Density-functional calculation of the static electronic polarizability of a small metal sphere

Abstract

The response of a small metal sphere to a uniform electrostatic field is calculated within density-functional theory. The gradient-expansion approximation is used for the kinetic-energy functional, while exchange and correlation are treated in the local-density approximation. A jellium model is assumed and the electron density is determined by a variational method. Calculations were performed for spheres containing between 5 and 8000 electrons. The applied field was found to induce a substantial amount of charge outside the sphere. This causes the electronic polarizability $\ensuremath{\alpha}$ to be larger than the classical value ${R}^{3}$, where $R$ is the radius of the sphere. Our calculations give $\ensuremath{\alpha}={(R+\ensuremath{\delta})}^{3}$, where $R+\ensuremath{\delta}$ plays the role of an effective radius, and $\ensuremath{\delta}$ is approximately 2.0 a.u. for the electron-radius parameter ${r}_{s}=2$ and is approximately 1.0 a.u. for ${r}_{s}=4$. Our calculated values of $\ensuremath{\delta}$ are nearly independent of $R$. As $R$ approaches infinity, $\ensuremath{\delta}$ approaches the image-plane distance for the problem of a flat surface. It is shown that there is a force sum rule that is not well satisfied by previous calculations, and is approximately satisfied by most of our calculations.