Second-Order Green's Function Approximation(GF2)

Self-consistent second-order perturbation theory (GF2) is a conserving diagrammatic approximation that provides a nonzero contribution to the correlated self-energy. Also known as Second Order Born, GF2 was introduced to molecular systems in recent times by Holleboom and Snijders ¹. This method is generally accurate for systems with large energy gaps and weak interactions but is known to fail for metallic systems. Unlike the GW approximation, GF2 incorporates a second-order exchange term but does not account for higher-order screening effects.

The second-order contribution to the self-energy in imaginary time ($\tau$) and momentum space ($\mathbf{k}$) is given by²’³:

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

Explanation of the Equation:

• $\Sigma^{(2)}_{ij}(\tau,\mathbf{k})$: This represents the second-order self-energy component for orbitals $i$ and $j$ at imaginary time $\tau$ and momentum $\mathbf{k}$.

• $N_{\mathbf{k}}$: The number of discrete momentum points considered in the finite cluster.

• Summations: The summations run over all possible orbital indices ($k, l, m, n, p, q$) and momentum indices ($\mathbf{k_1}, \mathbf{k_2}, \mathbf{k_3}$).

• $U^{\mathbf{k_1}\mathbf{k}\mathbf{k_2}\mathbf{k_3}}_{qjln}$ and $U^{\mathbf{k_2}\mathbf{k}\mathbf{k_1}\mathbf{k_3}}_{ljqn}$ : These are components of the Coulomb interaction tensor, representing the electron-electron interactions between different orbitals and momenta.

• $G^{\mathbf{k}}_{ij}(\tau)$: The imaginary-time Green’s function, describing the propagation of an electron from orbital $j$ to $i$ over time $\tau$.

• $\delta_{\mathbf{k}+\mathbf{k_3},\mathbf{k_1}+\mathbf{k_2}}$: The Kronecker delta ensures momentum conservation in the interaction process, meaning the total momentum before and after the interaction remains the same.

This equation effectively captures the second-order processes contributing to the self-energy, accounting for interactions between electrons mediated by the Coulomb tensor and the Green’s functions.