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GW approximation
GW approximation
GW approximation
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Electronic structure methods
Valence bond theory
Coulson–Fischer theory
Generalized valence bond
Modern valence bond theory
Molecular orbital theory
Hartree–Fock method
Semi-empirical quantum chemistry methods
Møller–Plesset perturbation theory
Configuration interaction
Coupled cluster
Multi-configurational self-consistent field
Quantum chemistry composite methods
Quantum Monte Carlo
Density functional theory
Time-dependent density functional theory
Thomas–Fermi model
Orbital-free density functional theory
Adiabatic connection fluctuation dissipation theorem
Görling-Levy pertubation theory
Optimized effective potential method
Linearized augmented-plane-wave method
Projector augmented wave method
Electronic band structure
Nearly free electron model
Tight binding
Muffin-tin approximation
k·p perturbation theory
Empty lattice approximation
GW approximation
Korringa–Kohn–Rostoker method

The GW approximation is a method used 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 function G 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 GW approximation.

Theory

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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 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)
  • The electron gas dielectric function in the simplest Thomas–Fermi screening model for a wave vector is

where is the screening wave number that scales as

  • 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.

History

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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.[4] Donald F. Dubois then used this method to obtain results for a very small Wigner–Seitz radius or very large electron densities in 1959.[4] The first full calculation using GW was done by Lars Hedin in 1965;[4][5] Hedin's equations for the GW method are named after him.[6]

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.[4]

Software

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  • 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
  • GPAW
  • 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
  • PySCF
  • QuantumATK - LCAO and PW methods.
  • 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 method

Sources

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

GW approximation

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from Grokipedia
The GW approximation is a cornerstone method in many-body for calculating the electronic in interacting many-electron systems, approximating the self-energy Σ\Sigma as the product of the single-particle GG and the dynamically screened interaction WW. Introduced by Lars Hedin in through a set of self-consistent equations for the , it provides a systematic framework to go beyond mean-field approximations like Hartree-Fock by incorporating screening effects and electron correlations via perturbation expansion. This approach yields energies and lifetimes, essential for understanding electronic excitations in solids, molecules, and nanostructures. The GW method excels in predicting accurate band structures and excitation spectra, particularly addressing the underestimation of band gaps in (DFT) calculations using local or semilocal approximations. For instance, in semiconductors like and , GW computations yield fundamental gaps close to experimental values (e.g., 1.17 eV for Si versus 1.24–1.29 eV from GW), outperforming DFT's typical errors of 0.5–1 eV. Its applications span insulators, transition metals (e.g., and ), surfaces (e.g., Si(100)), clusters, fullerenes like C60_{60}, and more recently, two-dimensional materials and molecular systems for ionization potentials and electron affinities. In , GW variants like G0_0W0_0@HF demonstrate high accuracy for weakly and strongly correlated molecules, with mean absolute errors for ionization potentials around 0.3 eV on the GW100 benchmark set. Despite its successes, the GW approximation has limitations, including computational cost scaling as O(N4)O(N^4) (where NN is the system size) in naive implementations, often mitigated by approximations like the plasmon-pole model or partial self-consistency. It neglects vertex corrections in the (setting the vertex function Γ≈1\Gamma \approx 1), which can lead to overestimation of self-energy magnitudes by 10–20% in some cases, and struggles with strongly correlated systems exhibiting structures or Mott transitions. Extensions like self-consistent GW (SCGW), GW+DMFT, or inclusion of vertex effects aim to address these, but the perturbative G0_0W0_0 variant—using a starting DFT or Hartree-Fock —remains the most practical and widely used. Ongoing developments focus on efficient algorithms for large systems and integration with for screening.

Fundamentals

Definition

The GW approximation is a method within many-body used to compute energies and wavefunctions for the electronic structure of solids and molecules, going beyond mean-field approaches such as Hartree-Fock. It approximates the operator Σ\Sigma, which accounts for electron-electron interactions, using the one-particle GG and the screened interaction WW, yielding Σ≈iGW\Sigma \approx iGW. This framework captures dynamic screening effects, where the Coulomb interaction is dressed by the response of the electron gas, providing a more accurate description of excitation energies compared to static mean-field treatments. The basic motivation for the GW approximation stems from the shortcomings of mean-field methods, which neglect many-body correlations and dynamic screening, often leading to underestimated band gaps in semiconductors and insulators. By incorporating these effects perturbatively, GW improves predictions for quasiparticle properties, such as ionization potentials and electron affinities, making it particularly valuable for materials where exchange-correlation interactions are significant. In the GW scheme, the self-energy is approximated as Σ≈iGW\Sigma \approx i G W, where the product denotes the convolution over space and time (or ) coordinates, and the vertex function Γ\Gamma—which accounts for higher-order vertex corrections—is set to unity for simplicity. The method is named after its key ingredients, GG (the ) and WW (the screened interaction), and was originally introduced by Lars Hedin in 1965 for the electron gas, with seminal applications to semiconductors following in the .

