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37
Manuscript/GW-srDFT.bib
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@ -1,13 +1,37 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-10-14 10:04:33 +0200
%% Created for Pierre-Francois Loos at 2019-10-14 10:43:41 +0200
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@article{Weigend_2003a,
Author = {Weigend, Florian and Furche, Filipp and Ahlrichs, Reinhart},
Date-Added = {2019-10-14 10:42:46 +0200},
Date-Modified = {2019-10-14 10:42:56 +0200},
Doi = {10.1063/1.1627293},
Journal = {J. Chem. Phys.},
Page = {12753-12762},
Title = {Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr},
Volume = {119},
Year = {2003},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1627293}}
@article{Weigend_2005a,
Author = {Weigend, Florian and Ahlrichs, Reinhart},
Date-Added = {2019-10-14 10:42:17 +0200},
Date-Modified = {2019-10-14 10:42:27 +0200},
Doi = {10.1039/b508541a},
Journal = {Phys. Chem. Chem. Phys.},
Page = {3297},
Title = {Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy},
Volume = {7},
Year = {2005},
Bdsk-Url-1 = {https://doi.org/10.1039/b508541a}}
@article{Jacquemin_2016,
Author = {D. Jacquemin and I. Duchemin and X. Blase},
Date-Added = {2019-10-14 10:02:38 +0200},
@ -17,10 +41,11 @@
Pages = {957},
Title = {Assessment Of The Convergence Of Partially Self- Consistent Bse/Gw Calculations},
Volume = {114},
Year = {2016}}
Year = {2016},
Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2015.1119901}}
@article{Marini_2009,
Author = {Andrea Marini and Conor Hogan and Myrta Gruning and Daniele Varsano },
Author = {Andrea Marini and Conor Hogan and Myrta Gruning and Daniele Varsano},
Date-Added = {2019-10-14 09:56:13 +0200},
Date-Modified = {2019-10-14 10:04:07 +0200},
Doi = {10.1016/j.cpc.2009.02.003},
@ -28,7 +53,8 @@
Pages = {1392},
Title = {Yambo: An Ab Initio Tool For Excited State Calculations},
Volume = {180},
Year = {2009}}
Year = {2009},
Bdsk-Url-1 = {https://doi.org/10.1016/j.cpc.2009.02.003}}
@article{Deslippe_2012,
Author = {Jack Deslippe and Georgy Samsonidze and David A. Strubbe and Manish Jain and Marvin L. Cohen and Steven G. Louie},
@ -39,7 +65,8 @@
Pages = {1269},
Title = {BerkeleyGW: A Massively Parallel Computer Package for the Calculation of the Quasiparticle and Optical Properties of Materials and Nanostructures},
Volume = {183},
Year = {2012}}
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1016/j.cpc.2011.12.006}}
@article{Lewis_2019a,
Author = {Alan M. Lewis and Timothy C. Berkelbach},

43
Manuscript/GW-srDFT.tex
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@ -540,8 +540,9 @@ Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
\hspace{1cm}
\includegraphics[width=0.45\linewidth]{IP_G0W0PBE0_H2O}
\caption{
IPs (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares) and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
IP (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares) and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets \cite{Dun-JCP-89} (cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
The thick black line represents the CBS value obtained by extrapolation (see text for more details).
The green area corresponds to chemical accuracy (\textit{i.e.}, error below $1$ {\kcal} or $0.043$ eV).
\label{fig:IP_G0W0_H2O}
}
\end{figure*}
@ -559,26 +560,26 @@ Later in this section, we also study the five canonical nucleobases (adenine, cy
\subsection{GW20}
\label{sec:GW20}
%%%%%%%%%%%%%%%%%%%%%%%%
The IPs of the GW20 obtained at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels with increasingly larger Dunning's basis sets cc-pVXZ (X $=$ D, T, Q and 5) are reported in Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0}, respectively.
The IPs of the GW20 set obtained at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels with increasingly larger Dunning's basis sets cc-pVXZ (X $=$ D, T, Q and 5) are reported in Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0}, respectively.
