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Julien Toulouse
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@ -168,8 +168,8 @@
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\includegraphics[width=\linewidth]{TOC}
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\end{wrapfigure}
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Similar to other electron correlation methods, many-body perturbation theory methods based on Green functions, such as the so-called $GW$ approximation, suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
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This displeasing feature is due to lack of explicit electron-electron terms modeling the infamous Kato electron-electron cusp and the correlation Coulomb hole around it.
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Here, we propose a computationally efficient density-based basis set correction based on short-range correlation density functionals which significantly speeds up the convergence of energetics towards the complete basis set limit.
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This displeasing feature is due to the lack of explicit electron-electron terms modeling the infamous Kato electron-electron cusp and the correlation Coulomb hole around it.
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Here, we propose a computationally efficient density-based basis-set correction based on short-range correlation density functionals which significantly speeds up the convergence of energetics towards the complete basis set limit.
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The performance of this density-based correction is illustrated by computing the ionization potentials of the twenty smallest atoms and molecules of the GW100 test set at the perturbative $GW$ (or $G_0W_0$) level using increasingly large basis sets.
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We also compute the ionization potentials of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) and show that, here again, a significant improvement is obtained.
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\end{abstract}
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@ -198,7 +198,7 @@ The {\GW} approximation stems from the acclaimed Hedin's equations \cite{Hedin_1
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& \qquad \qquad + \int \fdv{\Sig{}{}(12)}{\G{}{}(45)} \G{}{}(46) G(75) \Gam{}{}(673) d(4567),
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\\
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\label{eq:P}
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& \Po{}{}(12) = - i \int G(13) \Gam{}{}(324) G(41) d(34),
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& \Po{}{}(12) = - i \int G(13) G(41) \Gam{}{}(342) d(34),
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\\
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\label{eq:W}
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& \W{}{}(12) = \vc{}{}(12) + \int \vc{}{}(13) \Po{}{}(34) \W{}{}(42) d(34),
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@ -215,8 +215,8 @@ Within the {\GW} approximation, one bypasses the calculation of the vertex corre
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\Gam{}{}(123) \stackrel{GW}{\approx} \delta(12) \delta(13).
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\end{equation}
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Depending on the degree of self-consistency one is willing to perform, there exists several types of {\GW} calculations. \cite{Loos_2018}
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The simplest and most popular variant of {\GW} is perturbative {\GW}, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986}
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Although obviously starting-point dependent, \cite{Bruneval_2013, Jacquemin_2016, Gui_2018} it has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
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The simplest and most popular variant of {\GW} is perturbative {\GW} (or {\GOWO}). \cite{Hybertsen_1985a, Hybertsen_1986}
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Although obviously starting-point dependent, \cite{Bruneval_2013, Jacquemin_2016, Gui_2018} it has been widely used in the literature to study solids, atoms, and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
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For finite systems such as atoms and molecules, partially \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} or fully self-consistent \cite{Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b} {\GW} methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
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Similar to other electron correlation methods, MBPT methods suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
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@ -230,11 +230,11 @@ As shown in recent studies on both ground- and excited-state properties, \cite{L
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Explicitly correlated F12 correction schemes have been derived for second-order Green function methods (GF2) \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018} by Ten-no and coworkers \cite{Ohnishi_2016, Johnson_2018} and Valeev and coworkers. \cite{Pavosevic_2017, Teke_2019}
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However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
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In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} on ionization potentials obtained within {\GOWO}.
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Note that the present basis set correction can be straightforwardly applied to other properties (\eg, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green function-based methods, such as GF2 (and its higher-order variants).
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In the present manuscript, we illustrate the performance of the density-based basis-set correction developed in Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} on ionization potentials obtained within {\GOWO}.
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Note that the present basis-set correction can be straightforwardly applied to other properties (\eg, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green function-based methods, such as GF2 (and its higher-order variants).
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The paper is organized as follows.
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In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to {\GW} methods.
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In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis-set correction and its adaptation to {\GW} methods.
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Results for a large collection of molecular systems are reported and discussed in Sec.~\ref{sec:results}.
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Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
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Unless otherwise stated, atomic units are used throughout.
