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4
JCTC_revision/GW-srDFT.tex
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@ -536,7 +536,7 @@ with
\end{split}
\end{equation}
where $\bpot{}{\Bas}[\n{}{}](\br{})=\bpot{\srLDA}{\Bas}[\n{}{}](\br{})$ or $\bpot{\srPBE}{\Bas}[\n{}{}](\br{})$ and the density is calculated from the HF or KS orbitals.
The explicit expressions of these srLDA and srPBE correlation potentials are provided in the {\SI}.}
The expressions of these srLDA and srPBE correlation potentials are provided in the {\SI}.}
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.
@ -782,7 +782,7 @@ 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 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
See {\SI} for \titou{the 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).}
%%%%%%%%%%%%%%%%%%%%%%%%

142
JCTC_revision/SI/GW-srDFT-SI.tex
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@ -49,11 +49,16 @@
\newcommand{\QP}{\textsc{quantum package}}
% methods
\newcommand{\HF}{\text{HF}}
\newcommand{\PBEO}{\text{PBE0}}
\newcommand{\evGW}{ev$GW$}
\newcommand{\qsGW}{qs$GW$}
\newcommand{\GOWO}{$G_0W_0$}
\newcommand{\GW}{$GW$}
\newcommand{\GnWn}[1]{$G_{#1}W_{#1}$}
\newcommand{\srLDA}{\text{srLDA}}
\newcommand{\srPBE}{\text{srPBE}}
\newcommand{\Bas}{\mathcal{B}}
% operators
\newcommand{\hH}{\Hat{H}}
@ -72,6 +77,9 @@
\newcommand{\RH}{R_{\ce{H2}}}
\newcommand{\RF}{R_{\ce{F2}}}
\newcommand{\RBeO}{R_{\ce{BeO}}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
\newcommand{\bpot}[2]{\Bar{v}_{#1}^{#2}}
% orbital energies
\newcommand{\nDIIS}{N^\text{DIIS}}
@ -121,7 +129,7 @@
\newcommand{\bSigX}{\boldsymbol{\Sigma}^\text{x}}
\newcommand{\bSigC}{\boldsymbol{\Sigma}^\text{c}}
\newcommand{\bSigGW}{\boldsymbol{\Sigma}^\text{\GW}}
\newcommand{\be}{\boldsymbol{\epsilon}}
%\newcommand{\be}{\boldsymbol{\epsilon}}
\newcommand{\bDelta}{\boldsymbol{\Delta}}
\newcommand{\beHF}{\boldsymbol{\epsilon}^\text{HF}}
\newcommand{\beGW}{\boldsymbol{\epsilon}^\text{\GW}}
@ -150,6 +158,8 @@
\renewcommand{\bra}[1]{\ensuremath{\langle #1 \vert}}
\renewcommand{\ket}[1]{\ensuremath{\vert #1 \rangle}}
\renewcommand{\braket}[2]{\ensuremath{\langle #1 \vert #2 \rangle}}
\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
@ -189,102 +199,130 @@
\newcommand{\potpbeueg}[0]{\bar{v}_{\text{srPBE}}^{\basis}}
\newcommand{\potpbe}[0]{v^{\text{PBE}}_{\text{c}}}
\section{Complementary short-range correlation potentials}
\section{PBE-based complementary potential $\potpbeueg$}
Here, we provide the expressions of the complementary short-range LDA and PBE correlation potentials used in the present work in the case of closed-shell systems.
Here, we provide the explicit expression of the PBE-based complementary potential in the case of closed-shell systems such as the ones studied in the present paper.
The PBE-based correlation energy functional with multideterminant reference (ECMD) has been previously reported in Ref.~\onlinecite{Loos_2019} and is defined by the following equation:
\subsection{Complementary short-range LDA correlation potential}
The complementary short-range LDA correlation energy functional with multideterminant reference has the expression~\cite{Toulouse_2005,Paziani_2006}
\begin{equation}
\label{eq:def_lda_tot}
\bE{\srLDA}{\Bas}[\n{}{}] =
\int \n{}{}(\br{}) \be{\text{c,md}}{\srLDA}(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation}
with
\begin{equation}
\be{\text{c,md}}{\srLDA}(\n{}{},\rsmu{}{}) = \be{\text{c}}{\srLDA}(\n{}{},\rsmu{}{}) + \Delta^{\text{lr-sr}}(n,\mu),
\end{equation}
with $\be{\text{c,md}}{\srLDA}(\n{}{},\rsmu{}{})$ is the complementary short-range LDA correlation energy functional (with single-determinant reference) and $\Delta^{\text{lr-sr}}(n,\mu)$ is a mixed long-range/short-range contribution, both parametrized in Ref.~\onlinecite{Paziani_2006}.
