New Method for Calculating the OneParticle Green's Function with Application to the ElectronGas Problem
Abstract
A set of successively more accurate selfconsistent equations for the oneelectron Green's function have been derived. They correspond to an expansion in a screened potential rather than the bare Coulomb potential. The first equation is adequate for many purposes. Each equation follows from the demand that a corresponding expression for the total energy be stationary with respect to variations in the Green's function. The main information to be obtained, besides the total energy, is oneparticlelike excitation spectra, i.e., spectra characterized by the quantum numbers of a single particle. This includes the lowexcitation spectra in metals as well as configurations in atoms, molecules, and solids with one electron outside or one electron missing from a closedshell structure. In the latter cases we obtain an approximate description by a modified HartreeFock equation involving a "Coulomb hole" and a static screened potential in the exchange term. As an example, spectra of some atoms are discussed. To investigate the convergence of successive approximations for the Green's function, extensive calculations have been made for the electron gas at a range of metallic densities. The results are expressed in terms of quasiparticle energies E(k) and quasiparticle interactions f(k, k'). The very first approximation gives a good value for the magnitude of E(k). To estimate the derivative of E(k) we need both the first and the secondorder terms. The derivative, and thus the specific heat, is found to differ from the freeparticle value by only a few percent. Our correction to the specific heat keeps the same sign down to the lowest alkalimetal densities, and is smaller than those obtained recently by Silverstein and by Rice. Our results for the paramagnetic susceptibility are unreliable in the alkalimetaldensity region owing to poor convergence of the expansion for f. Besides the proof of a modified LuttingerWardKlein variational principle and a related selfconsistency idea, there is not much new in principle in this paper. The emphasis is on the development of a numerically manageable approximation scheme.
 Publication:

Physical Review
 Pub Date:
 August 1965
 DOI:
 10.1103/PhysRev.139.A796
 Bibcode:
 1965PhRv..139..796H