The GW approximation (GWA) is an approximation made in order to calculate the self-energy of a many-body system of electrons.[1][2][3] The approximation is that the expansion of the self-energy Σ in terms of the single particle Green's functionG and the screened Coulomb interaction W (in units of )
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 GWA.
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 GWA 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 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. (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
The average electron-electron repulsion from the bare (unscreened) Coulomb interaction scales as (simply the inverse of the typical separation)
Typical wave vectors scale as (again typical inverse separation)
Hence a typical screening value is
The screened Coulomb interaction is
Thus for the bare Coulomb interaction, the ratio of Coulomb to kinetic energy is of order 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 to the kinetic energy is greatly reduced by the screening and is of order which is well behaved and smaller than unity even for large : the screened interaction is much weaker and is more likely to give a rapidly converging perturbative series.
The key publications concerning the application of the GW approximation 2019-02-04 at the Wayback Machine
Picture of Lars Hedin, inventor of GW
GW100 - Benchmarking the GW approach for molecules.
References
^Hedin, Lars (1965). "New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem". Phys. Rev. 139 (3A): A796–A823. Bibcode:1965PhRv..139..796H. doi:10.1103/PhysRev.139.A796. S2CID 73720887.
^Aulbur, Wilfried G.; Jönsson, Lars; Wilkins, John W. (2000). "Quasiparticle Calculations in Solids". Solid State Physics. 54: 1–218. doi:10.1016/S0081-1947(08)60248-9. ISBN9780126077544. ISSN 0081-1947.
^Aryasetiawan, F; Gunnarsson, O (1998). "The GW method". Reports on Progress in Physics. 61 (3): 237–312. arXiv:cond-mat/9712013. Bibcode:1998RPPh...61..237A. doi:10.1088/0034-4885/61/3/002. ISSN 0034-4885. S2CID 119000468.
Further reading
Electron Correlation in the Solid State, Norman H. March (editor), World Scientific Publishing Company
Aryasetiawan, Ferdi. "Correlation effects in solids from first principles" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
August 21, 2023
approximation, approximation, made, order, calculate, self, energy, many, body, system, electrons, approximation, that, expansion, self, energy, terms, single, particle, green, function, screened, coulomb, interaction, units, displaystyle, hbar, displaystyle, . The GW approximation GWA is an approximation made in order to calculate the self energy of a many body system of electrons 1 2 3 The approximation is that the expansion of the self energy S in terms of the single particle Green s function G and the screened Coulomb interaction W in units of ℏ 1 displaystyle hbar 1 S i G W G W G W G displaystyle Sigma iGW GWGWG cdots can be truncated after the first term S i G W displaystyle Sigma approx iGW 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 Contents 1 Theory 2 Software implementing the GW approximation 3 Sources 4 References 5 Further readingTheory EditThe 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 S 1 2 i G 1 2 W 1 2 d 3 d 4 G 1 3 G 3 4 G 4 2 W 1 4 W 3 2 displaystyle 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 S 1 2 i G 1 2 W 1 2 displaystyle 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 displaystyle 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 displaystyle 1 r s 2 The average electron electron repulsion from the bare unscreened Coulomb interaction scales as 1 r s displaystyle 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 displaystyle q isϵ q 1 l 2 q 2 displaystyle epsilon q 1 lambda 2 q 2 where l displaystyle lambda is the screening wave number that scales as r s 1 2 displaystyle r s 1 2 Typical wave vectors q displaystyle q scale as 1 r s displaystyle 1 r s again typical inverse separation Hence a typical screening value is ϵ 1 r s displaystyle epsilon sim 1 r s The screened Coulomb interaction is W q V q ϵ q displaystyle W q V q epsilon q Thus for the bare Coulomb interaction the ratio of Coulomb to kinetic energy is of order r s displaystyle 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 displaystyle W to the kinetic energy is greatly reduced by the screening and is of order r s 1 r s displaystyle r s 1 r s which is well behaved and smaller than unity even for large r s displaystyle r s the screened interaction is much weaker and is more likely to give a rapidly converging perturbative series Software implementing the GW approximation EditABINIT plane wave pseudopotential method BerkeleyGW plane wave pseudopotential method CP2K Gaussian based low scaling all electron and pseudopotential method 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 GPAW Molgw small gaussian basis code PySCF Quantum ESPRESSO Wannier function pseudopotential method Questaal Full Potential FP LMTO method SaX Archived 2009 02 03 at the Wayback Machine 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 methodSources EditThe key publications concerning the application of the GW approximation Archived 2019 02 04 at the Wayback Machine Picture of Lars Hedin inventor of GW GW100 Benchmarking the GW approach for molecules References Edit Hedin Lars 1965 New Method for Calculating the One Particle Green s Function with Application to the Electron Gas Problem Phys Rev 139 3A A796 A823 Bibcode 1965PhRv 139 796H doi 10 1103 PhysRev 139 A796 S2CID 73720887 Aulbur Wilfried G Jonsson Lars Wilkins John W 2000 Quasiparticle Calculations in Solids Solid State Physics 54 1 218 doi 10 1016 S0081 1947 08 60248 9 ISBN 9780126077544 ISSN 0081 1947 Aryasetiawan F Gunnarsson O 1998 The GW method Reports on Progress in Physics 61 3 237 312 arXiv cond mat 9712013 Bibcode 1998RPPh 61 237A doi 10 1088 0034 4885 61 3 002 ISSN 0034 4885 S2CID 119000468 Further reading EditElectron Correlation in the Solid State Norman H March editor World Scientific Publishing Company Aryasetiawan Ferdi Correlation effects in solids from first principles PDF a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help Retrieved from https en wikipedia org w index php title GW approximation amp oldid 1118959345, wikipedia, wiki, book, books, library,
