diff --git a/Notes/Notes.tex b/Notes/Notes.tex index 5dcd9a7..06f1860 100644 --- a/Notes/Notes.tex +++ b/Notes/Notes.tex @@ -289,12 +289,12 @@ However, in order to use it we need to rely on approximations of the dynamical s In the following, we will focus on the GF(2), $GW$ and $GT$ approximations. The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory. -\begin{align} +\begin{equation} \label{eq:GF2_selfenergy} \Sigma_{pq}^{\text{GF(2)}}(\omega) - & = \sum_{ija} \frac{W_{pa,ij}^{\text{GF(2)}}W_{qa,ij}^{\text{GF(2)}}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\ - & + \sum_{iab} \frac{W_{pi,ab}^{\text{GF(2)}}W_{qi,ab}^{\text{GF(2)}}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag -\end{align} + = \sum_{ija} \frac{W_{pa,ij}^{\text{GF(2)}}W_{qa,ij}^{\text{GF(2)}}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} + + \sum_{iab} \frac{W_{pi,ab}^{\text{GF(2)}}W_{qi,ab}^{\text{GF(2)}}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} +\end{equation} with \begin{equation} \label{eq:GF2_sERI} @@ -303,7 +303,9 @@ with On the other hand, the $GW$ self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy. \begin{equation} \label{eq:GW_selfenergy} - \Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{W_{pi,v}^{\GW} W_{qi,v}^{\GW}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{av} \frac{W_{pa,v}^{\GW}W_{qa,v}^{\GW}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag + \Sigma_{pq}^{\GW}(\omega) + = \sum_{iv} \frac{W_{pi,v}^{\GW} W_{qi,v}^{\GW}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + + \sum_{av} \frac{W_{pa,v}^{\GW}W_{qa,v}^{\GW}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag \end{equation} with \begin{equation} @@ -314,13 +316,15 @@ Finally, the $GT$ approximation corresponds to another approximation to the pola The corresponding self-energies read as \begin{equation} \label{eq:GT_selfenergy} - \Sigma_{pq}^{\GT}(\omega) = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \epsilon_i - \Omega_{m}^{N+2} + \ii \eta} + \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \epsilon_a - \Omega_{m}^{N-2} - \ii \eta} \notag + \Sigma_{pq}^{\GT}(\omega) + = \sum_{iv} \frac{W_{pi,v}^{N+2} W_{qi,v}^{N+2}}{\omega + \epsilon_i - \Omega_{v}^{N+2} + \ii \eta} + + \sum_{av} \frac{W_{pa,v}^{N-2} W_{qa,v}^{N-2}}{\omega + \epsilon_a - \Omega_{v}^{N-2} - \ii \eta} \notag \end{equation} with \begin{align} \label{eq:GT_sERI} - \eri{pq}{\chi^{N+2}_m} &= \sum_{c