Working on the theory section
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2019-10-09 11:44:30 +0200
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%% Created for Pierre-Francois Loos at 2019-10-11 22:27:00 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Veril_2018,
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Author = {M. Veril and P. Romaniello and J. A. Berger and P. F. Loos},
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Date-Added = {2019-10-11 22:25:14 +0200},
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Date-Modified = {2019-10-11 22:26:57 +0200},
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Doi = {10.1021/acs.jctc.8b00745},
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Journal = {J. Chem. Theory Comput.},
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Pages = {5220},
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Title = {Unphysical Discontinuities in GW Methods},
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Volume = {14},
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Year = {2018}}
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@article{QP2,
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Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
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Date-Added = {2019-10-08 22:08:05 +0200},
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@ -1,11 +1,10 @@
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\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,wrapfig}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,wrapfig,txfonts}
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\usepackage{natbib}
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\usepackage[extra]{tipa}
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\bibliographystyle{achemso}
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\AtBeginDocument{\nocite{achemso-control}}
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\usepackage{mathpazo,libertine}
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\usepackage{hyperref}
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\hypersetup{
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@ -49,8 +48,8 @@
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\newcommand{\QP}{\textsc{quantum package}}
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% methods
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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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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@ -73,18 +72,16 @@
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\newcommand{\RF}{R_{\ce{F2}}}
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\newcommand{\RBeO}{R_{\ce{BeO}}}
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\newcommand{\CBS}{\text{CBS}}
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\newcommand{\LDA}{\text{LDA}}
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\newcommand{\PBE}{\text{PBE}}
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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{\beGOWO}[1]{\Bar{\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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@ -98,15 +95,17 @@
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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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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\F}[1]{F^{#1}}
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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{\G}[2]{G_{#1}^{#2}}
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\newcommand{\Gs}[1]{G_\text{s}^{#1}}
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\newcommand{\F}[2]{F_{#1}^{#2}}
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\newcommand{\Po}[2]{P_{#1}^{#2}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\Wc}[1]{W_{#1}^\text{c}}
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\newcommand{\vc}[2]{\varv_{#1}^{#2}}
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\newcommand{\pot}[2]{v_{#1}^{#2}}
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\newcommand{\bpot}[2]{\Bar{v}_{#1}^{#2}}
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\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
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\newcommand{\bSig}[2]{\Bar{\Sigma}_{#1}^{#2}}
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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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@ -114,24 +113,25 @@
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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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\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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% Matrices
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\newcommand{\bG}{\boldsymbol{G}}
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\newcommand{\bW}{\boldsymbol{W}}
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\newcommand{\bvc}{\boldsymbol{v}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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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{\bG}{\boldsymbol{G}}
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%\newcommand{\bW}{\boldsymbol{W}}
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%\newcommand{\bvc}{\boldsymbol{v}}
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%\newcommand{\bSig}{\boldsymbol{\Sigma}}
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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{\bA}{\boldsymbol{A}}
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\newcommand{\bB}{\boldsymbol{B}}
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@ -178,6 +178,13 @@
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\newcommand{\V}[2]{V_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\Hx}{\text{Hx}}
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\newcommand{\xc}{\text{xc}}
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%\newcommand{\ref}{\text{ref}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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@ -224,40 +231,40 @@ Here, we propose a density-based basis set correction based on short-range corre
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The purpose of many-body perturbation theory (MBPT) is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16}
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In MBPT, the \textit{screening} of the Coulomb interaction is a central quantity, and is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
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The so-called $GW$ approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} and is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2013, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}
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The $GW$ approximation stems from the acclaimed Hedin's equations \cite{Hedin_1965}
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The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} and is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2013, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}
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The {\GW} approximation stems from the acclaimed Hedin's equations \cite{Hedin_1965}
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\begin{subequations}
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\begin{align}
