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Manuscript/GW-srDFT.tex
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@ -48,32 +48,32 @@
\newcommand{\QP}{\textsc{quantum package}}
% methods
\newcommand{\evGW}{ev{\GW}}
\newcommand{\qsGW}{qs{\GW}}
\newcommand{\HF}{\text{HF}}
\newcommand{\KS}{\text{KS}}
\newcommand{\GOWO}{$G_0W_0$}
\newcommand{\evGW}{ev$GW$}
\newcommand{\qsGW}{qs$GW$}
\newcommand{\GW}{$GW$}
\newcommand{\GnWn}[1]{$G_{#1}W_{#1}$}
\newcommand{\Hxc}{\text{Hxc}}
\newcommand{\Hx}{\text{Hx}}
\newcommand{\xc}{\text{xc}}
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\Ne}{N}
% operators
\newcommand{\hH}{\Hat{H}}
% energies
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E_\text{HF}}
\newcommand{\EKS}{E_\text{KS}}
\newcommand{\EcK}{E_\text{c}^\text{Klein}}
\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
\newcommand{\EcGM}{E_\text{c}^\text{GM}}
\newcommand{\EcMP}{E_c^\text{MP2}}
\newcommand{\Egap}{E_\text{gap}}
\newcommand{\IP}{\text{IP}}
\newcommand{\EA}{\text{EA}}
\newcommand{\RH}{R_{\ce{H2}}}
\newcommand{\RF}{R_{\ce{F2}}}
\newcommand{\RBeO}{R_{\ce{BeO}}}
\newcommand{\CBS}{\text{CBS}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\PBE}{\text{PBE}}
\newcommand{\PBEO}{\text{PBE0}}
\newcommand{\srLDA}{\text{srLDA}}
\newcommand{\srPBE}{\text{srPBE}}
% orbital energies
\newcommand{\e}[1]{\epsilon_{#1}}
@ -82,17 +82,13 @@
\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
\newcommand{\beGOWO}[1]{\Bar{\epsilon}^\text{\GOWO}_{#1}}
\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
\newcommand{\de}[1]{\Delta\epsilon_{#1}}
\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
\newcommand{\Om}[1]{\Omega_{#1}}
\newcommand{\eHOMO}{\epsilon_\text{HOMO}}
\newcommand{\eLUMO}{\epsilon_\text{LUMO}}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}
% Matrix elements
% Matrix elements and operators
\newcommand{\A}[1]{A_{#1}}
\newcommand{\B}[1]{B_{#1}}
\newcommand{\G}[2]{G_{#1}^{#2}}
@ -110,76 +106,33 @@
\newcommand{\bSig}[2]{\Bar{\Sigma}_{#1}^{#2}}
\newcommand{\Z}[1]{Z_{#1}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
\newcommand{\fc}{f_\text{c}}
\newcommand{\Vc}{V_\text{c}}
\newcommand{\vne}{v_\text{ne}}
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\MO}[1]{\phi_{#1}}
% Matrices
%\newcommand{\bG}{\boldsymbol{G}}
%\newcommand{\bW}{\boldsymbol{W}}
%\newcommand{\bvc}{\boldsymbol{v}}
%\newcommand{\bSig}{\boldsymbol{\Sigma}}
%\newcommand{\bSigX}{\boldsymbol{\Sigma}^\text{x}}
%\newcommand{\bSigC}{\boldsymbol{\Sigma}^\text{c}}
%\newcommand{\bSigGW}{\boldsymbol{\Sigma}^\text{\GW}}
%\newcommand{\be}{\boldsymbol{\epsilon}}
%\newcommand{\bDelta}{\boldsymbol{\Delta}}
%\newcommand{\beHF}{\boldsymbol{\epsilon}^\text{HF}}
%\newcommand{\beGW}{\boldsymbol{\epsilon}^\text{\GW}}
%\newcommand{\beGnWn}[1]{\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
%\newcommand{\bdeGnWn}[1]{\Delta\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
%\newcommand{\bde}{\boldsymbol{\Delta\epsilon}}
%\newcommand{\bdeHF}{\boldsymbol{\Delta\epsilon}^\text{HF}}
%\newcommand{\bdeGW}{\boldsymbol{\Delta\epsilon}^\text{GW}}