The corresponding statistical deviations (with respect to the CBS values) are also reported: mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX).
These reference CBS values have been obtained with the usual X$^{-3}$ extrapolation procedure using the three largest basis sets. \cite{Bruneval_2012}
The convergence of the IP of the water molecule with respect to the basis set size is depicted in Fig.~\ref{fig:IP_G0W0_H2O}.
This represents a typical example.
Additional graphs reporting the convergence of the IPs of each molecule of the GW20 subset are reported in the {\SI}.
Additional graphs reporting the convergence of the IPs of each molecule of the GW20 subset at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels are reported in the {\SI}.
Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} and Fig.~\ref{fig:IP_G0W0_H2O} clearly shows that the present basis set correction significantly increase the rate of convergence of IPs.
At the {\GOWO}@{\HF} (see Table \ref{tab:GW20_HF}), the MAD of the conventional calculations (\text{i.e}, without basis set correction) is roughly divided by two each time one increases the basis set size (MADs of $0.60$, $0.24$, $0.10$ and $0.05$ eV going from cc-pVDZ to cc-pV5Z) with maximum errors higher $1$ eV for molecules such as \ce{HF}, \ce{H2O} and \ce{LiF} with the smallest basis set.
Even with the largest basis quintuple-$\zeta$ basis set, the MAD is still above chemical accuracy (\text{i.e.}, error below $1$ {\kcal} or $0.043$ eV).
Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} (as well as Fig.~\ref{fig:IP_G0W0_H2O}) clearly evidence that the present basis set correction significantly increase the rate of convergence of IPs.
At the {\GOWO}@{\HF} (see Table \ref{tab:GW20_HF}), the MAD of the conventional calculations (\textit{i.e}, without basis set correction) is roughly divided by two each time one increases the basis set size (MADs of $0.60$, $0.24$, $0.10$ and $0.05$ eV going from cc-pVDZ to cc-pV5Z) with maximum errors higher $1$ eV for molecules such as \ce{HF}, \ce{H2O} and \ce{LiF} with the smallest basis set.
Even with the largest quintuple-$\zeta$ basis, the MAD is still above chemical accuracy (\textit{i.e.}, error below $1$ {\kcal} or $0.043$ eV).
The correction brought by the short-range correlation functionals reduces by half the MAD, RMSD and MAX compared to the correction-free calculations.
For each basis set, the correction brought by the short-range correlation functionals reduces by (roughly) half the MAD, RMSD and MAX compared to the correction-free calculations.
For example, we obtain MADs of $0.27$, $0.12$, $0.04$ and $0.01$ eV at the {\GOWO}@HF+srPBE with increasingly larger basis sets.
Interestingly, in most cases, the srPBE correction is slightly larger than the srLDA one.
This observation is clear at the cc-pVDZ level but, for larger basis sets, the two RS-DFT-based corrections are basically equivalent.
Note also that, in some cases, the corrected IPs slightly overshoot the CBS values.
However, it is hard to know if it is not due to the extrapolation error.
In a nutshell, the present basis set correction provide cc-pVQZ quality results at the cc-pVTZ level.
It also allowed to reach chemical accuracy with the quadruple-$\zeta$ basis set, an accuracy that could not be reached even with the cc-pV5Z basis set for the conventional calculations.
In a nutshell, the present basis set correction provides cc-pVQZ quality results at the cc-pVTZ level.
Besides, it allows to reach chemical accuracy with the quadruple-$\zeta$ basis set, an accuracy that could not be reached even with the cc-pV5Z basis set for the conventional calculations.
Very similar conclusions are drawn at the {\GOWO}@{\PBEO} level (see Table \ref{tab:GW20_PBE0}) with a slightly faster convergence to the CBS limit.
For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV.
@ -586,12 +587,12 @@ For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV
It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a more significant basis set reduction.
For example, we evidenced in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets.
Here, the overall gain seems to be less important.
