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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{hyperref}
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\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem,physics,array}
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\usepackage{pifont}% http://ctan.org/pkg/pifont
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\newcommand{\cmark}{\ding{51}}%
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\newcommand{\xmark}{\ding{55}}%
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\usepackage{amsmath,amssymb,amsfonts,graphicx,xcolor,bm,microtype,wasysym,hyperref,tabularx,amscd,mhchem,physics}
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\definecolor{darkgreen}{RGB}{0, 180, 0}
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\definecolor{fooblue}{RGB}{0,153,255}
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\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}}
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\newcommand{\pub}[1]{\textcolor{purple}{#1}}
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\newcommand{\cdash}{\multicolumn{1}{c}{---}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\mr}{\multirow}
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\newcommand{\br}{\bm{r}}
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\newcommand{\ree}{r_{12}}
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\newcommand{\T}[1]{#1^{\intercal}}
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\newcommand{\la}{\lambda}
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\newcommand{\eps}{\epsilon}
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% operators
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hP}{\Hat{P}}
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\newcommand{\hQ}{\Hat{Q}}
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\newcommand{\hU}{\Hat{U}}
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\newcommand{\hI}{\Hat{1}}
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\newcommand{\hA}{\Hat{A}}
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hR}{\Hat{R}}
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\newcommand{\heta}{\Hat{\eta}}
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\newcommand{\hOm}{\Hat{\Omega}}
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\newcommand{\bH}{\Bar{H}}
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\newcommand{\hO}{\Hat{O}}
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% matrices
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\newcommand{\mA}{\boldsymbol{A}}
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\newcommand{\mB}{\boldsymbol{B}}
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\newcommand{\mx}{\boldsymbol{x}}
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\newcommand{\mS}{\boldsymbol{S}}
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\newcommand{\bC}{\boldsymbol{C}}
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\newcommand{\bF}{\boldsymbol{F}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bHd}{\boldsymbol{H}_\text{d}}
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\newcommand{\bHod}{\boldsymbol{H}_\text{od}}
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\bV}{\boldsymbol{V}}
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\newcommand{\bEta}{\boldsymbol{\eta}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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\newcommand{\bpsi}{\boldsymbol{\psi}}
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\newcommand{\bPsi}{\boldsymbol{\Psi}}
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\institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse \\
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\url{https://lcpq.github.io/pterosor}}
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\usetheme{pterosor}
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\author{Pierre-Fran\c{c}ois (Titou) Loos} % FIXME
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\date{3rd September 2021} % FIXME
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\title{Similarity Renormalization Group (SRG)} %FIXME
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\author{Antoine Marie \& Pierre-Fran\c{c}ois Loos}
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\date{14th November 2022}
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\title{Similarity Renormalization Group (SRG) Formalism Applied to Green's Function Methods}
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\begin{document}
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\maketitle
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%-----------------------------------------------------
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\begin{frame}{Effective Hamiltonian Theory}
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\begin{block}{Similarity transformation of the Hamiltonian}
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\begin{equation}
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\underbrace{\Omega}_{\text{wave operator}}:
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\underbrace{\hH}_{\text{bare Hamiltonian}} \rightarrow \underbrace{\hH^\text{eff}}_{\text{effective Hamiltonian}} = \Omega^{-1} \, \hH \, \Omega
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\end{equation}
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\end{block}
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\begin{block}{Examples of effective Hamiltonian theory}
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\begin{itemize}
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\item L\"owdin's partitioning technique
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\item Transcorrelated method
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\item (multireference) perturbation theory
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\item (multireference) coupled-cluster (CC) theory
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\item Fock-space CC
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\item Equation-of-motion CC
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\end{itemize}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Mathematical Definitions}
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\begin{block}{Matrix similarity}
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Two $n$-by-$n$ matrices $\mA$ and $\mB$ are called \alert{similar} if there exists an invertible $n$-by-$n$ matrix $\mS$ such that
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\begin{equation}
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\mB = \mS^{-1} \cdot \mA \cdot \mS
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\end{equation}
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\end{block}
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\begin{columns}
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\begin{column}{0.85\textwidth}
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\begin{block}{Properties}
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\begin{itemize}