Relation to Density Functional Theory

The GW approximation is frequently integrated with (DFT) as a post-processing step to enhance the accuracy of electronic structure predictions, particularly for excited-state properties where standard DFT falls short. In the commonly used G0W0 variant, the non-interacting Green's function G0 is constructed from the Kohn-Sham orbitals and eigenvalues obtained from a DFT ground-state calculation, typically employing local or semi-local functionals like PBE or LDA; this hybrid approach leverages DFT's computational efficiency while incorporating many-body to compute the . A primary motivation for adopting GW atop DFT is to rectify the notorious underestimation of band gaps in semiconductors and insulators by approximate DFT exchange-correlation functionals, which arise from their neglect of derivative discontinuities and inadequate treatment of long-range exchange. For instance, in bulk silicon, the PBE functional predicts an indirect band gap of approximately 0.6 eV, while G0W0@PBE yields about 1.2 eV, aligning closely with the experimental value of 1.17 eV. This correction stems from the GW self-energy's inclusion of screened exchange and correlation effects, which shift quasiparticle levels more accurately than DFT's static potential. The standard computational workflow begins with a DFT calculation to obtain the Kohn-Sham wave functions and densities, which serve as the basis for evaluating the and screened interaction W in the random-phase approximation; the is then computed perturbatively using these inputs, followed by diagonalization of the Hamiltonian to update the energies. Relative to pure DFT, this GW@DFT strategy provides superior spectra by accounting for non-local, dynamical exchange-correlation beyond semi-local s, though at a higher computational cost scaling as O(N4) in implementations, where N is the .

Theoretical Framework

Green's Function

In many-body perturbation theory, the one-electron Green's function serves as the central quantity for describing the propagation of electrons in interacting systems. It is defined as the time-ordered propagator G(1,2;ω)G(1,2;\omega), where 1=(r,τ)1 = (\mathbf{r}, \tau) denotes space-time coordinates with r\mathbf{r} the position and τ\tau the imaginary time, and ω\omega the frequency. Physically, GG represents the probability amplitude for adding an electron at position r2\mathbf{r}_2 and time τ2\tau_2 and subsequently removing it at r1\mathbf{r}_1 and τ1\tau_1, or vice versa, thereby capturing single-particle excitations that change the particle number by one. The spectral representation provides a Lehman expansion of the Green's function in terms of quasiparticle states, linking it directly to the excitation spectrum: G(r,r′;ω)=∑nψn(r)ψn∗(r′)ω−εn+iηsgn⁡(εn−μ),G(\mathbf{r},\mathbf{r}';\omega) = \sum_n \frac{\psi_n(\mathbf{r}) \psi_n^*(\mathbf{r}')}{\omega - \varepsilon_n + i\eta \operatorname{sgn}(\varepsilon_n - \mu)}, where {ψn,εn}\{\psi_n, \varepsilon_n\} are the quasiparticle wavefunctions and energies, μ\mu is the , and η\eta is a positive ensuring . The poles of GG at ω=εn\omega = \varepsilon_n correspond to the quasiparticle energies, while the residues give the quasiparticle strengths, quantifying the extent to which the excitations resemble non-interacting particles. This representation is particularly useful for interpreting the effects of interactions on the electronic structure. In practical calculations, the time-ordered Green's function is often evaluated using the Matsubara formalism for finite-temperature systems, where imaginary frequencies iωni\omega_n (with ωn=(2n+1)π/β\omega_n = (2n+1)\pi/\beta for fermions, β=1/kBT\beta = 1/k_B T) replace real frequencies to avoid singularities and enable to the retarded form via techniques like the . The Matsubara approach facilitates the summation over thermal Matsubara frequencies in perturbation expansions, making it suitable for equilibrium properties at nonzero temperatures. Within the GW approximation, computations typically begin with the non-interacting Green's function G0G_0, constructed from Kohn-Sham orbitals and eigenvalues obtained via (DFT), which provides an efficient starting point for the perturbative . Subsequent iterations may update GG through Dyson's equation to incorporate interactions, though the single-shot G0W0G_0 W_0 variant often suffices for energies in solids. This strategy leverages DFT's accuracy for ground-state properties while correcting band gaps and dispersions via many-body effects.

