sllides Sanibel
This commit is contained in:
parent
512f380e48
commit
b3a154df69
|
|
@ -0,0 +1,707 @@
|
|||
\documentclass[9pt,aspectratio=169]{beamer}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage{hyperref}
|
||||
\usepackage{mathtools,amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}
|
||||
\usepackage{emoji}
|
||||
\usepackage[normalem]{ulem}
|
||||
\newcommand\hcancel[2][black]{\setbox0=\hbox{$#2$}%
|
||||
\rlap{\raisebox{.45\ht0}{\textcolor{#1}{\rule{\wd0}{1pt}}}}#2}
|
||||
\urlstyle{same}
|
||||
|
||||
\definecolor{darkgreen}{RGB}{0, 180, 0}
|
||||
\definecolor{fooblue}{RGB}{0,153,255}
|
||||
\definecolor{fooyellow}{RGB}{234,180,0}
|
||||
\definecolor{lavender}{rgb}{0.71, 0.49, 0.86}
|
||||
\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36}
|
||||
\newcommand{\violet}[1]{\textcolor{lavender}{#1}}
|
||||
\newcommand{\orange}[1]{\textcolor{orange}{#1}}
|
||||
\newcommand{\purple}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
|
||||
\newcommand{\green}[1]{\textcolor{darkgreen}{#1}}
|
||||
\newcommand{\yellow}[1]{\textcolor{fooyellow}{#1}}
|
||||
\newcommand{\red}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\cyan}[1]{\textcolor{cyan}{#1}}
|
||||
\newcommand{\magenta}[1]{\textcolor{magenta}{#1}}
|
||||
\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}}
|
||||
\newcommand{\pub}[1]{\textcolor{purple}{#1}}
|
||||
|
||||
\newcommand{\mc}{\multicolumn}
|
||||
\newcommand{\fnm}{\footnotemark}
|
||||
\newcommand{\fnt}{\footnotetext}
|
||||
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
||||
\newcommand{\QP}{\textsc{quantum package}}
|
||||
\newcommand{\T}[1]{#1^{\intercal}}
|
||||
|
||||
% coordinates
|
||||
\newcommand{\br}{\boldsymbol{r}}
|
||||
\newcommand{\bx}{\boldsymbol{x}}
|
||||
|
||||
% methods
|
||||
\newcommand{\NO}[1]{\{#1\}}
|
||||
\newcommand{\GW}{GW}
|
||||
\newcommand{\SRGGW}{\text{SRG-$GW$}}
|
||||
\newcommand{\SRGqsGW}{\text{SRG-qs$GW$}}
|
||||
\newcommand{\qs}{\text{qs}}
|
||||
\newcommand{\qsGW}{\text{qs}GW}
|
||||
\newcommand{\KS}{KS}
|
||||
\newcommand{\GOWO}{G_0W_0}
|
||||
\newcommand{\dRPA}{\text{dRPA}}
|
||||
\newcommand{\RPAx}{\text{RPAx}}
|
||||
\newcommand{\BSE}{\text{BSE}}
|
||||
\newcommand{\CC}{\text{CC}}
|
||||
|
||||
%
|
||||
\newcommand{\Ne}{N}
|
||||
\newcommand{\Norb}{K}
|
||||
\newcommand{\Nocc}{O}
|
||||
\newcommand{\Nvir}{V}
|
||||
|
||||
% operators
|
||||
\newcommand{\hH}{\Hat{H}}
|
||||
\newcommand{\hP}{\Hat{P}}
|
||||
\newcommand{\hQ}{\Hat{Q}}
|
||||
\newcommand{\hU}{\Hat{U}}
|
||||
\newcommand{\hI}{\Hat{I}}
|
||||
\newcommand{\heta}{\Hat{\eta}}
|
||||
\newcommand{\sH}{\Bar{H}}
|
||||
\newcommand{\hHN}{\Hat{H}_{\text{N}}}
|
||||
\newcommand{\hFN}{\Hat{F}_{\text{N}}}
|
||||
\newcommand{\hVN}{\Hat{V}_{\text{N}}}
|
||||
\newcommand{\hh}{\Hat{h}}
|
||||
\newcommand{\hf}{\Hat{f}}
|
||||
\newcommand{\bHN}{\Bar{H}_{\text{N}}}
|
||||
\newcommand{\tHN}{\Tilde{H}_{\text{N}}}
|
||||
\newcommand{\hT}{\Hat{T}}
|
||||
\newcommand{\hS}{\Hat{S}}
|
||||
\newcommand{\cre}[1]{\Hat{a}_{#1}^{\dag}}
|
||||
\newcommand{\ani}[1]{\Hat{a}_{#1}}
|
||||
\newcommand{\ca}[2]{\Hat{a}^{#1}_{#2}}
|
||||
|
||||
\newcommand{\cF}{\mathcal{F}}
|
||||
\newcommand{\cW}{\mathcal{W}}
|
||||
|
||||
% energies
|
||||
\newcommand{\Enuc}{E^\text{nuc}}
|
||||
\newcommand{\Ec}{E_\text{c}}
|
||||
\newcommand{\EHF}{E_\text{HF}}
|
||||
\newcommand{\ECC}{E_\text{CC}}
|
||||
|
||||
% orbital energies
|
||||
\newcommand{\e}[2]{\epsilon_{#1}^{#2}}
|
||||
\newcommand{\eHF}[1]{\epsilon_{#1}^\text{HF}}
|
||||
\newcommand{\eKS}[1]{\epsilon_{#1}^\text{KS}}
|
||||
\newcommand{\eGW}[1]{\epsilon_{#1}^{GW}}
|
||||
\newcommand{\Om}{\Omega}
|
||||
\newcommand{\eHOMO}[1]{\epsilon_\text{HOMO}^{#1}}
|
||||
\newcommand{\eLUMO}[1]{\epsilon_\text{LUMO}^{#1}}
|
||||
|
||||
% Matrix elements
|
||||
\newcommand{\MO}[2]{\phi_{#1}^{#2}}
|
||||
\newcommand{\SO}[2]{\psi_{#1}^{#2}}
|
||||
\newcommand{\ERI}[2]{\braket{#1}{#2}}
|
||||
\newcommand{\sERI}[2]{(#1|#2)}
|
||||
\newcommand{\dbERI}[2]{\mel{#1}{}{#2}}
|
||||
\newcommand{\Sig}{\Sigma}
|
||||
|
||||
% Matrices
|
||||
\newcommand{\bO}{\boldsymbol{0}}
|
||||
\newcommand{\bH}{\boldsymbol{H}}
|
||||
\newcommand{\bI}{\boldsymbol{1}}
|
||||
\newcommand{\bpsi}{\boldsymbol{\psi}}
|
||||
\newcommand{\bPsi}{\boldsymbol{\Psi}}
|
||||
\newcommand{\bSig}{\boldsymbol{\Sigma}}
|
||||
\newcommand{\bbH}{\Bar{\Bar{H}}}
|
||||
\newcommand{\bOm}{\boldsymbol{\Omega}}
|
||||
\newcommand{\bom}{\boldsymbol{\omega}}
|
||||
\newcommand{\bA}{\boldsymbol{A}}
|
||||
\newcommand{\bB}{\boldsymbol{B}}
|
||||
\newcommand{\bC}{\boldsymbol{C}}
|
||||
\newcommand{\bc}{\boldsymbol{c}}
|
||||
\newcommand{\bD}{\boldsymbol{D}}
|
||||
\newcommand{\bF}{\boldsymbol{F}}
|
||||
\newcommand{\bR}{\boldsymbol{R}}
|
||||
\newcommand{\bT}{\boldsymbol{T}}
|
||||
\newcommand{\bU}{\boldsymbol{U}}
|
||||
\newcommand{\bV}{\boldsymbol{V}}
|
||||
\newcommand{\bW}{\boldsymbol{W}}
|
||||
\newcommand{\bX}{\boldsymbol{X}}
|
||||
\newcommand{\bY}{\boldsymbol{Y}}
|
||||
\newcommand{\bZ}{\boldsymbol{Z}}
|
||||
\newcommand{\bP}{\boldsymbol{P}}
|
||||
\newcommand{\bQ}{\boldsymbol{Q}}
|
||||
\newcommand{\be}{\boldsymbol{\epsilon}}
|
||||
|
||||
% orbitals, gaps, etc
|
||||
\newcommand{\IP}{I}
|
||||
\newcommand{\EA}{A}
|
||||
\newcommand{\HOMO}{\text{HOMO}}
|
||||
\newcommand{\LUMO}{\text{LUMO}}
|
||||
\newcommand{\Eg}[1]{E_\text{g}^{#1}}
|
||||
\newcommand{\EgFun}{\Eg^\text{fund}}
|
||||
\newcommand{\EgOpt}{\Eg^\text{opt}}
|
||||
\newcommand{\EB}{E_B}
|
||||
|
||||
\newcommand{\ii}{\mathrm{i}}
|
||||
\newcommand{\om}{\yellow{\omega}}
|
||||
\newcommand{\la}{\lambda}
|
||||
|
||||
|
||||
\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse\\
|
||||
\url{https://lcpq.github.io/pterosor}}
|
||||
\usetheme{pterosor}
|
||||
\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos}
|
||||
\date{Feb 13th 2023}
|
||||
\title{A similarity renormalization group (SRG) approach to $GW$}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{A SRG Approach to Green's Function Methods}
|
||||
\begin{columns}
|
||||
\begin{column}{0.75\textwidth}
|
||||
\centering
|
||||
\begin{figure}%[H]
|
||||
\includegraphics[width=0.8\textwidth]{fig/flow}
|
||||
\end{figure}
|
||||
\end{column}
|
||||
\begin{column}{0.25\textwidth}
|
||||
\centering
|
||||
\small
|
||||
\includegraphics[width=0.8\textwidth]{fig/AMarie}
|
||||
\\
|
||||
Antoine Marie (PhD)
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\bigskip
|
||||
See also our work on the connections between CC and Green's function methods
|
||||
\\
|
||||
\pub{Quintero-Monsebaiz, Monino, Marie \& Loos, JCP 157 (2022) 231102}
