More Advanced Topics in the GW Approximation

 

The introductory page presented the central idea of the GW approximation:

$$ \Sigma \approx iGW.$$

Here, $G$ is the one-particle Green’s function and $W$ is the dynamically screened Coulomb interaction.

This page develops the deeper structure behind that expression, including Hedin’s equations, the vertex approximation, screening, self-consistency, diagrammatic content, and the limitations of standard GW.

The vertex function

The vertex function $\Gamma$ describes additional correlations between the propagating particle and the excitations of the surrounding system.

Setting $\Gamma=1$ means that these explicit vertex corrections are omitted.

This simplifies the self-energy from

$$\Sigma=iGW\Gamma$$

to

$$\Sigma=iGW.$$

Physically, the electron interacts with the screened field $W$, but more complicated correlated scattering processes are not included explicitly.

Examples of processes associated with vertex corrections include:

  • repeated electron-hole scattering;
  • ladder diagrams;
  • certain short-range correlation effects;
  • corrections to the polarization beyond the independent bubble approximation;
  • corrections that reduce self-screening errors.

Standard GW therefore retains an important class of dynamical screening effects while omitting other higher-order correlations.

Bare and screened interactions

The bare Coulomb interaction is denoted by $v$.

It describes the interaction between charges before the surrounding electrons respond.

The screened interaction satisfies 

$$W = v+vPW.$$

Iterating this equation gives

$$ W = v + vPv + vPvPv + vPvPvPv + \cdots.$$

This is an infinite resummation of polarization insertions.

Each factor of $P$ represents an additional response of the electron system to the electric field produced by a charge.

Thus, $W$ is not a single interaction event. It is the bare Coulomb interaction dressed by repeated polarization processes.

Dielectric interpretation of screening

The screened interaction may also be written in terms of the dielectric response:

$$W = \varepsilon^{-1}v.$$

Here,

$$\varepsilon = 1-vP$$

is the dielectric operator in compact notation.

Therefore,

$$W = \left(1-vP\right)^{-1}v.$$

Expanding the inverse gives

$$\left(1-vP\right)^{-1} = 1+vP+vPvP+\cdots,$$

which reproduces the repeated screening series.

In a real material, $\varepsilon$ is generally a matrix in reciprocal lattice vectors, bands, orbitals, or spatial coordinates. Screening is therefore nonlocal and frequency dependent.

Dynamic screening

The screened interaction depends on frequency:

$$W=W(\omega).$$

This means that the electronic environment does not respond instantaneously.

At different frequencies, the material may screen the Coulomb interaction differently.

Dynamic screening allows $W$ to describe coupling to collective electronic excitations such as plasmons.

Because $\Sigma(1,2)=iG(1,2)W(1,2)$ is a direct product in the space-time representation, it becomes a convolution over internal momentum and frequency after Fourier transformation. Symbolically, this is often abbreviated as $\Sigma=iGW$, but it should not be interpreted as the pointwise product $G(\omega)W(\omega)$.

This frequency dependence is responsible for quasiparticle energy shifts, finite lifetimes, and satellite structures in the spectral function.

Exchange and correlation parts of GW

The screened interaction can be separated as

$$W = v + \left(W-v\right).$$

Substituting this into the self-energy gives

$$\Sigma = iGv + iG\left(W-v\right).$$

The first term,

$$\Sigma_x = iGv,$$

has the structure of an exchange contribution.

The second term,

$$\Sigma_c = iG\left(W-v\right),$$

contains correlation effects associated with dynamical screening.

Thus, the GW self-energy is often written as

$$\Sigma = \Sigma_x + \Sigma_c.$$

This shows that GW does not neglect exchange. It contains an exchange-like term together with a dynamical correlation contribution.

Diagrammatic content of GW

The expression

$$\Sigma=iGW$$

contains more diagrams than the lowest-order exchange diagram because $W$ itself contains an infinite series:

$$W = v + vPv + vPvPv + \cdots.$$

Substituting this expansion into the self-energy gives schematically

$$\Sigma = iGv + iGvPv + iGvPvPv + \cdots.$$

The first term is the exchange-like contribution.

The following terms describe interaction with increasingly complex polarization processes.

This is why it is misleading to say that GW ignores all multiple scattering or repeated interaction processes. It does resum repeated polarization insertions inside $W$.

What it does not include are all possible vertex-corrected diagrams.

Why higher-order expansions become complicated 

Although the GW equations can formally be expanded in powers of the bare Coulomb interaction $v$, the number of terms grows rapidly with perturbative order.

For a fixed self-energy, the Dyson equation generates the series

$$G = G_0 + G_0\Sigma G_0 + G_0\Sigma G_0\Sigma G_0 + \cdots,$$

while the screened interaction can be expanded as

$$W = v + vPv + vPvPv + \cdots.$$

In the GW approximation, however,

$$\Sigma=iGW$$

and

$$P=-iGG,$$

so both $\Sigma$ and $P$ must themselves be expanded recursively. Consequently, contributions of a given order in $v$ arise from several different substitutions and correspond to several distinct diagram topologies.

