slides

2022-11-09 13:45:34 +01:00 · 2022-11-09 13:45:34 +01:00 · 61a6813281
commit 61a6813281
parent 1af1b6ea33
13 changed files with 1040 additions and 2 deletions
--- a/Notes/Notes.tex
+++ b/Notes/Notes.tex
@ -402,7 +402,7 @@ Finding a second quantized effective Hamiltonian for MBPT is far from being triv
 A general upfolded MBPT matrix can be written as
 \begin{equation}
  \label{eq:H_MBPT}
-  H =
+  \bH =
         \begin{pmatrix}
         \bF{}{} & \bV{}{} \\
         \bV{}{\dagger} & \bC{}{} 
@ -411,7 +411,7 @@ A general upfolded MBPT matrix can be written as
 Using SRG language, we define the diagonal and off-diagonal parts as 
 \begin{equation}
  \label{eq:H_MBPT_partitioning}
-  H(0) =
+  \bH(0) =
  \begin{pmatrix}
    \bF{}{} & \bO \\
    \bO & \bC{\text{d}}{} 
--- a/Slides/SRG-GF.tex
+++ b/Slides/SRG-GF.tex
@ -0,0 +1,520 @@
+\documentclass[9pt,aspectratio=169]{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{hyperref}
+\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem,physics,array}
+\usepackage{pifont}% http://ctan.org/pkg/pifont
+\newcommand{\cmark}{\ding{51}}%
+\newcommand{\xmark}{\ding{55}}%
+
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+\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}}
+\newcommand{\pub}[1]{\textcolor{purple}{#1}}
+
+\newcommand{\cdash}{\multicolumn{1}{c}{---}}
+\newcommand{\mc}{\multicolumn}
+\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
+\newcommand{\mr}{\multirow}
+\newcommand{\br}{\bm{r}}
+\newcommand{\ree}{r_{12}}
+\newcommand{\T}[1]{#1^{\intercal}}
+
+\newcommand{\la}{\lambda}
+\newcommand{\eps}{\epsilon}
+
+
+% operators
+\newcommand{\hH}{\Hat{H}}
+\newcommand{\hP}{\Hat{P}}
+\newcommand{\hQ}{\Hat{Q}}
+\newcommand{\hU}{\Hat{U}}
+\newcommand{\hI}{\Hat{1}}
+\newcommand{\hA}{\Hat{A}}
+\newcommand{\hT}{\Hat{T}}
+\newcommand{\hR}{\Hat{R}}
+\newcommand{\heta}{\Hat{\eta}}
+\newcommand{\hOm}{\Hat{\Omega}}
+\newcommand{\bH}{\Bar{H}}
+\newcommand{\hO}{\Hat{O}}
+
+
+% matrices
+\newcommand{\mA}{\boldsymbol{A}}
+\newcommand{\mB}{\boldsymbol{B}}
+\newcommand{\mx}{\boldsymbol{x}}
+\newcommand{\mS}{\boldsymbol{S}}
+
+\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse \\
+\url{https://lcpq.github.io/pterosor}}
+\usetheme{pterosor}
+\author{Pierre-Fran\c{c}ois (Titou) Loos}  % FIXME
+\date{3rd September 2021} % FIXME
+\title{Similarity Renormalization Group (SRG)}   %FIXME
+
+
+\begin{document}
+
+\maketitle
+
+%-----------------------------------------------------
+\begin{frame}{Effective Hamiltonian Theory}
+	\begin{block}{Similarity transformation of the Hamiltonian}
+		\begin{equation}
+			\underbrace{\Omega}_{\text{wave operator}}: 
+			\underbrace{\hH}_{\text{bare Hamiltonian}} \rightarrow \underbrace{\hH^\text{eff}}_{\text{effective Hamiltonian}} = \Omega^{-1} \, \hH \, \Omega
+		\end{equation}
+	\end{block}
+	\begin{block}{Examples of effective Hamiltonian theory}
+		\begin{itemize}
+			\item L\"owdin's partitioning technique
+			\item Transcorrelated method
+			\item (multireference) perturbation theory
+			\item (multireference) coupled-cluster (CC) theory 
+			\item Fock-space CC
+			\item Equation-of-motion CC
+		\end{itemize}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Mathematical Definitions}