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\item If $\mA$ and $\mB$ are similar, they share the \alert{same eigenvalues}, but \alert{different eigenvectors}
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\begin{equation}
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\qif \mA \cdot \mx = \la \mx \qthen \mB \cdot (\mS^{-1} \cdot \mx) = \la (\mS^{-1} \cdot \mx)
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\end{equation}
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\item Similarity transformations aim at reducing the complexity of evaluating the eigenvalues
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\item If $\mA$ is diagonalizable, it is similar to a diagonal matrix (not unique)
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\item Even if $\mA$ is not diagonalizable, it is similar to a matrix in Jordan form (not unique)
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\item \alert{Unitary transformations} are a type of similarity transformation for which $\mS^{-1} = \mS^{\dag}$
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\item Every hermitian matrix is \alert{unitarily similar} to a diagonal real matrix
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\end{itemize}
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\end{block}
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\end{column}
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\begin{column}{0.15\textwidth}
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\includegraphics[width=\textwidth]{fig/JCF}
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\end{column}
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\end{columns}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Model And External Spaces}
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\begin{block}{Model/External space}
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\begin{align}
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\qq*{\underline{Model space projector:}} & \hP = \dyad*{\Psi_0}{\Psi_0} = \sum_I \dyad{I}{I}
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\begin{frame}{First-Quantized Form of SRG}
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\begin{block}{General upfolded many-body perturbation theory (MBPT) problem}
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\begin{align}
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\qty[ \bF + \bSig(\omega) ] \bpsi = \omega \bpsi
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& \qq{$\Leftrightarrow$}
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\bH \bPsi = \omega \bPsi
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\\
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\qq*{\underline{External space projector:}} & \hQ = \hI - \hP = \hI - \dyad*{\Psi_0}{\Psi_0}
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\end{align}
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\bSig(\omega) = \bV \qty(\omega \bI - \bC)^{-1} \bV^{\dag}
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& \qq{$\Leftrightarrow$}
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\bH =
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\begin{pmatrix}
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\bF & \bV
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\\
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\bV^{\dagger} & \bC
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\end{pmatrix}
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\end{align}
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\end{block}
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\begin{block}{L\"owdin's partitioning technique}
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\begin{equation}
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\hH \ket{\Psi} = E \ket{\Psi}
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\qq{$\Rightarrow$}
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%
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\begin{block}{Perturbative partitioning}
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\begin{equation}
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\bH \equiv \bH(s=0) =
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\underbrace{
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\begin{pmatrix}
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\bF & \bO
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\\
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\bO & \bC
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\end{pmatrix}
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}_{\bHd^{(0)}(s=0)}
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+ \lambda
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\underbrace{
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\begin{pmatrix}
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\hP \hH \hP & \hP \hH \hQ \\
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\hQ \hH \hP & \hQ \hH \hQ \\
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\bO & \bV
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\\
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\bV^{\dagger} & \bO
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\end{pmatrix}
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}_{\bHod^{(1)}(s=0)}
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\qq{with}
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\bHd^{(1)}(s=0) = \bHod^{(0)}(s=0) = \bO
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Perturbative Expansions}
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%
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\begin{block}{Perturbative partitioning in the SRG framework}
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\begin{equation}
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\bH(s) =
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\underbrace{
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\begin{pmatrix}
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\bF(s) & \bO
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\\
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\bO & \bC(s)
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\end{pmatrix}
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}_{\bHd{}(s)}
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+ \lambda
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\underbrace{
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\begin{pmatrix}
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\hP \ket{\Psi} \\
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\hQ \ket{\Psi} \\
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\bO & \bV(s)
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\\
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\bV^{\dagger}(s) & \bO
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\end{pmatrix}
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= E
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}_{\bHod(s)}
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\end{equation}
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\end{block}
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%
