diff --git a/JCTC_revision/GW-srDFT.tex b/JCTC_revision/GW-srDFT.tex index 7dde017..2744afc 100644 --- a/JCTC_revision/GW-srDFT.tex +++ b/JCTC_revision/GW-srDFT.tex @@ -56,6 +56,8 @@ \newcommand{\cD}{\mathcal{D}} \newcommand{\Ne}{N} \newcommand{\Nbas}{N_\text{bas}} +\newcommand{\Nocc}{N_\text{occ}} +\newcommand{\Nvirt}{N_\text{virt}} % energies @@ -365,9 +367,9 @@ Within the {\GW} approximation, the correlation part of the self-energy reads \Sig{\text{c},p}{\Bas}(\omega) & = \mel*{\MO{p}}{\Sig{\text{c}}{\Bas}(\omega)}{\MO{p}} \\ - & = 2 \sum_{i}^\text{occ} \sum_{m} \frac{[pi|m]^2}{\omega - \e{i} + \Om{m} - i \eta} + & = 2 \sum_{i}^{\Nocc} \sum_{m} \frac{[pi|m]^2}{\omega - \e{i} + \Om{m} - i \eta} \\ - & + 2 \sum_{a}^\text{virt} \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{equation} where $m$ labels excited states and $\eta$ is a positive infinitesimal. @@ -431,20 +433,36 @@ In particular, the ionization potential (IP) and electron affinity (EA) are extr where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies, respectively. %%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Short-range correlation functionals} -\label{sec:srDFT} +\subsection{Basis set correction} +\label{sec:BSC} %%%%%%%%%%%%%%%%%%%%%%%% -The fundamental idea behind the present basis set correction is to recognize that the two-electron interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$. -We can therefore define an effective two-electron interaction which equals the Coulomb operator in an incomplete basis, \ie, +\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 two-electron operator which equals its Coulomb counterpart in an incomplete basis, \ie, \begin{equation} \iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}' = - \iint \n{2}{\Bas}(\br{},\br{}') W(\br{},\br{}') d\br{} d\br{}'. + \iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'. \end{equation} -A convenient choice is, for example, +Because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can map this operator to a non-divergent, long-range interaction of the form +\begin{equation} + \label{eq:Wapprox} + W^{\Bas}(\br{},\br{}') + \approx \frac{1}{2} \qty{ + \frac{\erf[\rsmu{}{\Bas}(\br{}) \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}} + + \frac{\erf[\rsmu{}{\Bas}(\br{}') \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}} + }. +\end{equation} +The information about the finiteness of the basis set has been 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 $\abs*{\br{} - \br{}'} = 0$. +Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields +\begin{equation} + \rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} W^{\Bas}(\br{},\br{}). +\end{equation} +} + +\titou{A convenient choice for the on-top value of the two-electron effective operator is, for example, \begin{equation} \label{eq:W} - W(\br{},\br{}') = + W^{\Bas}(\br{},\br{}) = \begin{cases} f^{\Bas}(\br{})/\n{2}{\Bas}(\br{}), & \n{2}{\Bas}(\br{}) \neq 0, \\ \infty, & \text{otherwise}, \\ @@ -452,23 +470,23 @@ A convenient choice is, for example, \end{equation} where, in the case of a single-determinant method (such as HF and KS-DFT), \begin{equation} - f(\br{}) = \sum_{pq}^{\Nbas} \sum_{ij}^\text{occ} \MO{p}(\br{}) \MO{q}(\br{}) \braket{pq}{ij} \MO{i}(\br{}) \MO{j}(\br{}) + f^{\Bas}(\br{}) = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{}) \MO{q}(\br{}) (pi|qj) \MO{i}(\br{}) \MO{j}(\br{}), \end{equation} and \begin{equation} - \n{2}{\Bas}(\br{}) = \frac{[\n{}{\Bas}(\br{})]^2}{4} + \n{2}{\Bas}(\br{}) = \frac{1}{4} [\n{}{\Bas}(\br{})]^2 \end{equation} -is the opposite-spin on-top pair density. -The effective operator $W(\br{},\br{}')$ has some interesting properties. +is the opposite-spin on-top pair density in a closed-shell system. +} + +\titou{The effective operator $W^{\Bas}(\br{},\br{}')$ has some interesting properties. For example, we have \begin{equation} - \lim_{\Bas \to \CBS} W(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1} + \lim_{\Bas \to \CBS} W^{\Bas}(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1}, \end{equation} -which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb operator. -Consequently, the magnitude of the correction tends to zero (see Eq.~\eqref{eq:limSig}). -Note also that the divergence condition of $W(\br{},\br{}')$ in Eq.