GW & quasiparticles

The GW approximation is the workhorse many-body method for quasiparticle spectra — the band positions and fundamental gaps that Kohn–Sham eigenvalues systematically miss. It is the first term in Hedin’s expansion of the electron self-energy in the screened interaction $W$, and it lives in the language of single-particle Green’s functions.

Self-energy $\Sigma =iGW$

The interacting Green’s function obeys Dyson’s equation $G=G_0+G_0\Sigma G$, with the self-energy $\Sigma$ carrying all many-body exchange and correlation. Hedin’s equations close exactly through the vertex $\Gamma$; the GW approximation sets $\Gamma =1$, giving

$$\Sigma (\mathbf{r},{\mathbf{r}}^{\prime };\omega )=\frac{i}{2\pi }\int d{\omega }^{\prime }{e}^{i{\omega }^{\prime }\eta }G(\mathbf{r},{\mathbf{r}}^{\prime };\omega +{\omega }^{\prime })W(\mathbf{r},{\mathbf{r}}^{\prime };{\omega }^{\prime }),$$

i.e. $\Sigma =iGW$ — a screened-exchange-plus-Coulomb-hole self-energy. The screened interaction is $W={\epsilon }^{-1}v$ with $v$ the bare Coulomb interaction and $\epsilon$ the dielectric function, computed in the random-phase approximation from the polarizability $P=-iGG$. The physical content over Hartree–Fock (which is $\Sigma =iGv$, bare exchange) is exactly the dynamical screening of the exchange hole — the dynamic-correlation piece of the spectrum.

The quasiparticle energies are the poles of $G$, i.e. the solutions of

$${\epsilon }_{n\mathbf{k}}^{\mathrm{QP}}={\epsilon }_{n\mathbf{k}}^{\mathrm{KS}}+Z_{n\mathbf{k}}⟨{\psi }_{n\mathbf{k}}|\Sigma ({\epsilon }_{n\mathbf{k}}^{\mathrm{KS}})-V_{\mathrm{XC}}|{\psi }_{n\mathbf{k}}⟩,$$

with the renormalization (quasiparticle weight) $Z_{n\mathbf{k}}=(1-{\partial }_{\omega }\mathrm{Re}\Sigma {)}^{-1}<1$ accounting for spectral weight transferred to incoherent satellites.

$G_0{W}_0$ vs self-consistency

$G_0{W}_0$ evaluates $\Sigma$ once, perturbatively, on a fixed starting point $\{G_0,W_0\}$ taken from a prior DFT (or HF) calculation, and applies the correction above as a one-shot shift. It is cheap and accurate for many $sp$ semiconductors, but it is starting-point dependent: the result inherits the wavefunctions and the screening of the underlying functional, so $G_0{W}_0$@PBE, $G_0{W}_0$@HSE, and $G_0{W}_0$@HF differ — sometimes substantially for localized states and $d$/$f$ systems, where PBE’s delocalization error contaminates both $G_0$ and the screening in $W_0$.

Removing the dependence requires self-consistency:

• Eigenvalue-self-consistent GW ($\mathrm{ev}$-$GW$) updates the energies in $G$ (and optionally $W$) to convergence, keeping the DFT wavefunctions — cheap, removes the energy starting-point dependence but not the orbital one.
• Quasiparticle self-consistent GW (QSGW) constructs an optimal static non-local exchange–correlation potential whose eigenstates best reproduce the GW quasiparticles, iterating both energies and wavefunctions to a starting-point-independent fixed point. More expensive, and tends to slightly overestimate gaps because the RPA $W$ is under-screened — it omits the electron–hole (excitonic) vertex that would enhance the screening — often corrected pragmatically by reintroducing an electron–hole vertex in $W$ (or the empirical $0.8\Sigma$ scaling).

Quasiparticle gaps and the derivative discontinuity

The reason Kohn–Sham gaps are not the fundamental gap is the derivative discontinuity ${\Delta }_{xc}$ of the exact functional: the true gap $E_g=I-A=E_g^{\mathrm{KS}}+{\Delta }_{xc}$, and even the exact KS gap misses ${\Delta }_{xc}$, which semilocal functionals set to zero. GW supplies precisely this missing piece — the self-energy is non-local and frequency-dependent, exactly the structure $V_{\mathrm{XC}}$ lacks — so the GW gap is the quasiparticle (photoemission/inverse-photoemission) gap, the physically correct $I-A$, not an optical gap (excitonic binding requires solving the Bethe–Salpeter equation on top). This is the same exact condition that the delocalization-error discussion frames as restoring piecewise linearity, now seen in the spectrum: the integer-$N$ discontinuity is the gap.

DMFT — the local-correlation neighbour (context)

GW captures dynamic, non-local but weak-coupling correlation through RPA screening; it does not capture strong, local correlation — Mott physics, multiplets, heavy-fermion behaviour — because the $\Gamma =1$ (RPA) closure misses the local vertex. That regime is the province of dynamical mean-field theory (DMFT), which solves a local interacting impurity problem with a full frequency-dependent self-energy $\Sigma (\omega )$ but neglects its $\mathbf{k}$-dependence — complementary to GW’s $\mathbf{k}$-dependent-but-static-vertex content. The two are unified in GW+DMFT, with GW supplying the non-local screened exchange and DMFT the local correlations. Here DMFT is context only: the self-energy / spectral-function machinery is developed in Green’s functions & LKAG, and there is no standalone DMFT note in this map by design.

Prerequisites

Green’s functions & LKAG · Hartree–Fock · the RPA dielectric function · Kohn–Sham DFT as the GW starting point

Builds toward: the beyond-DFT ladder · exchange tensors from DFT

Key references

• The GW approximation — L. Hedin, Phys. Rev. 139, A796 (1965); M. S. Hybertsen & S. G. Louie, PRB 34, 5390 (1986); review: D. Golze, M. Dvorak & P. Rinke, Front. Chem. 7, 377 (2019).
• Self-consistency / QSGW — M. van Schilfgaarde, T. Kotani & S. Faleev, PRL 96, 226402 (2006); M. Shishkin & G. Kresse, PRB 75, 235102 (2007).
• Quasiparticle gap & derivative discontinuity — J. P. Perdew & M. Levy, PRL 51, 1884 (1983); L. J. Sham & M. Schlüter, PRL 51, 1888 (1983); optical gap / BSE: M. Rohlfing & S. G. Louie, PRB 62, 4927 (2000).
• GW+DMFT — S. Biermann, F. Aryasetiawan & A. Georges, PRL 90, 086402 (2003).