348 lines
14 KiB
TeX
348 lines
14 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\usepackage{hyperref}
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colorlinks=true,
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filecolor=blue,
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\QP}{\textsc{quantum package}}
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% methods
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\newcommand{\HF}{\text{HF}}
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\newcommand{\PBEO}{\text{PBE0}}
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\GW}{$GW$}
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\newcommand{\GnWn}[1]{$G_{#1}W_{#1}$}
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\newcommand{\srLDA}{\text{srLDA}}
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\newcommand{\srPBE}{\text{srPBE}}
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\newcommand{\Bas}{\mathcal{B}}
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% operators
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\newcommand{\hH}{\Hat{H}}
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% energies
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\EKS}{E_\text{KS}}
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\newcommand{\EcK}{E_\text{c}^\text{Klein}}
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\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
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\newcommand{\EcGM}{E_\text{c}^\text{GM}}
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\newcommand{\EcMP}{E_c^\text{MP2}}
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\newcommand{\Egap}{E_\text{gap}}
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\newcommand{\IP}{\text{IP}}
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\newcommand{\EA}{\text{EA}}
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\newcommand{\RH}{R_{\ce{H2}}}
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\newcommand{\RF}{R_{\ce{F2}}}
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\newcommand{\RBeO}{R_{\ce{BeO}}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
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\newcommand{\bpot}[2]{\Bar{v}_{#1}^{#2}}
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% orbital energies
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\newcommand{\nDIIS}{N^\text{DIIS}}
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\newcommand{\maxDIIS}{N_\text{max}^\text{DIIS}}
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\newcommand{\nSat}[1]{N_{#1}^\text{sat}}
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\newcommand{\eSat}[2]{\epsilon_{#1,#2}}
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\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
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\newcommand{\teHF}[1]{\Tilde{\epsilon}^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
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\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
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\newcommand{\de}[1]{\Delta\epsilon_{#1}}
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\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
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\newcommand{\Om}[1]{\Omega_{#1}}
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\newcommand{\eHOMO}{\epsilon_\text{HOMO}}
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\newcommand{\eLUMO}{\epsilon_\text{LUMO}}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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% Matrix elements
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\newcommand{\A}[1]{A_{#1}}
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\newcommand{\B}[1]{B_{#1}}
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\newcommand{\tA}{\Tilde{A}}
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\newcommand{\tB}{\Tilde{B}}
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\newcommand{\Po}[1]{P_{#1}}
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\newcommand{\W}[1]{W_{#1}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\tSigC}[1]{\Tilde{\Sigma}^\text{c}_{#1}}
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\newcommand{\SigCp}[1]{\Sigma^\text{p}_{#1}}
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\newcommand{\SigCh}[1]{\Sigma^\text{h}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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% Matrices
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\newcommand{\bSigX}{\boldsymbol{\Sigma}^\text{x}}
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\newcommand{\bSigC}{\boldsymbol{\Sigma}^\text{c}}
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\newcommand{\bSigGW}{\boldsymbol{\Sigma}^\text{\GW}}
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%\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bDelta}{\boldsymbol{\Delta}}
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\newcommand{\beHF}{\boldsymbol{\epsilon}^\text{HF}}
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\newcommand{\beGW}{\boldsymbol{\epsilon}^\text{\GW}}
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\newcommand{\beGnWn}[1]{\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bdeGnWn}[1]{\Delta\boldsymbol{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bde}{\boldsymbol{\Delta\epsilon}}
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\newcommand{\bdeHF}{\boldsymbol{\Delta\epsilon}^\text{HF}}
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\newcommand{\bdeGW}{\boldsymbol{\Delta\epsilon}^\text{GW}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\MO}[1]{\phi_{#1}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\renewcommand{\b}[1]{\mathbf{#1}}
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\renewcommand{\d}{\text{d}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\renewcommand{\bra}[1]{\ensuremath{\langle #1 \vert}}
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\renewcommand{\ket}[1]{\ensuremath{\vert #1 \rangle}}
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\renewcommand{\braket}[2]{\ensuremath{\langle #1 \vert #2 \rangle}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
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\newcommand{\IUF}{Institut Universitaire de France, Paris, France}
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\begin{document}
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\title{Supplementary Materials for ``A Density-Based Basis-Set Incompleteness Correction for GW Methods''}