|
||||
\\
|
||||
\bigskip
|
||||
$\hookrightarrow$ \pub{Scuseria et al. JCP 129 (2008) 231101}
|
||||
\\
|
||||
$\hookrightarrow$ \pub{Berkelbach, JCP 149 (2018) 041103; Lange \& Berkelbach, JCTC 14 (2018) 4224}
|
||||
\\
|
||||
$\hookrightarrow$ \pub{Tolle \& Chan, arXiv:2212.08982}
|
||||
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{The $GW$ Approximation}
|
||||
\begin{itemize}
|
||||
\item[\emoji{nerd-face}] The $GW$ approximation allows us to access \alert{charged} excitations (IPs \& EAs)
|
||||
\\
|
||||
\pub{Hedin, Phys. Rev. 139 (1965) A796}
|
||||
\bigskip
|
||||
\item[\emoji{face-with-monocle}] It yields accurate \alert{fundamental gaps} at an affordable price for \alert{solids} and \alert{molecules}
|
||||
\\
|
||||
\pub{Bruneval et al. Front. Chem. 9 (2021) 749779}
|
||||
\bigskip
|
||||
\item[\emoji{smiling-face-with-halo}] $GW$ corresponds to an elegant resummation of the direct ring diagrams
|
||||
\bigskip
|
||||
\item[\emoji{partying-face}] Hence, it is adequate for weak correlation or in the high-density regime
|
||||
\\
|
||||
\pub{Gell-Mann \& Brueckner, Phys. Rev. 106 (1957) 364}
|
||||
\bigskip
|
||||
\item[\emoji{confused}] \alert{Self-consistent} $GW$ calculations can be tricky to converge due to \alert{intruder states}
|
||||
\\
|
||||
\pub{Monino \& Loos JCP 156 (2022) 231101}
|
||||
\bigskip
|
||||
\item[\emoji{cry}] Going \alert{beyond} $GW$ is, let's say, difficult\ldots
|
||||
\\
|
||||
\pub{Mejuto-Zaera \& Vlcek, PRB 106 (2022) 165129}
|
||||
\end{itemize}
|
||||
% \pub{Martin, Reining \& Ceperley, \textit{Interacting Electrons: Theory and Computational Approaches}}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Hedin's Pentagon}
|
||||
\begin{columns}
|
||||
\begin{column}{0.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.8\linewidth]{fig/pentagon}
|
||||
\\
|
||||
\pub{Hedin, Phys Rev 139 (1965) A796}
|
||||
\end{column}
|
||||
\begin{column}{0.6\textwidth}
|
||||
\begin{block}{The wonderful equations of Hedin}
|
||||
\begin{align*}
|
||||
& \underbrace{\yellow{G}(12)}_{\text{Green's function}} = G_0(12) + \int G_0(13) \violet{\Sigma}(34) \yellow{G}(42) d(34)
|
||||
\\
|
||||
& \underbrace{\Gamma(123)}_{\text{vertex}} = \delta(12) \delta(13)
|
||||
+ \int \fdv{\violet{\Sigma}(12)}{\yellow{G}(45)} \yellow{G}(46) \yellow{G}(75) \Gamma(673) d(4567)
|
||||
\\
|
||||
& \underbrace{\orange{P}(12)}_{\text{polarizability}} = - i \int \yellow{G}(13) \Gamma(342) \yellow{G}(41) d(34)
|
||||
\\
|
||||
& \underbrace{\red{W}(12)}_{\text{screening}} = v(12) + \int v(13) \orange{P}(34) \red{W}(42) d(34)
|
||||
\\
|
||||
& \underbrace{\violet{\Sigma}(12)}_{\text{self-energy}} = i \int \yellow{G}(14) \red{W}(13) \Gamma(423) d(34)
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Hedin's \sout{Pentagon} Square}
|
||||
\begin{columns}
|
||||
\begin{column}{0.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.8\linewidth]{fig/square}
|
||||
\\
|
||||
\pub{Hedin, Phys Rev 139 (1965) A796}
|
||||
\end{column}
|
||||
\begin{column}{0.6\textwidth}
|
||||
\begin{block}{The $GW$ approximation}
|
||||
\begin{align*}
|
||||
& \underbrace{\yellow{G}(12)}_{\text{Green's function}} = G_0(12) + \int G_0(13) \violet{\Sigma}(34) \yellow{G}(42) d(34)
|
||||
\\
|
||||
& \underbrace{\Gamma(123)}_{\text{vertex}} = \delta(12) \delta(13)
|
||||
+ \hcancel[red]{\int \fdv{\violet{\Sigma}(12)}{\yellow{G}(45)} \yellow{G}(46) \yellow{G}(75) \Gamma(673) d(4567)}
|
||||
\\
|
||||
& \underbrace{\orange{P}(12)}_{\text{polarizability}} = - i \hcancel[red]{\int} \yellow{G}(\red{12}) \hcancel[red]{\Gamma(342)} \yellow{G}(\red{21}) \hcancel[red]{d(34)}
|
||||
= - i \yellow{G}(12)\yellow{G}(21)
|
||||
\\
|
||||
& \underbrace{\red{W}(12)}_{\text{screening}} = v(12) + \int v(13) \orange{P}(34) \red{W}(42) d(34)
|
||||
\\
|
||||
& \underbrace{\violet{\Sigma}(12)}_{\text{self-energy}} = i \hcancel[red]{\int} \yellow{G}(\red{12}) \red{W}(\red{12}) \hcancel[red]{\Gamma(423) d(34)}
|
||||
= i \yellow{G}(12) \red{W}(12)
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Dynamical Version of $GW$}
|
||||
\begin{columns}
|
||||
\begin{column}{0.6\textwidth}
|
||||
\begin{block}{Quasiparticle equation (in a general setting)}
|
||||
\begin{equation*}
|
||||
\qty[ \underbrace{\blue{\bF}}_{\text{\blue{Fock matrix}}} + \underbrace{\violet{\bSig^{\GW}} \qty(\om = \eGW{p})}_{\text{\violet{dynamic self-energy}}} ] \SO{p}{\GW}
|
||||
= \underbrace{\eGW{p}}_{\text{quasiparticle energies}} \SO{p}{\GW}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\end{column}
|
||||
\begin{column}{0.4\textwidth}
|
||||
\begin{block}{Practical issues}
|
||||
\begin{itemize}
|
||||
\bigskip
|
||||
\item dynamic
|
||||
\item highly non-linear
|
||||
\item non-Hermitian
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\begin{block}{$GW$ self-energy}
|
||||
\begin{equation*}
|
||||
\violet{\Sigma_{pq}^{\GW}}(\om)
|
||||
= \sum_{i\nu} \frac{\red{W_{pi}^{\nu}} \red{W_{qi}^{\nu}}}{\om - \eGW{i} + \orange{\Omega_{\nu}} - \underbrace{\ii \eta}_{\text{regularizer}}}
|
||||
+ \sum_{a\nu} \frac{\red{W_{pa}^{\nu}} \red{W_{qa}^{\nu}}}{\om - \eGW{a} - \underbrace{\orange{\Omega_{\nu}}}_{\text{\orange{RPA excitation}}} + \ii \eta}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\begin{block}{Screened two-electron integrals}
|
||||
\begin{equation*}
|
||||
\red{W_{pq}^{\nu}} = \sum_{ia}\ERI{pi}{qa}\underbrace{\orange{\qty(\bX+\bY)_{ia}^{\nu}}}_{\text{\orange{RPA eigenvectors}}}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{One-Shot $GW$ or $G_0W_0$}
|
||||
\begin{block}{$G_0W_0$ features}
|
||||
\begin{itemize}
|
||||
\item Diagonal approximation
|
||||
\item A single loop of Hedin's equations
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\begin{block}{Quasiparticle equation (assuming a HF starting point)}
|
||||
\begin{align*}
|
||||
\qq*{\underline{Dynamic version:}} & \om = \blue{\eHF{p}} + \underbrace{\violet{\Sig_{pp}^{\GW}}(\om)}_{\text{built with HF quantities}}
|
||||
\\
|
||||
\qq*{\underline{Linearized (static) version:}} & \eGW{p} = \blue{\eHF{p}} + Z_p \violet{\Sig_{pp}^{\GW}}(\om = \blue{\eHF{p}})
|
||||
\qq{with}
|
||||
Z_p = \underbrace{\qty[ 1 - \eval{\pdv{\violet{\Sig_{pp}^{\GW}}(\om)}{\om}}_{\om = \blue{\eHF{p}}} ]^{-1}}_{\text{renormalization factor}}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\begin{block}{$G_0W_0$ issues}
|
||||
\begin{itemize}
|
||||
\item Highly starting point dependent
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Eigenvalue-Only $GW$ or ev$GW$}
|
||||
\begin{block}{ev$GW$ features}
|
||||
\begin{itemize}