\newcommand{\bOm}{\boldsymbol{\Omega}}
\newcommand{\bA}{\boldsymbol{A}}
\newcommand{\bB}{\boldsymbol{B}}
\newcommand{\bX}{\boldsymbol{X}}
\newcommand{\bY}{\boldsymbol{Y}}
\newcommand{\bZ}{\boldsymbol{Z}}
\newcommand{\fc}{f_\text{c}}
\newcommand{\Vc}{V_\text{c}}
\newcommand{\MO}[1]{\phi_{#1}}
% coordinates
\renewcommand{\b}[1]{\mathbf{#1}}
\renewcommand{\d}{\text{d}}
% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
% coordinates
\newcommand{\br}[1]{\mathbf{r}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\Ne}{N}
\newcommand{\vne}{v_\text{ne}}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\E}[2]{E_{#1}^{#2}}
\newcommand{\DE}[2]{\Delta E_{#1}^{#2}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\DbE}[2]{\Delta \Bar{E}_{#1}^{#2}}
\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
%\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
%\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
\newcommand{\bec}[1]{\Bar{e}^{#1}}
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
%\newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\w}[2]{w_{#1}^{#2}}
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
\newcommand{\V}[2]{V_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
\newcommand{\HF}{\text{HF}}
\newcommand{\KS}{\text{KS}}
\newcommand{\Hxc}{\text{Hxc}}
\newcommand{\Hx}{\text{Hx}}
\newcommand{\xc}{\text{xc}}
% units
\newcommand{\IneV}[1]{#1 eV}
@ -277,10 +230,10 @@ Explicitly correlated F12 correction schemes have been derived for second-order
However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} on ionization potentials obtained within {\GOWO}.
Note that the the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavours of (self-consistent) {\GW} or Green's function-based methods, such as GF2 (and its higher-order variants).
Note that the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green's function-based methods, such as GF2 (and its higher-order variants).
Moreover, we are currently investigating the performances of the present approach for linear response theory, in order to speed up the convergence of excitation energies obtained within the random-phase approximation (RPA) \cite{Casida_1995, Dreuw_2005} and Bethe-Salpeter equation (BSE) formalism. \cite{Strinati_1988, Leng_2016, Blase_2018}
The paper is organised as follows.
The paper is organized as follows.
In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to {\GW} methods.
Results are reported and discussed in Sec.~\ref{sec:results}.
Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
@ -300,10 +253,10 @@ Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for
\E{0}{\Bas} = \min_{\n{}{} \in \cD^\Bas} \qty{ \F{}{}[n] + \int \vne(\br{}) \n{}{}(\br{}) \dbr{} },
\label{eq:E0B}
\end{equation}
where $\cD^\Bas$ is the set of $\Ne$-representable densities which can be extracted from a wave function $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$.
where $\cD^\Bas$ is the set of $\Ne$-representable densities which can be extracted from a wave function $\wf{}{\Bas}$ expandable in the Hilbert space generated by $\Bas$.
