Sangalli
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BSEdyn.bib
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BSEdyn.bib
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%% Created for Pierre-Francois Loos at 2020-06-10 22:40:48 +0200
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%% Created for Pierre-Francois Loos at 2020-06-17 15:22:32 +0200
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@article{Sangalli_2011,
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@article{Sangalli_2011,
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Author = {Sangalli, Davide and Romaniello, Pina and Onida, Giovanni and Marini, Andrea},
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Author = {Sangalli, Davide and Romaniello, Pina and Onida, Giovanni and Marini, Andrea},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-06-17 15:22:30 +0200},
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Doi = {10.1063/1.3518705},
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Doi = {10.1063/1.3518705},
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File = {/Users/loos/Zotero/storage/9S3XW2FJ/Sangalli et al. - 2011 - Double excitations in correlated systems A many--b.pdf},
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Issn = {0021-9606, 1089-7690},
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Journal = {J. Chem. Phys.},
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Journal = {J. Chem. Phys.},
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Language = {en},
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Month = jan,
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Number = {3},
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Pages = {034115},
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Pages = {034115},
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Shorttitle = {Double Excitations in Correlated Systems},
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Title = {Double Excitations in Correlated Systems: {{A}} Many\textendash{}Body Approach},
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Title = {Double Excitations in Correlated Systems: {{A}} Many\textendash{}Body Approach},
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Volume = {134},
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Volume = {134},
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Year = {2011},
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Year = {2011},
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\documentclass{article}
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\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-2}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
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\usepackage[version=4]{mhchem}
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\usepackage[version=4]{mhchem}
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%\usepackage[utf8]{inputenc}
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\usepackage[utf8]{inputenc}
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%\usepackage[T1]{fontenc}
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\usepackage[T1]{fontenc}
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%\usepackage{txfonts}
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\usepackage{txfonts}
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\usepackage[
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\usepackage[
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colorlinks=true,
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colorlinks=true,
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\newcommand{\bb}{\mathbf{b}}
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\newcommand{\bb}{\mathbf{b}}
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\newcommand{\bA}{\mathbf{A}}
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\newcommand{\bA}{\mathbf{A}}
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\newcommand{\bB}{\mathbf{B}}
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\newcommand{\bB}{\mathbf{B}}
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\newcommand{\bc}{\mathbf{c}}
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\newcommand{\bx}{\mathbf{x}}
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\newcommand{\bx}{\mathbf{x}}
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% units
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% units
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\title{Notes on the Dynamical Bethe-Salpeter Equation}
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\author{Pierre-Fran\c{c}ois Loos}
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\begin{document}
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\begin{document}
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\title{Dynamical Kernels}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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Similar to the ubiquitous adiabatic approximation in time-dependent density-functional theory, the static approximation, which substitutes a dynamical (\ie, frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the Bethe-Salpeter equation (BSE) formalism.
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Here, going beyond the static approximation, we compute the dynamical correction in the electron-hole screening for molecular excitation energies thanks to a renormalized first-order perturbative correction to the static BSE excitation energies.
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The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.
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Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approximation by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.
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%\\
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{The concept of dynamical quantities}
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\section{The concept of dynamical quantities}
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@ -155,11 +174,11 @@ To do so, let us consider the usual chemical scenario where one wants to get the
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In most cases, this can be done by solving a set of linear equations of the form
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In most cases, this can be done by solving a set of linear equations of the form
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\begin{equation}
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\begin{equation}
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\label{eq:lin_sys}
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\label{eq:lin_sys}
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\bA \bx = \omega \bx
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\bA \bc = \omega \bc
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\end{equation}
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\end{equation}
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where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
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where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bc$.
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If we assume that the operator $\bA$ has a matrix representation of size $K \times K$, this \textit{linear} set of equations yields $K$ excitation energies.
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If we assume that the operator $\bA$ has a matrix representation of size $N \times N$, this \textit{linear} set of equations yields $N$ excitation energies.