|
||||
\item Diagonal approximation
|
||||
\item Self-consistency on the quasiparticle energies only
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\begin{block}{Quasiparticle equation (assuming a HF starting point)}
|
||||
\begin{align*}
|
||||
\om = \blue{\eHF{p}} + \underbrace{\violet{\Sig_{pp}^{\GW}}(\om)}_{\text{built with $GW$ quantities}}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\begin{block}{ev$GW$ issues}
|
||||
\begin{itemize}
|
||||
\item Lack of self-consistency on the orbitals
|
||||
\item Challenging to converge (even with DIIS)
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Quasiparticle Self-Consistent $GW$ or qs$GW$}
|
||||
\begin{block}{qs$GW$ features}
|
||||
\begin{itemize}
|
||||
\item Static approximation of the self-energy
|
||||
\item Brute-force symmetrization
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\begin{block}{Quasiparticle equation}
|
||||
\begin{equation*}
|
||||
\qty[ \blue{\bF{}{}} + \underbrace{\purple{\bSig^{\qsGW}} }_{\text{static self-energy}} ] \SO{p}{\GW}
|
||||
= \eGW{p} \SO{p}{\GW}
|
||||
\qq{with}
|
||||
\purple{\Sig_{pq}^{\qsGW}} = \underbrace{\frac{\violet{\Sig_{pq}^{\GW}}(\eGW{p}) + \violet{\Sig_{pq}^{\GW}}(\eGW{q})}{2}}_{\text{symmetrization}}
|
||||
\end{equation*}
|
||||
\pub{Faleev et al. PRL 93 (2004) 126406}
|
||||
\end{block}
|
||||
\begin{block}{qs$GW$ issues}
|
||||
\begin{itemize}
|
||||
\item ``Empirical'' symmetrization \pub{[Ismail-Beigi, JPCM 29 (2017) 385501]}
|
||||
\item Very challenging to converge (even with DIIS)
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Intruder-State Problem}
|
||||
\begin{equation*}
|
||||
\begin{split}
|
||||
\text{\alert{Intruder-state problem}}
|
||||
& \Leftrightarrow \text{a determinant in $\bQ$ becomes near-degenerate with a determinant in $\bP$}
|
||||
\\
|
||||
& \Rightarrow \text{appearance of small denominators}
|
||||
\\
|
||||
& \Rightarrow \text{\alert{convergence issues!}}
|
||||
\\
|
||||
\\
|
||||
\text{How to avoid intruder states?}
|
||||
& \Rightarrow \text{do not enforce $\bQ \bH^\text{eff} \bP = \bO$}
|
||||
\\
|
||||
& \Leftrightarrow \text{near-degenerate determinants are not decoupled}
|
||||
\\
|
||||
\end{split}
|
||||
\end{equation*}
|
||||
\begin{center}
|
||||
\begin{tabular}{m{0.5\textwidth} b{0.5\textwidth}}
|
||||
\includegraphics[width=0.5\textwidth]{fig/Heff_SRG}
|
||||
&
|
||||
$\Leftarrow$ \alert{Continuous (unitary) SRG transformation}
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\alert{SRG decouples the Hamiltonian starting from states that have the largest energy separation and progressing to states with smaller energy separation}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Historical Overview of SRG}
|
||||
\begin{itemize}
|
||||
\item Introduced independently by
|
||||
\bigskip
|
||||
\begin{itemize}
|
||||
\item Glazek and Wilson in quantum field theory
|
||||
\pub{[PRD 48 (1993) 5863, ibid 49 (1994) 4214]}
|
||||
\bigskip
|
||||
\item Wegner in condensed matter systems
|
||||
\pub{[Ann. Phys. 506 (1994) 77]}
|
||||
\end{itemize}
|
||||
\bigskip
|
||||
\item (In-Medium) SRG is used a lot in nuclear physics
|
||||
\\
|
||||
\pub{[Hergert et al. Phys. Rep. 621 (2016) 165]}
|
||||
\bigskip
|
||||
\item First introduced in chemistry by Steven White
|
||||
\\
|
||||
\pub{[JCP 117 (2002) 7472]}
|
||||
\bigskip
|
||||
\item More recently developed by the group of Francesco Evangelista (SR/MR-DSRG)
|
||||
\\
|
||||
\pub{[JCP 141 (2014) 054109; Annu. Rev. Phys. Chem. 70 (2019) 275]}
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{SRG Fundamental Equation}
|
||||
\begin{block}{Unitary transformation of the Hamiltonian}
|
||||
\begin{equation*}
|
||||
\boxed{\bH \rightarrow \bH(s) = \bU(s) \, \bH \, \bU^\dag(s), \quad s \in [0,\infty)}
|
||||
\end{equation*}
|
||||
\begin{itemize}
|
||||
\item For $s > 0$, $\bH(s)$ has a more (block) diagonal form than $\bH$
|
||||
\bigskip
|
||||
\item The \alert{flow variable} $s$ is a time-like parameter that controls the extent of the transformation
|
||||
\bigskip
|
||||
\begin{itemize}
|
||||
\item If $s = 0$, then $\bU(s) = \bI$, i.e., $\bH(s=0) = \bH$
|
||||
\bigskip
|
||||
\item In the limit $s \to \infty$, $\bH(s)$ becomes (block) diagonal
|
||||
\bigskip
|
||||
\end{itemize}
|
||||
\end{itemize}
|
||||
\begin{equation*}
|
||||
\bH(s) = \underbrace{\bH_\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH_\text{od}(s)}_{\text{off-diagonal}}
|
||||
\qq{$\Rightarrow$}
|
||||
\lim_{s\to\infty} \bH_\text{od}(s) = 0
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{SRG Flow Equation}
|
||||
\begin{block}{The SRG flow equation}
|
||||
\begin{equation*}
|
||||
\label{eq:flow_eq}
|
||||
\boxed{\dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)}, \quad \bH(0) = \bH}
|
||||
\end{equation*}
|
||||
\begin{equation*}
|
||||
\qq*{where the \alert{flow generator}}
|
||||
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s)
|
||||
\qq{is an \alert{anti-Hermitian} operator}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\alert{Suitable parametrization of $\heta(s)$ allows to integrate the flow equation and find a numerical solution of $\hH(s)$ that satisfies the boundary conditions without having to explicitly construct $\hU(s)$}
|
||||
\begin{block}{Wegner's canonical generator}
|
||||
\begin{equation*}
|
||||
\boxed{\boldsymbol{\eta}^\text{W}(s) = \comm{\bH_\text{d}(s)}{\bH_\text{od}(s)}}
|
||||
\end{equation*}
|
||||
\begin{equation*}
|
||||
\qq*{As long as $\boldsymbol{\eta}^\text{W}(s) \neq 0$,} \dv{}{s} \Tr[\bH_\text{od}(s)^2] \le 0
|
||||
\qq{$\Rightarrow$ \alert{off-diagonal decreases in a monotonic way}}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Perturbative Analysis}
|
||||
\begin{block}{Partitionning of the initial problem}
|
||||
\begin{equation*}
|
||||
\bH(s=0)
|
||||
= \underbrace{\bH_\text{d}(s=0)}_{\text{\alert{zeroth order}}}
|
||||
+ \la \underbrace{\bH_\text{od}(s=0)}_{\text{\alert{first order}}}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\begin{block}{Perturbative analysis of the SRG equations}
|
||||
\begin{align*}
|
||||
\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
|
||||
\\
|
||||
\boldsymbol{\eta}(s)
|
||||
& = \boldsymbol{\eta}^{(0)}(s) + \la \boldsymbol{\eta}^{(1)}(s) + \la^2 \boldsymbol{\eta}^{(2)}(s) + \cdots
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\bigskip
|
||||
\alert{How to identify the diagonal and off-diagonal terms in $GW$?}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Static Version of $GW$}
|
||||
\begin{equation*}
|
||||
\left.