The corresponding complementary srLDA potential is
\begin{eqnarray}
\bpot{\srLDA}{\Bas}[\n{}{}](\br{}) &=& \frac{\delta \bE{\srLDA}{\Bas}[\n{}{}]}{\delta \n{}{}(\br{})}
\nonumber\\
&=& \be{\text{c,md}}{\srLDA}(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{}))
\nonumber\\
&&+ n(\br{}) \frac{\partial \be{\text{c,md}}{\srLDA}}{\partial n} (\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})).
\end{eqnarray}
The density derivative of $\be{\text{c,md}}{\srLDA}$ is calculated as
\begin{eqnarray}
\frac{\partial \be{\text{c,md}}{\srLDA}}{\partial n} = \frac{\partial \be{\text{c}}{\srLDA}}{\partial n} + \frac{\partial \Delta^{\text{lr-sr}}}{\partial n},
\end{eqnarray}
where $\partial \be{\text{c}}{\srLDA}/\partial n$ is given as a subroutine on Paola Gori-Giorgi's web site (\url{https://www.quantummatter.eu/source-codes-2}) and we have calculated $\partial \Delta^{\text{lr-sr}}/\partial n$ by taking the derivative of Eq. (42) of Ref.~\onlinecite{Paziani_2006}.
\subsection{Complementary short-range PBE correlation potential}
The complementary short-range PBE correlation energy functional with multideterminant reference has the expression~\cite{Loos_2019}
\begin{equation}
\label{eq:def_pbe}
\efuncbasispbe = \int n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) d\br{} ,
\efuncbasispbe = \int n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) d\br{},
\end{equation}
with,
with
\begin{equation}
\label{eq:def_epsipbeueg}
\epspbeueg(n,s,\mu) = \frac{\epspbe(n,s)}{1+\beta(n,s)\mu^3},
\epspbeueg(n,s,\mu) = \frac{\epspbe(n,s)}{1+\beta(n,s)\mu^3}.
\end{equation}
where $\epspbe(n,s)$ is the usual PBE correlation functional \cite{Perdew_1996}, $s=\nabla n/n^{4/3}$ is the reduced density gradient,
Here, $\epspbe(n,s)$ is the usual PBE correlation functional \cite{Perdew_1996}, $s$ is the reduced density gradient,
\begin{equation}
\beta(n,s) = \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\epspbe(n,s)}{n_2^{\text{UEG}}(n)/n},
\end{equation}
and
\begin{equation}
\label{eq:uegotop}
n_2^{\text{UEG}}(n)=n^2g_0(r_s)
n_2^{\text{UEG}}(n)=n^2g_0(r_\text{s})
\end{equation}
is the on-top pair density of the uniform electron gas (UEG). In Eq.~\eqref{eq:uegotop}, $r_s=(4\pi n/3)^{-1/3}$ the Wigner-Seitz radius and $g_0(r_s)$ is the UEG on-top pair-distribution function. The parametrization of $g_0(r_s)$ is given in Eq.~(46) of Ref.~\onlinecite{Gori-Giorgi_2006}.
is the on-top pair density of the uniform electron gas (UEG). In Eq.~\eqref{eq:uegotop}, $g_0(r_\text{s})$ is the UEG on-top pair-distribution function written as a function of the Wigner-Seitz radius $r_\text{s}=(4\pi n/3)^{-1/3}$. We use the parametrization of $g_0(r_\text{s})$ given in Eq.~(46) of Ref.~\onlinecite{Gori-Giorgi_2006}.
The potential of this GGA ECMD complementary functional has the following form:
\begin{equation}
\begin{split}
\potpbeueg[n]
& = \fdv{\efuncbasispbe}{n}
\\
% & = \frac{\partial n \epspbeueg }{\partial n}- \nabla . \frac{\partial n \epspbeueg }{\partial \nabla n}\\
& =\epspbeueg + n \pdv{\epspbeueg }{n}- \nabla \cdot \qty( n \pdv{\epspbeueg}{\nabla n} ).
