The GW Approximation

The $GW$ approximation is a fully self-consistent Green’s-function method in which the dynamical self-energy is built from the interacting one-particle Green’s function $G$ and a screened Coulomb interaction $W$. Green/WeakCoupling implements the finite-temperature, imaginary-axis, non-relativistic Gaussian Bloch-orbital formulation described in Ref. ¹ . The calculation does not use a quasiparticle approximation during the self-consistency loop, and all matrix elements of the self-energy are evaluated on the Matsubara axis.¹

The physical idea is different from GF2. GF2 expands the self-energy through second order in the bare Coulomb interaction. $GW$ instead keeps the self-energy first order in a screened interaction $W$, where $W$ itself contains an infinite random-phase-approximation-like series of polarization bubbles. This screening is essential for extended systems, especially when bare-interaction perturbation theory is poorly behaved.², ¹

Dyson Equation

In the non-relativistic periodic notation of Ref. ¹ , the interacting Green’s function satisfies

Here $S^{\mathbf{k}}$ is the overlap matrix, $H_0^{\mathbf{k}}$ is the one-electron Hamiltonian, $\mu$ is the chemical potential, and $\omega_n=(2n+1)\pi/\beta$ is a fermionic Matsubara frequency. The inverse is a matrix inverse in the orbital and spin space at fixed $\mathbf{k}$ and $i\omega_n$.

The $GW$ self-energy is split into static and dynamical pieces,

The static part is the Hartree-Fock self-energy built from the current correlated density matrix. In the notation of Ref. ¹ , it is the sum of a Hartree term $J$ and an exchange term $K$,

The new approximation specific to $GW$ is the dynamical part $\tilde{\Sigma}^{GW}$.

Screened Interaction

The screened interaction $W$ is the Coulomb interaction dressed by repeated particle-hole polarization bubbles. In four-index notation, the formal equation has the Dyson-like structure

where $U$ is the bare Coulomb interaction, $\Pi$ is the polarization, and $\Omega_n=2n\pi/\beta$ is a bosonic Matsubara frequency. This compact expression hides orbital and momentum sums, but it is the same equation written explicitly in Ref. ¹ .

The corresponding polarization bubble in imaginary time is built from two interacting Green’s functions,

This is why self-consistent $GW$ is nonlinear: $G$ determines $\Pi$, $\Pi$ determines $W$, $W$ determines $\Sigma$, and $\Sigma$ determines a new $G$ through the Dyson equation.

Density-Fitted Form

In Green/WeakCoupling, the four-index Coulomb tensor is represented by density-fitted three-index vertices

This is the practical form used in the non-relativistic Yeh implementation. It reduces memory and turns the screened-interaction problem into operations in the auxiliary basis indexed by $Q$.¹

For a momentum transfer $\mathbf{q}$, define the noninteracting auxiliary polarization $\tilde{P}_0^{\mathbf{q}}$. In imaginary time,

The bosonic Matsubara transform is

The RPA bubble series is then summed in the auxiliary basis:

This $\tilde{P}$ is the renormalized auxiliary quantity that contains the repeated bubble insertions. In the Yeh notation it generates the effective screened interaction $\tilde{W}=W-U$,

The frequency or time argument is inherited from $\tilde{P}$.

Dynamical Self-Energy

Using the effective screened interaction, the imaginary-time $GW$ self-energy is

Substituting the density-fitted expression for $\tilde{W}$ gives the implementation form

This is the central $GW$ equation used in the non-relativistic Green/WeakCoupling implementation.¹

Self-Consistency Loop

A self-consistent $GW$ iteration follows the structure below.

1. Start from a Green’s function, usually from HF, DFT, or the noninteracting Hamiltonian.
2. Build the correlated density matrix and the static Hartree-Fock self-energy $\Sigma_\infty^{GW}$.
3. Build $\tilde{P}_0$ from the current $G$.
4. Solve the auxiliary-basis RPA equation for $\tilde{P}$.
5. Construct $\tilde{\Sigma}^{GW}(\tau)$ and transform it to Matsubara frequency.
6. Adjust $\mu$ to obtain the target electron number.
7. Solve the Dyson equation for a new $G$.
8. Iterate until $G$, $\Sigma$, the energy, and the chemical potential are converged.

The key point is that the same interacting $G$ is used everywhere in the loop. This makes the method conserving and thermodynamically consistent when solved to self-consistency.³, ¹

Relation to Spectra

The calculation is performed entirely on the imaginary axis. After convergence, real-frequency information is obtained by analytic continuation of the Matsubara Green’s function. Ref. ¹ uses the relation

Because the Gaussian atomic-orbital basis is non-orthogonal, the Green’s function is transformed to an orthonormal basis before extracting normalized diagonal spectral functions.¹

What GW Omits

Fully self-consistent $GW$ contains infinitely many bubble-screening diagrams through $W$, but it omits vertex corrections in both the self-energy and the polarization. This is the main formal reason why self-consistent $GW$ can still deviate from exact results even after removing starting-point dependence. Later vertex-corrected methods add diagrams beyond this approximation.