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JCTC_revision/GW-srDFT.tex
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@ -168,8 +168,8 @@
\includegraphics[width=\linewidth]{TOC} \includegraphics[width=\linewidth]{TOC}
\end{wrapfigure} \end{wrapfigure}
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. 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.
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. 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.
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. 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.
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. 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.
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. 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.
\end{abstract} \end{abstract}
@ -198,7 +198,7 @@ The {\GW} approximation stems from the acclaimed Hedin's equations \cite{Hedin_1
& \qquad \qquad + \int \fdv{\Sig{}{}(12)}{\G{}{}(45)} \G{}{}(46) G(75) \Gam{}{}(673) d(4567), & \qquad \qquad + \int \fdv{\Sig{}{}(12)}{\G{}{}(45)} \G{}{}(46) G(75) \Gam{}{}(673) d(4567),
\\ \\
\label{eq:P} \label{eq:P}
& \Po{}{}(12) = - i \int G(13) \Gam{}{}(324) G(41) d(34), & \Po{}{}(12) = - i \int G(13) G(41) \Gam{}{}(342) d(34),
\\ \\
\label{eq:W} \label{eq:W}
& \W{}{}(12) = \vc{}{}(12) + \int \vc{}{}(13) \Po{}{}(34) \W{}{}(42) d(34), & \W{}{}(12) = \vc{}{}(12) + \int \vc{}{}(13) \Po{}{}(34) \W{}{}(42) d(34),
@ -215,8 +215,8 @@ Within the {\GW} approximation, one bypasses the calculation of the vertex corre
\Gam{}{}(123) \stackrel{GW}{\approx} \delta(12) \delta(13). \Gam{}{}(123) \stackrel{GW}{\approx} \delta(12) \delta(13).
\end{equation} \end{equation}
Depending on the degree of self-consistency one is willing to perform, there exists several types of {\GW} calculations. \cite{Loos_2018} Depending on the degree of self-consistency one is willing to perform, there exists several types of {\GW} calculations. \cite{Loos_2018}
The simplest and most popular variant of {\GW} is perturbative {\GW}, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986} The simplest and most popular variant of {\GW} is perturbative {\GW} (or {\GOWO}). \cite{Hybertsen_1985a, Hybertsen_1986}
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} 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}
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} 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}
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. 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.
@ -230,11 +230,11 @@ As shown in recent studies on both ground- and excited-state properties, \cite{L
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} 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}
However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet. 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{Giner_2018, Loos_2019, Giner_2019} on ionization potentials obtained within {\GOWO}. 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}.
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). 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).
The paper is organized 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. In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis-set correction and its adaptation to {\GW} methods.
Results for a large collection of molecular systems are reported and discussed in Sec.~\ref{sec:results}. Results for a large collection of molecular systems are reported and discussed in Sec.~\ref{sec:results}.
Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}. Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
Unless otherwise stated, atomic units are used throughout. Unless otherwise stated, atomic units are used throughout.
@ -245,7 +245,7 @@ Unless otherwise stated, atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{MBPT with DFT basis set correction} \subsection{MBPT with DFT basis-set correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 $\Bas$ 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 $\Bas$
@ -325,7 +325,7 @@ and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from the
with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
This is found from Eq.~\eqref{eq:stat} by using the chain rule, This is found from Eq.~\eqref{eq:stat} by using the chain rule,
\begin{equation} \begin{equation}
\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{}'', \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{}'',
\end{equation} \end{equation}
and and
\begin{equation} \begin{equation}
@ -347,9 +347,9 @@ The Dyson equation \eqref{eq:Dyson} can also be written with an arbitrary refere
where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$. where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$.
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. 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.
\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. \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.
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} 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}
Note, however, that the basis set correction is optimal for the \textit{exact} self-energy.} 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.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The {\GW} Approximation} \subsection{The {\GW} Approximation}
@ -358,7 +358,7 @@ Note, however, that the basis set correction is optimal for the \textit{exact} s
In this subsection, we provide the minimal set of equations required to describe {\GOWO}. In this subsection, we provide the minimal set of equations required to describe {\GOWO}.
More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}. More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
For the sake of simplicity, we only give the equations for closed-shell systems with a $\KS$ single-particle reference. For the sake of simplicity, we only give the equations for closed-shell systems with a $\KS$ single-particle reference.
