saving work

Manuscript/SRGGW.tex

@@ -480,9 +480,10 @@ Collecting every second-order terms in the flow equation and performing the bloc
   \dv{\bF^{(2)}}{s}  = \bF^{(0)}\bV^{(1)}\bV^{(1),\dagger} + \bV^{(1)}\bV^{(1),\dagger}\bF^{(0)} - 2 \bV^{(1)}\bC^{(0)}\bV^{(1),\dagger} .
 \end{equation}
 This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
-\begin{equation}
+\begin{align}
-  F_{pq}^{(2)}(s) = \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right).
+  &F_{pq}^{(2)}(s) =  \\
-\end{equation}
+  &\sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right), \notag
+\end{align}
 with $\Delta_{prv} = \epsilon_p - \epsilon_r - \text{sign}(\epsilon_r-\epsilon_F)\Omega_v$.
 
 At $s=0$, this second-order correction is null while for $s\to\infty$ it tends towards the following static limit
@@ -495,10 +496,11 @@ Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a st
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
 Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
-\begin{equation}
+\begin{align}
-  \label{eq:static_F2}
+  \label{eq:sym_qsGW}
-  \Sigma_{pq}^{\text{qs}GW}(\eta) =  \sum_{r,v}  \left( \frac{\Delta_{prv}}{\Delta_{prv}^2 + \eta^2} +\frac{\Delta_{qrv}}{\Delta_{qrv}^2 + \eta^2}  \right) W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
+  &\Sigma_{pq}^{\text{qs}GW}(\eta) = \\
-\end{equation}
+  &\frac{1}{2} \sum_{r,v}  \left( \frac{\Delta_{prv}}{\Delta_{prv}^2 + \eta^2} +\frac{\Delta_{qrv}}{\Delta_{qrv}^2 + \eta^2}  \right) W_{p,(r,v)} W^{\dagger}_{(r,v),q}. \notag 
+\end{align}
 Yet, both approximation are closely related as they share the same diagonal terms when $\eta=0$.
 Also, note that the SRG static approximation is naturally Hermitian as opposed to the symmetrized case where it is enforced by symmetrization.
 
@@ -508,7 +510,8 @@ Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\et
 Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
 \begin{align}
   \label{eq:SRG_qsGW}
-  \Sigma_{pq}^{\text{SRG}}(s) =  \frac{1}{2} \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W_{q,(r,v)}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right)
+  &\Sigma_{pq}^{\text{SRG}}(s) =  \\
+  &\sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W_{q,(r,v)}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right) \notag 
 \end{align}
 which depends on one regularising parameter $s$ analogously to $\eta$ in the usual case.
 The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
@@ -541,7 +544,6 @@ This means that the cations used an unrestricted HF reference while the neutral
 \label{sec:results}
 %=================================================================%
 
-
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 \section{Conclusion}
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