|
||||
\begin{array}{cc}
|
||||
\qty[ \blue{\bF} + \violet{\bSig^{\GW}} \qty(\om = \eGW{p}) ] \SO{p}{\GW} = \eGW{p} \SO{p}{\GW}
|
||||
\\
|
||||
\\
|
||||
\begin{split}
|
||||
\violet{\bSig^{\GW}}(\om)
|
||||
& = \red{\bW^{\text{2h1p}}} \qty(\om \bI - \orange{\bC^{\text{2h1p}}})^{-1} (\red{\bW^{\text{2h1p}}})^{\dag}
|
||||
\\
|
||||
& + \red{\bW^{\text{2p1h}}} \qty(\om \bI - \orange{\bC^{\text{2p1h}}})^{-1} (\red{\bW^{\text{2p1h}}})^{\dag}
|
||||
\end{split}
|
||||
\end{array}
|
||||
\right\}
|
||||
\qq{$\xleftrightharpoons[\text{upfolding}]{\text{downfolding}}$}
|
||||
\begin{cases}
|
||||
\bH \Psi_{p}^{\GW} = \eGW{p} \Psi_{p}^{\GW}
|
||||
\\
|
||||
\bH =
|
||||
\begin{pmatrix}
|
||||
\blue{\bF} & \red{\bW^{\text{2h1p}}} & \orange{\bW^{\text{2p1h}}}
|
||||
\\
|
||||
(\red{\bW^{\text{2h1p}}})^\dag & \red{\bC^{\text{2h1p}}} & \bO
|
||||
\\
|
||||
(\orange{\bW^{\text{2p1h}}})^\dag & \bO & \orange{\bC^{\text{2p1h}}}
|
||||
\end{pmatrix}
|
||||
\end{cases}
|
||||
\end{equation*}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.5\linewidth]{fig/upfolding}
|
||||
\end{center}
|
||||
\pub{Bintrim \& Berkelbach, JCP 154 (2021) 041101; Monino \& Loos JCP 156 (2022) 231101; Tolle \& Chan, arXiv:2212.08982}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Regularized Quasiparticle Equation}
|
||||
\begin{block}{Regularized $GW$ equations up to second order}
|
||||
\begin{equation*}
|
||||
\qty[ \blue{\widetilde{\bF}}(s) + \magenta{\widetilde{\bSig}^{\SRGGW}}(\om = \eGW{p} ;s) ] \SO{p}{\GW} = \eGW{p} \SO{p}{\GW}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\begin{block}{Energy-dependent regularization}
|
||||
\begin{equation*}
|
||||
\blue{\widetilde{\bF}_{pq}}(s) = \delta_{pq} \blue{\eHF{p}}
|
||||
+ \sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 }
|
||||
\qty[ \red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}} - \purple{W_{pr}^{\nu}}(s) \purple{W_{qr}^{\nu}}(s) ]
|
||||
\qq{with}
|
||||
\Delta_{pr}^{\nu} = \eGW{p} - \eGW{r} \pm \Om_\nu
|
||||
\end{equation*}
|
||||
\begin{equation*}
|
||||
\magenta{\widetilde{\Sigma}_{pq}^{\SRGGW}}(\om;s)
|
||||
= \sum_{i\nu} \frac{\purple{W_{pi}^{\nu}}(s) \purple{W_{qi}^{\nu}}(s)}{\om - \eGW{i} + \Om_{\nu}}
|
||||
+ \sum_{a\nu} \frac{\purple{W_{pa}^{\nu}}(s) \purple{W_{qa}^{\nu}}(s)}{\om - \eGW{a} - \Om_{\nu}}
|
||||
\qq{with}
|
||||
\boxed{\purple{W_{pr}^{\nu}}(s) = \red{W_{pr}^{\nu}} e^{-(\Delta_{pr}^{\nu})^2 s}}
|
||||
\end{equation*}
|
||||
For a fixed value of the \alert{energy cut-off} $\Lambda = s^{-1/2}$,
|
||||
\begin{align*}
|
||||
\qif* \abs*{\Delta_{pr}^{\nu}} \gg \Lambda & \qthen & \purple{W_{pr}^{\nu}}(s) & = \red{W_{pr}^{\nu}} e^{-(\Delta_{pr}^{\nu})^2 s} \approx 0 & \qq{(decoupled)}
|
||||
\\
|
||||
\qif* \abs*{\Delta_{pr}^{\nu}} \ll \Lambda & \qthen & \purple{W_{pr}^{\nu}}(s) & \approx \red{W_{pr}^{\nu}} & \qq{(remains coupled)}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Limiting Forms}
|
||||
\begin{block}{Limit as $s \to 0$}
|
||||
\begin{equation*}
|
||||
\blue{\widetilde{\bF}}(s=0) = \blue{\bF}
|
||||
\qq{and}
|
||||
\magenta{\widetilde{\bSig}^{\SRGGW}}(\om;s=0) = \violet{\bSig^{\GW}}(\om)
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\begin{block}{Limit as $s \to \infty$}
|
||||
\begin{equation*}
|
||||
\magenta{\widetilde{\bSig}^{\SRGGW}}(\om;s\to\infty) = \bO
|
||||
\qq{and}
|
||||
\blue{\widetilde{F}_{pq}}(s\to\infty)
|
||||
= \delta_{pq} \blue{\eHF{p}}
|
||||
+ \underbrace{\sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 }
|
||||
\red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}}}_{\text{static correction}}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\alert{By removing the coupling terms, SRG transforms continuously the dynamic problem into a static one}
|
||||
\begin{block}{SRG-qs$GW$ self-energy from first principles}
|
||||
\begin{equation*}
|
||||
\cyan{\widetilde{\bSig}^{\SRGqsGW}}(\om;s)
|
||||
= \sum_{r\nu} \frac{\Delta_{pr}^{\nu} + \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 }
|
||||
\qty[ \red{W_{pr}^{\nu}} \red{W_{qr}^{\nu}} - \purple{W_{pr}^{\nu}}(s) \purple{W_{qr}^{\nu}}(s) ]
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{SRG-qs$GW$}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.9\textwidth]{fig/flow}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{qs$GW$ vs SRG-qs$GW$ functional forms for $s=1/(2\eta^2)$}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.9\textwidth]{fig/qs_vs_SRG}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Example: Principal IP of water (aug-cc-pVTZ) wrt $\Delta$CCSD(T)}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.8\textwidth]{fig/water}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Principal IPs for a set of small molecules (aug-cc-pVTZ) wrt $\Delta$CCSD(T)}
|
||||
\begin{center}
|
||||
\includegraphics[width=\textwidth]{fig/IPs}
|
||||
\end{center}
|
||||
\begin{tabular}{p{1cm}p{3cm}p{3cm}p{3cm}p{3cm}}
|
||||
MSE & 0.64 eV & 0.26 eV & 0.24 eV & 0.17 eV \\
|
||||
MAE & 0.74 eV & 0.32 eV & 0.25 eV & 0.19 eV \\
|
||||
\end{tabular}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Principal EAs for a set of small molecules (aug-cc-pVTZ) wrt $\Delta$CCSD(T)}
|
||||
\begin{center}
|
||||
\includegraphics[width=\textwidth]{fig/EAs}
|
||||
\end{center}
|
||||
\begin{tabular}{p{1cm}p{3cm}p{3cm}p{3cm}p{3cm}}
|
||||
MSE & -0.30 eV & -0.02 eV & 0.00 eV & 0.00 eV \\
|
||||
MAE & 0.32 eV & 0.19 eV & 0.11 eV & 0.12 eV \\
|
||||
\end{tabular}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Acknowledgements \& Funding}
|
||||
\begin{columns}
|
||||
\begin{column}{0.5\textwidth}
|
||||
\begin{itemize}
|
||||
\item \textbf{Antoine Marie}
|
||||
\item \textbf{Francesco Evangelista}
|
||||
\item Enzo Monino
|
||||
\item Roberto Orlando
|
||||
\item Yann Damour
|
||||
\item Sara Giarrusso
|
||||
\item Ra\'ul Quintero-Monsebaiz
|
||||
\item F\'abris Kossoski
|
||||
\item Anthony Scemama
|
||||
\item Michel Caffarel
|
||||
\end{itemize}
|
||||
\end{column}
|
||||
\begin{column}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{fig/ERC}
|
||||
\bigskip
|
||||
\url{https://pfloos.github.io/WEB_LOOS}
|
||||
\url{https://lcpq.github.io/PTEROSOR}
|
||||
\bigskip
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 219 KiB |
|
|
@ -0,0 +1,33 @@
|
|||
\mode<presentation>
|
||||
|
||||
% Settings
|
||||
\definecolor{myBlue}{RGB}{97,148,201}
|
||||
\definecolor{myDarkBlue}{RGB}{73,110,150}
|
||||
\definecolor{myPink}{RGB}{200,149,147}
|
||||
\definecolor{myRed}{RGB}{206,51,45}
|
||||
|
||||
|
||||
\setbeamercolor*{title page header}{fg=myRed}
|
||||
\setbeamercolor*{author}{fg=myDarkBlue}
|
||||
\setbeamercolor*{institute}{fg=myDarkBlue}
|
||||
\setbeamercolor*{date}{fg=myBlue}
|
||||
|
||||
\setbeamercolor*{item}{fg=myBlue}
|
||||
\setbeamertemplate{items}[triangle]
|
||||
\setbeamertemplate{sections/subsections in toc}[square]
|
||||
|
||||
\setbeamercolor*{section title}{fg=myBlue}
|
||||
\setbeamercolor*{subsection title}{fg=myBlue}
|
||||
\setbeamercolor{block title}{fg=white,bg=myBlue}
|
||||
%\setbeamercolor{block body}{bg=block title.bg!30!bg}
|
||||
|
||||
% alertblock
|
||||
\setbeamercolor{block title alerted}{fg=white,bg=MyPink}
|
||||
%\setbeamercolor{block body alerted}{bg=block title alerted.bg!10!bg}
|
||||
|
||||
% exampleblock
|
||||
\setbeamercolor{block title example}{fg=myBlue}
|
||||
%\setbeamercolor{block body example}{bg=white}
|
||||
|
||||
\mode
|
||||
<all>
|
||||
|
|
@ -0,0 +1,66 @@
|
|||
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=pdflatex 2020.6.14) 10 DEC 2020 21:15
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
%&-line parsing enabled.