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\author{Pierre-Fran\c{c}ois Loos}
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\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Barth\'el\'emy Pradines}
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\affiliation{\LCT}
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\affiliation{\ISCD}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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\author{Emmanuel Giner}
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\affiliation{\LCT}
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\author{Julien Toulouse}
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\email[Corresponding author: ]{toulouse@lct.jussieu.fr}
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\affiliation{\LCT}
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\affiliation{\IUF}
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\begin{abstract}
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\end{abstract}
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\maketitle
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%Macros:
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\newcommand{\basis}[0]{\mathcal{B}}
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\newcommand{\efuncbasispbe}[0]{\bar{E}_{\text{srPBE}}^{\basis}[n]}
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\newcommand{\epspbeueg}[0]{\bar{\varepsilon}^{\text{sr},\text{PBE}}_{\text{c,md}}}
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\newcommand{\epspbe}[0]{\varepsilon^{\text{PBE}}_{\text{c}}}
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\newcommand{\potpbeueg}[0]{\bar{v}_{\text{srPBE}}^{\basis}}
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\newcommand{\potpbe}[0]{v^{\text{PBE}}_{\text{c}}}
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\section{Complementary short-range correlation potentials}
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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.
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\subsection{Complementary short-range LDA correlation potential}
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The complementary short-range LDA correlation energy functional with multideterminant reference has the expression~\cite{Toulouse_2005,Paziani_2006}
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\srLDA}{\Bas}[\n{}{}] =
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\int \n{}{}(\br{}) \be{\text{c,md}}{\srLDA}(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
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\end{equation}
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with
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\begin{equation}
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\be{\text{c,md}}{\srLDA}(\n{}{},\rsmu{}{}) = \be{\text{c}}{\srLDA}(\n{}{},\rsmu{}{}) + \Delta^{\text{lr-sr}}(n,\mu),
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\end{equation}
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where $\be{\text{c,md}}{\srLDA}(\n{}{},\rsmu{}{})$ 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}.
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The corresponding complementary srLDA potential is
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\begin{eqnarray}
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\bpot{\srLDA}{\Bas}[\n{}{}](\br{}) &=& \frac{\delta \bE{\srLDA}{\Bas}[\n{}{}]}{\delta \n{}{}(\br{})}
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\nonumber\\
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&=& \be{\text{c,md}}{\srLDA}(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{}))
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\nonumber\\
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&&+ n(\br{}) \frac{\partial \be{\text{c,md}}{\srLDA}}{\partial n} (\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})).
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\end{eqnarray}
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The density derivative of $\be{\text{c,md}}{\srLDA}$ is calculated as
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\begin{eqnarray}
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\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},
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\end{eqnarray}
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where $\partial \be{\text{c}}{\srLDA}/\partial n$ is given as a subroutine on Paola Gori-Giorgi's website (\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}.
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\subsection{Complementary short-range PBE correlation potential}
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The complementary short-range PBE correlation energy functional with multideterminant reference has the expression~\cite{Loos_2019}
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\begin{equation}
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\label{eq:def_pbe}
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\efuncbasispbe = \int n({\bf r})\epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) d\br{},
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\end{equation}
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with
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\begin{equation}
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\label{eq:def_epsipbeueg}
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\epspbeueg(n,s,\mu) = \frac{\epspbe(n,s)}{1+\beta(n,s)\mu^3}.
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\end{equation}
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Here, $\epspbe(n,s)$ is the usual PBE correlation functional,\cite{Perdew_1996} $s$ is the reduced density gradient,
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\begin{equation}
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\beta(n,s) = \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{\epspbe(n,s)}{n_2^{\text{UEG}}(n)/n},
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\end{equation}
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and
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\begin{equation}
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\label{eq:uegotop}
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n_2^{\text{UEG}}(n)=n^2g_0(r_\text{s})
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\end{equation}
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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}.