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However, in practice, $K$ might be very large, and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
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However, in practice, $N$ might be very large, and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
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\begin{equation}
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\begin{equation}
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\label{eq:lin_sys_split}
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\label{eq:lin_sys_split}
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\begin{pmatrix}
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\begin{pmatrix}
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@ -167,32 +186,32 @@ However, in practice, $K$ might be very large, and it might therefore be practic
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\bb & \bA_2 \\
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\bb & \bA_2 \\
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\end{pmatrix}
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\end{pmatrix}
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\begin{pmatrix}
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\begin{pmatrix}
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\bx_1 \\
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\bc_1 \\
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\bx_2 \\
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\bc_2 \\
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\end{pmatrix}
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\end{pmatrix}
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= \omega
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= \omega
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\begin{pmatrix}
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\begin{pmatrix}
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\bx_1 \\
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\bc_1 \\
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\bx_2 \\
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\bc_2 \\
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\end{pmatrix}
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\end{pmatrix}
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\end{equation}
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\end{equation}
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where the blocks $\bA_1$ and $\bA_2$, of sizes $K_1 \times K_1$ and $K_2 \times K_2$ (with $K_1 + K_2 = K$), can be associated with, for example, the single and double excitations of the system.
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where the blocks $\bA_1$ and $\bA_2$, of sizes $N_1 \times N_1$ and $N_2 \times N_2$ (with $N_1 + N_2 = N$), can be associated with, for example, the single and double excitations of the system.
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Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
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Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
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Solving separately each row of the system \eqref{eq:lin_sys_split} yields
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Solving separately each row of the system \eqref{eq:lin_sys_split} yields
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\begin{subequations}
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\begin{subequations}
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\begin{gather}
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\begin{gather}
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\label{eq:row1}
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\label{eq:row1}
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\bA_1 \bx_1 + \T{\bb} \bx_2 = \omega \bx_1
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\bA_1 \bc_1 + \T{\bb} \bc_2 = \omega \bc_1
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\\
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\\
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\label{eq:row2}
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\label{eq:row2}
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\bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1
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\bc_2 = (\omega \bI - \bA_2)^{-1} \bb \bc_1
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\end{gather}
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\end{gather}
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\end{subequations}
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\end{subequations}
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Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
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Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
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\begin{equation}
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\begin{equation}
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\label{eq:non_lin_sys}
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\label{eq:non_lin_sys}
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\Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1
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\Tilde{\bA}_1(\omega) \bc_1 = \omega \bc_1
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\end{equation}
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\end{equation}
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with
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with
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\begin{equation}
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\begin{equation}
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@ -240,16 +259,16 @@ where $p = v$ or $c$,
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\label{eq:SigC}
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\label{eq:SigC}
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\SigGW{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - \Omega}
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\SigGW{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - \Omega}
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\end{equation}
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\end{equation}
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are the correlation parts of the self-energy associated with the valence of conduction orbitals,
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are the correlation parts of the self-energy associated with wither the valence of conduction orbitals,
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\begin{equation}
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\begin{equation}
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Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
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Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
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\end{equation}
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\end{equation}
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is the renormalization factor, and
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is the renormalization factor, and
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\begin{equation}
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\begin{equation}
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\ERI{pq}{rs} = \iint p(\br) q(\br) \frac{1}{\abs{\br - \br'}} r(\br') s(\br') d\br d\br'
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\ERI{pq}{rs} = \iint \MO{p}(\br) \MO{q}(\br) \frac{1}{\abs{\br - \br'}} \MO{r}(\br') \MO{s}(\br') d\br d\br'
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\end{equation}
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\end{equation}
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are the usual (bare) two-electron integrals.
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are the usual (bare) two-electron integrals.
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In Eq.~\eqref{eq:SigC}, $\Omega = \Delta\e{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
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In Eq.~\eqref{eq:SigC}, $\Omega = \Delta\eGW{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
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One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads
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One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads
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\begin{equation} \label{eq:HBSE}
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\begin{equation} \label{eq:HBSE}
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@ -444,14 +463,12 @@ The perturbatively-corrected values are also reported, which shows that this sch
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Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy.