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\label{eq:G}
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& G(12) = G_0(12) + \int G_0(13) \Sigma(34) G(42) d(34),
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& \G{}{}(12) = \G{0}{}(12) + \int \G{0}{}(13) \Sig{}{}(34) \G{}{}(42) d(34),
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\\
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\label{eq:Gamma}
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& \Gamma(123) = \delta(12) \delta(13)
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& \Gam{}{}(123) = \delta(12) \delta(13)
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\notag
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\\
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& \qquad \qquad + \int \fdv{\Sigma(12)}{G(45)} G(46) G(75) \Gamma(673) d(4567),
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& \qquad \qquad + \int \fdv{\Sig{}{}(12)}{\G{}{}(45)} \G{}{}(46) G(75) \Gam{}{}(673) d(4567),
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\\
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\label{eq:P}
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& P(12) = - i \int G(13) \Gamma(324) G(41) d(34),
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& \Po{}{}(12) = - i \int G(13) \Gam{}{}(324) G(41) d(34),
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\\
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\label{eq:W}
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& W(12) = v(12) + \int v(13) P(34) W(42) d(34),
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& \W{}{}(12) = \vc{}{}(12) + \int \vc{}{}(13) \Po{}{}(34) \W{}{}(42) d(34),
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\\
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\label{eq:Sig}
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& \Sigma(12) = i \int G(13) W(14) \Gamma(324) d(34),
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& \Sig{}{}(12) = i \int \G{}{}(13) \W{}{}(14) \Gam{}{}(324) d(34),
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\end{align}
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\end{subequations}
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which connects the Green's function $G$, its non-interacting version $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$
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and the self-energy $\Sigma$, where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function \cite{NISTbook} and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\br{1},t_1)$.
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Within the $GW$ approximation, one bypasses the calculation of the vertex corrections by setting
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which connects the Green's function $\G{}{}$, its non-interacting version $\G{0}{}$, the irreducible vertex function $\Gam{}{}$, the irreducible polarizability $\Po{}{}$, the dynamically-screened Coulomb interaction $\W{}{}$
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and the self-energy $\Sig{}{}$, where $\vc{}{}$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function \cite{NISTbook} and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\br{1},t_1)$.
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Within the {\GW} approximation, one bypasses the calculation of the vertex corrections by setting
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\begin{equation}
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\label{eq:GW}
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\Gamma(123) \stackrel{GW}{\approx} \delta(12) \delta(13).
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\Gam{}{}(123) \stackrel{GW}{\approx} \delta(12) \delta(13).
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\end{equation}
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Depending on the degree of self-consistency one is willing to perform, there exists several types of $GW$ calculations. \cite{Loos_2018}
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The simplest and most popular variant of $GW$ is perturbative $GW$, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986}
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Depending on the degree of self-consistency one is willing to perform, there exists several types of {\GW} calculations. \cite{Loos_2018}
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The simplest and most popular variant of {\GW} is perturbative {\GW}, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986}
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Although obviously starting-point dependent, it has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
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For finite systems such as atoms and molecules, partially \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} or fully self-consistent \cite{Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b} $GW$ methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
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For finite systems such as atoms and molecules, partially \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} or fully self-consistent \cite{Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b} {\GW} methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
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Similar to other electron correlation methods, MBPT methods suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
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This can be tracked down to the lack of explicit electron-electron terms modeling the infamous electron-electron coalescence point (also known as Kato cusp \cite{Kat-CPAM-57}) and, more specifically, the Coulomb correlation hole around it.
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Pioneered by Hylleraas \cite{Hyl-ZP-29} in the 1930's and popularized in the 1990's by Kutzelnigg and coworkers \cite{Kut-TCA-85, NogKut-JCP-94, KutKlo-JCP-91} (and subsequently others \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17}), the so-called F12 methods overcome this slow convergence by employing geminal basis functions that closely resemble the correlation holes in electronic wave functions.
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@ -267,14 +274,14 @@ The basis-set correction presented here follow a different avenue, and relies on
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As shown in recent studies on both ground- and excited-state properties, \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of large auxiliary basis sets that are used in F12 methods to avoid the numerous three- and four-electron integrals. \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17, Barca_2018}
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Explicitly correlated F12 correction schemes have been derived for second-order Green's function methods (GF2) \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018} by Ten-no and coworkers \cite{Ohnishi_2016, Johnson_2018} and Valeev and coworkers. \cite{Pavosevic_2017, Teke_2019}
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However, to the best of our knowledge, a F12-based correction for $GW$ has not been designed yet.
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However, to the best of our knowledge, a F12-based correction for {\GW} has not been designed yet.
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In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} on ionization potentials (IPs) obtained within {\GOWO}.