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@ -245,7 +245,7 @@ Unless otherwise stated, atomic units are used throughout.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{MBPT with DFT basis set correction}
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\subsection{MBPT with DFT basis-set correction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Following Ref.~\onlinecite{Giner_2018}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\br{})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
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@ -325,7 +325,7 @@ and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from the
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with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
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This is found from Eq.~\eqref{eq:stat} by using the chain rule,
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\begin{equation}
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\pdv{\bE{}{\Bas}[\n{}{}]}{\G{}{}(\br{},\br{}',\omega)} = \int \pdv{\bE{}{\Bas}[\n{}{}]}{\n{}{}(\br{}'')} \frac{\delta \n{}{}(\br{}'')}{\delta \G{}{}(\br{},\br{}',\omega)} \dbr{}'',
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\frac{\delta \bE{}{\Bas}[\n{}{}]}{\delta \G{}{}(\br{},\br{}',\omega)} = \int \frac{\delta \bE{}{\Bas}[\n{}{}]}{\delta \n{}{}(\br{}'')} \frac{\delta \n{}{}(\br{}'')}{\delta \G{}{}(\br{},\br{}',\omega)} \dbr{}'',
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\end{equation}
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and
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\begin{equation}
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@ -347,9 +347,9 @@ The Dyson equation \eqref{eq:Dyson} can also be written with an arbitrary refere
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where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$.
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For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \Sig{\Hx}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham ($\KS$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \pot{\Hxc}{\Bas}(\br{}) \delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
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\titou{Note that the present basis set correction is applicable to any approximation of the self-energy (irrespectively of the diagrams included) without altering the CBS limit of such methods.
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Consequently, it can be applied, for example, to GF2 methods (also known as second Born approximation \cite{Stefanucci_2013} in the condensed matter community) or higher orders. \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018}
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Note, however, that the basis set correction is optimal for the \textit{exact} self-energy.}
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\titou{Note that the present basis-set correction is applicable to any approximation of the self-energy (irrespectively of the diagrams included) without altering the CBS limit of such methods.
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Consequently, it can be applied, for example, to GF2 methods (also known as second Born approximation \cite{Stefanucci_2013} in the condensed-matter community) or higher orders. \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018}
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Note, however, that the basis-set correction is optimal for the \textit{exact} self-energy within a given basis set, since it corrects only for the basis-set errors and not for the chosen approximate form of the self-energy within the basis set.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{The {\GW} Approximation}
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@ -358,7 +358,7 @@ Note, however, that the basis set correction is optimal for the \textit{exact} s
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In this subsection, we provide the minimal set of equations required to describe {\GOWO}.
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More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
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For the sake of simplicity, we only give the equations for closed-shell systems with a $\KS$ single-particle reference.
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The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals.
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The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then the $\KS$ orbitals and orbital energies.
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Within the {\GW} approximation, the correlation part of the self-energy reads
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\begin{equation}
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@ -372,10 +372,10 @@ Within the {\GW} approximation, the correlation part of the self-energy reads
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& + 2 \sum_{a}^{\Nvirt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta},
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\end{split}
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\end{equation}
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where $m$ labels excited states and $\eta$ is a positive infinitesimal.
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where $i$ runs over the $\Nocc$ occupied orbitals, $a$ runs over the $\Nvirt$ virtual orbitals, $m$ labels excited states (see below), and $\eta$ is a positive infinitesimal.
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The screened two-electron integrals
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\begin{equation}
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[pq|m] = \sum_{ia} (pq|ia) (\bX_m+\bY_m)_{ia}
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[pq|m] = \sum_{i}^{\Nocc} \sum_{a}^{\Nvirt} (pq|ia) (\bX_m+\bY_m)_{ia}
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\end{equation}
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are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994}
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\begin{equation}
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@ -407,7 +407,7 @@ with
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B_{ia,jb} & = 2 (ia|jb),
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\end{align}
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and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
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Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which represent the poles of the screened Coulomb interaction $\W{}{}(\omega)$.
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Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which correspond to the poles of the screened Coulomb interaction $\W{}{}(\omega)$.
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The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
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\begin{equation}
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@ -420,30 +420,34 @@ with the largest renormalization weight (or factor)
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\Z{p} = \qty[ 1 - \left. \pdv{\Re[\Sig{\text{c},p}{\Bas}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}.