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Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy.
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%%% TABLE I %%%
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%%% TABLE I %%%
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\begin{table}
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\begin{table*}
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\caption{BSE singlet and triplet excitation energies at various levels of theory.
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\caption{BSE singlet and triplet excitation energies at various levels of theory.
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\label{tab:BSE}
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\label{tab:BSE}
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}
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}
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\begin{center}
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\begin{ruledtabular}
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\footnotesize
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\begin{tabular}{|c|ccccccc|c|}
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\begin{tabular}{|c|ccccccc|c|}
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\hline
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Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
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Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
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\hline
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\hline
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$\omega_1$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\
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$\omega_1$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\
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@ -463,10 +480,9 @@ Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE
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$\omega_1$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\
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$\omega_1$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\
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$\omega_2$ & & & & 2.76178 & & & & \\
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$\omega_2$ & & & & 2.76178 & & & & \\
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$\omega_3$ & & & & 4.91545 & & & 4.91517 & \\
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$\omega_3$ & & & & 4.91545 & & & 4.91517 & \\
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\hline
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\end{tabular}
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\end{tabular}
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\end{center}
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\end{ruledtabular}
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\end{table}
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\end{table*}
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%%% %%% %%% %%%
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -529,48 +545,102 @@ and
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Note that the coupling blocks $B$ are frequency independent, as they should.
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Note that the coupling blocks $B$ are frequency independent, as they should.
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This has an important consequence as this lack of frequency dependence removes one of the spurious pole.
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This has an important consequence as this lack of frequency dependence removes one of the spurious pole.
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The singlet manifold has then the right number of excitations.
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The singlet manifold has then the right number of excitations.
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However, one spurious triplet excitation remains.
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However, one spurious triplet excitation remains (see Fig.~\ref{fig:dBSE2}).
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Numerical results for the two-level model are reported in Table \ref{tab:RPAx} with the usual approximations and perturbative treatmenets.
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Numerical results for the two-level model are reported in Table \ref{tab:RPAx} with the usual approximations and perturbative treatments.
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In the case of dRPAx, the perturbative partitioning is simply
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\begin{equation}
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\bH^{\dRPAx}(\omega) = \underbrace{\bH^{\RPAx}}_{\bH^{(0)}} + \underbrace{\qty[ \bH^{\dRPAx}(\omega) - \bH^{\RPAx} ]}_{\bH^{(1)}}
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\end{equation}
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This might not be the smartest way of decomposing the Hamiltonian though but it seems to work fine.
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%%% TABLE II %%%
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%%% TABLE II %%%
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\begin{table}
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\begin{table*}
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\caption{RPAx singlet and triplet excitation energies at various levels of theory.
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\caption{RPAx singlet and triplet excitation energies at various levels of theory.
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\label{tab:RPAx}
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\label{tab:RPAx}
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}
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}
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\begin{center}
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\begin{ruledtabular}
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\footnotesize
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\begin{tabular}{|c|ccccccc|c|}
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\begin{tabular}{|c|ccccccc|c|}
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\hline
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Singlets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
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Singlets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
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\hline
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\hline
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$\omega_1$ & 1.84903 & 1.90927 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
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$\omega_1$ & 1.84903 & 1.90941 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
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$\omega_2$ & & & & & & & & \\
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$\omega_2$ & & & & & & & & \\
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$\omega_3$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\
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$\omega_3$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\
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\hline
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\hline
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Triplets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
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Triplets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
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\hline
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\hline
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$\omega_1$ & 1.38912 & 1.44267 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
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$\omega_1$ & 1.38912 & 1.44285 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
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$\omega_2$ & & & & & & & & \\
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$\omega_2$ & & & & & & & & \\
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$\omega_3$ & & & & 4.47797 & & & 4.47767 & \\
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$\omega_3$ & & & & 4.47797 & & & 4.47767 & \\
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\hline
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\end{tabular}
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\end{tabular}
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\end{center}
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\end{ruledtabular}
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\end{table}
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\end{table*}
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%%% %%% %%% %%%
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%%% FIGURE 3 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{dBSE2}
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\caption{
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$\det[\bH^{\dRPAx}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds.
