCAO) model and the normal mode model of X-H (or X-D or other weakly coupled) stretching vibrations of polyatomic molecules. Here we consider more fully the relationship between the normal mode and the anharmonically coupled local mode models for X-H stretching vibrations. The local mode Hamiltonian is expressed in terms of boson shift operators and these operators are then transformed into the corresponding normal mode model operators. Such relationships have already been derived for the special cases of H₂X (Baggott, J. E., 1988, Molec. Phys., 65, 739), XH₃ (C_3v) and XH₄ (T_d) (Law, M. M., and Duncan, J. L., 1994, Molec. Phys., 83, 757) and here they are derived systematically for arbitrary local mode stretching systems. Explicit relationships are derived for molecules of the symmetry types of allene, ethylene and benzene respectively. These relationships between the unconstrained normal mode and anharmonically coupled local mode models allow for more reliable schemes for the relaxation of the HCAO x, K-relations than those suggested hitherto. In an application to the ethylene-d4 overtone spectrum a much improved fit to experimental data is obtained using a compact anharmonically coupled local mode model and the consequences for the corresponding normal mode model demonstrated. The generality of our results is emphasized by the application to systems involving non-equivalent bonds, illustrated with reference to HCN/DCN. Approximations in these local to normal mode transformations (such as the assumption of a unitary transformation) are discussed. The models presented here may be supplemented with appropriate matrix elements to take account of Fermi resonances in the same straightforward manner as the HCAO model.