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BSEdyn.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-10 22:40:48 +0200
%% Created for Pierre-Francois Loos at 2020-06-17 15:22:32 +0200
%% Saved with string encoding Unicode (UTF-8)
@ -2068,16 +2068,10 @@
@article{Sangalli_2011,
Author = {Sangalli, Davide and Romaniello, Pina and Onida, Giovanni and Marini, Andrea},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-06-17 15:22:30 +0200},
Doi = {10.1063/1.3518705},
File = {/Users/loos/Zotero/storage/9S3XW2FJ/Sangalli et al. - 2011 - Double excitations in correlated systems A many--b.pdf},
Issn = {0021-9606, 1089-7690},
Journal = {J. Chem. Phys.},
Language = {en},
Month = jan,
Number = {3},
Pages = {034115},
Shorttitle = {Double Excitations in Correlated Systems},
Title = {Double Excitations in Correlated Systems: {{A}} Many\textendash{}Body Approach},
Volume = {134},
Year = {2011},

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Notes/BSEdyn-notes.tex
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\documentclass{article}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig}
\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-2}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem}
%\usepackage[utf8]{inputenc}
%\usepackage[T1]{fontenc}
%\usepackage{txfonts}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{txfonts}
\usepackage[
colorlinks=true,
@ -111,6 +111,7 @@
\newcommand{\bb}{\mathbf{b}}
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bx}{\mathbf{x}}
% units
@ -137,14 +138,32 @@
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\title{Notes on the Dynamical Bethe-Salpeter Equation}
\author{Pierre-Fran\c{c}ois Loos}
\begin{document}
\title{Dynamical Kernels}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\begin{abstract}
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.
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.
The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.
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.
%\\
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The concept of dynamical quantities}
@ -155,11 +174,11 @@ To do so, let us consider the usual chemical scenario where one wants to get the
In most cases, this can be done by solving a set of linear equations of the form
\begin{equation}
\label{eq:lin_sys}
\bA \bx = \omega \bx
\bA \bc = \omega \bc
\end{equation}
where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
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.
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
where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bc$.
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.
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
\begin{equation}
\label{eq:lin_sys_split}
\begin{pmatrix}
@ -167,32 +186,32 @@ However, in practice, $K$ might be very large, and it might therefore be practic
\bb & \bA_2 \\
\end{pmatrix}
\begin{pmatrix}
\bx_1 \\
\bx_2 \\
\bc_1 \\
\bc_2 \\
\end{pmatrix}
= \omega
\begin{pmatrix}
\bx_1 \\
\bx_2 \\
\bc_1 \\
\bc_2 \\
\end{pmatrix}
\end{equation}
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.
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.
Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
Solving separately each row of the system \eqref{eq:lin_sys_split} yields
\begin{subequations}
\begin{gather}
\label{eq:row1}
\bA_1 \bx_1 + \T{\bb} \bx_2 = \omega \bx_1
\bA_1 \bc_1 + \T{\bb} \bc_2 = \omega \bc_1
\\
\label{eq:row2}
\bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1
\bc_2 = (\omega \bI - \bA_2)^{-1} \bb \bc_1
\end{gather}
\end{subequations}
Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
\begin{equation}
\label{eq:non_lin_sys}
\Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1
\Tilde{\bA}_1(\omega) \bc_1 = \omega \bc_1
\end{equation}
with
\begin{equation}
@ -240,16 +259,16 @@ where $p = v$ or $c$,
\label{eq:SigC}
\SigGW{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - \Omega}
\end{equation}
are the correlation parts of the self-energy associated with the valence of conduction orbitals,
are the correlation parts of the self-energy associated with wither the valence of conduction orbitals,
\begin{equation}
Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
\end{equation}
is the renormalization factor, and
\begin{equation}
\ERI{pq}{rs} = \iint p(\br) q(\br) \frac{1}{\abs{\br - \br'}} r(\br') s(\br') d\br d\br'
\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'
\end{equation}
are the usual (bare) two-electron integrals.
In Eq.~\eqref{eq:SigC}, $\Omega = \Delta\e{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
In Eq.~\eqref{eq:SigC}, $\Omega = \Delta\eGW{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads
\begin{equation} \label{eq:HBSE}
@ -444,14 +463,12 @@ The perturbatively-corrected values are also reported, which shows that this sch
Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy.
%%% TABLE I %%%
\begin{table}
\begin{table*}
\caption{BSE singlet and triplet excitation energies at various levels of theory.
\label{tab:BSE}
}
\begin{center}
\footnotesize
\begin{ruledtabular}
\begin{tabular}{|c|ccccccc|c|}
\hline
Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
\hline
$\omega_1$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\
@ -463,10 +480,9 @@ Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE
$\omega_1$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\
$\omega_2$ & & & & 2.76178 & & & & \\
$\omega_3$ & & & & 4.91545 & & & 4.91517 & \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{ruledtabular}
\end{table*}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -529,48 +545,102 @@ and
Note that the coupling blocks $B$ are frequency independent, as they should.
This has an important consequence as this lack of frequency dependence removes one of the spurious pole.
The singlet manifold has then the right number of excitations.
However, one spurious triplet excitation remains.
Numerical results for the two-level model are reported in Table \ref{tab:RPAx} with the usual approximations and perturbative treatmenets.
However, one spurious triplet excitation remains (see Fig.~\ref{fig:dBSE2}).
Numerical results for the two-level model are reported in Table \ref{tab:RPAx} with the usual approximations and perturbative treatments.
In the case of dRPAx, the perturbative partitioning is simply
\begin{equation}
\bH^{\dRPAx}(\omega) = \underbrace{\bH^{\RPAx}}_{\bH^{(0)}} + \underbrace{\qty[ \bH^{\dRPAx}(\omega) - \bH^{\RPAx} ]}_{\bH^{(1)}}
\end{equation}
This might not be the smartest way of decomposing the Hamiltonian though but it seems to work fine.
%%% TABLE II %%%
\begin{table}
\begin{table*}
\caption{RPAx singlet and triplet excitation energies at various levels of theory.
\label{tab:RPAx}
}
\begin{center}
\footnotesize
\begin{ruledtabular}
\begin{tabular}{|c|ccccccc|c|}
\hline
Singlets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
\hline
$\omega_1$ & 1.84903 & 1.90927 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
$\omega_1$ & 1.84903 & 1.90941 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
$\omega_2$ & & & & & & & & \\
$\omega_3$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\
\hline
Triplets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
\hline
$\omega_1$ & 1.38912 & 1.44267 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
$\omega_1$ & 1.38912 & 1.44285 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
$\omega_2$ & & & & & & & & \\
$\omega_3$ & & & & 4.47797 & & & 4.47767 & \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{ruledtabular}
\end{table*}
%%% %%% %%% %%%
%%% FIGURE 3 %%%
\begin{figure}
\includegraphics[width=\linewidth]{dBSE2}
\caption{
$\det[\bH^{\dRPAx}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds.
\label{fig:dBSE2}
}
\end{figure}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sangalli's kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Ref.~\cite{Sangalli_2011}, Sangalli proposed a norm-conserving kernel without (he claims) spurious excitations.
For the two-level model, this kernel (based on the second RPA) reads
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.
We will first start by writing down explicitly this kernel as it is given in obscure physicist notations.
The dynamical BSE Hamiltonian with Sangalli's kernel is
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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
\bibliographystyle{unsrt}
\bibliography{../BSEdyn}
\end{document}

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