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JCTC_revision/GW-srDFT.tex
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@ -58,6 +58,7 @@
\newcommand{\Nbas}{N_\text{bas}}
\newcommand{\Nocc}{N_\text{occ}}
\newcommand{\Nvirt}{N_\text{virt}}
\newcommand{\Ngrid}{N_\text{grid}}
% energies
@ -233,7 +234,7 @@ Explicitly correlated F12 correction schemes have been derived for second-order
However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
In the present manuscript, we illustrate the performance of the density-based basis-set correction developed in Refs.~\onlinecite{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 gaps), 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.
In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis-set correction and its adaptation to {\GW} methods.
@ -359,7 +360,7 @@ 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}.
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 (with a local potential).
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
@ -474,16 +475,17 @@ Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
\infty, & \text{otherwise}, \\
\end{cases}
\end{equation}
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,
where, in this work, $f^{\Bas}(\br{},\br{}')$ and $\n{2}{\Bas}(\br{},\br{}')$ are calculated using the opposite-spin two-electron density matrix of a spin-restricted single determinant (such as HF and KS), for a closed-shell system,
\begin{equation}
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{}'),
\label{fBsum}
f^{\Bas}(\br{},\br{}') = 2 \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{}) (pi|qj)\MO{q}(\br{}') \MO{j}(\br{}'),
\end{equation}
and
\begin{eqnarray}
\n{2}{\Bas}(\br{},\br{}') &=& 2 \sum_{ij}^{\Nocc} \MO{i}(\br{})^2 \MO{j}(\br{}')^2
= \frac{1}{2} n^{\Bas}(\br{}) n^{\Bas}(\br{}'),
\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.
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 with a single-determinant wave function. Note that in Eq.~\eqref{fBsum} the indices $p$ and $q$ run over all occupied and virtual orbitals ($\Nbas= \Nocc+\Nvirt$ is the total dimension of the basis set).
}
\titou{Thanks to this definition, the effective interaction $W^{\Bas}(\br{},\br{}')$ has the interesting property
@ -503,7 +505,7 @@ Since the present basis-set correction employs complementary short-range correla
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 $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
\jt{In this work, we have tested two complementary density functionals coming from two approximations to the short-range correlation functional with multideterminant reference of RS-DFT~\cite{Toulouse_2005}. The first one is a short-range local-density approximation ($\srLDA$)~\cite{Toulouse_2005,Paziani_2006}
\jt{In this work, we have tested two complementary density functionals coming from two approximations to the short-range correlation functional with multideterminant (md) reference of RS-DFT~\cite{Toulouse_2005}. The first one is a short-range local-density approximation ($\srLDA$)~\cite{Toulouse_2005,Paziani_2006}
\begin{equation}
\label{eq:def_lda_tot}
\bE{\srLDA}{\Bas}[\n{}{}] =
@ -635,8 +637,8 @@ The FC density-based basis-set correction~\cite{Loos_2019} is used consistently
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}.
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}
Compared to the conventional $\order*{\Nocc^3 \Nvirt^3}$ computational cost of {\GW}, the present basis-set correction represents a marginal $\order*{\Nocc^2 \Nbas^2 \Ngrid}$ additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}.
Note, however, that the formal $\order*{\Nocc^3 \Nvirt^3}$ computational scaling 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 %%%
\begin{figure*}
@ -679,7 +681,7 @@ Even with the largest quintuple-$\zeta$ basis, the MAD is still above chemical a
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.
This observation is clear at the cc-pVDZ level but, for larger basis sets, the two RS-DFT-based corrections are essentially 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.
@ -688,18 +690,18 @@ Besides, it allows to reach chemical accuracy with the quadruple-$\zeta$ basis s
Very similar conclusions are drawn at the {\GOWO}@{\PBEO} level (see Table \ref{tab:GW20_PBE0}) with a slightly faster convergence to the CBS limit.
For example, at the {\GOWO}@PBE0+srLDA/cc-pVQZ level, the MAD is only $0.02$ eV with a maximum error as small as $0.09$ eV.
It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a more significant basis set reduction.
It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a larger acceleration of the basis-set convergence.
For example, we evidenced in Ref.~\onlinecite{Loos_2019} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets.
Here, the overall gain seems to be less important.
The potential reasons for this could be: i) potential-based DFT corrections are usually less accurate than the ones based directly on energies, \cite{Kim_2013} and ii) because the present scheme only corrects the basis set incompleteness error originating from the electron-electron cusp, some incompleteness remains at the HF or KS level. \cite{Adler_2007}
The possible reasons for this could be: i) DFT approximations are usually less accurate for the potential than for the energy, \cite{Kim_2013} and ii) because the present scheme only corrects the basis-set incompleteness error originating from the electron-electron cusp, some incompleteness remains at the HF or KS level. \cite{Adler_2007}
%%% 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 and corrections.
The deviation with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are reported in square brackets.
The extrapolation error is reported in parenthesis.
\titou{Extrapolated {\GOWO}@PBE results obtained with plane wave basis sets, as well as CCSD(T) and experimental results are reported for comparison purposes.}
\titou{Extrapolated {\GOWO}@PBE results obtained with plane-wave basis sets, as well as CCSD(T) and experimental results are reported for comparison.}
\label{tab:DNA_IP}
}
\begin{ruledtabular}
@ -729,8 +731,8 @@ The extrapolation error is reported in parenthesis.
