van der Waals interaction between an atom and a surface at finite separations

A general expression for the van der Waals energy of an atom near a planar surface is derived in terms of the imaginary part of the retarded surface response function Im[g(q,ω]. This expression allows a direct calculation of the asymptotic coefficient $C_3$ of the van der Waals energy using response functions determined from optical data. For simple metal surfaces we introduce a dispersed surface plasmon-pole approximation to the response function. The van der Waals energy calculated with this response function reproduces the asymptotic coefficient $C_3$ and effective image-plane position $Z_{\mathrm{vdW}}$ calculated by Zaremba and Kohn, and is also finite at all atom-surface separations unlike the usual asymptotic series approach. Using this van der Waals energy we calculate the binding energies of helium on simple metal surfaces. We also take into account the velocity of the atom parallel to the surface, giving a velocity-dependent contribution to the van der Waals and, at high velocities, an imaginary part due to real excitations.