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Note that the the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavours of (self-consistent) $GW$ or Green's function-based methods, such as GF2 (and its higher-order variants).
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Note that the the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavours of (self-consistent) {\GW} or Green's function-based methods, such as GF2 (and its higher-order variants).
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Moreover, we are currently investigating the performances of the present approach for linear response theory, in order to speed up the convergence of excitation energies obtained within the random-phase approximation (RPA) \cite{Casida_1995, Dreuw_2005} and Bethe-Salpeter equation (BSE) formalism. \cite{Strinati_1988, Leng_2016, Blase_2018}
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The paper is organised as follows.
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In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to $GW$ methods.
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In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to {\GW} methods.
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Results are reported and discussed in Sec.~\ref{sec:results}.
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Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
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Unless otherwise stated, atomic units are used throughout.
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@ -288,72 +295,82 @@ Unless otherwise stated, atomic units are used throughout.
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\subsection{MBPT with DFT basis set correction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\b{r})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
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\begin{equation}
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\E{0}{\Bas} = \min_{\n{}{} \in \cD^\Bas} \qty{ F[n] + \int \vne(\br{}) \n{}{}(\br{}) \dbr{} },
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\label{E0B}
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\E{0}{\Bas} = \min_{\n{}{} \in \cD^\Bas} \qty{ \F{}{}[n] + \int \vne(\br{}) \n{}{}(\br{}) \dbr{} },
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\label{eq:E0B}
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\end{equation}
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where $\cD^\Bas$ is the set of $\Ne$-representable densities which can be extracted from a wave function $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$.
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In this expression, $F[n] = \min_{\Psi \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
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In this expression, $\F{}{}[n] = \min_{\Psi \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
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\begin{equation}
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F[n] = F^{\Bas}[\n{}{}] + \bE{}{\Bas}[\n{}{}],
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\label{Fn}
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\F{}{}[\n{}{}] = \F{}{\Bas}[\n{}{}] + \bE{}{\Bas}[\n{}{}],
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\label{eq:Fn}
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\end{equation}
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where $F^\Bas[n]$ is the Levy-Lieb density functional with wave functions $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$
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where $\F{}{\Bas}[\n{}{}]$ is the Levy-Lieb density functional with wave functions $\Psi^\Bas$ expandable in the Hilbert space generated by $\Bas$
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\begin{equation}
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F^\Bas[n] = \min_{\Psi^\Bas \to n} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
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\F{}{\Bas}[\n{}{}] = \min_{\Psi^\Bas \to \n{}{}} \mel*{\Psi^\Bas}{ \hT + \hWee{}}{\Psi^\Bas},
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\end{equation}
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and $\bar{E}^{\Bas}[n]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $F^\Bas[\n{}{}]$, we reexpress it with a contrained search over $N$-representable one-electron Green's functions $G^\Bas(\b{r},\b{r}',\omega)$ representable in the basis set $\Bas$
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and $\bE{}{\Bas}[\n{}{}]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we reexpress it with a constrained search over $\Ne$-representable one-electron Green's functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$