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The corresponding complementary srPBE potential is
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\begin{eqnarray}
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\potpbeueg[n](\br{})
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&=& \fdv{\efuncbasispbe}{n(\br{})}
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\nonumber\\
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&=& \epspbeueg(n({\bf r}),s({\bf r}),\mu^{\basis}(\br{}))
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\nonumber\\
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&+& n(\br{}) \pdv{\epspbeueg }{n} (n({\bf r}),s({\bf r}),\mu^{\basis}(\br{}))
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\nonumber\\
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&-& \nabla \cdot \qty( n(\br{}) \pdv{\epspbeueg}{\nabla n} (n({\bf r}),s({\bf r}),\mu^{\basis}(\br{})) ).\,\,\,
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\end{eqnarray}
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Hence, we have to compute the density derivative $\partial \epspbeueg/\partial n$ and the density-gradient derivative $\partial \epspbeueg/\partial \nabla n$.
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\subsubsection{Density derivative}
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From Eq.~\eqref{eq:def_epsipbeueg}, the density derivative is found to be
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\begin{equation}
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\pdv{\epspbeueg }{n}
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= \frac{1}{1+\beta\mu^3} \pdv{\epspbe}{n}
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- \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{n},
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\end{equation}
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where $\partial \epspbe/\partial n$ is the density derivative of the usual PBE correlation functional, and
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\begin{eqnarray}
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\pdv{\beta}{n}
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&=& \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}
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\Bigg[ \frac{1}{n_2^{\text{UEG}}/n} \pdv{\epspbe}{n}
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\nonumber\\
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&&\phantom{xxxxx} - \frac{\epspbe}{(n_2^{\text{UEG}}/n)^2} \frac{\partial (n_2^{\text{UEG}}/n)}{\partial n} \Bigg].
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\end{eqnarray}
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The only remaining missing part is the derivative of $n_2^{\text{UEG}}/n$ which is
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\begin{equation}
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\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},
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\end{equation}
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with
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\begin{equation}
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\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}}.
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\end{equation}
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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}
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\begin{equation}
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\begin{aligned}
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\pdv{g_0(r_\text{s})}{r_\text{s}}
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& = \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)
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\\
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& - F (1 - B r_\text{s} + C r_\text{s}^2 + D r_\text{s}^3 + E r_\text{s}^4) \big],
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\end{aligned}
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\end{equation}
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with $C = 0.0819306$, $F = 0.752411$, $D = -0.0127713$, $E =0.00185898$, and $B = 0.7317 - F$.
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\subsubsection{Density-gradient derivative}
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For the density-gradient derivative, we use the chain rule
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\begin{equation}
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\pdv{\epspbeueg}{\nabla n} = \pdv{\epspbeueg}{\epspbe}\pdv{\epspbe}{\nabla n},
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\end{equation}
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where $\partial \epspbe/\partial \nabla n$ is the density-gradient derivative of the usual PBE correlation functional, and
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\begin{equation}
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\pdv{\epspbeueg}{\epspbe}
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= \frac{1}{1+\beta\mu^3}
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- \frac{\epspbe \mu^3}{(1+\beta\mu^3)^2} \pdv{\beta}{\epspbe},
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\end{equation}
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with
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\begin{equation}
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\pdv{\beta}{\epspbe}= \frac{3}{2\sqrt{\pi}(1-\sqrt{2})}\frac{1}{n_2^{\text{UEG}}/n}.
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\end{equation}
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\section{Additional graphs of the convergence of the IPs of the GW20 subset}
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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 Figs.~\ref{fig:IP_G0W0HF} and~\ref{fig:IP_G0W0PBE0}, respectively.
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\begin{figure*}
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\includegraphics[width=\linewidth]{IP_G0W0HF}
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\caption{
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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.
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The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
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\label{fig:IP_G0W0HF}
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}
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\end{figure*}
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\begin{figure*}
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\includegraphics[width=\linewidth]{IP_G0W0PBE0}
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\caption{
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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.
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The thick black line represents the CBS value obtained by extrapolation with the three largest basis sets.
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\label{fig:IP_G0W0PBE0}
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}
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\end{figure*}
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\bibliography{../GW-srDFT,../GW-srDFT-control}
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\end{document}
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