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\begin{block}{Components of the Hamiltonian}
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\begin{equation}
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\bH(s) = \bH^{(0)}(s) + \lambda \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots
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\end{equation}
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\begin{equation}
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\bC(s) = \bC^{(0)}(s) + \lambda \bC^{(1)}(s) + \lambda^2 \bC^{(2)}(s) + \cdots
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\end{equation}
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\begin{equation}
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\bV(s) = \bV^{(0)}(s) + \lambda \bV^{(1)}(s) + \lambda^2 \bV^{(2)}(s) + \cdots
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\end{equation}
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\end{block}
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\begin{block}{Wegner generator}
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\begin{equation}
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\bEta(s) = \bEta^{(0)}(s) + \lambda \bEta^{(1)}(s) + \lambda^2 \bEta^{(2)}(s) + \cdots
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Zeroth-Order Terms}
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\begin{block}{Wegner generator}
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\begin{equation}
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\bEta^{(0)}(s)
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= \comm{\bHd^{(0)}(s)}{\bHod^{(0)}(s)}
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= \bO
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\end{equation}
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\end{block}
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%
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\begin{block}{Zeroth-order Hamiltonian}
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\begin{equation}
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\dv{\bH^{(0)}(s)}{s}
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= \comm{\bEta^{(0)}(s)}{\bH^{(0)}(s)}
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= \bO
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\qq{$\Rightarrow$}
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\bH^{(0)}(s) = \bH^{(0)}{(s=0)}
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\end{equation}
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\end{block}
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\alert{NB: we omit the $s$ dependency from hereon}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{First-Order Terms}
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\begin{block}{Wegner generator}
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\begin{equation}
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\bEta^{(1)} = \comm{\bHd^{(0)}}{\bHod^{(1)}}
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=
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\begin{pmatrix}
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\hP \ket{\Psi} \\
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\hQ \ket{\Psi} \\
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\end{pmatrix}
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\end{equation}
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\begin{center}
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\begin{tabular}{p{0.18\textwidth} m{0.3\textwidth} b{0.18\textwidth}}
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Traditional CC $\Rightarrow$
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&
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\includegraphics[width=0.3\textwidth]{fig/Heff}
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&
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$\Leftarrow$ Unitary CC
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\end{tabular}
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\end{center}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Intruder-State Problem}
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\begin{equation*}
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\begin{split}
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\text{\alert{Intruder-state problem}}
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& \Leftrightarrow \text{a determinant in $Q$ becomes near-degenerate with a determinant in $P$}
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\\
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& \Rightarrow \text{appearance of small denominators}
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\\
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& \Rightarrow \text{\alert{convergence issues!}}
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\\
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\\
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\text{How to avoid intruder states?}
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& \Rightarrow \text{do not enforce $\hQ H^\text{eff} \hP = 0$}
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\\
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& \Leftrightarrow \text{near-degenerate determinants are not decoupled}
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\\
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\end{split}
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\end{equation*}
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\begin{center}
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\begin{tabular}{m{0.5\textwidth} b{0.35\textwidth}}
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\includegraphics[width=0.5\textwidth]{fig/Heff_SRG}
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&
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$\Leftarrow$ \alert{Continuous SRG transformation}
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\end{tabular}
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\end{center}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Historical Overview of SRG}
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\begin{block}{SRG or flow equations}
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\begin{itemize}
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\item SRG is a \alert{continuous} (unitary) transformation of the Hamiltonian
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\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
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\item \alert{SRG decouples the Hamiltonian starting from states that have the largest energy separation and progressing to states with smaller energy separation}
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\item SRG does not enforce $\hQ H^\text{eff} \hP = 0$
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\item (MR-)SRG is used a lot in nuclear physics \pub{[Rep. Prog. Phys. 76, 126301 (2013)]}
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\item First introduced in chemistry by Steven White (father of DMRG) \pub{[J. Chem. Phys. 117, 7472 (2002)]}