In this expression,
\begin{equation}
\F{}{}[n] = \min_{\Psi \rightsquigarrow \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\Psi}
\F{}{}[n] = \min_{\wf{}{} \rightsquigarrow \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
\end{equation}
is the exact Levy-Lieb universal density functional, where the notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E0B} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
$\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
@ -311,9 +264,9 @@ $\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators,
\F{}{}[\n{}{}] = \F{}{\Bas}[\n{}{}] + \bE{}{\Bas}[\n{}{}],
\label{eq:Fn}
\end{equation}
where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional \cite{Lev-PNAS-79, Lev-PRA-82, Lie-IJQC-83} with wave functions $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$
where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional \cite{Lev-PNAS-79, Lev-PRA-82, Lie-IJQC-83} with wave functions $\wf{}{\Bas}$ expandable in the Hilbert space generated by $\Bas$
\begin{equation}
\F{}{\Bas}[\n{}{}] = \min_{\Psi^\Bas \rightsquigarrow \n{}{}} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
\F{}{\Bas}[\n{}{}] = \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}} \mel*{\wf{}{\Bas}}{ \hT + \hWee{}}{\wf{}{\Bas}},
\end{equation}
and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. \cite{GinPraFerAssSavTou-JCP-18}
In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we re-express it with a constrained search over $\Ne$-representable one-electron Green's functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$
@ -321,7 +274,7 @@ In the present work, instead of using wave-function methods for calculating $\F{
\F{}{\Bas}[\n{}{}] = \min_{\G{}{\Bas} \rightsquigarrow \n{}{}} \Omega^\Bas[\G{}{\Bas}],
\label{eq:FBn}
\end{equation}
where $\Omega^\Bas[G]$ is chosen to be a Klein-like energy functional of the Green's function (see, \textit{e.g.}, Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
where $\Omega^\Bas[G]$ is chosen to be a Klein-like energy functional of the Green's function (see, \textit{e.g.}, Refs.~\onlinecite{SteLee-BOOK-13, MarReiCep-BOOK-16, DahLee-JCP-05, DahLeeBar-IJQC-05, DahLeeBar-PRA-06})
\begin{equation}
\Omega^\Bas[\G{}{}] = \Tr[\ln( - \G{}{} ) ] - \Tr[ (\Gs{\Bas})^{-1} \G{}{} - 1 ] + \Phi_\Hxc^\Bas[\G{}{}],
\label{eq:OmegaB}
@ -334,7 +287,7 @@ and
\begin{equation}
\Tr[A B] = \frac{1}{2\pi i} \int_{-\infty}^{+\infty} d\omega \, e^{i \omega 0^+} \iint A(\br{},\br{}',\omega) B(\br{}',\br{}{},\omega) \dbr{} \dbr{}'.
\end{equation}
In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green's function such as its functional derivatives yields the Hxc self-energy in the basis
In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green's function such as its functional derivatives yields the $\Hxc$ self-energy in the basis
\begin{equation}
\fdv{\Phi_\Hxc^\Bas[\G{}{}]}{\G{}{}(\br{},\br{}',\omega)} = \Sig{\Hxc}{\Bas}[\G{}{}](\br{},\br{}',\omega).
\end{equation}
@ -462,7 +415,7 @@ where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies
%%%%%%%%%%%%%%%%%%%%%%%%
The frequency-independent local self-energy $\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}')$ originates from the functional derivative of complementary basis-correction density functionals $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a local-density approximation ($\LDA$) functional with multideterminant reference \cite{PazMorGorBac-PRB-06} and a Perdew-Burke-Ernzerhof ($\PBE$) inspired correlation functional \cite{FerGinTou-JCP-19} which interpolates between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) inspired correlation functional \cite{FerGinTou-JCP-19} which interpolates between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
Additionally to the one-electron density, these RS-DFT functionals requires a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error.
We refer the interested reader to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are provided.
The basis set corrected {\GOWO} quasiparticle energies are thus given by
@ -482,32 +435,6 @@ with
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis set correction is a non-self-consistent, \textit{post}-GW correction.
Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent GW calculations.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%
All the geometries have been extracted from the GW100 set. \cite{vanSetten_2015}
Unless otherwise stated, all the {\GOWO} calculations have been performed with MOLGW developed by Bruneval and coworkers. \cite{Bruneval_2016a}
The HF, PBE and PBE0 calculations as well as the srLDA and srPBE basis set corrections have been computed with Quantum Package, \cite{QP2} which by default uses the SG-2 quadrature grid for the numerical integrations.
Frozen-core (FC) calculations are systematically performed.
The FC density-based correction is used consistently with the FC approximation in the {\GOWO} calculations.
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \textit{i.e.}, by solving the non-linear, frequency-dependent quasiparticle equation (without linearization).