|
||||
**beamerfontthemepterosor.sty
|
||||
(./beamerfontthemepterosor.sty
|
||||
LaTeX2e <2020-02-02> patch level 5
|
||||
L3 programming layer <2020-06-03>
|
||||
|
||||
LaTeX Warning: You have requested package `',
|
||||
but the package provides `beamerfontthemepterosor'.
|
||||
|
||||
Package: beamerfontthemepterosor 2020/12/10 pterosor font theme
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/latex/etoolbox/etoolbox.sty
|
||||
Package: etoolbox 2019/09/21 v2.5h e-TeX tools for LaTeX (JAW)
|
||||
\etb@tempcnta=\count167
|
||||
)
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/generic/iftex/ifxetex.sty
|
||||
Package: ifxetex 2019/10/25 v0.7 ifxetex legacy package. Use iftex instead.
|
||||
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/generic/iftex/iftex.sty
|
||||
Package: iftex 2020/03/06 v1.0d TeX engine tests
|
||||
))
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/generic/iftex/ifluatex.sty
|
||||
Package: ifluatex 2019/10/25 v1.5 ifluatex legacy package. Use iftex instead.
|
||||
)
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/latex/pgfopts/pgfopts.sty
|
||||
Package: pgfopts 2014/07/10 v2.1a LaTeX package options with pgfkeys
|
||||
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex
|
||||
\pgfkeys@pathtoks=\toks15
|
||||
\pgfkeys@temptoks=\toks16
|
||||
|
||||
(/usr/local/texlive/2020/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.c
|
||||
ode.tex
|
||||
\pgfkeys@tmptoks=\toks17
|
||||
)))
|
||||
\pgfopts@list@add@a@toks=\toks18
|
||||
\pgfopts@list@add@b@toks=\toks19
|
||||
)
|
||||
|
||||
Package beamerthemepterosor Warning: You need to compile with XeLaTeX or LuaLaT
|
||||
eX to use the Fira fonts on input line 97.
|
||||
|
||||
./beamerfontthemepterosor.sty:98: Undefined control sequence.
|
||||
l.98 \setbeamerfont
|
||||
{title}{size=\Large,%
|
||||
?
|
||||
./beamerfontthemepterosor.sty:98: Emergency stop.
|
||||
l.98 \setbeamerfont
|
||||
{title}{size=\Large,%
|
||||
End of file on the terminal!
|
||||
|
||||
|
||||
Here is how much of TeX's memory you used:
|
||||
855 strings out of 480478
|
||||
17943 string characters out of 5899595
|
||||
254474 words of memory out of 5000000
|
||||
16952 multiletter control sequences out of 15000+600000
|
||||
532338 words of font info for 24 fonts, out of 8000000 for 9000
|
||||
1141 hyphenation exceptions out of 8191
|
||||
43i,0n,46p,812b,16s stack positions out of 5000i,500n,10000p,200000b,80000s
|
||||
./beamerfontthemepterosor.sty:98: ==> Fatal error occurred, no output PDF file
|
||||
produced!
|
||||
|
|
@ -0,0 +1,317 @@
|
|||
%%
|
||||
%% This is file `beamerfontthemepterosor.sty',
|
||||
%% generated with the docstrip utility.
|
||||
%%
|
||||
%% The original source files were:
|
||||
%%
|
||||
%% beamerfontthemepterosor.dtx (with options: `package')
|
||||
%% ---------------------------------------------------------------------------
|
||||
%% Copyright 2015 Matthias Vogelgesang and the LaTeX community. A full list of
|
||||
%% contributors can be found at
|
||||
%%
|
||||
%% https://github.com/matze/mtheme/graphs/contributors
|
||||
%%
|
||||
%% and the original template was based on the HSRM theme by Benjamin Weiss.
|
||||
%%
|
||||
%% This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
|
||||
%% International License (https://creativecommons.org/licenses/by-sa/4.0/).
|
||||
%% ---------------------------------------------------------------------------
|
||||
\NeedsTeXFormat{LaTeX2e}
|
||||
\ProvidesPackage{beamerfontthemepterosor}[2020/12/10 pterosor font theme]
|
||||
\RequirePackage{etoolbox}
|
||||
\RequirePackage{ifxetex}
|
||||
\RequirePackage{ifluatex}
|
||||
\RequirePackage{pgfopts}
|
||||
|
||||
%\RequirePackage{mathpazo}
|
||||
\RequirePackage{libertine}
|
||||
|
||||
\ifboolexpr{bool {xetex} or bool {luatex}}{
|
||||
\@ifpackageloaded{fontspec}{
|
||||
\PassOptionsToPackage{no-math}{fontspec}
|
||||
}{
|
||||
\RequirePackage[no-math]{fontspec}
|
||||
}
|
||||
\newcounter{fontsnotfound}
|
||||
\newcommand{\checkfont}[1]{%
|
||||
\suppressfontnotfounderror=1%
|
||||
\font\x = "#1" at 10pt
|
||||
\selectfont
|
||||
\ifx\x\nullfont%
|
||||
\stepcounter{fontsnotfound}%
|
||||
\fi%
|
||||
\suppressfontnotfounderror=0%
|
||||
}
|
||||
|
||||
\newcommand{\iffontsavailable}[3]{%
|
||||
\setcounter{fontsnotfound}{0}%
|
||||
\expandafter\forcsvlist\expandafter%
|
||||
\checkfont\expandafter{#1}%
|
||||
\ifnum\value{fontsnotfound}=0%
|
||||
#2%
|
||||
\else%
|
||||
#3%
|
||||
\fi%
|
||||
}
|
||||
\iffontsavailable{Fira Sans Light,%
|
||||
Fira Sans Light Italic,%
|
||||
Fira Sans,%
|
||||
Fira Sans Italic}%
|
||||
{%
|
||||
\setsansfont[ItalicFont={Fira Sans Light Italic},%
|
||||
BoldFont={Fira Sans},%
|
||||
BoldItalicFont={Fira Sans Italic}]%
|
||||
{Fira Sans Light}%
|
||||
}{%
|
||||
\iffontsavailable{Fira Sans Light OT,%
|
||||
Fira Sans Light Italic OT,%
|
||||
Fira Sans OT,%
|
||||
Fira Sans Italic OT}%
|
||||
{%
|
||||
\setsansfont[ItalicFont={Fira Sans Light Italic OT},%
|
||||
BoldFont={Fira Sans OT},%
|
||||
BoldItalicFont={Fira Sans Italic OT}]%
|
||||
{Fira Sans Light OT}%
|
||||
}{%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Could not find Fira Sans fonts%
|
||||
}
|
||||
}
|
||||
}
|
||||
\iffontsavailable{Fira Mono, Fira Mono Bold}{%
|
||||
\setmonofont[BoldFont={Fira Mono Medium}]{Fira Mono}%
|
||||
}{%
|
||||
\iffontsavailable{Fira Mono OT, Fira Mono Bold OT}{%
|
||||
\setmonofont[BoldFont={Fira Mono Medium OT}]{Fira Mono OT}%
|
||||
}{%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Could not find Fira Mono fonts%
|
||||
}
|
||||
}
|
||||
}
|
||||
\AtBeginEnvironment{tabular}{%
|
||||
\addfontfeature{Numbers={Monospaced}}%
|
||||
}
|
||||
}{%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
You need to compile with XeLaTeX or LuaLaTeX to use the Fira fonts%
|
||||
}
|
||||
}
|
||||
\setbeamerfont{title}{size=\Large,%
|
||||
series=\bfseries}
|
||||
\setbeamerfont{author}{size=\small}
|
||||
\setbeamerfont{date}{size=\small}
|
||||
\setbeamerfont{section title}{size=\Large,%
|
||||
series=\bfseries}
|
||||
\setbeamerfont{block title}{size=\normalsize,%
|
||||
series=\bfseries}
|
||||
\setbeamerfont{block title alerted}{size=\normalsize,%
|
||||
series=\bfseries}
|
||||
\setbeamerfont*{subtitle}{size=\large}
|
||||
\setbeamerfont{frametitle}{size=\large,%
|
||||
series=\bfseries}
|
||||
\setbeamerfont{caption}{size=\small}
|
||||
\setbeamerfont{caption name}{series=\bfseries}
|
||||
\setbeamerfont{description item}{series=\bfseries}
|
||||
\setbeamerfont{page number in head/foot}{size=\scriptsize}
|
||||
\setbeamerfont{bibliography entry author}{size=\normalsize,%
|
||||
series=\normalfont}
|
||||
\setbeamerfont{bibliography entry title}{size=\normalsize,%
|
||||
series=\bfseries}
|
||||
\setbeamerfont{bibliography entry location}{size=\normalsize,%
|
||||
series=\normalfont}
|
||||
\setbeamerfont{bibliography entry note}{size=\small,%
|
||||
series=\normalfont}
|
||||
\setbeamerfont{standout}{size=\Large,%
