GW for finite systems

Hedin's GW is a useful approximation for the so-called self-energy $\Sigma (r,r',\omega )$ that enters Dyson's equation for an interacting electronic propagator $G(r, r', \omega )$

(1)
\begin{align} (\omega-H(r,r'')-\Sigma_{xc}(r,r'',\omega)) G(r'',r',\omega) = \delta(r-r'). \end{align}

Hedin's GW approximation for the self-energy $\Sigma_{xc}(r,r',\omega)$ reads

(2)
\begin{align} \Sigma_{xc}(r,r',t) = \mathrm{i}G_0(r,r',t)W_0(r,r',t). \end{align}

It involves another, non interacting Green's function $G_{0}(r,r',t)$ and a screened Coulomb interaction $W_0(r,r',t)$. This approximation is a solution of a truncated version of Hedin's equations [1] [2]. The name of this approximation is taken from the simple form of the electronic self-energy $\Sigma =\mathrm{i}GW$.

The non interacting Green's function is resolvent of the non interacting Hamiltonian $H(r,r')$

(3)
\begin{align} (\omega-H(r,r'')) G_0(r'',r',\omega) = \delta(r-r'). \end{align}

The screened Coulomb interaction $W_0$ can be easily calculated in frequency domain using the so-called RPA approximation

(4)
\begin{align} W_0(r,r',\omega ) =\left[\delta(r-r''')-v(r,r'')\chi_0(r'',r''',\omega )\right]^{-1} v(r''',r'), \end{align}

where $v(r,r')\equiv |r-r'|^{-1}$ is the bare Coulomb interaction. Here and in the following we assume integration over repeated spatial coordinates ($r''$ and $r'''$ in equation (4)) on the right hand side of an equation if they do not appear on its left hand side. The screened interaction (4) is the sum of the bare Coulomb interaction created by a point charge at $r'$, plus a correction due to the redistribution of charge induced in response to the total field [2]. The non interacting response function $\chi_0(r,r',t)$ is related to the non-interacting Green's function

(5)
\begin{align} \mathrm{i}\chi_0(r,r',t) =2 G_0(r,r',t)G_0(r',r,-t), \end{align}

Further details can be found in [3].

Bibliography
1. L. Hedin, Phys. Rev. 139, A796 (1965). For a review see F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998).
2. C. Friedrich and A. Schindlmayr, Many-Body Perturbation Theory: The GW Approximation, NIC Series, 31, 335 (2006).
3. D. Foerster, P. Koval, and D. S\'anchez-Portal, The Journal of Chemical Physics 135, 074105 (2011). http://arxiv.org/abs/1101.2065, http://arxiv.org/abs/1105.3360.