The Image and Van der Waals Forces at a Metallic Surface

Abstract

Calculations based on a model which takes the structure of the metal into account indicate that the force on an electron outside the surface of a metal is just the classical image force, but that the van der Waals force on a neutral molecule may be smaller than that given by a semi-classical calculation based on the method of images. An approximate value for the van der Waals energy between two systems $A$ and $B$ is obtained by calculating the energy of $B$ in the field of $A$, supposing that the electrons of $A$ are in fixed positions. The result is then averaged over the coordinates of the electrons of $A$, giving an energy ${W}_{A}$. Systems $A$ and $B$ may be reversed in the above calculation, and an energy ${W}_{B}$ be found. The true van der Waals energy is given roughly by $\frac{{W}_{A}{W}_{B}}{({W}_{A}+{W}_{B})}$. This method gives a new approximate formula for the van der Waals interaction between two molecules, which reduces to a well-known result when further approximations are made. If system $A$ is the molecule and system $B$ a metallic surface, the energy ${W}_{A}$ is that found by the image method. The energy ${W}_{B}$ is also evaluated, and is shown to be of the same order of magnitude as ${W}_{A}$ for ordinary electron densities in the metal. Thus the true energy of interaction may be considerably smaller than that given by the image method.