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Notes/Notes.tex
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@ -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 A general upfolded MBPT matrix can be written as
\begin{equation} \begin{equation}
\label{eq:H_MBPT} \label{eq:H_MBPT}
H = \bH =
\begin{pmatrix} \begin{pmatrix}
\bF{}{} & \bV{}{} \\ \bF{}{} & \bV{}{} \\
\bV{}{\dagger} & \bC{}{} \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 Using SRG language, we define the diagonal and off-diagonal parts as
\begin{equation} \begin{equation}
\label{eq:H_MBPT_partitioning} \label{eq:H_MBPT_partitioning}
H(0) = \bH(0) =
\begin{pmatrix} \begin{pmatrix}
\bF{}{} & \bO \\ \bF{}{} & \bO \\
\bO & \bC{\text{d}}{} \bO & \bC{\text{d}}{}

520
Slides/SRG-GF.tex Normal file
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@ -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{\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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\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>

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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
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(/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!

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%%
%% 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'.

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\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>

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\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>

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\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>

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