diff --git a/JCTC_revision/GW-srDFT.tex b/JCTC_revision/GW-srDFT.tex index aaa3f6c..b490b93 100644 --- a/JCTC_revision/GW-srDFT.tex +++ b/JCTC_revision/GW-srDFT.tex @@ -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).} %%%%%%%%%%%%%%%%%%%%%%%% diff --git a/JCTC_revision/SI/GW-srDFT-SI.tex b/JCTC_revision/SI/GW-srDFT-SI.tex index ed55058..92ccbd9 100644 --- a/JCTC_revision/SI/GW-srDFT-SI.tex +++ b/JCTC_revision/SI/GW-srDFT-SI.tex @@ -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*} diff --git a/Response_Letter/Response_Letter.tex b/Response_Letter/Response_Letter.tex index 5e43acb..71a50d6 100644 --- a/Response_Letter/Response_Letter.tex +++ b/Response_Letter/Response_Letter.tex @@ -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. }