+	\begin{block}{Matrix similarity}
+	 	Two $n$-by-$n$ matrices $\mA$ and $\mB$ are called \alert{similar} if there exists an invertible $n$-by-$n$ matrix $\mS$ such that
+		\begin{equation}
+			\mB = \mS^{-1} \cdot \mA \cdot \mS
+		\end{equation}
+	\end{block}
+	\begin{columns}
+		\begin{column}{0.85\textwidth}
+			\begin{block}{Properties}
+				\begin{itemize}
+					\item If $\mA$ and $\mB$ are similar, they share the \alert{same eigenvalues}, but \alert{different eigenvectors}
+					\begin{equation}
+						\qif \mA \cdot \mx = \la \mx	\qthen \mB \cdot (\mS^{-1} \cdot \mx) = \la (\mS^{-1} \cdot \mx)
+					\end{equation}
+					\item Similarity transformations aim at reducing the complexity of evaluating the eigenvalues
+					\item If $\mA$ is diagonalizable, it is similar to a diagonal matrix (not unique)
+					\item Even if $\mA$ is not diagonalizable, it is similar to a matrix in Jordan form (not unique)
+					\item \alert{Unitary transformations} are a type of similarity transformation for which $\mS^{-1} = \mS^{\dag}$
+					\item Every hermitian matrix is \alert{unitarily similar} to a diagonal real matrix 
+				\end{itemize}
+			\end{block}
+		\end{column}
+		\begin{column}{0.15\textwidth}
+			\includegraphics[width=\textwidth]{fig/JCF}
+		\end{column}
+	\end{columns}
+\end{frame}
+%-----------------------------------------------------
+
+
+%-----------------------------------------------------
+\begin{frame}{Model And External Spaces}
+	\begin{block}{Model/External space}
+		\begin{align}
+			\qq*{\underline{Model space projector:}} & \hP = \dyad*{\Psi_0}{\Psi_0}	= \sum_I \dyad{I}{I}
+			\\
+			\qq*{\underline{External space projector:}} & \hQ = \hI - \hP = \hI - \dyad*{\Psi_0}{\Psi_0}
+		\end{align}
+	\end{block}
+	\begin{block}{L\"owdin's partitioning technique}
+		\begin{equation}
+			\hH \ket{\Psi} = E \ket{\Psi}
+			\qq{$\Rightarrow$}
+			\begin{pmatrix}
+				\hP \hH \hP	&	\hP \hH \hQ		\\
+				\hQ \hH \hP		&	\hQ \hH \hQ	\\
+			\end{pmatrix}
+			\begin{pmatrix}
+				\hP \ket{\Psi}	\\
+				\hQ \ket{\Psi}	\\
+			\end{pmatrix}
+			= E
+			\begin{pmatrix}
+				\hP \ket{\Psi}	\\
+				\hQ \ket{\Psi}  \\
+			\end{pmatrix}
+		\end{equation}
+		\begin{center}
+			\begin{tabular}{p{0.18\textwidth} m{0.3\textwidth} b{0.18\textwidth}}
+				Traditional CC $\Rightarrow$ 
+				&
+				\includegraphics[width=0.3\textwidth]{fig/Heff}
+				&
+				$\Leftarrow$ Unitary CC 
+			\end{tabular}
+		\end{center}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Intruder-State Problem}
+	\begin{equation*}
+	\begin{split}
+		\text{\alert{Intruder-state problem}}
+		& \Leftrightarrow \text{a determinant in $Q$ becomes near-degenerate with a determinant in $P$}
+		\\
+		& \Rightarrow \text{appearance of small denominators}
+		\\
+		& \Rightarrow \text{\alert{convergence issues!}}
+		\\
+		\\
+		\text{How to avoid intruder states?}
+		& \Rightarrow \text{do not enforce $\hQ H^\text{eff} \hP = 0$}
+		\\
+		& \Leftrightarrow \text{near-degenerate determinants are not decoupled}
+		\\
+	\end{split}
+	\end{equation*}
+	\begin{center}
+		\begin{tabular}{m{0.5\textwidth} b{0.35\textwidth}}
+			\includegraphics[width=0.5\textwidth]{fig/Heff_SRG}
+			&
+			$\Leftarrow$ \alert{Continuous SRG transformation}
