more modifs Sec IV

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<d} \aeri{pq}{cd} \bX_{cd,m}^{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N+2} \notag \\
-  \eri{pq}{\chi^{N-2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N-2}  + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N-2} \notag
+  W_{pq,v}^{N+2} & = \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N+2} \notag \\
+  W_{pq,v}^{N-2} & = \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N-2}  + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N-2} \notag
 \end{align}
 The two RPA problems giving the eigenvectors needed to build the $GW$ and $GT$ self-energies are given in Appendix \ref{sec:rpa}.