~\eqref{eq:W} evidences that one-electron systems are free of correction. - -Because the value of $W(\br{},\br{}')$ at coalescence, $W(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can map its value to a non-divergent, long-range interaction of the form $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$. +which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb operator, and that the magnitude of the energetic correction tends to zero [see Eq.~\eqref{eq:limSig}]. +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. +} %%%%%%%%%%%%%%%%%%%%%%%% \subsection{Short-range correlation functionals} @@ -478,7 +496,8 @@ Because the value of $W(\br{},\br{}')$ at coalescence, $W(\br{},\br{})$, is nece 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{Toulouse_2005, Paziani_2006} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{Ferte_2019, Loos_2019} which interpolates between the usual PBE functional \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior \cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density from the uniform-electron gas. \cite{Loos_2019} Additionally to the one-electron density calculated from the HF or KS orbitals, these RS-DFT functionals require a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial inhomogeneity of the basis-set incompleteness error and is computed using the HF or KS opposite-spin pair-density matrix in the basis set $\Bas$. -We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are provided. +We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} where our procedure is thoroughly detailed. +\titou{The explicit expressions of these two short-range correlation functionals, as well as their functional derivative, are provided in the {\SI}.} The basis set corrected {\GOWO} quasiparticle energies are thus given by @@ -739,8 +758,8 @@ We hope to report on this in the near future. %%%%%%%%%%%%%%%%%%%%%%%% \section*{Supporting Information} %%%%%%%%%%%%%%%%%%%%%%%% -See {\SI} for additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set. -\titou{The numerical data of Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} are provided in txt and json format.} +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 +the numerical data of Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} (provided in txt and json formats).} %%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} diff --git a/JCTC_revision/SI/GW-srDFT-SI.tex b/JCTC_revision/SI/GW-srDFT-SI.tex new file mode 100644 index 0000000..e69288e --- /dev/null +++ b/JCTC_revision/SI/GW-srDFT-SI.tex @@ -0,0 +1,205 @@ +\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,wrapfig} + +\usepackage{natbib} +\usepackage[extra]{tipa} +\bibliographystyle{achemso} +\AtBeginDocument{\nocite{achemso-control}} +\usepackage{mathpazo,libertine} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + citecolor=blue +} +\urlstyle{same} + +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{HTML}{009900} +\usepackage[normalem]{ulem} +\newcommand{\titou}[1]{\textcolor{red}{#1}} +\newcommand{\jt}[1]{\textcolor{purple}{#1}} +\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\toto}[1]{\textcolor{brown}{#1}} +\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} +\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}} +\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}} +\newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}} +\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}} +\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}} +\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} +\newcommand{\AS}[1]{\toto{(\underline{\bf TOTO}: #1)}} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + citecolor=blue +} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\SI}{\textcolor{blue}{supporting information}} +\newcommand{\QP}{\textsc{quantum package}} + +% methods +\newcommand{\evGW}{ev$GW$} +\newcommand{\qsGW}{qs$GW$} +\newcommand{\GOWO}{$G_0W_0$} +\newcommand{\GW}{$GW$} +\newcommand{\GnWn}[1]{$G_{#1}W_{#1}$} + +% operators +\newcommand{\hH}{\Hat{H}} + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EHF}{E_\text{HF}} +\newcommand{\EKS}{E_\text{KS}} +\newcommand{\EcK}{E_\text{c}^\text{Klein}} +\newcommand{\EcRPA}{E_\text{c}^\text{RPA}} +\newcommand{\EcGM}{E_\text{c}^\text{GM}} +\newcommand{\EcMP}{E_c^\text{MP2}} +\newcommand{\Egap}{E_\text{gap}} +\newcommand{\IP}{\text{IP}} +\newcommand{\EA}{\text{EA}} +\newcommand{\RH}{R_{\ce{H2}}} +\newcommand{\RF}{R_{\ce{F2}}} +\newcommand{\RBeO}{R_{\ce{BeO}}} + +% orbital