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Manuscript/SRGGW.tex
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@ -249,7 +249,7 @@ which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfg
The corresponding matrix elements are The corresponding matrix elements are
\begin{equation} \begin{equation}
\label{eq:sym_qsGW} \label{eq:sym_qsGW}
\Sigma_{pq}^{\titou{\text{qs}}}(\eta) = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}} ) W_ {p,r\nu} \titou{W_{q,r\nu}}. \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}} ) W_ {p,r\nu} W_{q,r\nu}.
\end{equation} \end{equation}
with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level). with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy. One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy.
@ -265,6 +265,7 @@ One can deal with them by introducing \textit{ad hoc} regularizers.
Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift. Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory. Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
This is the central aim of the present work. This is the central aim of the present work.
\PFL{SRG is energy-dependent.}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
\section{The similarity renormalization group} \section{The similarity renormalization group}
@ -332,30 +333,29 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation. By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward. However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms in the process. A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms: \cite{Bintrim_2021,Tolle_2022}
The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022}
\begin{equation} \begin{equation}
\label{eq:GWlin} \label{eq:GWlin}
\begin{pmatrix} \begin{pmatrix}
\bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\ \bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
(\bW^{\text{2h1p}})^\dag & \bC^{\text{2h1p}} & \bO \\ (\bW^{\text{2h1p}})^\dag & \bC^{\text{2h1p}} & \bO \\
(\bW^{\text{2p1h}})^\dag & \bO & \bC^{\text{2p1h}} \\ (\bW^{\text{2p1h}})^\dag & \bO & \bC^{\text{2p1h}} \\
\end{pmatrix} \end{pmatrix},
\begin{pmatrix} % \begin{pmatrix}
\bZ^{\text{1h/1p}} \\ % \bZ^{\text{1h/1p}} \\
\bZ^{\text{2h1p}} \\ % \bZ^{\text{2h1p}} \\
\bZ^{\text{2p1h}} \\ % \bZ^{\text{2p1h}} \\
\end{pmatrix} % \end{pmatrix}
= % =
\begin{pmatrix} % \begin{pmatrix}
\bZ^{\text{1h/1p}} \\ % \bZ^{\text{1h/1p}} \\
\bZ^{\text{2h1p}} \\ % \bZ^{\text{2h1p}} \\
\bZ^{\text{2p1h}} \\ % \bZ^{\text{2p1h}} \\
\end{pmatrix} % \end{pmatrix}
\boldsymbol{\epsilon}, % \boldsymbol{\epsilon},
\end{equation} \end{equation}
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, the 2h1p and 2p1h matrix elements are %where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies,
where the 2h1p and 2p1h matrix elements are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu}, C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
@ -380,7 +380,7 @@ The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitio
\end{equation} \end{equation}
which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}. which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} \titou{yield exactly the same energies} but one is linear and the other is not. Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} yield exactly the quasiparticle and satellite energies but one is linear and the other is not.
The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is only $\order{K}$ in the latter. The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is only $\order{K}$ in the latter.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Refs.~\onlinecite{Tolle_2022,Scott_2023}). We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Refs.~\onlinecite{Tolle_2022,Scott_2023}).
@ -524,7 +524,7 @@ which can be solved by simple integration along with the initial condition $\bF^
At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends towards the following static limit At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends towards the following static limit
\begin{equation} \begin{equation}
\label{eq:static_F2} \label{eq:static_F2}
F_{pq}^{(2)}(\infty) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} \titou{W_{q,r\nu}}. F_{pq}^{(2)}(\infty) = \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}.
\end{equation} \end{equation}
Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero. Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
\titou{Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}(s)$.} \titou{Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}(s)$.}
@ -607,7 +607,7 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s
%%% %%% %%% %%% %%% %%% %%% %%%
This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set. This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set.
Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to (w.r.t.) the CCSD(T) reference value. Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to the CCSD(T) reference value.
The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.} The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
\PFL{Check Szabo\&Ostlund, section on Koopman's theorem.} \PFL{Check Szabo\&Ostlund, section on Koopman's theorem.}
The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference. The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference.