From bdb215da239e2d6c41d70fb8c837c250be4fc999 Mon Sep 17 00:00:00 2001 From: pfloos Date: Wed, 9 Nov 2022 16:14:26 +0100 Subject: [PATCH] slides --- Slides/SRG-GF.tex | 699 ++++++++++++++++------------------------------ 1 file changed, 242 insertions(+), 457 deletions(-) diff --git a/Slides/SRG-GF.tex b/Slides/SRG-GF.tex index 0f11202..a8d0db0 100644 --- a/Slides/SRG-GF.tex +++ b/Slides/SRG-GF.tex @@ -2,10 +2,8 @@ \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}}% +\usepackage{amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics} + \definecolor{darkgreen}{RGB}{0, 180, 0} \definecolor{fooblue}{RGB}{0,153,255} @@ -22,499 +20,286 @@ \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}} +\newcommand{\bC}{\boldsymbol{C}} +\newcommand{\bF}{\boldsymbol{F}} +\newcommand{\bH}{\boldsymbol{H}} +\newcommand{\bHd}{\boldsymbol{H}_\text{d}} +\newcommand{\bHod}{\boldsymbol{H}_\text{od}} +\newcommand{\bO}{\boldsymbol{0}} +\newcommand{\bI}{\boldsymbol{1}} +\newcommand{\bV}{\boldsymbol{V}} +\newcommand{\bEta}{\boldsymbol{\eta}} +\newcommand{\bSig}{\boldsymbol{\Sigma}} +\newcommand{\bpsi}{\boldsymbol{\psi}} +\newcommand{\bPsi}{\boldsymbol{\Psi}} \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 - +\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos} +\date{14th November 2022} +\title{Similarity Renormalization Group (SRG) Formalism Applied to Green's Function Methods} \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} +\begin{frame}{First-Quantized Form of SRG} + \begin{block}{General upfolded many-body perturbation theory (MBPT) problem} + \begin{align} + \qty[ \bF + \bSig(\omega) ] \bpsi = \omega \bpsi + & \qq{$\Leftrightarrow$} + \bH \bPsi = \omega \bPsi \\ - \qq*{\underline{External space projector:}} & \hQ = \hI - \hP = \hI - \dyad*{\Psi_0}{\Psi_0} - \end{align} + \bSig(\omega) = \bV \qty(\omega \bI - \bC)^{-1} \bV^{\dag} + & \qq{$\Leftrightarrow$} + \bH = + \begin{pmatrix} + \bF & \bV + \\ + \bV^{\dagger} & \bC + \end{pmatrix} + \end{align} \end{block} - \begin{block}{L\"owdin's partitioning technique} - \begin{equation} - \hH \ket{\Psi} = E \ket{\Psi} - \qq{$\Rightarrow$} + % + \begin{block}{Perturbative partitioning} + \begin{equation} + \bH \equiv \bH(s=0) = + \underbrace{ + \begin{pmatrix} + \bF & \bO + \\ + \bO & \bC + \end{pmatrix} + }_{\bHd^{(0)}(s=0)} + + \lambda + \underbrace{ \begin{pmatrix} - \hP \hH \hP & \hP \hH \hQ \\ - \hQ \hH \hP & \hQ \hH \hQ \\ + \bO & \bV + \\ + \bV^{\dagger} & \bO \end{pmatrix} + }_{\bHod^{(1)}(s=0)} + \qq{with} + \bHd^{(1)}(s=0) = \bHod^{(0)}(s=0) = \bO + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Perturbative Expansions} + % + \begin{block}{Perturbative partitioning in the SRG framework} + \begin{equation} + \bH(s) = + \underbrace{ + \begin{pmatrix} + \bF(s) & \bO + \\ + \bO & \bC(s) + \end{pmatrix} + }_{\bHd{}(s)} + + \lambda + \underbrace{ \begin{pmatrix} - \hP \ket{\Psi} \\ - \hQ \ket{\Psi} \\ + \bO & \bV(s) + \\ + \bV^{\dagger}(s) & \bO \end{pmatrix} - = E + }_{\bHod(s)} + \end{equation} + \end{block} + % + \begin{block}{Components of the Hamiltonian} + \begin{equation} + \bH(s) = \bH^{(0)}(s) + \lambda \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots + \end{equation} + \begin{equation} + \bC(s) = \bC^{(0)}(s) + \lambda \bC^{(1)}(s) + \lambda^2 \bC^{(2)}(s) + \cdots + \end{equation} + \begin{equation} + \bV(s) = \bV^{(0)}(s) + \lambda \bV^{(1)}(s) + \lambda^2 \bV^{(2)}(s) + \cdots + \end{equation} + \end{block} + \begin{block}{Wegner generator} + \begin{equation} + \bEta(s) = \bEta^{(0)}(s) + \lambda \bEta^{(1)}(s) + \lambda^2 \bEta^{(2)}(s) + \cdots + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Zeroth-Order Terms} + \begin{block}{Wegner generator} + \begin{equation} + \bEta^{(0)}(s) + = \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)} + = \bO + \end{equation} + \end{block} + % + \begin{block}{Zeroth-order Hamiltonian} + \begin{equation} + \dv{\bH^{(0)}(s)}{s} + = \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)} + = \bO + \qq{$\Rightarrow$} + \bH^{(0)}(s) = \bH^{(0)}{(s=0)} + \end{equation} + \end{block} + \alert{NB: we omit the $s$ dependency from hereon} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{First-Order Terms} + \begin{block}{Wegner