diff --git a/Notes/Notes.tex b/Notes/Notes.tex index 03b849d..a8eba7c 100644 --- a/Notes/Notes.tex +++ b/Notes/Notes.tex @@ -406,7 +406,7 @@ 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{}{} @@ -429,7 +429,7 @@ Using SRG language, we define the diagonal and off-diagonal parts as we could also have defined it like this \begin{equation} \label{eq:H_MBPT_partitioning} - H(0) = + \bH(0) = \begin{pmatrix} \bF{}{} & \bO \\ \bO & \bC{\text{d}}{} diff --git a/Slides/SRG-GF.tex b/Slides/SRG-GF.tex new file mode 100644 index 0000000..0f11202 --- /dev/null +++ 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}}% + +\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} diff --git a/Slides/beamerbackgroundpterosor.png b/Slides/beamerbackgroundpterosor.png new file mode 100644 index 0000000..590eb5b Binary files /dev/null and b/Slides/beamerbackgroundpterosor.png differ diff --git a/Slides/beamercolorthemepterosor.sty b/Slides/beamercolorthemepterosor.sty new file mode 100644 index 0000000..8ac2304 --- /dev/null +++ b/Slides/beamercolorthemepterosor.sty @@ -0,0 +1,33 @@ +\mode + +% 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 + diff --git a/Slides/beamerfontthemepterosor.log b/Slides/beamerfontthemepterosor.log new file mode 100644 index 0000000..c8cad2a --- /dev/null +++ b/Slides/beamerfontthemepterosor.log @@ -0,0 +1,66 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=pdflatex 2020.6.14) 10 DEC 2020 21:15 +entering extended mode + restricted \write18 enabled. + file:line:error style messages enabled. + %&-line parsing enabled. +**beamerfontthemepterosor.sty +(./beamerfontthemepterosor.sty +LaTeX2e <2020-02-02> patch level 5 +L3 programming layer <2020-06-03> + +LaTeX Warning: You have requested package `', + but the package provides `beamerfontthemepterosor'. + +Package: beamerfontthemepterosor 2020/12/10 pterosor font theme +(/usr/local/texlive/2020/texmf-dist/tex/latex/etoolbox/etoolbox.sty +Package: etoolbox 2019/09/21 v2.5h e-TeX tools for LaTeX (JAW) +\etb@tempcnta=\count167 +) +(/usr/local/texlive/2020/texmf-dist/tex/generic/iftex/ifxetex.sty +Package: ifxetex 2019/10/25 v0.7 ifxetex legacy package. Use iftex instead. + +(/usr/local/texlive/2020/texmf-dist/tex/generic/iftex/iftex.sty +Package: iftex 2020/03/06 v1.0d TeX engine tests +)) +(/usr/local/texlive/2020/texmf-dist/tex/generic/iftex/ifluatex.sty +Package: ifluatex 2019/10/25 v1.5 ifluatex legacy package. Use iftex instead. +) +(/usr/local/texlive/2020/texmf-dist/tex/latex/pgfopts/pgfopts.sty +Package: pgfopts 2014/07/10 v2.1a LaTeX package options with pgfkeys + +(/usr/local/texlive/2020/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +(/usr/local/texlive/2020/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks15 +\pgfkeys@temptoks=\toks16 + +(/usr/local/texlive/2020/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.c +ode.tex +\pgfkeys@tmptoks=\toks17 +))) +\pgfopts@list@add@a@toks=\toks18 +\pgfopts@list@add@b@toks=\toks19 +) + +Package beamerthemepterosor Warning: You need to compile with XeLaTeX or LuaLaT +eX to use the Fira fonts on input line 97. + +./beamerfontthemepterosor.sty:98: Undefined control sequence. +l.98 \setbeamerfont + {title}{size=\Large,% +? +./beamerfontthemepterosor.sty:98: Emergency stop. +l.98 \setbeamerfont + {title}{size=\Large,% +End of file on the terminal! + + +Here is how much of TeX's memory you used: + 855 strings out of 480478 + 17943 string characters out of 5899595 + 254474 words of memory out of 5000000 + 16952 multiletter control sequences out of 15000+600000 + 532338 words of font info for 24 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 43i,0n,46p,812b,16s stack positions out of 5000i,500n,10000p,200000b,80000s +./beamerfontthemepterosor.sty:98: ==> Fatal error occurred, no output PDF file + produced! diff --git a/Slides/beamerfontthemepterosor.sty b/Slides/beamerfontthemepterosor.sty new file mode 100644 index 0000000..5e237da --- /dev/null +++ b/Slides/beamerfontthemepterosor.sty @@ -0,0 +1,317 @@ +%% +%% This is file `beamerfontthemepterosor.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% beamerfontthemepterosor.dtx (with options: `package') +%% --------------------------------------------------------------------------- +%% Copyright 2015 Matthias Vogelgesang and the LaTeX community. A full list of +%% contributors can be found at +%% +%% https://github.com/matze/mtheme/graphs/contributors +%% +%% and the original template was based on the HSRM theme by Benjamin Weiss. +%% +%% This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 +%% International License (https://creativecommons.org/licenses/by-sa/4.0/). +%% --------------------------------------------------------------------------- +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{beamerfontthemepterosor}[2020/12/10 pterosor font theme] +\RequirePackage{etoolbox} +\RequirePackage{ifxetex} +\RequirePackage{ifluatex} +\RequirePackage{pgfopts} + +%\RequirePackage{mathpazo} +\RequirePackage{libertine} + +\ifboolexpr{bool {xetex} or bool {luatex}}{ + \@ifpackageloaded{fontspec}{ + \PassOptionsToPackage{no-math}{fontspec} + }{ + \RequirePackage[no-math]{fontspec} + } + \newcounter{fontsnotfound} + \newcommand{\checkfont}[1]{% + \suppressfontnotfounderror=1% + \font\x = "#1" at 10pt + \selectfont + 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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'. diff --git a/Slides/beamerinnerthemepterosor.sty b/Slides/beamerinnerthemepterosor.sty new file mode 100644 index 0000000..7f657f3 --- /dev/null +++ b/Slides/beamerinnerthemepterosor.sty @@ -0,0 +1,53 @@ +\mode + +\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 diff --git a/Slides/beamerouterthemepterosor.sty b/Slides/beamerouterthemepterosor.sty new file mode 100644 index 0000000..57362a0 --- /dev/null +++ b/Slides/beamerouterthemepterosor.sty @@ -0,0 +1,20 @@ +\mode + +% 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 + diff --git a/Slides/beamerthemepterosor.sty b/Slides/beamerthemepterosor.sty new file mode 100644 index 0000000..8aad8a8 --- /dev/null +++ b/Slides/beamerthemepterosor.sty @@ -0,0 +1,29 @@ +\mode + + +% 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 diff --git a/Slides/beamertitlepterosor.png b/Slides/beamertitlepterosor.png new file mode 100644 index 0000000..a851df0 Binary files /dev/null and b/Slides/beamertitlepterosor.png differ diff --git a/Slides/fig/Heff.png b/Slides/fig/Heff.png new file mode 100644 index 0000000..5665e1c Binary files /dev/null and b/Slides/fig/Heff.png differ diff --git a/Slides/fig/Heff_SRG.png b/Slides/fig/Heff_SRG.png new file mode 100644 index 0000000..fe2219d Binary files /dev/null and b/Slides/fig/Heff_SRG.png differ diff --git a/Slides/fig/JCF.png b/Slides/fig/JCF.png new file mode 100644 index 0000000..440c8f4 Binary files /dev/null and b/Slides/fig/JCF.png differ