Electron-electron interaction in simple metals

We calculate $V_{\mathrm{ee}}$, the effective screened interaction between two electrons in a simple metal. The result is derived by a self-consistent perturbation theory based on Hartree-Fock theory. Exchange and correlation are treated in local approximation. We consider the lattice to be a smooth but elastically deformable, charged medium in which the electrons move. Our results is a generalization of the Fröhlich interaction to include exchange and correlation. The effective interaction is spin dependent and may be written $V_{\mathrm{ee}}(q,\omega )=V_0(q,\omega )-J(q,\omega ){\overrightarrow{\sigma }}_1·{\overrightarrow{\sigma }}_2$. Explicit expressions for $V_0$ and $J$ are given in terms of the lattice stiffness and the same local exchange and correlation potentials of the electron gas which are needed to calculate the dielectric function and the spin susceptibility.