|
||||
series=\bfseries}
|
||||
\pgfkeys{
|
||||
/pterosor/font/titleformat title/.cd,
|
||||
.is choice,
|
||||
regular/.code={%
|
||||
\let\pterosor@titleformat\@empty%
|
||||
\setbeamerfont{title}{shape=\normalfont}%
|
||||
},
|
||||
smallcaps/.code={%
|
||||
\let\pterosor@titleformat\@empty%
|
||||
\setbeamerfont{title}{shape=\scshape}%
|
||||
},
|
||||
allsmallcaps/.code={%
|
||||
\let\pterosor@titleformat\lowercase%
|
||||
\setbeamerfont{title}{shape=\scshape}%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat title=allsmallcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
allcaps/.code={%
|
||||
\let\pterosor@titleformat\uppercase%
|
||||
\setbeamerfont{title}{shape=\normalfont}
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat title=allcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
}
|
||||
\pgfkeys{
|
||||
/pterosor/font/titleformat subtitle/.cd,
|
||||
.is choice,
|
||||
regular/.code={%
|
||||
\let\pterosor@subtitleformat\@empty%
|
||||
\setbeamerfont{subtitle}{shape=\normalfont}%
|
||||
},
|
||||
smallcaps/.code={%
|
||||
\let\pterosor@subtitleformat\@empty%
|
||||
\setbeamerfont{subtitle}{shape=\scshape}%
|
||||
},
|
||||
allsmallcaps/.code={%
|
||||
\let\pterosor@subtitleformat\lowercase%
|
||||
\setbeamerfont{subtitle}{shape=\scshape}%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat subtitle=allsmallcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
allcaps/.code={%
|
||||
\let\pterosor@subtitleformat\uppercase%
|
||||
\setbeamerfont{subtitle}{shape=\normalfont}%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat subtitle=allcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
}
|
||||
\pgfkeys{
|
||||
/pterosor/font/titleformat section/.cd,
|
||||
.is choice,
|
||||
regular/.code={%
|
||||
\let\pterosor@sectiontitleformat\@empty%
|
||||
\setbeamerfont{section title}{shape=\normalfont}%
|
||||
},
|
||||
smallcaps/.code={%
|
||||
\let\pterosor@sectiontitleformat\@empty%
|
||||
\setbeamerfont{section title}{shape=\scshape}%
|
||||
},
|
||||
allsmallcaps/.code={%
|
||||
\let\pterosor@sectiontitleformat\MakeLowercase%
|
||||
\setbeamerfont{section title}{shape=\scshape}%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat section=allsmallcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
allcaps/.code={%
|
||||
\let\pterosor@sectiontitleformat\MakeUppercase%
|
||||
\setbeamerfont{section title}{shape=\normalfont}%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat section=allcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
}
|
||||
\pgfkeys{
|
||||
/pterosor/font/titleformat frame/.cd,
|
||||
.is choice,
|
||||
regular/.code={%
|
||||
\let\pterosor@frametitleformat\@empty%
|
||||
\setbeamerfont{frametitle}{shape=\normalfont}%
|
||||
},
|
||||
smallcaps/.code={%
|
||||
\let\pterosor@frametitleformat\@empty%
|
||||
\setbeamerfont{frametitle}{shape=\scshape}%
|
||||
},
|
||||
allsmallcaps/.code={%
|
||||
\let\pterosor@frametitleformat\MakeLowercase%
|
||||
\setbeamerfont{frametitle}{shape=\scshape}%
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat frame=allsmallcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
allcaps/.code={%
|
||||
\let\pterosor@frametitleformat\MakeUppercase%
|
||||
\setbeamerfont{frametitle}{shape=\normalfont}
|
||||
\PackageWarning{beamerthemepterosor}{%
|
||||
Be aware that titleformat frame=allcaps can lead to problems%
|
||||
}
|
||||
},
|
||||
}
|
||||
\pgfkeys{
|
||||
/pterosor/font/.cd,
|
||||
titleformattitle/.code=\pgfkeysalso{titleformat title=#1},
|
||||
titleformatsubtitle/.code=\pgfkeysalso{titleformat subtitle=#1},
|
||||
titleformatsection/.code=\pgfkeysalso{titleformat section=#1},
|
||||
titleformatframe/.code=\pgfkeysalso{titleformat frame=#1},
|
||||
}
|
||||
\newcommand{\pterosor@font@setdefaults}{
|
||||
\pgfkeys{/pterosor/font/.cd,
|
||||
titleformat title=regular,
|
||||
titleformat subtitle=regular,
|
||||
titleformat section=regular,
|
||||
titleformat frame=regular,
|
||||
}
|
||||
}
|
||||
\def\pterosor@titleformat#1{#1}
|
||||
\def\pterosor@subtitleformat#1{#1}
|
||||
\def\pterosor@sectiontitleformat#1{#1}
|
||||
\def\pterosor@frametitleformat#1{#1}
|
||||
\patchcmd{\beamer@title}%
|
||||
{\def\inserttitle{#2}}%
|
||||
{\def\inserttitle{\pterosor@titleformat{#2}}}%
|
||||
{}%
|
||||
{\PackageError{beamerfontthemepterosor}{Patching title failed}\@ehc}
|
||||
\patchcmd{\beamer@subtitle}%
|
||||
{\def\insertsubtitle{#2}}%
|
||||
{\def\insertsubtitle{\pterosor@subtitleformat{#2}}}%
|
||||
{}%
|
||||
{\PackageError{beamerfontthemepterosor}{Patching subtitle failed}\@ehc}
|
||||
\patchcmd{\sectionentry}
|
||||
{\def\insertsectionhead{#2}}
|
||||
{\def\insertsectionhead{\pterosor@sectiontitleformat{#2}}}
|
||||
{}
|
||||
{\PackageError{beamerfontthemepterosor}{Patching section title failed}\@ehc}
|
||||
\@tempswafalse
|
||||
\patchcmd{\beamer@section}
|
||||
{\def\insertsectionhead{\hyperlink{Navigation\the\c@page}{#1}}}
|
||||
{\def\insertsectionhead{\hyperlink{Navigation\the\c@page}{%
|
||||
\pterosor@sectiontitleformat{#1}}}}
|
||||
{\@tempswatrue}
|
||||
{}
|
||||
\patchcmd{\beamer@section}
|
||||
{\protected@edef\insertsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{#1}}}
|
||||
{\protected@edef\insertsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{%
|
||||
\noexpand\pterosor@sectiontitleformat{#1}}}}
|
||||
{\@tempswatrue}
|
||||
{}
|
||||
\if@tempswa\else
|
||||
\PackageError{beamerfontthemepterosor}{Patching section title failed}\@ehc
|
||||
\fi
|
||||
\@tempswafalse
|
||||
\patchcmd{\beamer@subsection}
|
||||
{\def\insertsubsectionhead{\hyperlink{Navigation\the\c@page}{#1}}}
|
||||
{\def\insertsubsectionhead{\hyperlink{Navigation\the\c@page}{%
|
||||
\pterosor@sectiontitleformat{#1}}}}
|
||||
{\@tempswatrue}
|
||||
{}
|
||||
\patchcmd{\beamer@subsection}
|
||||
{\protected@edef\insertsubsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{#1}}}
|
||||
{\protected@edef\insertsubsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{%
|
||||
\noexpand\pterosor@sectiontitleformat{#1}}}}
|
||||
{\@tempswatrue}
|
||||
{}
|
||||
\if@tempswa\else
|
||||
\PackageError{beamerfontthemepterosor}{Patching section title failed}\@ehc
|
||||
\fi
|
||||
\patchcmd{\beamer@@frametitle}
|
||||
{{%
|
||||
\gdef\insertframetitle{{#2\ifnum\beamer@autobreakcount>0\relax{}\space%
|
||||
\usebeamertemplate*{frametitle continuation}\fi}}%
|
||||
\gdef\beamer@frametitle{#2}%
|
||||
\gdef\beamer@shortframetitle{#1}%
|
||||
}}
|
||||
{{%
|
||||
\gdef\insertframetitle{{\pterosor@frametitleformat{#2}\ifnum%
|
||||
\beamer@autobreakcount>0\relax{}\space%
|
||||
\usebeamertemplate*{frametitle continuation}\fi}}%
|
||||
\gdef\beamer@frametitle{#2}%
|
||||
\gdef\beamer@shortframetitle{#1}%
|
||||
}}
|
||||
{}
|
||||
{\PackageError{beamerfontthemepterosor}{Patching frame title failed}\@ehc}
|
||||
\pterosor@font@setdefaults
|
||||
\ProcessPgfPackageOptions{/pterosor/font}
|
||||
\endinput
|
||||
%%
|
||||
%% End of file `beamerfontthemepterosor.sty'.