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\begin{equation}
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F^\Bas[n] = \min_{G^\Bas\to n} \Omega^\Bas[G^\Bas],
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\label{FBn}
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\F{}{\Bas}[\n{}{}] = \min_{\G{}{\Bas} \to \n{}{}} \Omega^\Bas[\G{}{\Bas}],
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\label{eq:FBn}
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||||
\end{equation}
|
||||
where $\Omega^\Bas[G]$ is chosen to be a Klein-like energy functional of the Green's function (see, \textit{e.g.}, Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
|
||||
\begin{equation}
|
||||
\Omega^\Bas[G] = \Tr[\ln ( - G ) ] - \Tr[ (G_\text{s}^\Bas)^{-1} G -1 ] + \Phi_\text{Hxc}^\Bas[G],
|
||||
\label{OmegaB}
|
||||
\Omega^\Bas[\G{}{}] = \Tr[\ln( - \G{}{} ) ] - \Tr[ (\Gs{\Bas})^{-1} \G{}{} - 1 ] + \Phi_\Hxc^\Bas[\G{}{}],
|
||||
\label{eq:OmegaB}
|
||||
\end{equation}
|
||||
where $(G_\text{s}^\Bas)^{-1}$ is the projection into $\Bas$ of the inverse free-particle Green's function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 )\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr [A B] = 1/(2\pi i) \int_{-\infty}^{+\infty} \! \d \omega \, e^{i \omega 0^+} \! \iint \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^\Bas[G]$ is a Hartree-exchange-correlation (Hxc) functional of the Green's functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta \Phi_\text{Hxc}^\Bas[G]/\delta G(\b{r},\b{r}',\omega) = \Sigma_\text{Hxc}^\Bas[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
|
||||
where $(\Gs{\Bas})^{-1}$ is the projection into $\Bas$ of the inverse free-particle Green's function
|
||||
\begin{equation}
|
||||
E_0^\Bas = \min_{G^\Bas} \left\{ \Omega^\Bas[G^\Bas] + \int v_\text{ne}(\b{r}) n_{G^\Bas}(\b{r}) \d\b{r} + \bar{E}^\Bas[n_{G^\Bas}] \right\},
|
||||
\label{E0BGB}
|
||||
(\Gs{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla^2_{\br{}}}{2} ) \delta(\br{}-\br{}'),
|
||||
\end{equation}
|
||||
where the minimization is over $N$-representable one-electron Green's functions $G^\Bas(\b{r},\b{r}',\omega)$ representable in the basis set $\Bas$.
|
||||
and we have used the notation
|
||||
\begin{equation}
|
||||
\Tr[A B] = \frac{1}{2\pi i} \int_{-\infty}^{+\infty} d\omega \, e^{i \omega 0^+} \iint \dbr{} \dbr{}' A(\br{},\br{}',\omega) B(\br{}',\br{}{},\omega).
|
||||
\end{equation}
|
||||
In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green's functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta \Phi_\Hxc^\Bas[\G{}{}] / \delta \G{}{}(\br{},\br{}',\omega) = \Sig{\Hxc}{\Bas}[\G{}{}](\br{},\br{}',\omega)$.
|
||||
Inserting Eqs.~\eqref{eq:Fn} and \eqref{eq:FBn} into Eq.~\eqref{eq:E0B}, we finally arrive at
|
||||
\begin{equation}
|
||||
\E{0}{\Bas} = \min_{\G{}{\Bas}} \qty{ \Omega^\Bas[\G{}{\Bas}] + \int \vne(\br{}) \n{\G{}{\Bas}}{}(\br{}) \dbr{} + \bE{}{\Bas}[\n{\G{}{\Bas}}{}] },
|
||||
\label{eq:E0BGB}
|
||||
\end{equation}
|
||||
where the minimization is over $\Ne$-representable one-electron Green's functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$.
|
||||
|
||||
The stationary condition from Eq.~(\ref{E0BGB}) gives the following Dyson equation
|
||||
The stationary condition from Eq.~\eqref{eq:E0BGB} gives the following Dyson equation
|
||||
\begin{equation}
|
||||
(G^\Bas)^{-1} = (G_\text{0}^\Bas)^{-1}- \Sigma_\text{Hxc}^\Bas[G^\Bas]- \bar{\Sigma}^\Bas[n_{G^\Bas}],
|
||||
\label{Dyson}
|
||||
(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\text{Hxc}}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
|
||||
\label{eq:Dyson}
|
||||
\end{equation}
|
||||
where $(G_\text{0}^\Bas)^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $v_\text{ne}(\b{r})$,
|
||||
$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\text{ne}(\b{r}) + \lambda)\delta(\b{r}-\b{r}')$ with the chemical potential $\lambda$, and $\bar{\Sigma}^\Bas$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
|
||||
\begin{equation}
|
||||
\bar{\Sigma}^\Bas[n](\b{r},\b{r}') = \bar{v}^\Bas[n](\b{r}) \delta(\b{r}-\b{r}'),
|
||||
(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} + \vne(\br{}) + \lambda) \delta(\br{}-\br{}')
|
||||
\end{equation}
|
||||
with $\bar{v}^\Bas[n](\br{}) = \delta \bE{}{\Bas}[n] / \delta \n{}{}(\br{})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green's function $G^\Bas(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^\Bas[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^\Bas$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{\Bas\to \CBS} = 0$, and the Green's function becomes exact, $G^{\Bas\to \CBS}=G$.