\fnt[1]{This work.}
\fnt[2]{Unpublished data taken from \url{https://gw100.wordpress.com} obtained with TURBOMOLE v7.0.}
\fnt[3]{Extrapolated values obtained from the def2-TZVP and def2-QZVP values.}
\fnt[4]{\titou{Extrapolated plane wave results from Ref.~\onlinecite{Maggio_2017} obtained with WEST.}}
\fnt[5]{\titou{Extrapolated plane wave results from Ref.~\onlinecite{Govoni_2018} obtained with VASP.}}
\fnt[4]{\titou{Extrapolated plane-wave results from Ref.~\onlinecite{Maggio_2017} obtained with WEST.}}
\fnt[5]{\titou{Extrapolated plane-wave results from Ref.~\onlinecite{Govoni_2018} obtained with VASP.}}
\fnt[6]{\titou{CCSD(T)//CCSD/aug-cc-pVDZ results from Ref.~\onlinecite{Roca-Sanjuan_2006}.}}
\fnt[7]{Reference \onlinecite{Krause_2015}.}
\fnt[8]{Experimental values are taken from Ref.~\onlinecite{vanSetten_2015} and correspond to vertical ionization energies.}
@ -740,7 +742,7 @@ The extrapolation error is reported in parenthesis.
\begin{figure*}
\includegraphics[width=\linewidth]{fig2}
\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 and corrections.
\label{fig:DNA_IP}
}
\end{figure*}
@ -751,12 +753,12 @@ The extrapolation error is reported in parenthesis.
%%%%%%%%%%%%%%%%%%%%%%%%
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}.
\titou{Table \ref{tab:DNA_IP} also contains extrapolated IPs obtained with plane wave basis sets with two different software packages. \cite{Maggio_2017,Govoni_2018}
\titou{Table \ref{tab:DNA_IP} also contains extrapolated IPs obtained with plane-wave basis sets with two different software packages. \cite{Maggio_2017,Govoni_2018}
The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.} \cite{Krause_2015} on the same geometries, the CCSD(T)//CCSD/aug-cc-pVDZ results from Ref.~\onlinecite{Roca-Sanjuan_2006}, as well as the experimental results extracted from Ref.~\onlinecite{vanSetten_2015} are reported for comparison purposes.}
For these five systems, the IPs are all of the order of $8$ or $9$ eV with an amplitude of roughly $1$ eV between the smallest basis set (def2-SVP) and the CBS 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).
@ -768,9 +770,9 @@ In the present manuscript, we have shown that the density-based basis-set correc
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.}
Therefore, because srPBE is only slightly more computationally expensive than srLDA, we do recommend the use of srPBE.}
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 gaps.
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.
@ -778,7 +780,7 @@ We hope to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for \titou{the explicit expression of the short-range correlation functionals (and their functional derivatives)}, additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set, \titou{and
See {\SI} for \titou{the explicit expression of the short-range correlation potentials}, additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set, \titou{and
the numerical data of Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} (provided in txt and json formats).}
%%%%%%%%%%%%%%%%%%%%%%%%

6
Response_Letter/Response_Letter.tex
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@ -52,11 +52,11 @@ We look forward to hearing from you.
{As a corollary to this comment, the referee is still surprised that one may build a ``universal'' correction, in a sens that the same correction would apply to any approximation to the self-energy (if the referee understands correctly ...) whatever the diagrams used. If this is a correct statement, this should be emphasised and probably better commented.}
\\
\alert{This is indeed the case: the present basis-set correction can be apply to any self-energy.
However, the more accurate the self-energy, the faster the convergence, hence the choice of the GW self-energy in the present work.
However, the basis-set correction only corrects for the basis-set errors and not for the self-energy errors within the basis set.
We have clarified this point in the revised version of the manuscript (end of Section II.A).}
\item
{Minor: The referee is somehow surprised by the IP CCSD(T) values for cytosine and uracil in table III which are noticeably much larger than the experiment, in contrast with the other nucleobases. As a matter of fact the CCSD(T) values by Roca-Sanjuan et al (JCP 2006) agree reasonably with the values reported by the authors for adenine, guanine, thymine, but are completely off for cytosine and uracil. Could the authors check and potentially comment.}
{Minor: The referee is somehow surprised by the IP CCSD(T) values for cytosine and uracil in Table III which are noticeably much larger than the experiment, in contrast with the other nucleobases. As a matter of fact the CCSD(T) values by Roca-Sanjuan et al (JCP 2006) agree reasonably with the values reported by the authors for adenine, guanine, thymine, but are completely off for cytosine and uracil. Could the authors check and potentially comment.}
\\
\alert{After double checking the CCSD(T) IP values of cytosine and uracil, we can affirm that they have been correctly extracted from the work of Krause, Harding, and Klopper (see Table III).
An important point is that these CCSD(T) IPs have been obtained with the def2-TZVPP basis set (with the geometries of the GW100 test set which have been optimized at the PBE/def2-QZVP level of theory), a basis set which is larger than the aug-cc-pVDZ basis considered by Roca-Sanjuan et al.
@ -103,7 +103,7 @@ We look forward to hearing from you.
It may be interesting to include this in the comparison.}
\\
\alert{The work of Govoni and Galli ({10.1021/acs.jctc.7b00952}) reports indeed CBS values with a plane wave basis.
However, their data are computed with the PBE functional while table I and II are computed with HF and PBE0 respectively.
However, their data are computed with the PBE functional while Table I and II are computed with HF and PBE0 respectively.
Therefore, these data are not directly comparable.
However, we have included their values for the nucleobases (alongside other plane wave values obtained with VASP) in Table III and cited the corresponding references.}