GW approximation of Hedin's equations

Green's functions

The GW method can be understood in terms of the following eigenvalue equation

$(T+V_{ext}+V_h)\phi_{n{\bf k}}({\bf r})+\int d{\bf r}\Sigma({\bf r},{\bf r}',\omega=E_{n{\bf k}})\phi_{n{\bf k}}({\bf r}') = E_{n{\bf k}}\phi_{n{\bf k}}({\bf r})$

Here $T$ is the kinetic energy, $V_{ext}$ the external potential of the nuclei, $V_h$ the Hartree potential and $E_{n{\bf k}}$ the quasiparticle energies with orbitals $\phi_{n{\bf k}}$. In contrast to DFT, the exchange-correlation potential is replaced by the many-body self-energy $\Sigma$ and should be obtained together with the Green's function $G$, the irreducible polarizability $\chi$, the screened Coulomb interaction $W$ and the irreducible vertex function $\Gamma$ in a self-consistent procedure. For completeness, these equations are

$G(1,2)=G_0(1,2)+\int d(3,4) G_0(1,3)\Sigma(3,4)G(4,2)$

$\chi(1,2)=\int d(3,4) G(1,3)G(4,1)\Gamma(3,4;2)$

$W(1,2)=V(1,2)+\int d(3,4) V(1,3)\chi(3,4)W(4,2)$

$\Sigma(1,2)=\int d(3,4) G(1,3)\Gamma(3,2;4)W(4,1)$

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

Here the common notation $1=({\bf r}_1,t_1)$ was adopted and $V$ denotes the bare Coulomb interaction. Note, that these equations are exact and provide an alternative to the Schrödinger equation for the many-body problem. Nevertheless, approximations are necessary for realistic systems. The most popular one is the GW approximation and is obtained by neglecting the equation for the vertex function and using the bare vertex instead:

$\Gamma(1,2;3)=\delta(1,2)\delta(1,3)$

This means that the equations for the polarizability and self-energy reduce to

$\chi(1,2)=G(1,2)G(2,1)$

$\Sigma(1,2)=G(1,2)W(2,1)$

while the equations for the Green's function and the screened potential remain the same.

However, in practice, these equations are usually solved in reciprocal space in the frequency domain

$W_{{\bf G}{\bf G}'}({\bf q},\omega)=\left[\delta_{{\bf G}{\bf G}'}-\chi_{{\bf G}{\bf G}'}({\bf q},\omega)V_{{\bf G}{\bf G}'}({\bf q})\right]^{-1}V_{{\bf G}{\bf G}'}({\bf q})$

$G_{{\bf G}{\bf G}'}({\bf q},\omega)=\left[\delta_{{\bf G}{\bf G}'}-\Sigma_{{\bf G}{\bf G}'}({\bf q},\omega)G^{(0)}_{{\bf G}{\bf G}'}({\bf q})\right]^{-1}G^{(0)}_{{\bf G}{\bf G}'}({\bf q})$

In principle Hedin's equations have to be solved self-consistently, where in the first iteration $G^{(0)}$ is the non-interacting Green's function

$G^{(0)}({\bf r},{\bf r}',\omega)=\sum_{n{\bf k}}\frac{\phi_{n{\bf k}}^{*(0)} ({\bf r})\phi^{(0)}_{n{\bf k}} ({\bf r}')}{\omega-E^{(0)}_{n{\bf k}}}$

with $\phi^{(0)}_{n{\bf k}}$ being a set of one-electron orbitals and $E_{n{\bf k}}^{(0)}$ the corresponding energies. Afterwards the polarizability $\chi^{(0)}$ is determined, followed by the screened potential $W^{(0)}$ and the self-energy $\Sigma^{(0)}$. This means that GW calculations require a first guess for the one-electron eigensystem, which is usually taken from a preceding DFT step.

In principle, one has to repeat all steps by the updating the Green's function with the Dyson equation given above in each iteration cycle until self-consistency is reached. In practice, this is hardly ever done due to computational complexity on the one hand (in fact fully self-consistent GW calculations are available as of VASP 6 only).

On the other hand, one observes that by keeping the screened potential $W$ in the first iteration to the DFT level one benefits from error cancelling, which is the reason why often the screening is kept on the DFT level and one aims at self-consistency in Green's function only.

Following possible approaches are applied in practice.

Single Shot: G₀W₀

Performing only one GW iteration step is commonly referred to the G₀W₀ method. Here the self-energy $\Sigma^{(0)}$ is determined and the eigenvalue equation

$(T+V_{ext}+V_h)\phi_{n{\bf k}}({\bf r})+\int d{\bf r}\Sigma^{(0)}\left({\bf r},{\bf r}',\omega=E^{(0)}_{n{\bf k}}\right)\phi_{n{\bf k}}({\bf r}') = E^{(0)}_{n{\bf k}}\phi_{n{\bf k}}({\bf r})$

is solved for the quasi-particle energies $E^{(0)}_{n\mathbf{k}}$. The orbitals, in contrast, remain unchanged.

Although self-consistency is completely neglected, G₀W₀ often yields satisfactory results with band-gaps close to experimental measurements.

Low-scaling GW: The Space-time Formalism

The scaling with system size $N$ (number of electrons) of GW calculations can be reduced by performing a so-called Wick-rotation to imaginary time $t\to -i\tau$.

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