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\end{equation}
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Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
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In Eq.~\eqref{eq:QP-G0W0}, $\Sig{\text{x},p}{\Bas} = \mel*{\MO{p}}{\Sig{\text{x}}{\Bas}}{\MO{p}}$ is the (static) exchange part of the self-energy and
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In Eq.~\eqref{eq:QP-G0W0}, $\Sig{\text{x},p}{\Bas} = \mel*{\MO{p}}{\Sig{\text{x}}{\Bas}}{\MO{p}}$ is the (static) HF exchange part of the self-energy and
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\begin{equation}
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\Pot{\xc,p}{\Bas} = \int \MO{p}(\br{}) \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{}) \dbr{}.
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\Pot{\xc,p}{\Bas} = \int \MO{p}(\br{}) \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{}) \dbr{},
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\end{equation}
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where $\pot{\xc}{\Bas}(\br{})$ is the KS exchange-correlation potential.
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In particular, the ionization potential (IP) and electron affinity (EA) are extracted thanks to the following relationships: \cite{SzaboBook}
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\begin{align}
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\IP & = -\eGOWO{\HOMO},
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&
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\EA & = -\eGOWO{\LUMO},
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\end{align}
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where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies, respectively.
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where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO quasiparticle energies, respectively.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Basis set correction}
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\subsection{Basis-set correction}
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\label{sec:BSC}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\titou{The fundamental idea behind the present basis set correction is to recognize that the singular two-electron Coulomb interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$, which ``resembles'' the long-range interaction operator $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$ used within RS-DFT. \cite{Giner_2018}
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We can therefore define an effective non-divergent two-electron interaction $W^{\Bas}(\br{},\br{}')$ which reproduces the expectation value of the Coulomb interaction over a given pair density $\n{2}{\Bas}(\br{},\br{}')$, \ie,
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\titou{The fundamental idea behind the present basis-set correction is to recognize that the singular two-electron Coulomb interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$, which ``resembles'' the long-range interaction operator $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$ used within RS-DFT. \cite{Giner_2018}
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}
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\titou{
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We start therefore by considering an effective non-divergent two-electron interaction $W^{\Bas}(\br{},\br{}')$ within the basis set which reproduces the expectation value of the Coulomb interaction over a given pair density $\n{2}{\Bas}(\br{},\br{}')$, \ie,
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\begin{equation}
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\iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
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\frac{1}{2} \iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
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=
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\iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'.
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\frac{1}{2} \iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'.
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\end{equation}
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The derivation and properties of $W^{\Bas}(\br{},\br{}')$ are detailed in Ref.~\onlinecite{Giner_2018}, but a key aspect that is worth mentioning here is that because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can fit $W^{\Bas}(\br{},\br{}')$ by a non-divergent, long-range interaction of the form
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The properties of $W^{\Bas}(\br{},\br{}')$ are detailed in Ref.~\onlinecite{Giner_2018}. A key aspect is that because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can approximate $W^{\Bas}(\br{},\br{}')$ by a non-divergent, long-range interaction of the form
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\begin{equation}
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\label{eq:Wapprox}
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W^{\Bas}(\br{},\br{}')
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@ -452,42 +456,42 @@ The derivation and properties of $W^{\Bas}(\br{},\br{}')$ are detailed in Ref.~\
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+ \frac{\erf[\rsmu{}{\Bas}(\br{}') \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}}
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}.
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\end{equation}
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The information about the finiteness of the basis set is then transferred to the range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$, and its value can be set by ensuring that the two sides of Eq.~\eqref{eq:Wapprox} are strictly equal at $\abs*{\br{} - \br{}'} = 0$.
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The information about the finiteness of the basis set is then transferred to the range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$, and its value can be determined by ensuring that the two sides of Eq.~\eqref{eq:Wapprox} are strictly equal at $\abs*{\br{} - \br{}'} = 0$.
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Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
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\begin{equation}
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\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} W^{\Bas}(\br{},\br{}).