|
||||
|
|
@ -0,0 +1,53 @@
|
|||
\mode<presentation>
|
||||
|
||||
\setbeamertemplate{background}{
|
||||
\ifnum\thepage>1
|
||||
\includegraphics[height=\paperheight]{beamerbackgroundpterosor.png}
|
||||
\else
|
||||
\includegraphics[height=\paperheight]{beamertitlepterosor.png}
|
||||
\fi
|
||||
}
|
||||
|
||||
% Title page
|
||||
\defbeamertemplate*{title page}{pterosor}[1][]
|
||||
{
|
||||
\vskip3cm%
|
||||
\begin{beamercolorbox}[wd=6.cm,leftskip=-0.5cm,#1]{title page header}
|
||||
\usebeamerfont{title}\inserttitle\par%
|
||||
\end{beamercolorbox}%
|
||||
\vskip0.2cm%
|
||||
\begin{beamercolorbox}[wd=6.cm,leftskip=-0.5cm,#1]{author}
|
||||
\usebeamerfont{author}\insertauthor%
|
||||
\end{beamercolorbox}
|
||||
\vskip0.3cm%
|
||||
\begin{beamercolorbox}[wd=6.cm,leftskip=-0.5cm,#1]{date}
|
||||
\usebeamerfont{author}\insertdate%
|
||||
\end{beamercolorbox}
|
||||
\vskip0.3cm%
|
||||
\begin{beamercolorbox}[wd=7.cm,leftskip=-0.5cm,#1]{institute}
|
||||
\usebeamerfont{author}{\tiny \insertinstitute}%
|
||||
\end{beamercolorbox}
|
||||
\vfill
|
||||
}
|
||||
|
||||
% Section page
|
||||
\defbeamertemplate*{section page}{pterosor}{
|
||||
\centering
|
||||
\begin{minipage}{22em}
|
||||
\raggedright
|
||||
\usebeamercolor[fg]{section title}
|
||||
\usebeamerfont{section title}
|
||||
\insertsectionhead\\[-1ex]
|
||||
\par
|
||||
\ifx\insertsubsectionhead\@empty\else%
|
||||
\usebeamercolor[fg]{subsection title}%
|
||||
\usebeamerfont{subsection title}%
|
||||
\insertsubsectionhead
|
||||
\fi
|
||||
\end{minipage}
|
||||
\par
|
||||
\vspace{\baselineskip}
|
||||
}
|
||||
|
||||
|
||||
\mode<all>
|
||||
|
|
@ -0,0 +1,20 @@
|
|||
\mode<presentation>
|
||||
|
||||
% Frame title
|
||||
\defbeamertemplate*{frametitle}{pterosor}[1][]
|
||||
{
|
||||
\begin{beamercolorbox}[wd=\paperwidth,ht=0.95cm]{frametitle}
|
||||
\begin{tikzpicture}
|
||||
% \useasboundingbox[fill=white](0,0) rectangle(\the\paperwidth,1.);
|
||||
%\fill[red!80!black] (0,0) rectangle(2.95,1.);
|
||||
%\fill[blue!50!cyan!80!black] (3.05,0) rectangle(\the\paperwidth,1.);
|
||||
%\fill[white!50] (3.05,0) rectangle(\the\paperwidth,1.);
|
||||
\useasboundingbox[](0,0) rectangle(\the\paperwidth,1.);
|
||||
\node[anchor=east, white,font=\large] at
|
||||
(\paperwidth,0.25){\insertframetitle \phantom{m}};
|
||||
\end{tikzpicture}
|
||||
\end{beamercolorbox}
|
||||
}
|
||||
|
||||
\mode<all>
|
||||
|
||||
|
|
@ -0,0 +1,29 @@
|
|||
\mode<presentation>
|
||||
|
||||
|
||||
% Requirement
|
||||
\RequirePackage{tikz}
|
||||
|
||||
% Settings
|
||||
\useinnertheme{pterosor}
|
||||
\useoutertheme{pterosor}
|
||||
\usecolortheme{pterosor}
|
||||
\usefonttheme{pterosor}
|
||||
|
||||
|
||||
% we remove the navigation symbols and define blocks aspect.
|
||||
\setbeamertemplate{navigation symbols}{}
|
||||
\setbeamertemplate{blocks}[rounded][shadow=true]
|
||||
|
||||
\AtBeginSection[]{%
|
||||
\begin{frame}[plain]
|
||||
\usebeamertemplate{section page}
|
||||
\end{frame}
|
||||
}
|
||||
|
||||
\setbeamertemplate{footline}{
|
||||
\vspace{0.6cm}
|
||||
}
|
||||
|
||||
|
||||
\mode<all>
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 1.0 MiB |
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 56 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 26 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 48 KiB |
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 9.7 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 20 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 31 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 20 KiB |
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 204 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 29 KiB |
|
|
@ -0,0 +1,61 @@
|
|||
\documentclass{standalone}
|
||||
|
||||
\usepackage{graphicx,bm,microtype,hyperref,algpseudocode,subfigure,algorithm,algorithmicx,multirow,footnote,xcolor,physics,lipsum,wasysym}
|
||||
|
||||
\definecolor{darkgreen}{RGB}{0, 180, 0}
|
||||
\newcommand{\red}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\purple}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\orange}[1]{\textcolor{orange}{#1}}
|
||||
\newcommand{\green}[1]{\textcolor{darkgreen}{#1}}
|
||||
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
|
||||
\newcommand{\pub}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\violet}[1]{\textcolor{violet}{#1}}
|
||||
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows,positioning,shapes.geometric}
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{decorations.pathreplacing}
|
||||
|
||||
\tikzset{snake it/.style={
|
||||
decoration={snake,
|
||||
amplitude = .4mm,
|
||||
segment length = 2mm},decorate}
|
||||
}
|
||||
|
||||
\begin{document}
|
||||
|
||||
%%% FIGURE 1 %%%
|
||||
\begin{tikzpicture}
|
||||
\begin{scope}[very thick,
|
||||
node distance=2.5cm,on grid,>=stealth',
|
||||
GOp/.style={circle,draw,fill=yellow!40},
|
||||
POp/.style={circle,draw,fill=orange!40},
|
||||
WOp/.style={circle,draw,fill=red!40},
|
||||
SigOp/.style={circle,draw,fill=violet!40},
|
||||
DeadOp/.style={circle,draw,fill=gray!40},
|
||||
Input/.style={fill=white!40},
|
||||
Output/.style={fill=white!40}]
|
||||
|
||||
\node [GOp, align=center] (G) at (3*0.587785, 3*0.809017) {$G$};
|
||||
\node [DeadOp, align=center] (Gamma) at (3*0.951057, -3*0.309017) {$\Gamma$};
|
||||
\node [POp, align=center] (P) at (3*0, -3*1.00000) {$P$};
|
||||
\node [WOp, align=center] (W) at (-3*0.951057, -3*0.309017) {$W$};
|
||||
\node [SigOp, align=center] (Sigma) at (-3*0.587785, 3*0.809017) {$\Sigma$};
|
||||
\node [Input, align=center] (In) [above=of G] {};
|
||||
\node [Output, align=center] (Out) [above=of Sigma] {};
|
||||
\node [Input, align=center] (In) [above=of G] {In};
|
||||
\node [Output, align=center] (Out) [above=of Sigma] {Out};
|
||||
\path
|
||||
(G) edge [->,color=black] node [above,sloped,black] {$\Gamma = 1 + \fdv{\Sigma}{G} GG \Gamma$} (Gamma)
|
||||
(Gamma) edge [->,color=black] node [below,sloped,black] {$P = - i GG \Gamma$} (P)
|
||||
(P) edge [->,color=black] node [above,sloped,black] {$W = v + vPW$} (W)
|
||||
(W) edge [->,color=black] node [above,sloped,black] {$\Sigma = i GW\Gamma$} (Sigma)
|
||||
(Sigma) edge [->,color=black] node [above,sloped,black] {$G = G_0 + G_0 \Sigma G$} (G)
|
||||
(In) edge [->,color=black] node [above,sloped,black] {$\varepsilon^\text{HF/KS}$} (G)
|
||||
(Sigma) edge [->,color=black] node [above,sloped,black] {$\varepsilon^{GW\Gamma}$} (Out)
|
||||
;
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
%%% %%%
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 40 KiB |
Binary file not shown.
