[1][2][3] The approximation is that the expansion of the self-energy Σ in terms of the single particle Green's function G and the screened Coulomb interaction W (in units of
More precisely, if we label an electron coordinate with its position, spin, and time and bundle all three into a composite index (the numbers 1, 2, etc.
), we have where the "+" superscript means the time index is shifted forward by an infinitesimal amount.
Therefore, loosely speaking, the GWA represents a type of dynamically screened Hartree–Fock self-energy.
This is because the screening of the medium reduces the effective strength of the Coulomb interaction: for example, if one places an electron at some position in a material and asks what the potential is at some other position in the material, the value is smaller than given by the bare Coulomb interaction (inverse distance between the points) because the other electrons in the medium polarize (move or distort their electronic states) so as to screen the electric field.
Therefore, W is a smaller quantity than the bare Coulomb interaction so that a series in W should have higher hopes of converging quickly.
(We only present a scaling argument and will not compute numerical prefactors that are order unity.)
which is of order 2-5 for a typical metal and not small at all: in other words, the bare Coulomb interaction is rather strong and makes for a poor perturbative expansion.
The first GWA calculation for Hartree–Fock method was in 1958 by John Quinn and Richard Allan Ferrell but with many approximation and limited approach.
[4] Donald F. Dubois used this method to obtain results at for very small Wigner–Seitz radius or very large electron densities in 1959.
[6] With the advanced of computational resources, real materials were first studied using GWA in the 1980s, with the works of Mark S. Hybertsen and Steven Gwon Sheng Louie.