|
||||
with the chemical potential $\lambda$, and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
|
||||
\begin{equation}
|
||||
\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}'),
|
||||
\end{equation}
|
||||
with $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$. The solution of the Dyson equation \eqref{eq:Dyson} gives the Green's function $\G{}{\Bas}(\br{},\br{}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bSig{}{\Bas}[\n{}{}]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bSig{}{\Bas}$. Of course, in the CBS limit, the basis-set correction vanishes, $\bSig{}{\Bas \to \CBS} = 0$, and the Green's function becomes exact, $\G{}{\Bas\to \CBS} = \G{}{}$.
|
||||
|
||||
The Dyson equation \eqref{eq:Dyson} can be written with an arbitrary reference
|
||||
\begin{equation}
|
||||
(\G{}{\Bas})^{-1} = (\G{\text{ref}}{\Bas})^{-1} - \qty( \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \Sig{\text{ref}}{\Bas} ) - \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
|
||||
\end{equation}
|
||||
where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$.
|
||||
For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \Sig{\Hx,\HF}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham, $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \pot{\Hxc}{\Bas}(\br{}) \delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{The $GW$ Approximation}
|
||||
\subsection{The {\GW} Approximation}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The Dyson equation can be written with an arbitrary reference
|
||||
\begin{equation}
|
||||
(G^\Bas)^{-1} = (G_\text{ref}^\Bas)^{-1}- \left( \Sigma_\text{Hxc}^\Bas[G^\Bas]- \Sigma_\text{ref}^\Bas \right) - \bar{\Sigma}^\Bas[n_{G^\Bas}],
|
||||
\end{equation}
|
||||
where $(G_\text{ref}^\Bas)^{-1} = (G_\text{0}^\Bas)^{-1} - \Sigma_\text{ref}^\Bas$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^\Bas(\b{r},\b{r}') = \Sigma_\text{Hx,HF}^\Bas(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^\Bas(\b{r},\b{r}') = v_\text{Hxc}^\Bas(\b{r}) \delta(\b{r}-\b{r}')$ is the local Hxc potential.
|
||||
|
||||
|
||||
Here, we provide self-contained summary of the main equations and quantities behind {\GOWO}.
|
||||
In this section, 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 a given (occupied or virtual) orbital $p$, the correlation part of the self-energy is conveniently split in its hole (h) and particle (p) contributions
|
||||
For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy can decomposed in its hole (h) and particle (p) contributions
|
||||
\begin{equation}
|
||||
\label{eq:SigC}
|
||||
\SigC{p}(\omega) = \SigCp{p}(\omega) + \SigCh{p}(\omega),
|
||||
\end{equation}
|
||||
which, within the $GW$ approximation, read
|
||||
which, within the {\GW} approximation, read
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\label{eq:SigCh}
|
||||
@ -370,12 +387,16 @@ The screened two-electron integrals
|
||||
\begin{equation}
|
||||
[pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x}
|
||||
\end{equation}
|
||||
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994} $(pq|rs)$ and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
|
||||
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994}
|
||||
\begin{equation}
|
||||
(pq|rs) = \iint \MO{p}(\br{}) \MO{q}(\br{}) \frac{1}{r_{12}} \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}',
|
||||
\end{equation}
|
||||
and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
|
||||
\begin{equation}
|
||||
\label{eq:LR}
|
||||
\begin{pmatrix}
|
||||
\bA & \bB \\
|
||||
\bB & \bA \\
|
||||
-\bB & -\bA \\
|
||||
\end{pmatrix}
|
||||
\begin{pmatrix}
|
||||
\bX \\
|
||||
@ -383,10 +404,6 @@ are obtained via the contraction of the bare two-electron integrals \cite{Gill_1
|
||||
\end{pmatrix}
|
||||
=
|
||||
\bOm
|
||||
\begin{pmatrix}
|
||||
\boldsymbol{1} & \boldsymbol{0} \\
|
||||
\boldsymbol{0} & \boldsymbol{-1} \\
|
||||
\end{pmatrix}
|
||||
\begin{pmatrix}
|
||||
\bX \\
|
||||
\bY \\
|
||||
@ -395,28 +412,44 @@ are obtained via the contraction of the bare two-electron integrals \cite{Gill_1
|
||||
with
|
||||
\begin{align}
|
||||
\label{eq:RPA}
|
||||
A_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb),
|
||||
A_{ia,jb} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 (ia|jb),
|
||||
&
|
||||
B_{ia,jb} & = 2 (ia|bj),
|
||||
\end{align}
|
||||
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
|
||||
The one-electron energies $\epsilon_p$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} are either the HF or KS orbital energies.