Moreover, the infinitesimal $\eta$ has been set to zero.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Discussion}
\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we study a subset of atoms and molecules from the GW100 test set.
In particular, we study the 20 smallest molecules of the GW100 set, a subset that we label as GW20.
This subset has been recently considered by Lewis and Berkelbach to study the effect of vertex corrections to $\W{}{}$ on IPs of molecules. \cite{Lewis_2019a}
Later in this section, we also study the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) which are also part of the GW100 test set.
The IPs of the GW20 obtained at the {\GOWO}@HF and {\GOWO}@PBE0 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}
%%% TABLE I %%%
\begin{squeezetable}
\begin{table*}
@ -551,7 +478,6 @@ The mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximu
\end{table*}
\end{squeezetable}
%%% TABLE II %%%
\begin{squeezetable}
\begin{table*}
@ -595,10 +521,17 @@ The mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximu
\end{table*}
\end{squeezetable}
The convergence of the IPs of the water molecule with respect to the size of the basis set are depicted in Fig.~\ref{fig:IP_G0W0_H2O}.
This represents a typical example.
Additional graphs reporting the convergence of the IPs of the GW20 subset with respect to the size of the basis set are reported in the {\SI}.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%
All the geometries have been extracted from the GW100 set. \cite{vanSetten_2015}
Unless otherwise stated, all the {\GOWO} calculations have been performed with the MOLGW software developed by Bruneval and coworkers. \cite{Bruneval_2016a}
The HF, PBE and PBE0 calculations as well as the srLDA and srPBE basis set corrections have been computed with Quantum Package, \cite{QP2} which by default uses the SG-2 quadrature grid for the numerical integrations.
Frozen-core (FC) calculations are systematically performed.
The FC density-based correction is used consistently with the FC approximation in the {\GOWO} calculations.
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \textit{i.e.}, by solving the non-linear, frequency-dependent quasiparticle equation \eqref{eq:QP-G0W0} (without linearization).
Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
%%% FIG 1 %%%
\begin{figure*}
@ -612,8 +545,50 @@ Additional graphs reporting the convergence of the IPs of the GW20 subset with r
}
\end{figure*}
The values of the IPs of the five canonical nucleobases computed at the {\GOWO}@PBE level of theory for various basis sets are reported in Table \ref{tab:DNA_IP}.
Their error with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are shown in Fig.~\ref{fig:DNA_IP}.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Discussion}
\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we study a subset of atoms and molecules from the GW100 test set. \cite{vanSetten_2015}
In particular, we study the 20 smallest molecules of the GW100 set, a subset that we label as GW20.
This subset has been recently considered by Lewis and Berkelbach to study the effect of vertex corrections to $\W{}{}$ on IPs of molecules. \cite{Lewis_2019a}
Later in this section, we also study the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) which are also part of the GW100 test set.
%%%%%%%%%%%%%%%%%%%%%%%%
\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 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}.
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).
The correction brought by the short-range correlation functionals reduces by 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.
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.
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.
