GW Approximation

Assuming we already know the Green's function, now I will start with Dyson equation and then will introduce Hedin's equation. Lastly, I will introduce practical workflow of a GW calculation, and parameters and convergence tests related to this.

Dyson equation

\[ G(\omega) = G_0(\omega) + G_0(\omega) \Sigma(\omega) G(\omega) = \left[ G_0^{-1}(\omega) - \Sigma(\omega) \right]^{-1} \]

It links the dressed/quasi-particle solution \(G(\omega)\) to the bare particle solution \(G_0(\omega)\) through a kernel \(\Sigma(\omega)\) which contains the effect of coupling (called the self-energy). For interacting electrons, the coupling is due to the Coulomb interaction. It can be also written as:

\[ G = G_0 + G_0 \Sigma G_0 + G_0 \Sigma G_0 \Sigma G_0 + \ldots \]

This relation is not limited for one-particle Green's function and the self energy. Instead it holds for any type of Green's functions and any derived quantities such as the screened interaction \(W\) and the polarization \(P\),

\[ W(\omega) = v_c + v_c P(\omega) W(\omega) = \left[ v_c^{-1} - P(\omega) \right]^{-1}\]

or, in the expanded form:

\[ W = v_c + v_c P v_c + v_c P v_c P v_c + \ldots \]

Here, the coupling is denoted by the irreducible or proper polarizability \(P(\omega)\) instead of the reducible polarizability \(\chi(\omega)\) that only includes terms connected by bare Coulomb lines.

The self energy

\(\Sigma\) is the self-energy defined as the energy difference between the dressed/clothed/renormalized or quasi-particle and the bare/real particle, i.e.,

\[ \epsilon_{_{\text{self-energy}}} = \epsilon_{_{\text{real particle}}} - \epsilon_{_{\text{quasi particle}}} \]

In the propagator language, \(\Sigma(\omega) = G_0^{-1}(\omega) - G^{-1}(\omega)\). Sometimes \(\tilde{\Sigma}\) is used in the literature where Hartree potential is included, i.e., \(\tilde{\Sigma}=\Sigma + v_{\mathrm{H}}\)

Coupled Dyson equation

Any type of Dyson equation can be presented by a set of coupled equation which has practical importance. For example, if we can divide the self energy as a composition of two distinct parts, i.e.,

\[ \Sigma = \Sigma_1 + \Sigma_2 \]

Now we can introduce a modified Green's function \(G_1\) such as:

\[ G_1 = G_0 + G_0\Sigma_1G_1\]

and then the Dyson equation can be written as:

\[ G = G_1 + G_1\Sigma_2G \]

In this case, if we already know/approximate \(G_1\), then we only need to consider \(\Sigma_2\) part of the overall self-energy. This holds for any number of division. The art is to divide the self-energy into some useful compositions. For the screened interaction, if we have \(P=P_1 + P_2\),

\[ W_1 = v_c + v_cP_1W_1 \]

\[ W = W_1 + W_1 P_2 W\]

This shows that when we consider some smaller space, the quantities are frequency independent such as \(v_c\) but it becomes frequency dependent as we include contributions from larger space. In practice, one might separate \(\Sigma\) into a static \(\Sigma_1\) part from a KS potential and a dynamic \(\Sigma_2\) part evaluated perturbatively using KS wavefunctions. There is also COHSEX approximation or quasi-particle self-consistent \(GW\) where \(\Sigma_1\) is obtained as a static approximation.

The dynamical screening

The GW approximation is different than the HF theory in terms of the dynamical screening. This is the only difference, i.e., in the GWA,

\[ \Sigma_{\mathrm{xc}} = iGW\]

while in the HF,

\[ \Sigma_{\mathrm{x}} = iGv_c\]

where the interaction \(v_c\) is unscreened.

Hedin's equation

Manipulating the Dyson equation for \(G\), \(W\) yields a closed set of equation that can be solved self-consistently. The key quantities in the equations are: 1. \(G\) is the one-particle Green's function (i.e. the propagator for adding/removing an electron in an interacting many-body system) 2. \(G_0\) is the independent-particle Green's function (i.e. in a non-interacting many-body system) 3. \(\Sigma\) is the self-energy 4. \(W\) is the screened Coulomb interaction and \(v_c\) is the bare Coulomb interaction 5. \(P\) is the polarization function 6. \(\Gamma\) is the vertex function (the 3-point vertex in the Feynman diagram)

Then Hedin's equations are:

\[ \Sigma = iGW\Gamma \]

\[ G = G_0 + G_0 \Sigma G\]

\[ \Gamma = 1 + \frac{\delta\Sigma}{\delta G}GG\Gamma \]

\[ P = -i GG\Gamma \]

\[ W = v + vPW \]

The GW approximation:

When the vertex term becomes unity, i.e.,

\[\Gamma = 1 \]

and therefore,

\[\Sigma =iGW \]

and

\[P = -i GG\]

This is called the \(GW\) approximation. Depending on how we are calculating each of the term in Hedin's equation, the methods are named differently such as:

• \(G_0W_0\) : single shot
• \(GW_0\) : only the screened Coulomb interaction part is calculated non self-consistently
• \(\mathrm{sc}GW\) : self-consistent \(GW\) but setting \(\Gamma\) to a single spacetime point
• \(\mathrm{ev}GW\) : eigenvalue self-consistent updates the energies only, not the wave function
• \(\mathrm{qs}GW\) : quasi-particle self-consistent \(GW\)

Comparison

Theory	KS-DFT	\(G-\Sigma\) functional	Hartree-Fock approximation
Quantity	Static density \(n(r)\)	Dynamic Green’s function \(G(r, r', ω)\)	Static density matrix \(ρ_{\mathrm{HF}} (r, r' )\)
Approach	Auxiliary system of independent particles with static local potential \(v_{\mathrm{xc}}(r)\)	System described by dynamic non-local self-energy \(\Sigma(r,r',\omega)\)	Approximation of independent particles with static non-local potential \(\Sigma_x(r,r')\)
Exact formulation	Universal \(\Omega_{\mathrm{xc}}[n]\), \(v_{\mathrm{xc}}(r) = \frac{\delta \Omega_{\mathrm{xc}}}{\delta n}\)	Luttinger-Ward \(\Phi[G]\), \(\Sigma(r,r',\omega) = \frac{\delta \Phi}{\delta G}\)	None
Canonical approximation	LDA	\(GW\) approximation \(\Sigma=iGW\)	HF equations
Other approximations	GGA, meta-GGA, hybrid, RPA	Vertex correction, T-matrix	Multireference HF