Screened Coulomb Interaction W

The screened Coulomb interaction WW, a central quantity in the GW approximation, describes the effective electron-electron interaction in a many-body , incorporating dielectric screening effects from the surrounding electrons. It is defined in space, time, and frequency as W(r,r′;ω)=v(r,r′)+∫dr1dr2 v(r,r1)P(r1,r2;ω)v(r2,r′)W(\mathbf{r},\mathbf{r}';\omega) = v(\mathbf{r},\mathbf{r}') + \int d\mathbf{r}_1 d\mathbf{r}_2 \, v(\mathbf{r},\mathbf{r}_1) P(\mathbf{r}_1,\mathbf{r}_2;\omega) v(\mathbf{r}_2,\mathbf{r}'), where v(r,r′)=1/∣r−r′∣v(\mathbf{r},\mathbf{r}') = 1/|\mathbf{r} - \mathbf{r}'| is the bare potential and PP is the irreducible function representing the density response of the . This form arises from the Dyson equation for the screened interaction, truncating higher-order terms in the polarization expansion. In practice, the polarizability PP is often evaluated within the random phase approximation (RPA), where it is approximated as the independent-particle response P(r,r′;ω)≈−i∫dω′2πG(r,r′;ω+ω′)G(r′,r;ω′)P(\mathbf{r},\mathbf{r}';\omega) \approx -i \int \frac{d\omega'}{2\pi} G(\mathbf{r},\mathbf{r}';\omega + \omega') G(\mathbf{r}',\mathbf{r};\omega'), using the non-interacting Green's function G0G_0 and neglecting vertex corrections. This RPA level for PP simplifies computations while capturing the leading screening effects through ring diagrams in the perturbation expansion. The interaction WW is dynamically screened and frequency-dependent, reflecting the response of the electronic medium to perturbations at different scales. At high frequencies (ω→∞\omega \to \infty), screening diminishes, and W(ω)W(\omega) approaches the bare potential vv, as rapid oscillations prevent charge rearrangement. At low frequencies, W(ω)W(\omega) incorporates collective excitations such as plasmons in metals or excitonic effects tied to the bandgap in insulators, leading to enhanced screening and a softened interaction. In metallic systems, such as , the imaginary part of WW exhibits a characteristic plasma edge, manifesting as peaks around 20–30 eV corresponding to satellites that broaden and renormalize bands. In insulators like , screening is weaker and bandgap-dependent, with WW contributing to gap corrections through effects near atomic bonds, increasing the fundamental gap by approximately 0.7 eV relative to local-density approximations.

Self-Energy Operator

In the GW approximation, the self-energy operator Σ\Sigma encapsulates the effects of many-body electron-electron interactions and serves as an effective, non-local, and frequency-dependent potential in the description of quasiparticles. It is defined in the space-time representation as Σ(1,2)≈i G(1,2)[W](/page/W)(1+,2)\Sigma(1,2) \approx i \, G(1,2) [W](/page/W)(1^+,2), where 1=(r1,t1)1 = (\mathbf{r}_1, t_1) and 1+=(r1,t1+0+)1^+ = (\mathbf{r}_1, t_1 + 0^+) denotes an infinitesimal positive time shift to ensure ordering, GG is the one-particle , and [W](/page/W)[W](/page/W) is the screened interaction. This form arises from a perturbative expansion where the vertex function is approximated by the identity, reducing higher-order diagrams. The self-energy Σ\Sigma can be decomposed into an exchange part Σx\Sigma_x and a correlation part Σc\Sigma_c, such that Σ=Σx+Σc\Sigma = \Sigma_x + \Sigma_c. The exchange component Σx\Sigma_x is static and resembles the Fock operator but uses the bare Coulomb interaction vv, obtained by setting W→vW \to v in the GW expression, capturing the non-local exchange effects akin to Hartree-Fock theory. In contrast, the correlation part Σc\Sigma_c is dynamic and stems from the screening effects in W−vW - v, accounting for the frequency-dependent response of the electron gas to perturbations and including phenomena like plasmons. This enters the equation, which linearizes the Dyson equation for practical computation: [h0+Vxc+Re⁡Σ(ε)]ψ=εψ[h_0 + V_{xc} + \operatorname{Re} \Sigma(\varepsilon)] \psi = \varepsilon \psi, where h0h_0 is the non-interacting Hamiltonian (kinetic plus external potential), VxcV_{xc} is the exchange-correlation potential from , and the real part of Σ\Sigma at the ε\varepsilon corrects the single-particle eigenvalues. Due to the dependence of Σ\Sigma, the wave functions and are renormalized by the factor Z=[1−∂Re⁡Σ∂ω∣ω=ε]−1Z = \left[1 - \left. \frac{\partial \operatorname{Re} \Sigma}{\partial \omega} \right|_{\omega = \varepsilon} \right]^{-1}
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