The potential reasons for this could be: i) potential-based DFT correction are usually less accurate than the ones based directly on energies, and ii) because the present scheme only corrects the basis set incompleteness error originating from the electron-electron cusp, some incompleteness remains at the HF or KS level.
The potential reasons for this could be: i) potential-based DFT corrections are usually less accurate than the ones based directly on energies, and ii) because the present scheme only corrects the basis set incompleteness error originating from the electron-electron cusp, some incompleteness remains at the HF or KS level.
%%% TABLE III %%%
\begin{table*}
\caption{
IPs (in eV) of the five canonical nucleobases computed at the {\GOWO}@PBE level of theory for various basis sets.
IPs (in eV) of the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
The deviation with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are reported in square brackets.
The extrapolation error is reported in parenthesis.
The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purposes.
@ -670,20 +671,21 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
\subsection{Nucleobases}
\label{sec:DNA}
%%%%%%%%%%%%%%%%%%%%%%%%
In order to check the transferability of the present observations to larger systems, we have computed the values of the IPs of the five canonical nucleobases at the {\GOWO}@PBE level of theory with a different family of basis sets.
The numerical values are reported in Table \ref{tab:DNA_IP}, and their error with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values (obtained via extrapolation of the def2-TZVP and def2-QZVP results) are shown in Fig.~\ref{fig:DNA_IP}.
In order to check the transferability of the present observations to larger systems, we have computed the values of the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) at the {\GOWO}@PBE level of theory with a different basis set family. \cite{Weigend_2003a, Weigend_2005a}
The numerical values are reported in Table \ref{tab:DNA_IP}, and their error with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values \cite{vanSetten_2015} (obtained via extrapolation of the def2-TZVP and def2-QZVP results) are shown in Fig.~\ref{fig:DNA_IP}.
The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.} \cite{Krause_2015} as well as the experimental results extracted from Ref.~\onlinecite{vanSetten_2015} are reported for comparison purposes.
For these five systems, the IPs are all of the order of $8$ or $9$ eV with an amplitude of roughly $1$ eV between the smallest basis set (def2-SVP) and the CBS values.
The conclusions that we have drawn in the previous section do apply here as well.
For these five systems, the IPs are all of the order of $8$ or $9$ eV with an amplitude of roughly $1$ eV between the smallest basis set (def2-SVP) and the CBS value.
The conclusions that we have drawn in the previous subsection do apply here as well.
For the smallest double-$\zeta$ basis def2-SVP, the basis set correction reduces by roughly half an eV the basis set incompleteness error.
It is particularly interesting to note that the basis-set corrected def2-TZVP results are on par with the correction-free def2-QZVP numbers.
This is quite remarkable as the number of basis functions jumps from $371$ for a def2-TZVP calculation to $777$ for the largest system guanine.
This is quite remarkable as the number of basis functions jumps from $371$ to $777$ for the largest system guanine.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
In the present manuscript, we have shown that the density-based basis set correction developed by some of the authors in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and applied to ground- and excited-state properties \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} can also be successfully applied to Green's function methods such as {\GW}.
In the present manuscript, we have shown that the density-based basis set correction developed by some of the authors in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and applied recently to ground- and excited-state properties \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} can also be successfully applied to Green's function methods such as {\GW}.
In particular, we have evidenced that the present basis set correction (which relies on LDA- or PBE-based short-range correlation functionals) significantly speeds up the convergence of IPs for small and larger molecules towards the CBS limit.
These findings have been observed for different {\GW} starting points (HF, PBE or PBE0).
@ -693,13 +695,14 @@ We are currently investigating the performances of the present approach within l
We hope to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information Available}
\section*{Supporting Information}
%%%%%%%%%%%%%%%%%%%%%%%%
Additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set.
See {\SI} for additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
PFL would like to thank Fabien Bruneval for technical assistance. He also would like to thank Arjan Berger and Pina Romaniello for stimulating discussions.
PFL would like to thank Fabien Bruneval for technical assistance.
PFL and JT would like to thank Arjan Berger and Pina Romaniello for stimulating discussions.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.

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Manuscript/biblio.bib
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