\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}