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\end{equation}
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}
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\titou{Following Refs.~\onlinecite{Giner_2018,Loos_2019,Giner_2019}, we adopt the following definition:
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\titou{Following Refs.~\onlinecite{Giner_2018,Loos_2019,Giner_2019}, we adopt the following definition for $W^{\Bas}(\br{},\br{}')$
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\begin{equation}
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\label{eq:W}
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W^{\Bas}(\br{},\br{}') =
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\begin{cases}
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f^{\Bas}(\br{},\br{}')/\n{2}{\Bas}(\br{},\br{}'), & \n{2}{\Bas}(\br{},\br{}') \neq 0, \\
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f^{\Bas}(\br{},\br{}')/\n{2}{\Bas}(\br{},\br{}'), &\text{if } \n{2}{\Bas}(\br{},\br{}') \neq 0, \\
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\infty, & \text{otherwise}, \\
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\end{cases}
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\end{equation}
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where, in the case of a single-determinant method (such as HF and KS-DFT),
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where, in this work, $f^{\Bas}(\br{},\br{}')$ and $\n{2}{\Bas}(\br{},\br{}')$ are calculated with a spin-restricted single determinant (such as HF and KS) for a closed-shell system,
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\begin{equation}
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f^{\Bas}(\br{},\br{}') = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{}) (pi|qj)\MO{q}(\br{}') \MO{j}(\br{}'),
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f^{\Bas}(\br{},\br{}') = 2 \sum_{pq}^{\Nocc+\Nvirt} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{}) (pi|qj)\MO{q}(\br{}') \MO{j}(\br{}'),
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\end{equation}
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and
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\begin{equation}
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\n{2}{\Bas}(\br{},\br{}') = n_{\uparrow}^{\Bas}(\br{}) n_{\downarrow}^{\Bas}(\br{}')
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\end{equation}
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is the opposite-spin pair density in a closed-shell system, and $n_{\sigma}^{\Bas}(\br{})$ is the spin-$\sigma$ one-electron density.}
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\begin{eqnarray}
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\n{2}{\Bas}(\br{},\br{}') &=& 2 \sum_{ij}^{\Nocc} \MO{i}(\br{})^2 \MO{j}(\br{}')^2
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= \frac{1}{2} n^{\Bas}(\br{}) n^{\Bas}(\br{}'),
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\end{eqnarray}
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where $n^{\Bas}(\br{})$ is the one-electron density. The quantity $\n{2}{\Bas}(\br{},\br{}')$ represents the opposite-spin pair density of a closed-shell system.
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}
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\titou{Because of this definition, the effective interaction $W^{\Bas}(\br{},\br{}')$ has some interesting properties.
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For example, we have
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\titou{Thanks to this definition, the effective interaction $W^{\Bas}(\br{},\br{}')$ has the interesting property
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\begin{equation}
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\lim_{\Bas \to \CBS} W^{\Bas}(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1},
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\end{equation}
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which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb interaction.
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which means that in the CBS limit one recovers the genuine (divergent) Coulomb interaction.
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Therefore, in the CBS limit, the coalescence value $W^{\Bas}(\br{},\br{})$ goes to infinity, and so does $\rsmu{}{\Bas}(\br{})$.
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As the present basis set correction employs complementary short-range potentials from RS-DFT which have the property of going to zero when $\mu$ goes to infinity,
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the present basis set correction vanishes in the CBS limit.
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Note also that the divergence condition of $W^{\Bas}(\br{},\br{}')$ in Eq.~\eqref{eq:W} ensures that one-electron systems are free of correction.
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Since the present basis-set correction employs complementary short-range correlation potentials from RS-DFT which have the property of going to zero when $\mu$ goes to infinity, the present basis-set correction properly vanishes in the CBS limit.
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%Note also that the divergence condition of $W^{\Bas}(\br{},\br{}')$ in Eq.~\eqref{eq:W} ensures that one-electron systems are free of correction.
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}
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -502,7 +506,7 @@ We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner
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\titou{The explicit expressions of these two short-range correlation functionals, as well as their corresponding potentials, are provided in the {\SI}.}
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The basis set corrected {\GOWO} quasiparticle energies are thus given by
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The basis-set corrected {\GOWO} quasiparticle energies are thus given by
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\begin{equation}
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\beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas}
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\label{eq:QP-corrected}
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@ -516,7 +520,7 @@ with
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& = \int \MO{p}(\br{}) \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{}) \dbr{}.
