The above formulae are schematic in nature and show the overall idea of the approximation. 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 : \Sigma(1,2) = iG(1,2)W(1^+,2) - \int d3 \int d4 \, G(1,3)G(3,4)G(4,2)W(1,4)W(3,2) + ... where the "+" superscript means the time index is shifted forward by an infinitesimal amount. The
GW approximation is then : \Sigma(1,2) \approx iG(1,2)W(1^+,2) To put this in context, if one replaces
W by the bare Coulomb interaction (i.e. the usual 1/r interaction), one generates the standard perturbative series for the self-energy found in most many-body textbooks. The
GW approximation with
W replaced by the bare Coulomb yields nothing other than the
Hartree–Fock exchange potential (self-energy). Therefore, loosely speaking, the
GW approximation represents a type of dynamically screened Hartree–Fock self-energy. In a solid state system, the series for the self-energy in terms of
W should converge much faster than the traditional series in the bare Coulomb interaction. 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. To see the more rapid convergence, we can consider the simplest example involving the
homogeneous or uniform electron gas which is characterized by an electron density or equivalently the average electron-electron separation or
Wigner–Seitz radius r_s . (We only present a scaling argument and will not compute numerical prefactors that are order unity.) Here are the key steps: • The kinetic energy of an electron scales as 1/r_s^2 • The average electron-electron repulsion from the bare (
unscreened) Coulomb interaction scales as 1/r_s (simply the inverse of the typical separation) • The electron gas
dielectric function in the simplest
Thomas–Fermi screening model for a wave vector q is : \epsilon(q) = 1 + \lambda^2/q^2 where \lambda is the screening wave number that scales as r_s^{-1/2} • Typical wave vectors q scale as 1/r_s (again typical inverse separation) • Hence a typical screening value is \epsilon \sim 1 + r_s • The screened Coulomb interaction is W(q) = V(q)/\epsilon(q) Thus for the bare Coulomb interaction, the ratio of Coulomb to kinetic energy is of order r_s 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. On the other hand, the ratio of a typical W to the kinetic energy is greatly reduced by the screening and is of order r_s/(1+r_s) which is well behaved and smaller than unity even for large r_s : the screened interaction is much weaker and is more likely to give a rapidly converging perturbative series. == History ==