+		\end{tabular}
+	\end{center}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Historical Overview of SRG}
+	\begin{block}{SRG or flow equations}
+		\begin{itemize}
+			\item SRG is a \alert{continuous} (unitary) transformation of the Hamiltonian
+			\item Introduced independently by Glazek and Wilson \pub{[Phys. Rev. D 48, 5863 (1993), ibid 49, 4214 (1994)]} and Wegner \pub{[Ann. Phys. 506, 77 (1994)]} in quantum field theory
+			\item \alert{SRG decouples the Hamiltonian starting from states that have the largest energy separation and progressing to states with smaller energy separation}
+			\item SRG does not enforce $\hQ H^\text{eff} \hP = 0$
+			\item (MR-)SRG is used a lot in nuclear physics \pub{[Rep. Prog. Phys. 76, 126301 (2013)]}
+			\item First introduced in chemistry by Steven White (father of DMRG) \pub{[J. Chem. Phys. 117, 7472 (2002)]}
+			\item More recently developed by the group of Evangelista (SR/MR-DSRG) \pub{[J. Chem. Phys. 141, 054109 (2014); Annu. Rev. Phys. Chem. 70, 275 (2019)]}
+		\end{itemize}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{SRG Fundamental Equation}
+	\begin{block}{Unitary transformation of the Hamiltonian}
+		\begin{equation}
+			\boxed{\hH \rightarrow \hH(s) = \hU(s) \, \hH \, \hU^\dag(s), \quad s \in [0,\infty)}
+		\end{equation}
+		with 
+		\begin{equation}
+			\hH = 
+			\underbrace{\sum_{pq} h_{p}^{q} \Hat{a}_{p}^{q}}_{\text{one-body terms}}
+			+ \frac{1}{4} \underbrace{\sum_{pqrs} v_{pq}^{rs} \Hat{a}_{rs}^{pq}}_{\text{two-body terms}}
+			\qq{and}
+			\Hat{a}_{rs\cdots}^{pq\cdots} = \Hat{a}_{p}^\dag \Hat{a}_{q}^\dag \cdots \Hat{a}_{s} \Hat{a}_{r}
+		\end{equation}		
+		\begin{itemize}
+			\item For $s > 0$, $\hH(s)$ has a more (block) diagonal form than $\hH$
+			\item The \alert{flow variable} $s$ is a time-like parameter that controls the extent of the transformation
+			\begin{itemize}
+				\item If $s = 0$, then $\hU(s) = \hI$, i.e., $\hH(s=0) = \hH$
+				\item In the limit $s \to \infty$, $\hH(s)$ becomes (block) diagonal
+			\end{itemize}		
+		\end{itemize}		
+		\begin{equation}
+			\hH(s) = \underbrace{\hH_\text{d}(s)}_{\text{diagonal}} + \underbrace{\hH_\text{od}(s)}_{\text{off-diagonal}}
+			\qq{$\Rightarrow$}
+			\lim_{s\to\infty} \hH_\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{\hH(s)}{s} = \comm{\heta(s)}{\hH(s)}, \quad \hH(0) = \hH}
+		\end{equation}
+		\begin{equation*}
+			\qq*{where the \alert{flow generator}}
+			\heta(s) = \dv{\hU(s)}{s}	\hU^\dag(s)	= - \heta^\dag(s) 
+			\qq{is an \alert{anti-Hermitian} operator}
+		\end{equation*}
+	\end{block}
+	\begin{block}{Take-home message}
+		Suitable parametrization of $\heta(s)$ allows to integrate Eq.~\eqref{eq:flow_eq} and find a numerical solution of $\hH(s)$ that satisfies the boundary conditions without having to explicitly construct $\hU(s)$.
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Canonical Generator \& SRG Hamiltonian}
+	\begin{block}{Wegner's canonical generator}
+		\begin{equation}
+			\boxed{\eta^\text{W}(s) = \comm{\hH_\text{d}(s)}{\hH(s)} = \comm{\hH_\text{d}(s)}{\hH_\text{od}(s)}}
+		\end{equation}
+		As long as $\eta^\text{W}(s) \neq 0$, $\dv{}{s} \Tr[\hH_\text{od}^\dag(s)\hH_\text{od}(s)] \le 0$.
