New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem

1965/08/02 by Lars Hedin · 5,497 citations
Chemistry · Physics and Astronomy · #Advanced Chemical Physics Studies #Advanced Physical and Chemical Molecular Interactions #Atomic and Molecular Physics #Atomic physics #Coulomb #Electron #Fermi gas #Function (biology) #Green's function #Materials science #Physics #Quantum mechanics #Quasiparticle #Range (aeronautics) #Spectral line

paper · open access · doi:10.1103/physrev.139.a796

published in Physical Review 139(3A), A796-A823 (American Institute of Physics)

openalex publication_date 1965/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

A set of successively more accurate self-consistent equations for the one-electron 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 one-particle-like excitation spectra, i.e., spectra characterized by the quantum numbers of a single particle. This includes the low-excitation spectra in metals as well as configurations in atoms, molecules, and solids with one electron outside or one electron missing from a closed-shell structure. In the latter cases we obtain an approximate description by a modified Hartree-Fock 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^\ensuremath'). 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 second-order terms. The derivative, and thus the specific heat, is found to differ from the free-particle value by only a few percent. Our correction to the specific heat keeps the same sign down to the lowest alkali-metal densities, and is smaller than those obtained recently by Silverstein and by Rice. Our results for the paramagnetic susceptibility are unreliable in the alkali-metal-density region owing to poor convergence of the expansion for f. Besides the proof of a modified Luttinger-Ward-Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. The emphasis is on the development of a numerically manageable approximation scheme.

Citations

• On the Interaction of Electrons in Metals
• On the Constitution of Metallic Sodium

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• One-Particle Properties of an Inhomogeneous Interacting Electron Gas
• Quantum-Mechanical Analysis of Single-Particle Level Density
• Green's Function for Magnetically Incoherent Interacting Electrons in One Dimension
• Density-Functional Green Function Theory: Dynamical exchange-correlation field in lieu of self-energy
• Quasiparticle wavefunction and its equation of motion