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. 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.
Within the {\GW} approximation, the correlation part of the self-energy reads Within the {\GW} approximation, the correlation part of the self-energy reads
\begin{equation} \begin{equation}
@ -372,10 +372,10 @@ Within the {\GW} approximation, the correlation part of the self-energy reads
& + 2 \sum_{a}^{\Nvirt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta}, & + 2 \sum_{a}^{\Nvirt} \sum_{m} \frac{[pa|m]^2}{\omega - \e{a} - \Om{m} + i \eta},
\end{split} \end{split}
\end{equation} \end{equation}
where $m$ labels excited states and $\eta$ is a positive infinitesimal. 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.
The screened two-electron integrals The screened two-electron integrals
\begin{equation} \begin{equation}
[pq|m] = \sum_{ia} (pq|ia) (\bX_m+\bY_m)_{ia} [pq|m] = \sum_{i}^{\Nocc} \sum_{a}^{\Nvirt} (pq|ia) (\bX_m+\bY_m)_{ia}
\end{equation} \end{equation}
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994} are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994}
\begin{equation} \begin{equation}
@ -407,7 +407,7 @@ with
B_{ia,jb} & = 2 (ia|jb), B_{ia,jb} & = 2 (ia|jb),
\end{align} \end{align}
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook} and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which represent the poles of the screened Coulomb interaction $\W{}{}(\omega)$. Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which correspond to the poles of the screened Coulomb interaction $\W{}{}(\omega)$.
The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018} The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
\begin{equation} \begin{equation}
@ -420,30 +420,34 @@ with the largest renormalization weight (or factor)
\Z{p} = \qty[ 1 - \left. \pdv{\Re[\Sig{\text{c},p}{\Bas}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}. \Z{p} = \qty[ 1 - \left. \pdv{\Re[\Sig{\text{c},p}{\Bas}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}.
\end{equation} \end{equation}
Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight. Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
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 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
\begin{equation} \begin{equation}
\Pot{\xc,p}{\Bas} = \int \MO{p}(\br{}) \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{}) \dbr{}. \Pot{\xc,p}{\Bas} = \int \MO{p}(\br{}) \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{}) \dbr{},
\end{equation} \end{equation}
where $\pot{\xc}{\Bas}(\br{})$ is the KS exchange-correlation potential.
In particular, the ionization potential (IP) and electron affinity (EA) are extracted thanks to the following relationships: \cite{SzaboBook} In particular, the ionization potential (IP) and electron affinity (EA) are extracted thanks to the following relationships: \cite{SzaboBook}
\begin{align} \begin{align}
\IP & = -\eGOWO{\HOMO}, \IP & = -\eGOWO{\HOMO},
& &
\EA & = -\eGOWO{\LUMO}, \EA & = -\eGOWO{\LUMO},
\end{align} \end{align}
where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies, respectively. where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO quasiparticle energies, respectively.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Basis set correction} \subsection{Basis-set correction}
\label{sec:BSC} \label{sec:BSC}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\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} \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}
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, }
\titou{
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,
\begin{equation} \begin{equation}
\iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}' \frac{1}{2} \iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
= =
\iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'. \frac{1}{2} \iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'.
\end{equation} \end{equation}
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 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
\begin{equation} \begin{equation}
\label{eq:Wapprox} \label{eq:Wapprox}
W^{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}')
@ -452,42 +456,42 @@ The derivation and properties of $W^{\Bas}(\br{},\br{}')$ are detailed in Ref.~\
+ \frac{\erf[\rsmu{}{\Bas}(\br{}') \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}} + \frac{\erf[\rsmu{}{\Bas}(\br{}') \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}}
}. }.
\end{equation} \end{equation}
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 $\abs*{\br{} - \br{}'} = 0$. 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 $\abs*{\br{} - \br{}'} = 0$.
Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
\begin{equation} \begin{equation}
\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} W^{\Bas}(\br{},\br{}). \rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} W^{\Bas}(\br{},\br{}).