energies +\newcommand{\nDIIS}{N^\text{DIIS}} +\newcommand{\maxDIIS}{N_\text{max}^\text{DIIS}} +\newcommand{\nSat}[1]{N_{#1}^\text{sat}} +\newcommand{\eSat}[2]{\epsilon_{#1,#2}} +\newcommand{\e}[1]{\epsilon_{#1}} +\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}} +\newcommand{\teHF}[1]{\Tilde{\epsilon}^\text{HF}_{#1}} +\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}} +\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}} +\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}} +\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}} +\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}} +\newcommand{\de}[1]{\Delta\epsilon_{#1}} +\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}} +\newcommand{\Om}[1]{\Omega_{#1}} +\newcommand{\eHOMO}{\epsilon_\text{HOMO}} +\newcommand{\eLUMO}{\epsilon_\text{LUMO}} +\newcommand{\HOMO}{\text{HOMO}} +\newcommand{\LUMO}{\text{LUMO}} + +% Matrix elements +\newcommand{\A}[1]{A_{#1}} +\newcommand{\B}[1]{B_{#1}} +\newcommand{\tA}{\Tilde{A}} +\newcommand{\tB}{\Tilde{B}} +\renewcommand{\S}[1]{S_{#1}} +\newcommand{\G}[1]{G_{#1}} +\newcommand{\Po}[1]{P_{#1}} +\newcommand{\W}[1]{W_{#1}} +\newcommand{\Wc}[1]{W^\text{c}_{#1}} +\newcommand{\vc}[1]{v_{#1}} +\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} +\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} +\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}} +\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}} +\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}} +\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}} +\newcommand{\Z}[1]{Z_{#1}} + +% Matrices +\newcommand{\bG}{\boldsymbol{G}} +\newcommand{\bW}{\boldsymbol{W}} +\newcommand{\bvc}{\boldsymbol{v}} +\newcommand{\bSig}{\boldsymbol{\Sigma}} +\newcommand{\bSigX}{\boldsymbol{\Sigma}^\text{x}} +\newcommand{\bSigC}{\boldsymbol{\Sigma}^\text{c}} +\newcommand{\bSigGW}{\boldsymbol{\Sigma}^\text{\GW}} +\newcommand{\be}{\boldsymbol{\epsilon}} +\newcommand{\bDelta}{\boldsymbol{\Delta}} +\newcommand{\beHF}{\boldsymbol{\epsilon}^\text{HF}} +\newcommand{\beGW}{\boldsymbol{\epsilon}^\text{\GW}} +\newcommand{\beGnWn}[1]{\boldsymbol{\epsilon}^\text{\GnWn{#1}}} +\newcommand{\bdeGnWn}[1]{\Delta\boldsymbol{\epsilon}^\text{\GnWn{#1}}} +\newcommand{\bde}{\boldsymbol{\Delta\epsilon}} +\newcommand{\bdeHF}{\boldsymbol{\Delta\epsilon}^\text{HF}} +\newcommand{\bdeGW}{\boldsymbol{\Delta\epsilon}^\text{GW}} +\newcommand{\bOm}{\boldsymbol{\Omega}} +\newcommand{\bA}{\boldsymbol{A}} +\newcommand{\bB}{\boldsymbol{B}} +\newcommand{\bX}{\boldsymbol{X}} +\newcommand{\bY}{\boldsymbol{Y}} +\newcommand{\bZ}{\boldsymbol{Z}} + +\newcommand{\fc}{f_\text{c}} +\newcommand{\Vc}{V_\text{c}} + +\newcommand{\MO}[1]{\phi_{#1}} + +% coordinates +\newcommand{\br}[1]{\mathbf{r}_{#1}} +\renewcommand{\b}[1]{\mathbf{#1}} +\renewcommand{\d}{\text{d}} +\newcommand{\dbr}[1]{d\br{#1}} +\renewcommand{\bra}[1]{\ensuremath{\langle #1 \vert}} +\renewcommand{\ket}[1]{\ensuremath{\vert #1 \rangle}} +\renewcommand{\braket}[2]{\ensuremath{\langle #1 \vert #2 \rangle}} + + +\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France} +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France} +\newcommand{\IUF}{Institut Universitaire de France, Paris, France} + +\begin{document} + +\title{Supplementary Materials for ``A Density-Based Basis-Set Incompleteness Correction for GW Methods''} + +\author{Pierre-Fran\c{c}ois Loos} +\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr} +\affiliation{\LCPQ} +\author{Barth\'el\'emy Pradines} +\affiliation{\LCT} +\affiliation{\ISCD} +\author{Anthony Scemama} +\affiliation{\LCPQ} +\author{Emmanuel Giner} +\affiliation{\LCT} +\author{Julien Toulouse} +\email[Corresponding author: ]{toulouse@lct.jussieu.fr} +\affiliation{\LCT} +\affiliation{\IUF} + +\begin{abstract} +\end{abstract} + +\maketitle + +\begin{figure*} + \includegraphics[width=\linewidth]{IP_G0W0HF} + \caption{ + IPs (in eV) computed at the {\GOWO}@HF (black circles), {\GOWO}@HF+srLDA (red squares), and {\GOWO}@HF+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} + } +\end{figure*} + +\begin{figure*} + \includegraphics[width=\linewidth]{IP_G0W0PBE0} + \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} + } +\end{figure*} + + +\bibliography{../GW-srDFT,../GW-srDFT-control} + +\end{document} diff --git 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