\end{split}
\end{equation}
Hence, we have to compute two main contributions: the scalar part $\pdv{\epspbeueg}{n}$ and the gradient part $\pdv{\epspbeueg }{\nabla n}$.
The corresponding complementary srPBE potential is
\begin{eqnarray}
\potpbeueg[n](\br{})
&=& \fdv{\efuncbasispbe}{n(\br{})}
\nonumber\\
&=& \epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{}))
\nonumber\\
&+& n(\br{}) \pdv{\epspbeueg }{n} (n({\bf r}),s({\bf r}),\mu^{\basis}(\br{}))
\nonumber\\
&-& \nabla \cdot \qty( n(\br{}) \pdv{\epspbeueg}{\nabla n} (n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) ).\,\,\,
\end{eqnarray}
Hence, we have to compute the density derivative $\partial \epspbeueg/\partial n$ and the density-gradient derivative $\partial \epspbeueg/\partial \nabla n$.
\subsection{Scalar contribution}
\subsubsection{Density derivative}
For the scalar contribution, we simply differenciate Eq.~\eqref{eq:def_epsipbeueg} with respect to the density:
From Eq.~\eqref{eq:def_epsipbeueg}, the density derivative is found to be
\begin{equation}
\pdv{\epspbeueg }{n}
= \frac{\potpbe}{1+\beta\mu^3}
= \frac{1}{1+\beta\mu^3} \pdv{\epspbe}{n}
- \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{n},
\end{equation}
where $\potpbe = \pdv{\epspbe}{n}$ and
\begin{equation}
where $\partial \epspbe/\partial n$ is the density derivative of the usual PBE correlation functional, and
\begin{eqnarray}
\pdv{\beta}{n}
= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}
\Bigg[ \frac{\potpbe}{n_2^{\text{UEG}}/n}
- \frac{\epspbe}{(n_2^{\text{UEG}}/n)^2} \frac{\partial (n_2^{\text{UEG}}/n)}{\partial n} \Bigg].
\end{equation}
The only remaining missing part is the derivative of $n_2^{\text{UEG}}/n$ with respect to the density:
&=& \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}
\Bigg[ \frac{1}{n_2^{\text{UEG}}/n} \pdv{\epspbe}{n}
\nonumber\\
&&\phantom{xxxxx} - \frac{\epspbe}{(n_2^{\text{UEG}}/n)^2} \frac{\partial (n_2^{\text{UEG}}/n)}{\partial n} \Bigg].
\end{eqnarray}
The only remaining missing part is the derivative of $n_2^{\text{UEG}}/n$ which is
\begin{equation}
\pdv{(n_2^{\text{UEG}}/n)}{n} = \pdv{[n g_0(r_s)]}{n} = g_0(r_s)+ n \pdv{g_0(r_s)}{n}.
\pdv{(n_2^{\text{UEG}}/n)}{n} = \pdv{[n g_0(r_\text{s})]}{n} = g_0(r_\text{s})+ n \pdv{g_0(r_\text{s})}{n},
\end{equation}
with
\begin{equation}
\pdv{g_0(r_s)}{n} = \pdv{r_s}{n} \pdv{g_0(r_s)}{r_s} = -(6 n^{2}\sqrt{\pi})^{-2/3} \pdv{g_0(r_s)}{r_s}.
\pdv{g_0(r_\text{s})}{n} = \pdv{r_\text{s}}{n} \pdv{g_0(r_\text{s})}{r_\text{s}} = -(6 n^{2}\sqrt{\pi})^{-2/3} \pdv{g_0(r_\text{s})}{r_\text{s}}.
\end{equation}
The derivative with respect to $r_s$ can be expressed as
Finally, we calculate $\partial g_0(r_\text{s}) /\partial r_\text{s}$ by taking the derivative of Eq.~(46) of Ref.~\onlinecite{Gori-Giorgi_2006}
\begin{equation}
\begin{aligned}
\pdv{g_0(r_s)}{r_s}
& = \frac{e^{-F\,r_s}}{2} \big[ (-B + 2 C r_s + 3 D r_s^2 + 4 E r_s^3)
\pdv{g_0(r_\text{s})}{r_\text{s}}
& = \frac{e^{-F\,r_\text{s}}}{2} \big[ (-B + 2 C r_\text{s} + 3 D r_\text{s}^2 + 4 E r_\text{s}^3)
\\
& - F (1 - B r_s + C r_s^2 + D r_s^3 + E r_s^4) \big],
& - F (1 - B r_\text{s} + C r_\text{s}^2 + D r_\text{s}^3 + E r_\text{s}^4) \big],
\end{aligned}
\end{equation}
with
\begin{align}
C & = 0.0819306, \\
F & = 0.752411, \\
D & = -0.0127713,\\
E & =0.00185898,\\
B & = 0.7317 - F.
