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 $$ \begin{align*} \Sigma^{(2)}_{ij}(\tau,\mathbf{k}) = - \frac{1}{N_{\mathbf{k}}^3}\sum\limits_{\substack{klmnpq\\ \mathbf{k_1}\mathbf{k_2}\mathbf{k_3} }} & (2U^{\mathbf{k_1}\mathbf{k}\mathbf{k_2}\mathbf{k_3}}_{qjln} - U^{\mathbf{k_2}\mathbf{k}\mathbf{k_1}\mathbf{k_3}}_{ljqn}) \times U^{\mathbf{k}\mathbf{k_1}\mathbf{k_3}\mathbf{k_2}}_{ipmk} \\ \times &G^{\mathbf{k_1}}_{pq}(\tau)G^{\mathbf{k_2}}_{kl}(\tau)G^{\mathbf{k_3}}_{nm}(-\tau)\delta_{\mathbf{k}+\mathbf{k_3},\mathbf{k_1}+\mathbf{k_2} }, \end{align*} $$

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.