Electron Self-Energy Approach to Correlation in a Degenerate Electron Gas

Abstract

A new method of computing the correlation energy of a degenerate electron gas is presented in which the interactions are studied by considering the self-energy of a lone particle impurity in the system. The self-energy results as in quantum electro-dynamics from the action of the proper field set up by the charged particle back on itself; the Feynman space-time formulation of quantum mechanics is employed in the self-energy calculation, which is carried out along lines already laid out by Lindhard. The Feynman propagator, which takes the particle from one point in space-time to another, is derived. A slight but essential change in the particle propagator is needed to allow for exchange effects when the particle impurity is an additional electron in the degenerate electron gas. This gives the electron gas a dual role: it acts as a dielectric medium which can be polarized and also as a vacuum from which electron-hole pairs can be created and undergo exchange with incident electrons. The polarization propagator for the effective potential set up by the impurity in the electron gas, considered as a dielectric medium, is derived heuristically in the text from Lindhard's dynamic dielectric constant and more rigorously in an Appendix from the momentum-exciton model. The electron self-energy is a Feynman integral involving the particle and polarization propagators and defines an optical potential which is found to have both real and imaginary parts. For momenta less than the Fermi momentum, it is shown in a second Appendix that the optical potential is simply the negative of the self-energy of a hole in the Fermi sea. The imaginary part of the optical potential for an electron of momentum $p$ is proportional to ${(\frac{p}{{p}_{0}}\ensuremath{-}1)}^{2}$ (where ${p}_{0}$ is the Fermi momentum), and gives rise to damping. Thus the concept of a one-electron state is only valid for small excitation energies and breaks down when the electron is appreciably far removed from the Fermi surface. The mean free path for high electron density is given (in units of $\frac{\ensuremath{\hbar}}{{p}_{0}}$) by $3.98{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}$ times the above function of momentum. (${r}_{s}$ is the unit-sphere radius in Bohr radii.) The derivative of the real part of the optical potential with respect to momentum, evaluated at the Fermi surface, gives a correction to the specific heat in agreement with Gell-Mann. The value of the optical potential itself is related by Seitz's theorem to the derivative of the correlation energy with respect to density. Integration over density yields an expression for the ground state energy which agrees with the results of other investigators. Finally a brief discussion is given of Bethe's theorem, which directly relates the optical potential to the ground state correlation energy per particle. Although Bethe's theorem is not valid for the idealized electron gas with uniform positive background, it does apply to actual metals in equilibrium.