\end{align}
with $C = 0.0819306$, $F = 0.752411$, $D = -0.0127713$, $E =0.00185898$, and $B = 0.7317 - F$.
\subsubsection{Density-gradient derivative}
\subsection{Gradient contribution}
For the gradient part, we also used the chain rule:
For the density-gradient derivative, we use the chain rule
\begin{equation}
\pdv{\epspbeueg}{\nabla n} = \pdv{\epspbeueg}{\epspbe}\pdv{\epspbe}{\nabla n}.
\pdv{\epspbeueg}{\nabla n} = \pdv{\epspbeueg}{\epspbe}\pdv{\epspbe}{\nabla n},
\end{equation}
The term $\pdv{\epspbe}{\nabla n}$ is already known (\textbf{ref??}), and the partial derivative of $\epspbeueg$ with respect to $\epspbe$ is
where $\partial \epspbe/\partial \nabla n$ is the density-gradient derivative of the usual PBE correlation functional, and
\begin{equation}
\pdv{\epspbeueg}{\epspbe}
= \frac{1}{1+\beta\mu^3}
- \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{\epspbe},
\end{equation}
where
with
\begin{equation}
\pdv{\beta}{\epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
\end{equation}
\section{Additional graphs of the convergence of the IPs of the GW20 subset}
Graphs reporting the convergence of the IPs of each molecule of the GW20 subset at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels are given in Figure~\ref{fig:IP_G0W0HF} and~\ref{fig:IP_G0W0PBE0}, respectively.
\begin{figure*}
\includegraphics[width=\linewidth]{IP_G0W0HF}
\caption{
@ -299,7 +337,7 @@ where
\caption{
IPs (in eV) computed at the {\GOWO}@PBE0 (black circles), {\GOWO}@PBE0+srLDA (red squares), and {\GOWO}@PBE0+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) for the 20 smallest molecules of the GW100 set.
The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
\label{fig:IP_G0W0HF}
\label{fig:IP_G0W0PBE0}
}
\end{figure*}

4
Response_Letter/Response_Letter.tex
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@ -41,9 +41,9 @@ We look forward to hearing from you.
{The main criticism as a reader is that all details of the construction of the total energy correction to the ``finite-size basis difference'' with respect to the CBS limit is absent from the paper (very short Section II-C). The authors refer the reader to previous publications (mainly [57]) dealing with total energies in a CCSD(T) quantum chemistry wavefunction framework with which the Green's function community may not be very familiar with. In particular the construction of a local range-separation parameter related to the diagonal of the ``effective'' 2-electron-operator-in-a-basis ($W^{B}$) would deserve to be somehow explained in the present paper.}
\\
\alert{We have included a new subsection (Section II.C.) to include additional details about the present basis set correction.
In particular, the construction of the range-separation function $\mu(\mathbf{r})$ is detailed as well as the corresponding effective two-electron operator $W(\mathbf{r}_1,\mathbf{r}_2)$.}
In particular, the construction of the range-separation function $\mu(\mathbf{r})$ is detailed as well as the corresponding effective two-electron operator $W(\mathbf{r}_1,\mathbf{r}_2)$.
We have also expanded Section II.D. to add more details about the short-range correlation functionals.
Their corresponding potentials are reported in the Supporting Information.
Their corresponding potentials are reported in the Supporting Information.}
\item
{Following the previous question, and from a pragmatic point of view, what is needed as an input to construct this basis-set-incompleteness correction, namely this effective local potential of Eq. [31] ? Again the answer is present in equations 4-9 of Ref. [57] but could be summarised in the present paper and possibly simplified in the present case of a perturbation theory based on a input mono-determinental Kohn-Sham or HF description of the many-body wavefunction. This may also give an hint on the cost (scaling) and complexity of the approach. }