The compact notation also hides integrations over internal space-time variables. For example,

$$(G_0\Sigma G_0)(1,2) = \int d3 d4 G_0(1,3) \Sigma(3,4) G_0(4,2).$$

At higher orders, one must consistently account for internal integrations, time or frequency arguments, fermionic minus signs, symmetry factors, and the distinction between the reference propagator $G_0$ and the interacting propagator $G$.

Care is also required when combining explicit perturbative diagrams with partially or fully self-consistent Green’s functions, because some classes of diagrams may already be implicitly resummed.

For these reasons, explicit high-order algebraic expansions are mainly useful for identifying the diagrammatic content of the approximation. In practical GW calculations, the coupled equations are usually evaluated numerically rather than expanded term by term to high order.

One-shot $G_0W_0$

The simplest practical form is one-shot $G_0W_0$.

A reference Green’s function is constructed from a mean-field calculation:

$$G_0.$$

The corresponding polarization is

$$P_0 = -iG_0G_0.$$

The screened interaction is then

$$W_0 = v + vP_0W_0.$$

Finally, the self-energy is evaluated as

$$\Sigma^{G_0W_0} = iG_0W_0.$$

The quasiparticle correction is calculated once, without updating the full cycle.

The result may depend significantly on the chosen starting point, such as the exchange-correlation functional used in the initial DFT calculation.

Partially self-consistent GW

Several intermediate forms of self-consistency are possible.

For example, one may update quasiparticle energies in G while leaving the wave functions or W fixed.

A common schematic form is

$$G_nW_0,$$

where $G$ is updated iteratively but $W_0$ remains fixed.

Other approaches update both quasiparticle energies and screening while keeping some parts of the reference system unchanged.

These methods attempt to reduce starting-point dependence without the full computational cost of self-consistent GW.

Self-consistent GW

In fully self-consistent GW, the cycle

$$G \rightarrow P \rightarrow W \rightarrow \Sigma \rightarrow G $$

is repeated until convergence.

That is,

$$G^{(n)} \longrightarrow P^{(n)} \longrightarrow W^{(n)} \longrightarrow \Sigma^{(n)} \longrightarrow G^{(n+1)}.$$

The final $G$ is then consistent with the $P$, $W$, and $\Sigma$ obtained from it.

Full self-consistency removes much of the ambiguity associated with the starting point, but it is computationally more expensive and does not automatically compensate for neglected vertex corrections.

Quasiparticle equation in GW

In practical calculations, the GW self-energy is used to correct reference one-particle energies.

For a Kohn–Sham starting point, the quasiparticle equation is

$$E_{n\mathbf k}^{\mathrm{QP}} = \varepsilon_{n\mathbf k}^{\mathrm{KS}} + \left\langle n\mathbf k \left|
\Sigma\left(E_{n\mathbf k}^{\mathrm{QP}}\right) - V_{\mathrm{xc}} \right| n\mathbf k \right\rangle.$$

The self-energy is evaluated at the quasiparticle energy because it is frequency dependent.

This makes the equation nonlinear: the unknown energy appears both on the left-hand side and inside $\Sigma$.

A common linearized form is

$$E_{n\mathbf k}^{\mathrm{QP}} \approx \varepsilon_{n\mathbf k}^{\mathrm{KS}} + Z_{n\mathbf k} \left\langle n\mathbf k
\left| \Sigma\left(\varepsilon_{n\mathbf k}^{\mathrm{KS}}\right) - V_{\mathrm{xc}} \right| n\mathbf k \right\rangle,$$

where

$$Z_{n\mathbf k} = \left[ 1- \left. \frac{\partial \operatorname{Re}\Sigma_{n\mathbf k}(\omega)} {\partial\omega}
\right|_{\omega=\varepsilon_{n\mathbf k}^{\mathrm{KS}}} \right]^{-1}.$$

The factor $Z_{n\mathbf k}$ accounts for the frequency dependence of the self-energy near the reference energy.

Charged and neutral excitations

The GW approximation is primarily designed for charged one-particle excitations.

These correspond to adding an electron or removing an electron:

$$N \longrightarrow N+1 $$

or

$$N \longrightarrow N-1.$$

The resulting excitation energies determine quantities such as:

  • electron affinities;
  • ionization energies;
  • quasiparticle band gaps;
  • band alignments;
  • quasiparticle dispersions.

Neutral optical excitations are different. They involve simultaneous creation of an electron and a hole without changing the total number of electrons.

Although electron-hole propagation contributes to the polarization $P$, the attractive ladder interaction responsible for bound excitons is not included in a basic quasiparticle GW calculation.

Such excitations are usually treated with the Bethe–Salpeter equation.