|
||||
Equation \eqref{eq:LR} also provides the neutral excitation energies $\Om{x}$.
|
||||
The one-electron energies $\e{p}$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} and their corresponding orbitals $\MO{p}(\br{})$ are either the HF or KS energies and orbitals depending on the chosen reference.
|
||||
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which are used to build the screened Coulomb potential $\W{}{}$.
|
||||
|
||||
The {\GOWO} QP energies are provided by one of the many solutions of the (non-linear) QP equation \cite{Hybertsen_1985a, vanSetten_2013}
|
||||
The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
|
||||
\begin{equation}
|
||||
\label{eq:QP-G0W0}
|
||||
\omega = \epsilon_{p} + \Re[\SigC{p}(\omega)].
|
||||
\omega = \e{p} + \Re[\SigC{p}(\omega)].
|
||||
\end{equation}
|
||||
In this case, special care has to be taken in order to select the ``right'' solution, known as the QP solution. \cite{Veril_2019}
|
||||
In particular, it is usually worth calculating its renormalization weight (or factor)
|
||||
with the largest renormalization weight (or factor)
|
||||
\begin{equation}
|
||||
\label{eq:Z}
|
||||
\Z{p} = \qty[ 1 - \left. \pdv{\Re[\SigC{p}(\omega)]}{\omega} \right|_{\omega = \epsilon_{p}}]^{-1}.
|
||||
\Z{p} = \qty[ 1 - \left. \pdv{\Re[\SigC{p}(\omega)]}{\omega} \right|_{\omega = \e{p}}]^{-1}.
|
||||
\end{equation}
|
||||
Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
|
||||
In a well-behaved case, the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Short-range correlation functionals}
|
||||
\label{sec:srDFT}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
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 local-density approximation ($\LDA$) functional with multideterminant reference \cite{PazMorGorBac-PRB-06} and a Perdew-Burke-Ernzerhof ($\PBE$) inspired correlation functional \cite{FerGinTou-JCP-19} which interpolate between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
|
||||
Additionally to the one-electron density, these RS-DFT functionals requires a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error.
|
||||
We refer the interested reader to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are given.
|
||||
The basis set corrected {\GOWO} quasiparticle energies $\beGOWO{p}$ are thus given by
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
\beGOWO{p}
|
||||
& = \eGOWO{p} + \int \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}) \MO{p}(\br{}') \dbr{} \dbr{}'
|
||||
\\
|
||||
& = \eGOWO{p} + \int \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{})^2 \dbr{}.
|
||||
\end{split}
|
||||
\end{equation}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Computational details}
|
||||
@ -572,7 +605,7 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\fnt[1]{This work.}
|
||||
\fnt[2]{Unpublished data taken from \url{https://gw100.wordpress.com} obtained with Turbomole v7.0.}
|
||||
\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]{Reference \onlinecite{Krause_2015}.}
|
||||
\fnt[5]{Experimental values are taken from Ref.~\onlinecite{vanSetten_2015} and correspond to vertical ionization energies.}
|
||||
@ -606,7 +639,7 @@ The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purp
|
||||
% \end{tabular}
|
||||
% \end{ruledtabular}
|
||||
% \fnt[1]{This work.}
|
||||
% \fnt[2]{Unpublished data taken from \url{https://gw100.wordpress.com} obtained with MolGW 2.B.}
|
||||
% \fnt[2]{Unpublished data taken from \url{https://gw100.wordpress.com} obtained with MOLGW 2.B.}
|
||||
% \fnt[3]{Extrapolated values obtained from the def2-TZVP and def2-QZVP values.}
|
||||
% \fnt[4]{Reference \onlinecite{Krause_2015}.}
|
||||
% \fnt[5]{Experimental values are taken from Ref.~\onlinecite{vanSetten_2015} and correspond to laser photoelectron spectroscopy values.}
|
||||
|
||||
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Reference in New Issue
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