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\end{split}
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\end{equation}
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As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis set correction is a non-self-consistent, \textit{post}-{\GW} correction.
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As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis-set correction is a non-self-consistent, \textit{post}-{\GW} correction.
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Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent {\GW} calculations.
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%%% TABLE I %%%
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@ -611,13 +615,13 @@ The mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximu
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%%%%%%%%%%%%%%%%%%%%%%%%
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All the geometries have been extracted from the GW100 set. \cite{vanSetten_2015}
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Unless otherwise stated, all the {\GOWO} calculations have been performed with the MOLGW software developed by Bruneval and coworkers. \cite{Bruneval_2016a}
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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.
|
||||
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.
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The FC density-based basis set correction~\cite{Loos_2019} is used consistently with the FC approximation in the {\GOWO} calculations.
|
||||
The FC density-based basis-set correction~\cite{Loos_2019} is used consistently with the FC approximation in the {\GOWO} calculations.
|
||||
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \ie, 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.
|
||||
|
||||
Compared to the conventional $\order*{\Nbas^6}$ computational cost of {\GW} (where $\Nbas$ is the number of basis functions), the present basis set correction represents a marginal additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}.
|
||||
Compared to the conventional $\order*{\Nbas^6}$ computational cost of {\GW} (where $\Nbas$ is the number of basis functions), the present basis-set correction represents a marginal additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}.
|
||||
Note, however, that the formal $\order*{\Nbas^6}$ cost of {\GW} can be significantly reduced thanks to resolution-of-the-identity techniques \cite{vanSetten_2013, Bruneval_2016, Duchemin_2017} and other tricks. \cite{Rojas_1995, Duchemin_2019}
|
||||
|
||||
%%% FIG 1 %%%
|
||||
@ -654,8 +658,8 @@ The convergence of the IP of the water molecule with respect to the basis set si
|
||||
This represents a typical example.
|
||||
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} (as well as Fig.~\ref{fig:IP_G0W0_H2O}) clearly evidence that the present basis set correction significantly increases 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 than $1$ eV for molecules such as \ce{HF}, \ce{H2O}, and \ce{LiF} with the smallest basis set.
|
||||
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 increases 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 than $1$ 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 (\ie, error below $1$ {\kcal} or $0.043$ eV).
|
||||
|
||||
For each basis set, the correction brought by the short-range correlation functionals reduces by roughly half or more the MAD, RMSD, and MAX compared to the correction-free calculations.
|
||||
@ -664,7 +668,7 @@ Interestingly, in most cases, the srPBE correction is slightly larger than the s
|
||||
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 provides cc-pVQZ quality results at the cc-pVTZ level.
|
||||
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.
|
||||
@ -738,7 +742,7 @@ The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.} \cite{Krause_2015} on
|
||||
|
||||
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.
|
||||
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 $371$ to $777$ for the largest system (guanine).
|
||||
|
||||
@ -746,13 +750,13 @@ This is quite remarkable as the number of basis functions jumps from $371$ to $7
|
||||
\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{Giner_2018} and applied recently to ground- and excited-state properties \cite{Loos_2019, Giner_2019} can also be successfully applied to Green 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.
|
||||
In the present manuscript, we have shown that the density-based basis-set correction developed by some of the authors in Ref.~\onlinecite{Giner_2018} and applied recently to ground- and excited-state properties \cite{Loos_2019, Giner_2019} can also be successfully applied to Green 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, and PBE0).
|
||||
\titou{We have observed that the performance of the two short-range correlation functionals (srLDA and srPBE) are quite similar with a slight edge for srPBE over srLDA.
|
||||
Therefore, because srPBE is only slightly more computationally expensive than srLDA, we do recommend the use of the former.}
|
||||
|
||||
As mentioned earlier, the present basis set correction can be straightforwardly applied to other properties of interest such as electron affinities or fundamental gap.
|
||||
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, \titou{and more generally to any approximation of the self-energy.}
|
||||
We are currently investigating the performance of the present approach within linear response theory in order to speed up the convergence of excitation energies obtained within the RPA and Bethe-Salpeter equation (BSE) \cite{Strinati_1988, Leng_2016, Blase_2018} formalisms.
|
||||
We hope to report on this in the near future.
|
||||
|
||||
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