GW approximation

The GW approximation (GWA) is an approximation made in order to calculate the self-energy of a many-body system of electrons.{{Cite journal|first=Lars |last=Hedin|s2cid=73720887|title=New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem|doi=10.1103/PhysRev.139.A796|journal= Phys. Rev.|volume=139|year=1965|pages=A796–A823|issue=3A|bibcode = 1965PhRv..139..796H |url=https://lup.lub.lu.se/record/8777834}}{{cite journal|last1=Aulbur|first1=Wilfried G.|last2=Jönsson|first2=Lars|last3=Wilkins|first3=John W.|title=Quasiparticle Calculations in Solids|journal=Solid State Physics|volume=54|year=2000|pages=1–218|issn=0081-1947|doi=10.1016/S0081-1947(08)60248-9|isbn=9780126077544}}{{cite journal|last1=Aryasetiawan|first1=F|last2=Gunnarsson|first2=O|title=The GW method|journal=Reports on Progress in Physics|volume=61|issue=3|year=1998|pages=237–312|issn=0034-4885|doi=10.1088/0034-4885/61/3/002|arxiv=cond-mat/9712013|bibcode = 1998RPPh...61..237A|s2cid=119000468}} 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 $\hbar=1$ )

: $\Sigma = iGW - GWGWG + \cdots$

can be truncated after the first term:

: $\Sigma \approx iG W$

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 GWA.

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

: $\Sigma(1,2) = iG(1,2)W(1^+,2) - \int d3 \int d4 \, G(1,3)G(3,4)G(4,2)W(1,4)W(3,2) + ...$

where the "+" superscript means the time index is shifted forward by an infinitesimal amount. The GWA 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 GWA with W replaced by the bare Coulomb yields nothing other than the Hartree–Fock exchange potential (self-energy). Therefore, loosely speaking, the GWA 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

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.{{Cite journal |last=Aryasetiawan |first=F |last2=Gunnarsson |first2=O |date=1998-03-01 |title=The GW method |url=https://iopscience.iop.org/article/10.1088/0034-4885/61/3/002 |journal=Reports on Progress in Physics |volume=61 |issue=3 |pages=237–312 |doi=10.1088/0034-4885/61/3/002 |issn=0034-4885|arxiv=cond-mat/9712013 }} Donald F. Dubois used this method to obtain results at for very small Wigner–Seitz radius or very large electron densities in 1959. The first full calculation using GWA was done by Lars Hedin in 1965.{{Cite book |last=March |first=Norman H. |url=https://www.google.fr/books/edition/Electron_Correlations_In_The_Solid_State/gM42DwAAQBAJ?hl=en&gbpv=1&dq=gw+approximation+history&pg=PT225&printsec=frontcover |title=Electron Correlations In The Solid State |date=1999-12-13 |publisher=World Scientific Publishing Company |isbn=978-1-911298-94-6 |language=en}} Hedin equations for GWA are named after him.{{Cite journal |last=Rehr |first=J J |date=2005-01-01 |title=Lars Hedin and the quest for a theory of excited states |url=https://iopscience.iop.org/article/10.1088/0031-8949/2005/T115/001 |journal=Physica Scripta |volume=T115 |pages=19–23 |doi=10.1088/0031-8949/2005/T115/001 |issn=0031-8949|url-access=subscription }}

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.

Software implementing the GW approximation

ABINIT - plane-wave pseudopotential method
ADF - Slater basis set method
[http://www.berkeleygw.org BerkeleyGW] - plane-wave pseudopotential method
CP2K - Gaussian-based low-scaling all-electron and pseudopotential method
[http://elk.sourceforge.net ELK] - full-potential (linearized) augmented plane-wave (FP-LAPW) method
FHI-aims - numeric atom-centered orbitals method
[http://perso.neel.cnrs.fr/xavier.blase/fiesta/ Fiesta] - Gaussian all-electron method
[https://web.archive.org/web/20160821032231/http://www.chem.pku.edu.cn/jianghgroup/codes/GAP2.html GAP] - an all-electron GW code based on augmented plane-waves, currently interfaced with WIEN2k
[https://wiki.fysik.dtu.dk/gpaw/ GPAW]
[https://green-phys.org GREEN] - fully self-consistent GW in Gaussian basis for molecules and solids
[https://github.com/bruneval/molgw Molgw] - small gaussian basis code
[https://real-space.org/downloads/ NanoGW] - real-space wave functions and Lanczos iterative methods
PySCF
[https://www.synopsys.com/manufacturing/quantumatk.html QuantumATK] - LCAO and PW methods.
Quantum ESPRESSO - Wannier-function pseudopotential method
[https://questaal.org/docs/code/gwoverview/ Questaal] - Full Potential (FP-LMTO) method
[http://www.sax-project.org/ SaX] {{Webarchive|url=https://web.archive.org/web/20090203223750/http://www.sax-project.org/ |date=2009-02-03 }} - plane-wave pseudopotential method
[http://www.flapw.de/master/spex Spex] - full-potential (linearized) augmented plane-wave (FP-LAPW) method
TURBOMOLE - Gaussian all-electron method
VASP - projector-augmented-wave (PAW) method
[https://west-code.org/ West] - large scale GW
YAMBO code - plane-wave pseudopotential method

Sources

[http://www.tcm.phy.cam.ac.uk/~mv230/gw/gw.html The key publications concerning the application of the GW approximation] {{Webarchive|url=https://web.archive.org/web/20190204172923/http://www.tcm.phy.cam.ac.uk/~mv230/gw/gw.html |date=2019-02-04 }}
[http://iopscience.iop.org/1402-4896/2005/T115/001 Picture of Lars Hedin, inventor of GW]
[https://gw100.wordpress.com/ GW100] - Benchmarking the GW approach for molecules.

GW approximation

Theory

History

Software implementing the GW approximation

Sources

References

Further reading