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\label{fig:dBSE2}
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}
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\end{figure}
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%%% %%% %%% %%%
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Sangalli's kernel}
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\subsection{Sangalli's kernel}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In Ref.~\cite{Sangalli_2011}, Sangalli proposed a norm-conserving kernel without (he claims) spurious excitations.
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In Ref.~\cite{Sangalli_2011}, Sangalli proposed a dynamical kernel (based on the second RPA) without (he claims) spurious excitations thanks to the design of a number-conserving approach which correctly describes particle indistinguishability and Pauli exclusion principle.
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For the two-level model, this kernel (based on the second RPA) reads
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We will first start by writing down explicitly this kernel as it is given in obscure physicist notations.
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The dynamical BSE Hamiltonian with Sangalli's kernel is
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|
\begin{equation}
|
||||||
|
\bH^\text{NC}(\omega) =
|
||||||
|
\begin{pmatrix}
|
||||||
|
H(\omega) & K(\omega)
|
||||||
|
\\
|
||||||
|
-K(-\omega) & -H(-\omega)
|
||||||
|
\end{pmatrix}
|
||||||
|
\end{equation}
|
||||||
|
with
|
||||||
|
\begin{align}
|
||||||
|
H_{ia,jb}(\omega) & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \Xi_{ia,jb} (\omega)
|
||||||
|
\\
|
||||||
|
K_{ia,jb}(\omega) & = \Xi_{ia,bj} (\omega)
|
||||||
|
\end{align}
|
||||||
|
and
|
||||||
|
\begin{gather}
|
||||||
|
\Xi_{ia,jb} (\omega) = \sum_{m \neq n} \frac{ C_{ia,mn} C_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n} + 2i\eta)}
|
||||||
|
\\
|
||||||
|
C_{ia,mn} = \frac{1}{2} \sum_{jb,kc} \qty{ \qty[ \ERI{ij}{kc} \delta_{ab} + \ERI{kc}{ab} \delta_{ij} ] \qty[ R_{m,jc} R_{n,kb}
|
||||||
|
+ R_{m,kb} R_{n,jc} ] }
|
||||||
|
\end{gather}
|
||||||
|
where $R_{m,ia}$ are the elements of the RPA eigenvectors.
|
||||||
|
Here $i$, $j$, and $k$ are occupied orbitals, $a$, $b$, and $c$ are unoccupied orbitals, and $m$ and $n$ label single excitations.
|
||||||
|
|
||||||
|
For the two-level model, Sangalli's kernel reads
|
||||||
|
\begin{align}
|
||||||
|
H(\omega) & = \Delta\eGW{} + \Xi_H (\omega)
|
||||||
|
\\
|
||||||
|
K(\omega) & = \Xi_K (\omega)
|
||||||
|
\end{align}
|
||||||
|
|
||||||
|
\begin{align}
|
||||||
|
\Xi_H (\omega) = \sum_{m \neq n} \frac{ C_{ia,mn} C_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n} + 2i\eta)}
|
||||||
|
\end{align}
|
||||||
|
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
\section{Take-home messages}
|
\section{Take-home messages}
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
What have we learnt here?
|
What have we learnt here?
|
||||||
|
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
\acknowledgements{
|
||||||
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
|
The author thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
|
||||||
|
|
||||||
|
|
||||||
% BIBLIOGRAPHY
|
% BIBLIOGRAPHY
|
||||||
\bibliographystyle{unsrt}
|
|
||||||
\bibliography{../BSEdyn}
|
\bibliography{../BSEdyn}
|
||||||
|
|
||||||
\end{document}
|
\end{document}
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