Brownian motion model of the interactions between chemical species and metallic electrons: Bootstrap derivation and parameter evaluation

We develop a Brownian motion model of the interaction between reacting chemicals on metal surfaces and the substrate electrons. The basic equation of such a model is the transport equation for the description of the chemicals' kinetics. We assume here that this equation has the functional form of the classical Fokker-Planck equation. The parameters of this equation---the average force and the friction coefficient---are determined by a bootstrap procedure, which requires that equilibrium and linear-response properties of the solution of the Fokker-Planck equation be identical to those calculated from the full quantum-mechanical equations. This derivation has the advantage that there is no need to explicitly require the electron-adparticle mass ratio to be small. We show further that our general expressions for the parameters reduce to those derived previously for systems in which the electron-adparticle mass ratio is formally made to tend to zero. These small mass-ratio expressions for the average force and the friction coefficient are then evaluated for several model metal plus adparticles systems. Simple systems and/or approximations are used in order to clarify the physical magnitude and dependences of these parameters.