+		\\
+		Therefore, as $s \to \infty$, 
+		\begin{itemize}
+			\item $E_0(s) = \mel{\Psi_0}{\hH(s)}{\Psi_0}$ evolves towards one of the eigenvalues of $\hH$
+			\item The state $\hU^\dag(s) \ket{\Psi_0}$ approaches one of its eigenvectors
+		\end{itemize}
+	\end{block}
+	\begin{block}{The many-body SRG Hamiltonian}
+		\begin{equation}
+			 \hH(s) 
+			 = E_0(s) 
+			 + \underbrace{\sum_{pq} f_{p}^{q}(s) \{\Hat{a}_{p}^{q}\}}_{\text{one-body terms}}
+			 + \frac{1}{4} \underbrace{\sum_{pqrs} v_{pq}^{rs}(s) \overbrace{\{\Hat{a}_{rs}^{pq}\}}^{\text{normal ordered}}}_{\text{two-body terms}} 
+			 + \frac{1}{36} \underbrace{\sum_{pqrstu} w_{pqr}^{stu}(s) \{\Hat{a}_{stu}^{pqr}\}}_{\text{three-body terms}} 
+			 + \cdots
+		\end{equation}
+		In practice, $\hH(s)$ and $\heta(s)$ must be truncated to a given order $\Rightarrow$ \alert{SRG($n$)}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Perturbative Analysis}
+	\begin{block}{Perturbative analysis of the SRG equations }
+		\begin{align}
+			 \hH(s) & = \hH^{(0)}(s) + \la \hH^{(1)}(s) +  \la^2 \hH^{(2)}(s) + \cdots
+			 \\
+			 E_0(s) & = E_0^{(0)}(s) + \la E_0^{(1)}(s) +  \la^2 E_0^{(2)}(s) + \cdots
+		\end{align}
+	\end{block}
+	\begin{block}{First-order off-diagonal components}
+		For a fixed value of the \alert{energy cut-off} $\Lambda = s^{-1/2}$,
+		\begin{align}
+			\qif* \abs*{\Delta_{ij}^{ab}} \gg \Lambda	& \qthen	v_{ij}^{ab,(1)}(s) = \mel{ij}{}{ab} e^{-s (\Delta_{ij}^{ab})^2} \approx 0 \qq{(decoupled)}
+			\\ 
+			\qif* \abs*{\Delta_{ij}^{ab}} \ll \Lambda	& \qthen	v_{ij}^{ab,(1)}(s) \approx \mel{ij}{}{ab} \qq{(remains coupled)}
+		\end{align}
+	\end{block}
+	\begin{block}{Second-order energy contribution $\equiv$ renormalized MP2}
+		\begin{equation}
+			E_0^{(2)}(s) = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\Delta_{ij}^{ab}} \qty[ 1 - e^{-2s (\Delta_{ij}^{ab})^2}]
+			\qq{with}
+			\Delta_{ij}^{ab} = \eps_i + \eps_j - \eps_a - \eps_b
+		\end{equation}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Driven Similarity Renormalization Group (DSRG)}
+	\begin{block}{Main drawbacks of SRG}
+		\begin{itemize}
+			\item More challenging and less numerically robust to solve ODEs than polynomial equations
+			\item Lacks the nice exponential form of the CC expansion
+		\end{itemize}
+	\end{block}
+	\begin{block}{DSRG Hamiltonian = SRG \`a la CC \pub{[Evangelista, J. Chem. Phys. 141, 054109 (2014)]}}
+		\begin{equation}
+			\boxed{\bH(s) = e^{-\hA(s)} \, \hH \, e^{\hA(s)}, \quad s \in [0,\infty)}
+		\end{equation}
+	\end{block}
+	\begin{block}{Unitary CC inspired transformation}
+		\begin{gather}
+			\underbrace{\hA(s)}_{\text{anti-Hermitian operator}} = \hT(s) - \hT^{\dag}(s)
+			\qq{with}
+			\hT(s) = \hT_1(s) + \hT_2(s) + \cdots + \hT_n(s) 
+			\\
+			\hT_k(s) = \frac{1}{(k!)^2} \sum_{ij\cdots} \sum_{ab\cdots} \underbrace{t_{ij\cdots}^{ab\cdots}(s)}_{\text{cluster amplitudes}} \{ \Hat{a}_{ij\cdots}^{ab\cdots}\}
+		\end{gather}	
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{DSRG Equation}
+	\begin{block}{DSRG equation and source operator}
+		\begin{equation}
+			\qty[ e^{-\hA(s)}  \hH e^{\hA(s)} ]_\text{od} = \underbrace{\hR(s)}_{\text{(Hermitian) source operator}}
+			\qq{$\Rightarrow$}
+			\underbrace{\hOm(s)}_{\text{residual operator}} = \qty[ e^{-\hA(s)} \, \hH \, e^{\hA(s)} ]_\text{od} - \hR(s)
+		\end{equation}
+		$\hR(s)$ drives the off-diagonal components of $\bH(s)$ to zero 
+	\end{block}
+	\begin{block}{Many-body expansion of the residual operator}