generator} + \begin{equation} + \bEta^{(1)} = \comm{\bHd^{(0)}}{\bHod^{(1)}} + = \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} + \bO & \bF^{(0)} \bV^{(1)} - \bV^{(1)} \bC^{(0)} + \\ + \bC^{(0)} \bV^{(1),\dagger} - \bV^{(1),\dagger} \bF^{(0)} & \bO + \end{pmatrix} + \end{equation} + \end{block} + % + \begin{block}{First-order Hamiltonian} \begin{equation} - \boxed{\hH \rightarrow \hH(s) = \hU(s) \, \hH \, \hU^\dag(s), \quad s \in [0,\infty)} + \dv{\bH^{(1)}}{s} = \comm{\bEta^{(1)}}{\bHd^{(0)}} + = + \begin{pmatrix} + \dv{\bF^{(1)}}{s} & \dv{\bV^{(1)}}{s} + \\ + \dv{\bV^{(1),\dagger}}{s} & \dv{\bC^{(1)}}{s} + \end{pmatrix} \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} + with \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) + \dv{\bF^{(1)}}{s} + = \dv{\bC^{(1)}}{s} + = \bO \\ - \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} + \dv{\bV^{(1)}}{s} + = 2 \bF^{(0)} \bV^{(1)} \bC^{(0)} + - \qty[\bF^{(0)}]^2 \bV^{(1)} + - \bV^{(1)} \qty[\bC^{(0)}]^2 + \end{gather} \end{block} \end{frame} %----------------------------------------------------- %----------------------------------------------------- -\begin{frame}{DSRG Equation} - \begin{block}{DSRG equation and source operator} +\begin{frame}{Second-Order Terms} + \begin{block}{Wegner generator} + \begin{equation} + \bEta^{(2)} + = \comm{\bHd^{(0)}}{\bHod^{(2)}} + + \underbrace{\comm{\bHd^{(1)}}{\bHod^{(1)}}}_{=\bO} + = + \begin{pmatrix} + \bO & \bF^{(0)} \bV^{(2)} - \bV^{(2)} \bC^{(0)} + \\ + \bC^{(0)} \bV^{(2),\dagger} - \bV^{(2),\dagger} \bF^{(0)} & \bO + \end{pmatrix} + \end{equation} + \end{block} + % + \begin{block}{Second-order Hamiltonian} \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) + \dv{\bH^{(2)}}{s} = \comm{\bEta^{(2)}}{\bHd^{(0)}} + \comm{\bEta^{(1)}}{\bHd^{(1)}} + = + \begin{pmatrix} + \dv{\bF^{(2)}}{s} & \dv{\bV^{(2)}}{s} + \\ + \dv{\bV^{(2),\dagger}}{s} & \dv{\bC^{(2)}}{s} + \end{pmatrix} \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} + with \begin{gather} - \hOm(s) = \hOm_1(s) + \hOm_2(s) + \cdots + \hOm_n(s) + \dv{\bF^{(2)}}{s} + = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} + + \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} + - 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag} \\ - \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} + \dv{\bC^{(2)}}{s} + = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} + + \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} + - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag} + \\ + \dv{\bV^{(2)}}{s} + = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)} + - \qty[\bF^{(0)}]^2 \bV^{(2)} + - \bV^{(2)} \qty[\bC^{(0)}]^2 + \end{gather} \end{block} - \begin{block}{DSRG equations} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Integration of the First-Order Terms} \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($n$)}} + \dv{\bF^{(1)}}{s} = \bO + \land + \bF^{(1)}(0) = \bO + \Rightarrow + \bF^{(1)}(s) = \bO \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)} + \dv{\bC^{(1)}}{s} + \land + \bC^{(1)}(0) = \bO + \Rightarrow + \bC^{(1)}(s) = \bO \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) + \dv{\bV^{(1)}}{s} + = 2 \bF^{(0)} \bV^{(1)} \bC^{(0)} + - \qty[\bF^{(0)}]^2 \bV^{(1)} + - \bV^{(1)} \qty[\bC^{(0)}]^2 + \land + \Rightarrow + \bV^{(1)}(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} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Integration of the Second-Order Terms} \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 + \dv{\bF^{(2)}}{s} + = \bF^{(0)} \bV^{(1)} \bV^{(1),\dag} + + \bV^{(1)} \bV^{(1),\dag} \bF^{(0)} + - 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag} + \land + \bF^{(2)}(0) = ? + \Rightarrow + \bF^{(2)}(s) = ? + \end{equation} + \begin{equation} + \dv{\bC^{(2)}}{s} + = \bC^{(0)} \bV^{(1)} \bV^{(1),\dag} + + \bV^{(1)} \bV^{(1),\dag} \bC^{(0)} + - 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag} + \land + \bC^{(2)}(0) = ? + \Rightarrow + \bC^{(2)}(s) = ? + \end{equation} + \begin{equation} + \dv{\bV^{(2)}}{s} + = 2 \bF^{(0)} \bV^{(2)} \bC^{(0)} + - \qty[\bF^{(0)}]^2 \bV^{(2)} + - \bV^{(2)} \qty[\bC^{(0)}]^2 + \land + \Rightarrow + \bV^{(2)}(s) = ? \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}