Wikipedia / GW approximation
Wikipedia / GW approximation
The GW approximation is a method used to calculate the self-energy of a many-body system of electrons. 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
can be truncated after the first term:
In other words, the self-energy is expanded in a formal Taylor series in powers of the screened interaction W and the lowest order term is kept in the expansion in GW approximation.
Theory
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
where the "+" superscript means the time index is shifted forward by an infinitesimal amount. The GW approximation is then
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. (We only present a scaling argument and will not compute numerical prefactors that are order unity.) Here are the key steps:
The average electron-electron repulsion from the bare (unscreened (simply the inverse of the typical separation)
The electron gas dielectric function in the simplest Thomas–Fermi screening model is
(again typical inverse separation)
: the screened interaction is much weaker and is more likely to give a rapidly converging perturbative series.
History
The first GW calculation with the Hartree–Fock method was performed in 1958 by John Quinn and Richard Allan Ferrell, but with many approximations and using a limited approach. Donald F. Dubois then used this method to obtain results for a very small Wigner–Seitz radius or very large electron densities in 1959. The first full calculation using GW was done by Lars Hedin in 1965; Hedin's equations for the GW method are named after him.
With the advancement of computational resources, it became possible to study real materials using GW in the 1980s, with the works of Mark S. Hybertsen and Steven Gwon Sheng Louie.
Software
ABINIT - plane-wave pseudopotential method
ADF - Slater basis set method
BerkeleyGW - plane-wave pseudopotential method
CP2K - Gaussian-based low-scaling all-electron and pseudopotential method
ComDMFT - full-potential linearized augmented plane-wave (FP-LAPW) method with optionally additional DMFT functionality support, whereas the GW part was originally developed as LqsgwFlapw
ELK - full-potential (linearized) augmented plane-wave (FP-LAPW) method
FHI-aims - numeric atom-centered orbitals method
Fiesta - Gaussian all-electron method
GAP - an all-electron GW code based on augmented plane-waves, currently interfaced with WIEN2k
GREEN - fully self-consistent GW in Gaussian basis for molecules and solids with optionally SEET support
Molgw - small gaussian basis code
momentGW - gaussian basis code with moment decomposition of the Green function
NanoGW - real-space wave functions and Lanczos iterative methods
QuantumATK - LCAO and PW methods.
Quantum ESPRESSO - Wannier-function pseudopotential method
Questaal - Full Potential (FP-LMTO) method
SaX - plane-wave pseudopotential method
Spex - full-potential (linearized) augmented plane-wave (FP-LAPW) method
TURBOMOLE - Gaussian all-electron method
VASP - projector-augmented-wave (PAW) method
West - large scale GW
YAMBO code - plane-wave pseudopotential method
Sources
GW100 - Benchmarking the GW approach for molecules.