Localized vibrational modes in Fermi liquids. General theory

There exist an abundance of physically realized situations in atomic, chemical, molecular, nuclear, and solid-state physics which are modeled as a discrete state interacting with a continuum. One of these situations, a localized oscillator coupled to be the particle-hole excitation continuum of a Fermi liquid, forms the area of inquiry here. Special emphasis is placed on the electron-hole pair excitations of a metal, although the results are more general. In particular, an exactly soluble model, in which a localized harmonic oscillator is coupled to the pair-density fluctuations of a random-phase-approximation Fermi liquid by an interaction linear in both oscillator displacements and density operators, is considered. It is shown that this model is equivalent to a system of coupled harmonic oscillators, and, as such, the eigenstates are obtained via a straightforward normal-mode analysis. Expressions for the local spectral function of the oscillator are obtained in terms of level-shift and broadening functions which are evaluated in closed form in the density-of-states limit. Parametric calculations for vibrational line shapes are made, and the results are discussed in terms of "exponential-decay theory." The general results presented here have direct bearing in areas such as surface vibrational spectroscopy of adsorbed molecules, metal-hydride spectroscopy, and surface chemical-reactivity theory.