The GW–BSE workflow

A common many-body workflow is

$$\mathrm{DFT} \longrightarrow GW \longrightarrow \mathrm{BSE}.$$

The roles of the three steps are:

$$\mathrm{DFT} \quad\longrightarrow\quad \text{reference orbitals and energies},$$

$$GW \quad\longrightarrow\quad \text{charged quasiparticle energies},$$

$$\mathrm{BSE} \quad\longrightarrow\quad \text{neutral electron-hole excitations}.$$

The GW step corrects the energies of the electron and hole entering the BSE Hamiltonian.

The BSE then introduces their mutual interaction and produces exciton energies and optical spectra.

What standard GW includes

Standard GW includes:

  • dynamical screening;
  • an exchange-like self-energy contribution;
  • correlation through repeated polarization processes;
  • quasiparticle energy corrections;
  • finite lifetimes when the imaginary self-energy is evaluated;
  • coupling to collective electronic excitations;
  • plasmon-related satellite structures at an approximate level.

These effects represent a substantial improvement over static mean-field descriptions.

What standard GW does not include

Standard GW neglects explicit vertex corrections:

$$\Gamma\approx1.$$

Therefore, it does not include every many-body interaction process.

Important missing or approximate effects may include:

  • electron-hole ladder diagrams;
  • some short-range correlation effects;
  • self-screening corrections;
  • strong local multiplet physics;
  • certain satellite structures;
  • strong-correlation effects associated with Mott physics;
  • vertex corrections to both $P$ and $\Sigma$.

The importance of these effects depends strongly on the material and the observable.

Self-screening

In an approximate GW calculation, an electron may contribute to the polarization that screens its own interaction.

This unphysical effect is called self-screening.

In the exact theory, vertex corrections cancel such contributions appropriately.

Because standard GW neglects the vertex, self-screening errors may remain.

These errors can be especially important in localized systems, molecules, and situations where screening is weak.

Frequency dependence and numerical treatment

The correlation part of the self-energy requires an integration over internal frequency.

Schematically,

$$\Sigma_c(\omega) \sim \int d\omega’ G(\omega+\omega’) W_c(\omega’).$$

This integration can be computationally demanding because both $G$ and $W$ contain poles and other frequency-dependent structures.

Common numerical approaches include:

  • direct real-frequency integration;
  • contour deformation;
  • imaginary-frequency methods;
  • analytic continuation;
  • plasmon-pole models.

Each method balances computational cost, stability, and accuracy differently.

Plasmon-pole models

A plasmon-pole model approximates the frequency dependence of the dielectric response using one or a small number of effective poles.

This replaces the full function

$$\varepsilon^{-1}(\omega)$$

with a simpler analytic representation.

Such models can greatly reduce computational cost.

However, they may become less accurate when the material has several important screening channels or a complicated excitation spectrum.

Full-frequency GW avoids this approximation but is more expensive.

Matrix structure of screening

In periodic systems, the dielectric function and screened interaction are matrices in reciprocal lattice vectors:

$$\varepsilon_{\mathbf G\mathbf G’}(\mathbf q,\omega),$$

$$W_{\mathbf G\mathbf G’}(\mathbf q,\omega).$$

The off-diagonal matrix elements describe local-field effects.

These effects arise because the induced charge density is not spatially uniform within a unit cell.

Neglecting off-diagonal elements corresponds to treating the screening as more homogeneous than it actually is.

Local-field effects may be important in low-dimensional, ionic, molecular, and strongly inhomogeneous systems.

Screening in low-dimensional materials

Screening behaves differently in two-dimensional materials than in bulk three-dimensional solids.

In reduced dimensions:

  • screening is generally weaker;
  • it is more strongly nonlocal;
  • it depends sensitively on the surrounding dielectric environment;
  • supercell calculations require careful treatment of interactions between periodic images;
  • Coulomb truncation may be necessary.

As a result, GW quasiparticle corrections in two-dimensional materials can be large and highly dependent on substrate or environmental screening.

Beyond standard GW

Several extensions aim to include physics missing from standard GW.

Examples include:

  • $GW\Gamma$,
  • where vertex corrections are introduced explicitly;
  • self-consistent GW;
  • quasiparticle self-consistent GW;
  • cumulant expansions for improved satellite spectra;
  • combined GW + DMFT approaches for both nonlocal screening and strong local correlations;
  • and GW-BSE for neutral optical excitations.

The appropriate method depends on whether the main goal is quasiparticle energies, spectral functions, excitons, satellites, or strongly correlated behavior.

Summary

The GW approximation is based on

$$\Sigma \approx iGW, $$

with

$$P \approx -iGG $$

and

$$W = v+vPW.$$

It is obtained from Hedin’s equations by using

$$\Gamma\approx1.$$

The method includes dynamical screening through repeated polarization processes but neglects explicit vertex corrections.

Different practical levels include

$$G_0W_0,$$

partially self-consistent GW, and fully self-consistent GW.

The central physical role of GW is to calculate charged quasiparticle excitations, while neutral electron-hole excitations are commonly treated subsequently using the Bethe–Salpeter equation.