|
|
@ -0,0 +1,62 @@
|
|||
\documentclass{standalone}
|
||||
|
||||
\usepackage{graphicx,bm,microtype,hyperref,algpseudocode,subfigure,algorithm,algorithmicx,multirow,footnote,xcolor,physics,lipsum,wasysym}
|
||||
|
||||
\definecolor{darkgreen}{RGB}{0, 180, 0}
|
||||
\newcommand{\red}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\purple}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\orange}[1]{\textcolor{orange}{#1}}
|
||||
\newcommand{\green}[1]{\textcolor{darkgreen}{#1}}
|
||||
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
|
||||
\newcommand{\pub}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\violet}[1]{\textcolor{violet}{#1}}
|
||||
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows,positioning,shapes.geometric}
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{decorations.pathreplacing}
|
||||
|
||||
\tikzset{snake it/.style={
|
||||
decoration={snake,
|
||||
amplitude = .4mm,
|
||||
segment length = 2mm},decorate}
|
||||
}
|
||||
|
||||
\begin{document}
|
||||
|
||||
%%% FIGURE 1 %%%
|
||||
\begin{tikzpicture}
|
||||
\begin{scope}[very thick,
|
||||
node distance=2.5cm,on grid,>=stealth',
|
||||
GOp/.style={circle,draw,fill=yellow!40},
|
||||
POp/.style={circle,draw,fill=orange!40},
|
||||
WOp/.style={circle,draw,fill=red!40},
|
||||
SigOp/.style={circle,draw,fill=violet!40},
|
||||
DeadOp/.style={circle,draw,fill=gray!40,opacity=0.4},
|
||||
Input/.style={fill=white!40},
|
||||
Output/.style={fill=white!40}]
|
||||
|
||||
\node [GOp, align=center] (G) at (3*0.587785, 3*0.809017) {$G$};
|
||||
\node [DeadOp, align=center] (Gamma) at (3*0.951057, -3*0.309017) {$\Gamma$};
|
||||
\node [POp, align=center] (P) at (3*0, -3*1.00000) {$P$};
|
||||
\node [WOp, align=center] (W) at (-3*0.951057, -3*0.309017) {$W$};
|
||||
\node [SigOp, align=center] (Sigma) at (-3*0.587785, 3*0.809017) {$\Sigma$};
|
||||
\node [Input, align=center] (In) [above=of G] {};
|
||||
\node [Output, align=center] (Out) [above=of Sigma] {};
|
||||
\node [Input, align=center] (In) [above=of G] {In};
|
||||
\node [Output, align=center] (Out) [above=of Sigma] {Out};
|
||||
\path
|
||||
(G) edge [->,color=gray!50] node [above,sloped,gray!50] {$\Gamma = 1 + \fdv{\Sigma}{G} GG \Gamma$} (Gamma)
|
||||
(Gamma) edge [->,color=gray!50] node [below,sloped,gray!50] {$P = - i GG \Gamma$} (P)
|
||||
(P) edge [->,color=black] node [above,sloped,black] {$W = v + vPW$} (W)
|
||||
(W) edge [->,color=black] node [above,sloped,black] {$\Sigma = i GW$} (Sigma)
|
||||
(Sigma) edge [->,color=black] node [above,sloped,black] {$G = G_0 + G_0 \Sigma G$} (G)
|
||||
(G) edge [->,color=black] node [above,sloped,black] {$P = - i GG$} (P)
|
||||
(In) edge [->,color=black] node [above,sloped,black] {$\varepsilon^\text{HF/KS}$} (G)
|
||||
(Sigma) edge [->,color=black] node [above,sloped,black] {$\varepsilon^\text{GW}$} (Out)
|
||||
;
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
%%% %%%
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,77 @@
|
|||
\documentclass{standalone}
|
||||
|
||||
\usepackage{graphicx,bm,microtype,hyperref,algpseudocode,subfigure,algorithm,algorithmicx,multirow,footnote,xcolor,physics,lipsum,wasysym}
|
||||
|
||||
\definecolor{darkgreen}{RGB}{0, 180, 0}
|
||||
\newcommand{\red}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\purple}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\orange}[1]{\textcolor{orange}{#1}}
|
||||
\newcommand{\green}[1]{\textcolor{darkgreen}{#1}}
|
||||
\newcommand{\black}[1]{\textcolor{black}{#1}}
|
||||
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
|
||||
\newcommand{\pub}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\violet}[1]{\textcolor{violet}{#1}}
|
||||
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows,positioning,shapes.geometric}
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{decorations.pathreplacing}
|
||||
|
||||
\tikzset{snake it/.style={
|
||||
decoration={snake,
|
||||
amplitude = .4mm,
|
||||
segment length = 2mm},decorate}
|
||||
}
|
||||
|
||||
\begin{document}
|
||||
|
||||
%%% FIGURE 1 %%%
|
||||
\begin{tikzpicture}
|
||||
\begin{scope}[very thick,
|
||||
node distance=2.5cm,on grid,>=stealth',
|
||||
Operator/.style={circle,draw,fill=yellow!40},
|
||||
DeadOp/.style={circle,draw,fill=gray!40},
|
||||
Input/.style={fill=white!40},
|
||||
Output/.style={fill=white!40}]
|
||||
|
||||
% model space
|
||||
\draw [blue] (0,3.3) -- (1.1,3.3) -- (1.1,2.2) -- (0,2.2) -- (0,3.3);
|
||||
\node [blue,thick] (model) at (0.55,2.75) {$\boldsymbol{F}$};
|
||||
% perturbative space
|
||||
\draw [red] (0,3.3) -- (2.2,3.3) -- (2.2,1.1) -- (0,1.1) -- (0,3.3);
|
||||
\node [thick] (model) at (1.65,1.65) {$\red{\boldsymbol{C}^{\text{2h1p}}}$};
|
||||
\node [thick] (model) at (1.65,2.75) {$\red{\boldsymbol{W}^{\text{2h1p}}}$};
|
||||
\node [thick] (model) at (0.55,1.65) {$\red{\boldsymbol{W}^{\text{2h1p}}}$};
|
||||
% FCI space
|
||||
\draw [orange] (0,3.3) -- (3.3,3.3) -- (3.3,0) -- (0,0);
|
||||
\draw [black] (0,0) -- (0,3.3) -- (3.3,3.3);
|
||||
\node [thick] (model) at (2.75,0.55) {$\orange{\boldsymbol{C}^{\text{2p1h}}}$};
|
||||
\node [thick] (model) at (2.75,2.75) {$\orange{\boldsymbol{W}^{\text{2p1h}}}$};
|
||||
\node [thick] (model) at (0.55,0.55) {$\orange{\boldsymbol{W}^{\text{2p1h}}}$};
|
||||
\node [thick] (model) at (2.75,1.65) {$\black{\boldsymbol{0}}$};
|
||||
\node [thick] (model) at (1.65,0.55) {$\black{\boldsymbol{0}}$};
|
||||
\draw[step=1.1,black,thin] (0,0) grid (3.3,3.3);
|
||||
|
||||
\draw [decorate,decoration={brace,amplitude=3pt},xshift=3pt,yshift=0]
|
||||
(3.3,3.3) -- (3.3,2.2) node [black,midway,xshift=1.3cm] {\footnotesize internal space $\boldsymbol{P}$};
|
||||
|
||||
\draw [decorate,decoration={brace,mirror,amplitude=3pt},xshift=-3pt,yshift=0]
|
||||
(0,3.3) -- (0,2.2) node [blue,midway,xshift=-1.3cm] {\footnotesize \blue{1h \& 1p conf.}};
|
||||
|
||||
\draw [decorate,decoration={brace,mirror,amplitude=3pt},xshift=-3pt,yshift=0]
|
||||
(0,2.2) -- (0,1.1) node [black,midway,xshift=-1.3cm] {\footnotesize \red{2h1p conf.}};
|
||||
|
||||
\draw [decorate,decoration={brace,mirror,amplitude=3pt},xshift=-3pt,yshift=0]
|
||||
(0,1.1) -- (0,0) node [black,midway,xshift=-1.3cm] {\footnotesize \orange{2p1h conf.}};
|
||||
|
||||
\draw [decorate,decoration={brace,amplitude=3pt},xshift=3pt,yshift=0pt]
|
||||
(3.3,2.2) -- (3.3,0) node [black,midway,xshift=1.3cm] {\footnotesize external space $\boldsymbol{Q}$};
|
||||
|
||||
% \draw [decorate,decoration={brace,mirror,amplitude=3pt},xshift=0pt,yshift=-3pt]
|
||||
% (0,0) -- (3,0) node [black,midway,yshift=-0.3cm] {\footnotesize FCI space};
|
||||
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
%%% %%%
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
Loading…
Reference in New Issue