\end{equation} \end{equation}
} }
\titou{Following Refs.~\onlinecite{Giner_2018,Loos_2019,Giner_2019}, we adopt the following definition: \titou{Following Refs.~\onlinecite{Giner_2018,Loos_2019,Giner_2019}, we adopt the following definition for $W^{\Bas}(\br{},\br{}')$
\begin{equation} \begin{equation}
\label{eq:W} \label{eq:W}
W^{\Bas}(\br{},\br{}') = W^{\Bas}(\br{},\br{}') =
\begin{cases} \begin{cases}
f^{\Bas}(\br{},\br{}')/\n{2}{\Bas}(\br{},\br{}'), & \n{2}{\Bas}(\br{},\br{}') \neq 0, \\ f^{\Bas}(\br{},\br{}')/\n{2}{\Bas}(\br{},\br{}'), &\text{if } \n{2}{\Bas}(\br{},\br{}') \neq 0, \\
\infty, & \text{otherwise}, \\ \infty, & \text{otherwise}, \\
\end{cases} \end{cases}
\end{equation} \end{equation}
where, in the case of a single-determinant method (such as HF and KS-DFT), 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,
\begin{equation} \begin{equation}
f^{\Bas}(\br{},\br{}') = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{}) (pi|qj)\MO{q}(\br{}') \MO{j}(\br{}'), 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{}'),
\end{equation} \end{equation}
and and
\begin{equation} \begin{eqnarray}
\n{2}{\Bas}(\br{},\br{}') = n_{\uparrow}^{\Bas}(\br{}) n_{\downarrow}^{\Bas}(\br{}') \n{2}{\Bas}(\br{},\br{}') &=& 2 \sum_{ij}^{\Nocc} \MO{i}(\br{})^2 \MO{j}(\br{}')^2
\end{equation} = \frac{1}{2} n^{\Bas}(\br{}) n^{\Bas}(\br{}'),
is the opposite-spin pair density in a closed-shell system, and $n_{\sigma}^{\Bas}(\br{})$ is the spin-$\sigma$ one-electron density.} \end{eqnarray}
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.
}
\titou{Because of this definition, the effective interaction $W^{\Bas}(\br{},\br{}')$ has some interesting properties. \titou{Thanks to this definition, the effective interaction $W^{\Bas}(\br{},\br{}')$ has the interesting property
For example, we have
\begin{equation} \begin{equation}
\lim_{\Bas \to \CBS} W^{\Bas}(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1}, \lim_{\Bas \to \CBS} W^{\Bas}(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1},
\end{equation} \end{equation}
which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb interaction. which means that in the CBS limit one recovers the genuine (divergent) Coulomb interaction.
Therefore, in the CBS limit, the coalescence value $W^{\Bas}(\br{},\br{})$ goes to infinity, and so does $\rsmu{}{\Bas}(\br{})$. Therefore, in the CBS limit, the coalescence value $W^{\Bas}(\br{},\br{})$ goes to infinity, and so does $\rsmu{}{\Bas}(\br{})$.
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, 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.
the present basis set correction vanishes in the CBS limit. %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.
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.
} }
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
@ -502,7 +506,7 @@ We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner
\titou{The explicit expressions of these two short-range correlation functionals, as well as their corresponding potentials, are provided in the {\SI}.} \titou{The explicit expressions of these two short-range correlation functionals, as well as their corresponding potentials, are provided in the {\SI}.}
The basis set corrected {\GOWO} quasiparticle energies are thus given by The basis-set corrected {\GOWO} quasiparticle energies are thus given by
\begin{equation} \begin{equation}
\beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas} \beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas}
\label{eq:QP-corrected} \label{eq:QP-corrected}
@ -516,7 +520,7 @@ with
& = \int \MO{p}(\br{}) \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{}) \dbr{}. & = \int \MO{p}(\br{}) \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{}) \dbr{}.
\end{split} \end{split}
\end{equation} \end{equation}
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis set correction is a non-self-consistent, \textit{post}-{\GW} correction. 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. Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent {\GW} calculations.
%%% TABLE I %%% %%% TABLE I %%%
@ -611,13 +615,13 @@ The mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximu
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
All the geometries have been extracted from the GW100 set. \cite{vanSetten_2015} 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} 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. 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. Frozen-core (FC) calculations are systematically performed.
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). 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. 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} 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 %%% %%% 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. 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}. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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} \section{Conclusion}
\label{sec: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 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 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). 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. \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.} 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.} 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 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. We hope to report on this in the near future.