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Manuscript/GW-srDFT.tex
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@ -152,7 +152,7 @@
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Bath\'elemy Pradines}
\author{Barth\'el\'emy Pradines}
\affiliation{\LCT}
\affiliation{\ISCD}
\author{Anthony Scemama}
@ -173,7 +173,7 @@ Similar to other electron correlation methods, many-body perturbation theory met
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.
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.
We also compute the ionization potentials of the five canonical nucleobase (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}
\maketitle
@ -352,7 +352,7 @@ Within the {\GW} approximation, the correlation part of the self-energy reads
& + 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
\end{split}
\end{equation}
\jt{x should be defined} where $\eta$ is a positive infinitesimal.
\jt{x should be defined. If it is just an index for the eigenvector/eigenvalue, maybe we should call it $m$ instead?} where $\eta$ is a positive infinitesimal.
The screened two-electron integrals
\begin{equation}
[pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x}
@ -420,7 +420,7 @@ 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 short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{TouGorSav-TCA-05,PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) 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{TouGorSav-TCA-05,PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{FerGinTou-JCP-19,LooPraSceTouGin-JPCL-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} using the on-top pair density from the uniform-electron gas~\cite{LooPraSceTouGin-JPCL-19}.
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 and is computed using the opposite-spin on-top pair density.
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
@ -532,9 +532,9 @@ 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}
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.
The FC density-based correction is used consistently with the FC approximation in the {\GOWO} calculations.
The FC density-based basis set correction~\cite{LooPraSceTouGin-JPCL-19} 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.
@ -544,7 +544,7 @@ Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
\hspace{1cm}
\includegraphics[width=0.45\linewidth]{IP_G0W0PBE0_H2O}
\caption{
IP (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares) and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets \cite{Dun-JCP-89} (cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
IP (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares), and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets \cite{Dun-JCP-89} (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
The thick black line represents the CBS value obtained by extrapolation (see text for more details).
The green area corresponds to chemical accuracy (\textit{i.e.}, error below $1$ {\kcal} or $0.043$ eV).
\label{fig:IP_G0W0_H2O}
@ -558,13 +558,13 @@ Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
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.
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 set obtained at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels with increasingly larger Dunning's basis sets cc-pVXZ (X $=$ D, T, Q and 5) are reported in Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0}, respectively.
The IPs of the GW20 set obtained at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels with increasingly larger Dunning's basis sets cc-pVXZ (X $=$ D, T, Q, and 5) are reported in Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0}, respectively.
The corresponding statistical deviations (with respect to the CBS values) are also reported: mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX).
These reference CBS values have been obtained with the usual X$^{-3}$ extrapolation procedure using the three largest basis sets. \cite{Bruneval_2012}
@ -573,11 +573,11 @@ 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 increase the rate of convergence of IPs.
At the {\GOWO}@{\HF} (see Table \ref{tab:GW20_HF}), the MAD of the conventional calculations (\textit{i.e}, without basis set correction) is roughly divided by two each time one increases the basis set size (MADs of $0.60$, $0.24$, $0.10$ and $0.05$ eV going from cc-pVDZ to cc-pV5Z) with maximum errors higher $1$ eV for molecules such as \ce{HF}, \ce{H2O} and \ce{LiF} with the smallest basis set.
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 (\textit{i.e.}, 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 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.
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 example, we obtain MADs of $0.27$, $0.12$, $0.04$, and $0.01$ eV at the {\GOWO}@HF+srPBE level 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.
@ -586,7 +586,7 @@ In a nutshell, the present basis set correction provides cc-pVQZ quality results
Besides, it allows to reach chemical accuracy with the quadruple-$\zeta$ basis set, an accuracy that could not be reached even with the cc-pV5Z basis set for the conventional calculations.
Very similar conclusions are drawn at the {\GOWO}@{\PBEO} level (see Table \ref{tab:GW20_PBE0}) with a slightly faster convergence to the CBS limit.
For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV.
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. \jt{these numbers do not exactly correspond to the ones in the Table II. Is it because of round off?}
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.
@ -596,7 +596,7 @@ The potential reasons for this could be: i) potential-based DFT corrections are
%%% TABLE III %%%
\begin{table*}
\caption{
IPs (in eV) of the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
IPs (in eV) of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
The deviation with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are reported in square brackets.
The extrapolation error is reported in parenthesis.
The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purposes.
@ -665,7 +665,7 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
\begin{figure*}
\includegraphics[width=\linewidth]{DNA_IP}
\caption{
Error (in eV) with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values for the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
Error (in eV) with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values for the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
\label{fig:DNA_IP}
}
\end{figure*}
@ -674,7 +674,7 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
\subsection{Nucleobases}
\label{sec:DNA}
%%%%%%%%%%%%%%%%%%%%%%%%
In order to check the transferability of the present observations to larger systems, we have computed the values of the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine and uracil) at the {\GOWO}@PBE level of theory with a different basis set family. \cite{Weigend_2003a, Weigend_2005a}
In order to check the transferability of the present observations to larger systems, we have computed the values of the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) at the {\GOWO}@PBE level of theory with a different basis set family. \cite{Weigend_2003a, Weigend_2005a}
The numerical values are reported in Table \ref{tab:DNA_IP}, and their error with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values \cite{vanSetten_2015} (obtained via extrapolation of the def2-TZVP and def2-QZVP results) are shown in Fig.~\ref{fig:DNA_IP}.
The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.} \cite{Krause_2015} as well as the experimental results extracted from Ref.~\onlinecite{vanSetten_2015} are reported for comparison purposes.
@ -682,7 +682,7 @@ For these five systems, the IPs are all of the order of $8$ or $9$ eV with an am
The conclusions that we have drawn in the previous subsection do apply here as well.
For the smallest double-$\zeta$ basis def2-SVP, the basis set correction reduces by roughly half an eV the basis set incompleteness error.
It is particularly interesting to note that the basis-set corrected def2-TZVP results are on par with the correction-free def2-QZVP numbers.
This is quite remarkable as the number of basis functions jumps from $371$ to $777$ for the largest system guanine.
This is quite remarkable as the number of basis functions jumps from $371$ to $777$ for the largest system (guanine).
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
@ -690,7 +690,7 @@ This is quite remarkable as the number of basis functions jumps from $371$ to $7
%%%%%%%%%%%%%%%%%%%%%%%%
In the present manuscript, we have shown that the density-based basis set correction developed by some of the authors in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and applied recently to ground- and excited-state properties \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} can also be successfully applied to Green 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).
These findings have been observed for different {\GW} starting points (HF, PBE, and 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.