Hedin's Equations, Screened Interactions, and the GW Approximation

Hedin’s equations are an exact closed hierarchy for the one-particle propagator, screened interaction, polarization, self-energy, and three-point vertex. They expose where the GW approximation enters: GW replaces the exact vertex by its bare value in the self-energy and polarization. The compact notation hides demanding integral equations, so every practical calculation must state which objects are iterated and which vertex information is omitted. The five-equation hierarchy appears in Hedin 1965, pp. A796–A802.

Required background. Dyson Equations and Self-Energy defines the dressed one-particle line, Coulomb Screening, Dielectric Response, and RPA defines $W$ , and Irreducible Vertices and Bethe–Salpeter Equations supplies the vertex equation.

Helpful background. Schwinger–Dyson Hierarchies and Renormalization Inputs develops the general hierarchy and closure problem.

The five coupled equations

Use condensed labels such as $1=(\mathbf r_1,t_1,\sigma_1)$ and let repeated labels be integrated. With conventional real-time factors displayed schematically, Hedin’s equations are

G=G_0+G_0\Sigma G, \qquad W=v+vPW,

\Sigma(1,2)=i\,G(1,3)W(1^+,4)\Gamma(3,2;4),

P(1,2)=-i\,G(1,3)G(4,1^+)\Gamma(3,4;2),

\Gamma(1,2;3)=\delta(1,2)\delta(1,3) +\frac{\delta\Sigma(1,2)}{\delta G(4,5)} G(4,6)G(7,5)\Gamma(6,7;3).

Imaginary-time conventions move factors of $i$ and signs but not the structure. Dyson dresses $G$ ; the polarization dresses $v$ into $W$ ; $GW\Gamma$ produces $\Sigma$ ; and the functional derivative of $\Sigma$ generates the vertex kernel. No equation is optional in the exact hierarchy.

The GW truncation

Set $\Gamma$ to the bare local vertex. Then

\Sigma_{GW}=iGW, \qquad P_{GW}=-iGG.

Different algorithms bearing the name GW are not equivalent:

$G_0W_0$ evaluates both from a chosen reference propagator once;
eigenvalue-only or partially self-consistent schemes update selected pole energies;
quasiparticle self-consistent GW constructs an updated static effective one-particle Hamiltonian;
fully self-consistent GW iterates $G$ , $P$ , $W$ , and $\Sigma$ to convergence with the bare vertex.

Starting-point dependence is therefore part of a one-shot result. Self-consistency removes that particular dependence but does not restore the missing vertex, and it can shift spectral weight or broaden satellites differently from experiment.

Screening, exchange, and correlation

Writing $W=v+(W-v)$ separates the instantaneous exchange term $iGv$ from a dynamic correlation self-energy $iG(W-v)$ . The frequency dependence of $W$ produces quasiparticle energy shifts, finite lifetimes, and satellite structures. A plasmon-pole model can reduce the numerical cost, but it replaces the full dielectric spectrum by a parameterized analytic form; its moment and causality constraints must be checked.

The approximation is often effective when screening is appreciable and quasiparticles remain identifiable. It is not a controlled expansion for all solids. Strong local multiplet physics, excitonic vertex effects, low-dimensional screening, and near-degenerate reference states can require explicit vertex or embedding corrections.

Conservation and response

Fully self-consistent GW can be generated from a skeleton functional and is conserving when solved self-consistently. A response calculated with the corresponding functional derivative kernel is required to inherit the associated Ward identities. One-shot GW is not automatically conserving, and a conserving approximation is not automatically quantitatively accurate or guaranteed to produce a positive spectrum in every implementation.

Useful checks include causality of $G^R$ and $W^R$ , particle number, spectral normalization, high-frequency moments, the $f$ -sum for dielectric response, convergence in frequency/basis/cell size, and sensitivity to the reference for non-self-consistent variants.

Common pitfalls

Writing “GW” without the iteration scheme. $G_0W_0$ , partial updates, quasiparticle self-consistency, and full self-consistency answer different equations.

Assuming self-consistency supplies the vertex. Iterating $G$ and $W$ changes internal lines but leaves $\Gamma=1$ unless a vertex correction is added.

Identifying every satellite with a separate quasiparticle. Satellites arise from the convolution with neutral excitations and must be interpreted through residues and spectral weight.

Exercises

Recover RPA screening

Show that the bare-vertex polarization in Hedin’s equations produces the RPA series for $W$ .

Solution

With $\Gamma=1$ , $P=-iGG$ is the bubble built from the declared propagators. The equation $W=v+vPW$ gives $W=(1-vP)^{-1}v=v+vPv+\cdots$ . With $G=G_0$ this is ordinary RPA; with dressed $G$ it is a self-consistent bubble screening approximation.

Separate exchange from correlation

Insert $W=v+W_c$ into $\Sigma=iGW$ . What two terms result?