+		\begin{gather}
+			\hOm(s) = \hOm_1(s) + \hOm_2(s) + \cdots + \hOm_n(s) 
+			\\
+			\hOm_k(s) = \frac{1}{(k!)^2} \sum_{ij\cdots} \sum_{ab\cdots} \omega_{ij\cdots}^{ab\cdots}(s) \{ \Hat{a}_{ij\cdots}^{ab\cdots}\} + \text{h.c.}
+		\end{gather}	
+	\end{block}
+	\begin{block}{DSRG equations}
+		\begin{equation}
+			\boxed{\omega_{ij\cdots}^{ab\cdots}(s) = 0}	\qq{$\Rightarrow$}	t_{ij\cdots}^{ab\cdots}(s)  \qq{\green{\checkmark}}
+			\qq{DSRG up to $n$-body terms $\Rightarrow$ \alert{DSRG($n$)}}
+		\end{equation}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+	
+%-----------------------------------------------------
+\begin{frame}{Many-Body Expansions}
+	\begin{block}{Many-body expansion of the source opertor}
+		\begin{gather}
+			\hR(s) = \hR_1(s) + \hR_2(s) + \cdots + \hR_n(s) 
+			\\
+			\hR_k(s) = \frac{1}{(k!)^2} \sum_{ij\cdots} \sum_{ab\cdots} r_{ij\cdots}^{ab\cdots}(s) \{ \Hat{a}_{ij\cdots}^{ab\cdots}\} + \text{h.c.}
+		\end{gather}	
+	\end{block}
+	\begin{block}{Many-body expansion of the DSRG Hamiltonian}
+		\begin{gather}
+			\bH(s) = E_0(s) + \hH_1(s) + \hH_2(s) + \cdots + \hH_n(s) 
+			\\
+			\bH_k(s) = \frac{1}{(k!)^2} \sum_{pqrs\cdots} \bH_{pq\cdots}^{rs\cdots}(s) \{ \Hat{a}_{pq\cdots}^{rs\cdots}\} 
+		\end{gather}	
+	\end{block}
+	\begin{block}{DSRG equations}
+		\begin{equation}
+			\boxed{\bH_{ij\cdots}^{ab\cdots}(s) = r_{ij\cdots}^{ab\cdots}(s)}	
+		\end{equation}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Elements of The DSRG Hamiltonian and Source Operator}
+	\begin{block}{Baker--Campbell--Hausdorff (BCH) formula}
+		\begin{equation}
+			\bH(s) 
+			\equiv e^{-\hA(s)} \hH e^{\hA(s)}
+			= \hH + \comm{\hH}{\hA(s)} + \frac{1}{2!} \comm{\comm{\hH}{\hA(s)}}{\hA(s)} + \frac{1}{3!} \comm{\comm{\comm{\hH}{\hA(s)}}{\hA(s)}}{\hA(s)} + \cdots
+		\end{equation}	
+		Because $\hA(s) = \hT(s) - \hT^{\dag}(s)$, the BCH expression \alert{does not terminate!}
+	\end{block}
+	\begin{block}{Yanai--Chan linear truncation scheme \pub{[J. Chem. Phys. 124, 194106 (2006); ibid 127, 104107 (2007)]}}
+		\begin{equation}
+			\comm{\cdot}{\hA} \approx \sum_{k=0}^m \underbrace{\comm{\cdot}{\hA}_k}_{\text{$k$-body component}} \equiv \comm{\cdot}{\hA}_{\{m\}}
+		\end{equation}	
+	\end{block}
+	\begin{block}{Parametrization of the source operator based on a perturbative analysis}
+		 \begin{equation}
+			r_{ij\cdots}^{ab\cdots}(s) = \qty[ \bH_{ij\cdots}^{ab\cdots}(s)  + \Delta_{ij\cdots}^{ab\cdots} t_{ij\cdots}^{ab\cdots}(s) ] e^{-s (\Delta_{ij\cdots}^{ab\cdots})^2}
+		\end{equation}	
+		 which satisfies the boundary conditions: (i) $\bH(s) = \hH$ when $s = 0$; (ii) $\bH_{ij\cdots}^{ab\cdots}(s) = 0$ when $s \to \infty$
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Truncated DSRG}
+	\begin{block}{DSRG(2) equations}
+		\begin{equation}
+			\text{\alert{DSRG(2)}}	\qq{$\Leftrightarrow$}	\hT(s) = \hT_1(s) + \hT_2(s)
+		\end{equation}
+		\begin{align}
+			\bH_{i}^{a}(s) & =  r_{i}^{a}(s)
+			&
+			\bH_{ij}^{ab}(s) & = r_{ij}^{ab}(s)
+		\end{align}
+	\end{block}
+	\begin{block}{Source operator}
+		\begin{align}
+			r_{i}^{a}(s) & =  \qty[ \bH_{i}^{a}(s)  + \Delta_{i}^{a} t_{i}^{a}(s) ] e^{-s (\Delta_{i}^{a})^2}
+			&
+			r_{ij}^{ab}(s) & =  \qty[ \bH_{ij}^{ab}(s)  + \Delta_{ij}^{ab} t_{ij}^{ab}(s) ] e^{-s (\Delta_{ij}^{ab})^2}
+		\end{align}
+	\end{block}
+	\begin{block}{Recursive evaluation of the approximate BCH expansion}
+		\begin{equation}
+			\bH_{\{ 2 \}}(s) = \sum_{k=0}^{N_\text{com}} \hO^{(k)}(s)
