modifs in IVD

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Pierre-Francois Loos 2023-02-14 12:12:50 -05:00
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@ -585,44 +585,42 @@ Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and her
Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible. Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible.
However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm}. However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm}.
This yields a $s$-dependent static self-energy which matrix elements read This yields a $s$-dependent static self-energy which matrix elements read
\begin{equation} \begin{multline}
\label{eq:SRG_qsGW} \label{eq:SRG_qsGW}
\begin{split}
\Sigma_{pq}^{\SRGqsGW}(s) \Sigma_{pq}^{\SRGqsGW}(s)
& = F_{pq}^{(2)}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}
\\ \\
& = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu} \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ].
\qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ], \end{multline}
\end{split} Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally Hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization.
\end{equation} Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator.
Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by brute-force symmetrization. Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms.
Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every denominator of the self-energy.
Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
It is well-known that in traditional qs$GW$ calculations, increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable. It is well-known that in traditional qs$GW$ calculations, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
Similarly, in SRG-qs$GW$, one might need to decrease the value of $s$ to ensure convergence. Similarly, in SRG-qs$GW$, one might need to decrease the value of $s$ to ensure convergence.
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 $\tilde{\bF}$. Indeed, the fact that SRG-qs$GW$ calculations do not always converge in the large-$s$ limit is expected as, in this limit, potential intruder states have been included.
Therefore, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states. Therefore, one should use a value of $s$ large enough to include as many states as possible but small enough to avoid intruder states.
It is instructive to plot the functional form of both regularizing functions, this is done in Fig.~\ref{fig:plot}. It is instructive to plot the functional form of both regularizing functions (see Fig.~\ref{fig:plot}).
The surfaces are plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first order Taylor expansion around $(0,0)$ of both functional forms is equal. These have been plotted for a regularizing parameter value of $\eta=1$, where we have set $s=1/2\eta^2$ such that the first-order Taylor expansion around $(x,y) = (0,0)$ of both functional forms is equal.
One can observe that the SRG surface is much smoother than its qs counterpart. One can observe that the SRG-qs$GW$ surface is much smoother than its qs$GW$ counterpart.
This is due to the fact that the SRG functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities. This is due to the fact that the SRG-qs$GW$ functional at $\eta=0$, $f^{\SRGqsGW}(x,y;0)$, has fewer irregularities.
In fact, there is a single singularity at $x=y=0$. In fact, there is a single singularity at $x=y=0$.
On the other hand, the function $f^{\qsGW}(x,y;0)$ is singular on the two entire axes, $x=0$ and $y=0$. On the other hand, the function $f^{\qsGW}(x,y;0)$ is singular on the two entire axes, $x=0$ and $y=0$.
The smoothness of the SRG surface might induce a smoother convergence of SRG-qs$GW$ compared to qs$GW$. We believe that the smoothness of the SRG-qs$GW$ surface is the key feature that explains the smoother convergence of SRG-qs$GW$ compared to qs$GW$.
The convergence properties and the accuracy of both static approximations will be quantitatively gauged in Sec.~\ref{sec:results}. The convergence properties and the accuracy of both static approximations are quantitatively gauged in Sec.~\ref{sec:results}.
To conclude this section, the case of discontinuities will be briefly discussed. 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 at the $\GOWO$ level. 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}
So is it possible to use the SRG machinery developed above 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 while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the 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 $s$ values are suitable to avoid intruder states in its static part.
However, doing a change of variable such that However, performing the following bijective transformation
\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},
\end{align} % s = t/2 - \ln 2 - \ln[\sinh(t/2)]
will reverse the situation and now a finite value of $t$ will be well-designed to avoid discontinuities in the renormalized dynamic part. \end{align}}
The dynamic part after the change of variable is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}. reverse the situation and makes finite values of $t$ suitable to avoid discontinuities in the regularized dynamic part of the quasiparticle equation.
Note that, after this transformation, the form of the regularizer is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
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\section{Computational details} \section{Computational details}