\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}}%

\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{\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}