%%% TABLE III %%%
\begin{table*}
@ -630,17 +605,17 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
\cline{3-7}
Method & Basis & \tabc{Adenine} & \tabc{Cytosine} & \tabc{Guanine} & \tabc{Thymine} & \tabc{Uracil} \\
\hline
{\GOWO}@PBE\fnm[1] & def2-SVP & 7.27[-0.88] & 7.53[-0.92] & 6.95[-0.92] & 8.02[-0.85] & 8.38[-1.00] \\
{\GOWO}@PBE+srLDA\fnm[1] & def2-SVP & 7.60[-0.55] & 7.95[-0.50] & 7.29[-0.59] & 8.36[-0.51] & 8.80[-0.58] \\
{\GOWO}@PBE+srPBE\fnm[1] & def2-SVP & 7.64[-0.51] & 8.06[-0.39] & 7.34[-0.54] & 8.41[-0.45] & 8.91[-0.47] \\
{\GOWO}@PBE\fnm[1] & def2-TZVP & 7.74[-0.41] & 8.06[-0.39] & 7.45[-0.42] & 8.48[-0.38] & 8.86[-0.52] \\
{\GOWO}@PBE+srLDA\fnm[1] & def2-TZVP & 7.92[-0.23] & 8.26[-0.19] & 7.64[-0.23] & 8.67[-0.20] & 9.25[-0.13] \\
{\GOWO}@PBE+srPBE\fnm[1] & def2-TZVP & 7.92[-0.23] & 8.27[-0.18] & 7.64[-0.23] & 8.68[-0.19] & 9.27[-0.11] \\
{\GOWO}@PBE\fnm[2] & def2-QZVP & 7.98[-0.18] & 8.29[-0.16] & 7.69[-0.18] & 8.71[-0.16] & 9.22[-0.16] \\
{\GOWO}@PBE\fnm[3] & def2-TQZVP & 8.16(1) & 8.44(1) & 7.87(1) & 8.87(1) & 9.38(1) \\
{\GOWO}@PBE\fnm[1] & def2-SVP & $7.27$[$-0.88$] & $7.53$[$-0.92$] & $6.95$[$-0.92$] & $8.02$[$-0.85$] & $8.38$[$-1.00$] \\
{\GOWO}@PBE+srLDA\fnm[1] & def2-SVP & $7.60$[$-0.55$] & $7.95$[$-0.50$] & $7.29$[$-0.59$] & $8.36$[$-0.51$] & $8.80$[$-0.58$] \\
{\GOWO}@PBE+srPBE\fnm[1] & def2-SVP & $7.64$[$-0.51$] & $8.06$[$-0.39$] & $7.34$[$-0.54$] & $8.41$[$-0.45$] & $8.91$[$-0.47$] \\
{\GOWO}@PBE\fnm[1] & def2-TZVP & $7.74$[$-0.41$] & $8.06$[$-0.39$] & $7.45$[$-0.42$] & $8.48$[$-0.38$] & $8.86$[$-0.52$] \\
{\GOWO}@PBE+srLDA\fnm[1] & def2-TZVP & $7.92$[$-0.23$] & $8.26$[$-0.19$] & $7.64$[$-0.23$] & $8.67$[$-0.20$] & $9.25$[$-0.13$] \\
{\GOWO}@PBE+srPBE\fnm[1] & def2-TZVP & $7.92$[$-0.23$] & $8.27$[$-0.18$] & $7.64$[$-0.23$] & $8.68$[$-0.19$] & $9.27$[$-0.11$] \\
{\GOWO}@PBE\fnm[2] & def2-QZVP & $7.98$[$-0.18$] & $8.29$[$-0.16$] & $7.69$[$-0.18$] & $8.71$[$-0.16$] & $9.22$[$-0.16$] \\
{\GOWO}@PBE\fnm[3] & def2-TQZVP & $8.16(1)$ & $8.44(1)$ & $7.87(1)$ & $8.87(1)$ & $9.38(1)$ \\
\hline
CCSD(T)\fnm[4] & def2-TZVPP & 8.33 & 9.51 & 8.03 & 9.08 & 10.13 \\
Experiment\fnm[5] & & 8.48 & 8.94 & 8.24 & 9.20 & 9.68 \\
CCSD(T)\fnm[4] & def2-TZVPP & $8.33$ & $9.51$ & $8.03$ & $9.08$ & $10.13$ \\
Experiment\fnm[5] & & $8.48$ & $8.94$ & $8.24$ & $9.20$ & $9.68$ \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{This work.}
@ -693,10 +668,32 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%
\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}.
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.
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.
%%%%%%%%%%%%%%%%%%%%%%%%
\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 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).
As mentioned earlier, the present basis set correction can be straightforwardly applied to other properties of interest such as electron affinities or fundamental gap.
It is also applicable to other flavors of {\GW} such as the partially self-consistent {\evGW} or {\qsGW} methods.
We are currently investigating the performances of the present approach within linear response theory in order to speed up the convergence of excitation energies obtained within the RPA and BSE formalisms.
We hope to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information Available}

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