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Pierre-Francois Loos 2022-02-23 14:24:28 +01:00
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12
Manuscript/ufGW.tex
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@ -353,7 +353,7 @@ Inspection of their corresponding eigenvectors reveals that the $(\Ne+1)$-electr
By construction, the quasiparticle solution diabatically follows the reference determinant $\ket*{1\Bar{1}3}$ through the avoided crossing (thick lines in Fig.~\ref{fig:H2_zoom}) which is precisely the origin of the energetic discontinuity.
A similar scenario is at play in the region around $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ (right panel of Fig.~\ref{fig:H2_zoom}) but it now involves three solutions ($s = 5$, $s = 6$, and $s = 7$).
The electronic configurations of the Slater determinant involved are the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration \titou{$\ket*{1\Bar{?}?}$} and \titou{$\ket*{1\Bar{?}?}$}.
The electronic configurations of the Slater determinant involved are the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration $\ket*{1\Bar{2}3}$ and $\ket*{12\Bar{3}}$.
These forms two avoided crossings in rapid successions, which create two discontinuities in the energy surface (see Fig.~\ref{fig:H2}).
In this region, although the ground-state wave function is well described by the $\Ne$-electron HF determinant, a situation that can be safely labeled as single-reference, one can see that the $(\Ne+1)$-electron state involves three Slater determinants and can then be labeled as a multi-reference (or strongly-correlated) situation with near-degenerate electronic configurations.
Therefore, one can conclude that this downfall of $GW$ is a key signature of strong correlation in the $(\Ne\pm1)$-electron states that yields a significant redistribution of weights amongst electronic configurations.
@ -380,7 +380,6 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
% FIGURE 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
% \includegraphics[width=\linewidth]{fig4a}
\includegraphics[width=\linewidth]{fig4}
\caption{
\label{fig:H2reg}
@ -438,16 +437,17 @@ Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasipa
It clearly shows how the regularization of the $GW$ self-energy diabatically linked the two solutions to get rid of the discontinuities.
However, this diabatization is more or less accurate depending on the value of $\eta$.
For $\eta = 10$, the value is clearly too large inducing a large difference between the two sets of quasiparticle energies (purple curves).
For $\eta = 0.1$, we have the opposite scenario where the value is too small and some irregularities remain (green curves).
For $\eta = 0.1$, we have the opposite scenario where $\eta$ is too small and some irregularities remain (green curves).
We have found that $\eta = 1$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions.
\titou{This values could be further refined for specfici applications.}
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 $\RHH$.
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 .
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.

25
Notebooks/GW_conundrum.nb
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