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ok with Sec III

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Pierre-Francois Loos 2023-02-20 11:24:19 +01:00
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Manuscript/SRGGW.tex
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@ -488,9 +488,13 @@ and
% \end{align} % \end{align}
%\end{subequations} %\end{subequations}
Equation \eqref{eq:F0_C0} implies Equation \eqref{eq:F0_C0} implies
\begin{subequations}
\begin{align} \begin{align}
\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO, \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, &
\\
\bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
\end{align} \end{align}
\end{subequations}
and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
\begin{equation} \begin{equation}
W_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^2 s} W_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^2 s}
@ -499,7 +503,7 @@ At $s=0$, $W_{pq}^{\nu(1)}(s)$ reduces to the screened two-electron integrals de
\begin{equation} \begin{equation}
\lim_{s\to\infty} W_{pq}^{\nu(1)}(s) = 0. \lim_{s\to\infty} W_{pq}^{\nu(1)}(s) = 0.
\end{equation} \end{equation}
Therefore, $W_{pq}^{\nu(1)}(s)$ are genuine renormalized two-electron screened integrals. Therefore, $W_{pq}^{\nu(1)}(s)$ is a genuine renormalized two-electron screened integral.
It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}). It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}).
%///////////////////////////% %///////////////////////////%
@ -620,7 +624,7 @@ The convergence properties and the accuracy of both static approximations are qu
To conclude this section, we briefly discussed the case of discontinuities mentioned in Sec.~\ref{sec:intro}. To conclude this section, we briefly discussed the case of discontinuities mentioned in Sec.~\ref{sec:intro}.
Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023} Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
Is it then possible to rely on the SRG machinery to remove discontinuities? Is it then possible to rely on the SRG machinery to remove discontinuities?
Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite $s$ values are suitable to avoid intruder states in its static part. Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite value of $s$ is suitable to avoid intruder states in its static part.
However, performing the following bijective transformation However, performing the following bijective transformation
\titou{\begin{align} \titou{\begin{align}
e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t}, e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t},