Vibrational self-consistent field method for many-mode systems: A new approach and application to the vibrations of CO adsorbed on Cu(100)

Abstract

We report calculations of the vibrational energies of CO–Cu(100) using a new code to perform vibrational self-consistent field (VSCF) and state-mixing calculations for many-mode systems. The major new feature of the code is the representation of the potential. Unlike recent implementations of the VSCF method, the potential is not expanded in terms of normal coordinates as a multinomial series about a minimum. The full potential, in normal coordinates, is used in the Watson Hamiltonian. This approach, while rigorous, can lead to prohibitively large numerical quadratures, and so we suggest a novel representation of the potential as an expansion in all two-mode, or all three-mode, or all four-mode coupling terms. The new code is tested against previous exact calculations of vibrational states of HCO, and also against previous VSCF calculations that used a fourth-order, normal coordinate force field representation of the global HCO potential. The new code is applied to calculations of the vibrations of CO adsorbed to Cu(100). We explicitly treat nine modes corresponding to the motion of the C and O atoms and the Cu atom that is bonded to C. The potential used is a semi-empirical one developed by Tully and co-workers [J. C. Tully, M. Gomez, and M. Head-Gordon, J. Vac. Sci. Technol. A 11, 1914 (1993)], and is used fully, i.e., without recourse to multinomial expansion in displacement coordinates. We test the convergence of the results with respect to the number of modes coupled and find that the errors in the two-mode coupling representation vary from 0.6 to 6 cm−1 for the fundamentals but grow to 30 cm−1 for overtone and combination states. The errors in the three-mode representation of the potential are less than 0.2 cm−1 for the fundamentals and no larger than 2.5 cm−1 for high overtone/combination states with as much as 9 quanta of excitation. We calculate the thermally broadened spectra of the CO-stretch fundamental, the CO–Cu frustrated rotation and the CO–Cu frustrated translation over the temperature range 50–350 K. We compare the temperature dependence of the average frequency and standard deviation of these modes with experiment, and find semiquantitative agreement.