GF2 approximation

Second Order approximation (GF2)

Self-consistent second-order perturbation theory is a conserving diagrammatic approximation with nonzero contribution to the correlated self-energy. It is also known as Second Order Born. The method has been introduced to molecules in modern times by Holleboom and Snijders 1. Generally, it is considered accurate whenever gaps are large and interactions are weak but is known to fail for metals. Unlike GW, it contains a second-order exchange term but does not include higher-order screening contributions.

The second-order contribution to the self-energy in imaginary time and momentum space is2 $$ \begin{align*} \Sigma^{(2)}_{ij}(\tau,\mathbf{k}) = - \frac{1}{N_{\mathbf{k}}^3}\sum\limits_{\substack{klmnop\\ \mathbf{k_1}\mathbf{k_2}\mathbf{k_3} }} & (2U^{\mathbf{k_1}\mathbf{k}\mathbf{k_2}\mathbf{k_3}}_{qjln} - U^{\mathbf{k_2}\mathbf{k}\mathbf{k_1}\mathbf{k_3}}_{ljqn}) \times U^{\mathbf{k}\mathbf{k_1}\mathbf{k_3}\mathbf{k_2}}_{ipmk} \\ \times &G^{\mathbf{k_1}}_{pq}(\tau)G^{\mathbf{k_2}}_{kl}(\tau)G^{\mathbf{k_3}}_{nm}(-\tau)\delta_{\mathbf{k}+\mathbf{k_3},\mathbf{k_1}+\mathbf{k_2} }, \end{align*} $$

here $G^\mathbf{k}_{ij}(\tau)$ is imaginary time Green's function, $U^{\mathbf{k}\mathbf{k_1}\mathbf{k_2}\mathbf{k_3}}_{ijkl}$ is the Coloumb interaction tensor, and $\delta_{\mathbf{k}+\mathbf{k_3},\mathbf{k_1}+\mathbf{k_2} }$ guarantees the momentum conservation.

Implementation details for Green’s software are given in this 3 paper.