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Pierre-Francois Loos 2022-02-23 14:34:09 +01:00
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Manuscript/ufGW.tex
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@ -148,29 +148,6 @@ The idea behind the $GW$ approximation is to recast the many-body problem into a
G = G_0 + G_0 \Sigma G
\end{equation}
Electron correlation is then explicitly incorporated into one-body quantities via a sequence of self-consistent steps known as Hedin's equations. \cite{Hedin_1965}
%which connect $G$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$, and $\Sigma$ through a set of five equations.
%\begin{subequations}
%\begin{align}
% \label{eq:G}
% & G(12) = G_0(12) + \int G_\text{H}(13) \Sigma(34) G(42) d(34),
% \\
% \label{eq:Gamma}
% & \Gamma(123) = \delta(12) \delta(13)
% \notag
% \\
% & \qquad \qquad + \int \fdv{\Sigma(12)}{G(45)} G(46) G(75) \Gamma(673) d(4567),
% \\
% \label{eq:P}
% & P(12) = - i \int G(13) \Gamma(324) G(41) d(34),
% \\
% \label{eq:W}
% & W(12) = v(12) + \int v(13) P(34) W(42) d(34),
% \\
% \label{eq:Sig}
% & \Sigma(12) = i \int G(13) W(14) \Gamma(324) d(34),
%\end{align}
%\end{subequations}
%where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\boldsymbol{r}_1,t_1)$.
In recent studies, \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021} we discovered that one can observe (unphysical) irregularities and/or discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, fundamental and optical gaps, total and correlation energies, as well as excitation energies) even in the weakly-correlated regime.
These issues were discovered in Ref.~\onlinecite{Loos_2018b} while studying a model two-electron system \cite{Seidl_2007,Loos_2009a,Loos_2009c} and they were further investigated in Ref.~\onlinecite{Veril_2018}, where we provided additional evidences and explanations of these undesirable features in real molecular systems.
@ -442,14 +419,14 @@ We have found that $\eta = 1$ is a good compromise that does not alter significa
This value can be certainly refined for specific applications.
To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized and non-regularized quasiparticle energies as functions of $\RHH$ for each orbital.
The principal observation is that, in the absence of intruder states, the regularization induces an error below \SI{10}{\milli\eV} ($p = 1$ and $p = 2$), which is practically variable .
The principal observation is that, in the absence of intruder states, the regularization induces an error below \SI{10}{\milli\eV} for the HOMO and LUMO ($p = 1$ and $p = 2$), which is practically viable.
Of course, in the troublesome regions ($p = 3$ and $p = 4$), the correction brought by the regularization procedure is larger but it has the undeniable advantage to provide smooth curves.
As a final example, we report in Fig.~\ref{fig:F2} the ground-state potential energy surface of the \ce{F2} molecule obtained at various levels of theory with the cc-pVDZ basis.
In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol detailed in Ref.~\onlinecite{Loos_2020e}.
These results are compared to high-level coupled-cluster calculations \cite{Purvis_1982,Christiansen_1995b} extracted from the same work.
As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' and it is clear that the regularization scheme (black curve) allows to smooth it out without significantly altering the overall accuracy.
Morevoer, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface.
As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' around the equilibrium bond length and it is clear that the regularization scheme (black curve) allows to smooth it out without significantly altering the overall accuracy.
Moreover, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface (gray curve).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concluding remarks}
@ -471,7 +448,6 @@ This project has received funding from the European Research Council (ERC) under
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article.% and its supplementary material.
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\bibliography{ufGW}
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