+			\qq{with}
+			\hO^{(k)}(s) = \frac{1}{k} \comm{\hO^{(k-1)}(s)}{\hA(s)}_{\{2\}}
+			\qq{and}
+			\hO^{(0)} = \hH 
+		\end{equation}
+		Computing $\bH_{\{ 2 \}}(s)$ is the computational bottleneck and scales as $\order*{O^2 V^2 N^2}$
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{DSRG(2) in Practice}
+	\begin{block}{Updating scheme}
+		One solves iteratively the DSRG(2) equations following
+		\begin{align}
+			\qq*{\underline{Singles amplitudes:}} & t_{i}^{a}(s) \leftarrow  \qty[ \bH_{i}^{a}(s)  + \Delta_{i}^{a} t_{i}^{a}(s) ] \frac{1 - e^{-s (\Delta_{i}^{a})^2}}{\Delta_{i}^{a}}
+			\\
+			\qq*{\underline{Doubles amplitudes:}} & t_{ij}^{ab}(s) \leftarrow  \qty[ \bH_{ij}^{ab}(s)  + \Delta_{ij}^{ab} t_{ij}^{ab}(s) ] \frac{1 - e^{-s (\Delta_{ij}^{ab})^2}}{\Delta_{ij}^{ab}}
+		\end{align}
+		with starting values
+		\begin{align}
+			t_{i}^{a}(s) & = 0
+			\\
+			t_{ij}^{ab}(s) & = v_{ij}^{ab,(1)} \frac{1 - e^{-s (\Delta_{ij}^{ab})^2}}{\Delta_{ij}^{ab}}
+		\end{align}
+		
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Further reading}
+	\small
+	\begin{itemize}
+%		\item \textit{A driven similarity renormalization group approach to quantum many-body problems}
+%		\\
+%		Evangelista, J. Chem. Phys. 141, 054109 (2014)
+		
+		\item \textit{Multireference driven similarity renormalization group: a second-order perturbative analysis}
+		\\
+		Li \& Evangelista, J. Chem. Theory Comput. 11, 2097 (2015)
+		
+		\item \textit{An integral-factorized implementation of the driven similarity renormalization group second-order multireference perturbation theory}
+		\\
+		Hannon et al. J. Chem. Phys. 144, 204111 (2016)
+		
+		\item \textit{Towards numerically robust multireference theories: The driven similarity renormalization group truncated to one- and two-body operators}
+		\\
+		Li \& Evangelista, J. Chem. Phys. 144, 164114 (2016)
+		
+		\item \textit{A low-cost approach to electronic excitation energies based on the driven similarity renormalization group}
+		\\
+		Li et al. J. Chem. Phys. 147, 074107 (2017)
+		
+		\item \textit{Driven similarity renormalization group: Third-order multireference perturbation theory}
+		\\
+		Li \& Evangelista J. Chem. Phys. 146, 124132 (2017);
+		
+		\item \textit{Driven similarity renormalization group for excited states: A state-averaged perturbation theory}
+		\\
+		Li \& Evangelista, J. Chem. Phys. 148, 124106 (2018);
+		
+%		\item \textit{Multireference Theories of Electron Correlation Based on the Driven Similarity Renormalization Group}
+%		\\
+%		Li \& Evangelista, Annu. Rev. Phys. Chem. 2019. 70:275?303
+		
+%		\item \textit{Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation}
+%		\\
+%		Zhang et al. J. Chem. Theory Comput., 15, 4399 (2019)
+		
+		\item \textit{Connected three-body terms in single-reference unitary many-body theories: Iterative and perturbative approximations}
+		\\
+		Li \& Evangelista, J. Chem. Phys. 152, 234116 (2020)
+		
+		\item \textit{Analytic gradients for the single-reference driven similarity renormalization group second-order perturbation theory}
+		\\
+		Wang, J. Chem. Phys. 151, 044118 (2019)
+		
+%		\item \textit{Spin-free implementation of the multireference driven similarity renormalization group: A benchmark study of open-shell diatomic molecules and spin-crossover energetics}
+%		\\
+%		Li \& Evangelista, arXiv:2106.07097
+	\end{itemize}
+\end{frame}
+%-----------------------------------------------------
+
+
+\end{document}
