In previous publications (especially by Hohenberg, Kohn, and Sham), a theory of the ground state of an inhomogeneous interacting electron gas was developed, in which the electronic density $n\left(\mathrm{r}\right)$ played a dominant role. The present paper extends this approach to the one-particle Green's function and physical properties related to it, such as single-particle-like excitations and, in the case of metals, the Fermi surface. The Dyson mass operator $\Sigma$ is studied as a function of its spatial arguments and as a functional of $n\left(\mathrm{r}\right)$, and, in both senses, it is found to have important short-range properties. An approximation for $\Sigma$, which is exact for systems of slowly varying density, is proposed. This leads to simple, explicit, Schrödinger-like equations for the single-particle-like excitations, whose solution determines their energies and lifetimes. In particular, we show how to apply this procedure to metals.

• Received 20 December 1965

DOI:https://doi.org/10.1103/PhysRev.145.561