Merge branch 'master' of git.irsamc.ups-tlse.fr:loos/SRGGW

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Antoine Marie 2023-02-21 09:41:09 +01:00
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@ -195,12 +195,12 @@ where $\eta$ is a positive infinitesimal and the screened two-electron integrals
with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie without exchange) RPA problem defined as with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie without exchange) RPA problem defined as
\begin{equation} \begin{equation}
\label{eq:full_dRPA} \label{eq:full_dRPA}
\mqty( \bA & \bB \\ -\bB & -\bA ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ), \mqty( \bA & \bB \\ -\bB & -\bA ) \mqty( \bX & \bY \\ \bY & \bX ) = \mqty( \bX & \bY \\ \bY & \bX ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
\end{equation} \end{equation}
with with
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
A_{ia,jb} & = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, A_{ia,jb} & = (\epsilon_a - \epsilon_i) \delta_{ij}\delta_{ab} + \eri{ib}{aj},
\\ \\
B_{ia,jb} & = \eri{ij}{ab}, B_{ia,jb} & = \eri{ij}{ab},
\end{align} \end{align}
@ -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},
@ -635,7 +639,7 @@ Note that, after this transformation, the form of the regularizer is actually cl
%=================================================================% %=================================================================%
% Reference comp det % Reference comp det
Our set is composed by closed-shell organic molecules that correspond to the 50 smallest atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015} Our set of molecules is composed by closed-shell organic compounds that correspond to the 50 smallest atoms and molecules of the $GW$100 benchmark set. \cite{vanSetten_2015}
Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR} Following the same philosophy as the \textsc{quest} database for neutral excited states, \cite{Loos_2020d,Veril_2021} their geometries have been optimized at the CC3 level \cite{Christiansen_1995b,Koch_1997} in the aug-cc-pVTZ basis set using the \textsc{cfour} program. \cite{CFOUR}
The reference CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively. The reference CCSD(T) principal ionization potentials (IPs) and electron affinities (EAs) have been obtained using Gaussian 16 \cite{g16} with default parameters, that is, within the restricted and unrestriced HF formalism for the neutral and charged species, respectively.
@ -647,7 +651,7 @@ We use (restricted) HF guess orbitals and energies for all self-consistent $GW$
The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively. The maximum size of the DIIS space \cite{Pulay_1980,Pulay_1982} and the maximum number of iterations were set to 5 and 64, respectively.
In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme. In practice, one may achieve convergence, in some cases, by adjusting these parameters or by using an alternative mixing scheme.
However, in order to perform a black-box comparison, these parameters have been fixed to these default values. However, in order to perform a black-box comparison, these parameters have been fixed to these default values.
\titou{The $\eta$ value used in the convetional $G_0W_0$ and $qsGW$ calculations corresponds to the largest value where one succesfully converges all systems.} \titou{The $\eta$ value used in the convetional $G_0W_0$ and qs$GW$ calculations corresponds to the largest value where one successfully converges all systems.}
%=================================================================% %=================================================================%
\section{Results} \section{Results}