Third order perturbation theory and lattice dynamics of simple metals

The dynamic matrix of simple metals is set up by evaluating the total electron energy to the third order in the electron-ion potential. In this way three body unpaired non central forces, arising from the off diagonal elements of the dielectric matrix are explicitly introduced.Since the usual on-Fermi-sphere approximation to a non local pseudopotential form factor gives a completely unreliable estimate of third order contributions and a full non local calculation is not feasible, we give a procedure to construct an average local potential, which disposes of most on the effects of non-locality and allows for a realistic contribution of the third-order corrections to lattice dynamics. We evaluate phonon frequencies and elastic constants for Li, Na, Al and Pb.