Variational Calculation of the Image Potential near a Metal Surface

We have calculated the interaction energy of a static point charge with a metal surface. The metal surface is treated in the jellium approximation with the positive ions replaced by a uniform background and the electrons characterized parametrically by their total charge density. The energy of the system is written as a functional of the total electron density, including that induced by the point charge external to the metal. The distribution of this induced charge is varied to minimize the energy and thus calculate the image potential. We find that we can write the image potential for $D>\mathrm{}2\mathrm{}$ Å as $-\frac{Q^2}{4(D-d)\mathrm{}}$, where $Q$ is the size of the point charge, $D$ the distance of the point charge from that jellium discontinuity, and $d$ an origin shift which is a function of the electron density at the metal surface. For smaller distances the potential no longer scales as $Q^2$ and saturation sets in.