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\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)]}
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\end{itemize}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{SRG Fundamental Equation}
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\begin{block}{Unitary transformation of the Hamiltonian}
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\bO & \bF^{(0)} \bV^{(1)} - \bV^{(1)} \bC^{(0)}
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\\
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\bC^{(0)} \bV^{(1),\dagger} - \bV^{(1),\dagger} \bF^{(0)} & \bO
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\end{pmatrix}
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\end{equation}
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\end{block}
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%
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\begin{block}{First-order Hamiltonian}
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\begin{equation}
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\boxed{\hH \rightarrow \hH(s) = \hU(s) \, \hH \, \hU^\dag(s), \quad s \in [0,\infty)}
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\dv{\bH^{(1)}}{s} = \comm{\bEta^{(1)}}{\bHd^{(0)}}
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=
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\begin{pmatrix}
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\dv{\bF^{(1)}}{s} & \dv{\bV^{(1)}}{s}
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\\
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\dv{\bV^{(1),\dagger}}{s} & \dv{\bC^{(1)}}{s}
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\end{pmatrix}
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\end{equation}
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with
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\begin{equation}
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\hH =
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\underbrace{\sum_{pq} h_{p}^{q} \Hat{a}_{p}^{q}}_{\text{one-body terms}}
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+ \frac{1}{4} \underbrace{\sum_{pqrs} v_{pq}^{rs} \Hat{a}_{rs}^{pq}}_{\text{two-body terms}}
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\qq{and}
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\Hat{a}_{rs\cdots}^{pq\cdots} = \Hat{a}_{p}^\dag \Hat{a}_{q}^\dag \cdots \Hat{a}_{s} \Hat{a}_{r}
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\end{equation}
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\begin{itemize}
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\item For $s > 0$, $\hH(s)$ has a more (block) diagonal form than $\hH$
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\item The \alert{flow variable} $s$ is a time-like parameter that controls the extent of the transformation
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\begin{itemize}
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\item If $s = 0$, then $\hU(s) = \hI$, i.e., $\hH(s=0) = \hH$
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\item In the limit $s \to \infty$, $\hH(s)$ becomes (block) diagonal
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\end{itemize}
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\end{itemize}
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\begin{equation}
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\hH(s) = \underbrace{\hH_\text{d}(s)}_{\text{diagonal}} + \underbrace{\hH_\text{od}(s)}_{\text{off-diagonal}}
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\qq{$\Rightarrow$}
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\lim_{s\to\infty} \hH_\text{od}(s) = 0
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{SRG Flow Equation}
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\begin{block}{The SRG flow equation}
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\begin{equation}
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\label{eq:flow_eq}
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\boxed{\dv{\hH(s)}{s} = \comm{\heta(s)}{\hH(s)}, \quad \hH(0) = \hH}
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\end{equation}
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\begin{equation*}
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\qq*{where the \alert{flow generator}}
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\heta(s) = \dv{\hU(s)}{s} \hU^\dag(s) = - \heta^\dag(s)
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\qq{is an \alert{anti-Hermitian} operator}
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\end{equation*}
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\end{block}
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\begin{block}{Take-home message}
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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 $\hU(s)$.
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Canonical Generator \& SRG Hamiltonian}
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\begin{block}{Wegner's canonical generator}
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\begin{equation}
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\boxed{\eta^\text{W}(s) = \comm{\hH_\text{d}(s)}{\hH(s)} = \comm{\hH_\text{d}(s)}{\hH_\text{od}(s)}}
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\end{equation}
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As long as $\eta^\text{W}(s) \neq 0$, $\dv{}{s} \Tr[\hH_\text{od}^\dag(s)\hH_\text{od}(s)] \le 0$.
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\\
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Therefore, as $s \to \infty$,
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\begin{itemize}
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\item $E_0(s) = \mel{\Psi_0}{\hH(s)}{\Psi_0}$ evolves towards one of the eigenvalues of $\hH$
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\item The state $\hU^\dag(s) \ket{\Psi_0}$ approaches one of its eigenvectors
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\end{itemize}
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\end{block}
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\begin{block}{The many-body SRG Hamiltonian}
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\begin{equation}
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\hH(s)
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= E_0(s)
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+ \underbrace{\sum_{pq} f_{p}^{q}(s) \{\Hat{a}_{p}^{q}\}}_{\text{one-body terms}}
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+ \frac{1}{4} \underbrace{\sum_{pqrs} v_{pq}^{rs}(s) \overbrace{\{\Hat{a}_{rs}^{pq}\}}^{\text{normal ordered}}}_{\text{two-body terms}}
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+ \frac{1}{36} \underbrace{\sum_{pqrstu} w_{pqr}^{stu}(s) \{\Hat{a}_{stu}^{pqr}\}}_{\text{three-body terms}}
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+ \cdots
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\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}
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user