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+%% generated with the docstrip utility.
+%%
+%% The original source files were:
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+%% beamerfontthemepterosor.dtx  (with options: `package')
+%% ---------------------------------------------------------------------------
+%% Copyright 2015 Matthias Vogelgesang and the LaTeX community. A full list of
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+%% and the original template was based on the HSRM theme by Benjamin Weiss.
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+%% This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
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+  {\def\inserttitle{#2}}%
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+    \gdef\beamer@frametitle{#2}%
+    \gdef\beamer@shortframetitle{#1}%
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+  {{%
+      \gdef\insertframetitle{{\pterosor@frametitleformat{#2}\ifnum%
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+  {\PackageError{beamerfontthemepterosor}{Patching frame title failed}\@ehc}
+\pterosor@font@setdefaults
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--- a/Slides/beamerinnerthemepterosor.sty
+++ b/Slides/beamerinnerthemepterosor.sty
@ -0,0 +1,53 @@
+\mode<presentation>
+
+\setbeamertemplate{background}{
+\ifnum\thepage>1
+\includegraphics[height=\paperheight]{beamerbackgroundpterosor.png}
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+\includegraphics[height=\paperheight]{beamertitlepterosor.png}
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+{
+   \vskip3cm%
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+    \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}
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+      \usebeamerfont{author}{\tiny \insertinstitute}%
+    \end{beamercolorbox}
+  \vfill
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+
+% Section page
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+  \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>
--- a/Slides/beamerouterthemepterosor.sty
+++ b/Slides/beamerouterthemepterosor.sty
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+\mode<presentation>
+
+% Frame title
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+{
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+ \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.);
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+  (\paperwidth,0.25){\insertframetitle \phantom{m}};
+ \end{tikzpicture}
+ \end{beamercolorbox}
+}
+
+\mode<all>
+
--- a/Slides/beamerthemepterosor.sty
+++ b/Slides/beamerthemepterosor.sty
@ -0,0 +1,29 @@
+\mode<presentation>
+
+
+% Requirement
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+
+% Settings
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+\usecolortheme{pterosor}
+\usefonttheme{pterosor}
+
+
+% we remove the navigation symbols and define blocks aspect.
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+\setbeamertemplate{blocks}[rounded][shadow=true]
+
+\AtBeginSection[]{%
+    \begin{frame}[plain]
+        \usebeamertemplate{section page}
+    \end{frame}
+}
+
+\setbeamertemplate{footline}{
+    \vspace{0.6cm}
+}
+
+
+\mode<all>
--- a/Slides/beamertitlepterosor.png
+++ b/Slides/beamertitlepterosor.png
--- a/Slides/fig/Heff.png
+++ b/Slides/fig/Heff.png
--- a/Slides/fig/Heff_SRG.png
+++ b/Slides/fig/Heff_SRG.png
--- a/Slides/fig/JCF.png
+++ b/Slides/fig/JCF.png