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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 2020-06-17 15:22:32 +0200
%% Created for Pierre-Francois Loos at 2020-06-19 15:03:36 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Maitra_2016,
Author = {N. T. Maitra},
Date-Added = {2020-06-19 14:18:29 +0200},
Date-Modified = {2020-06-19 14:19:14 +0200},
Doi = {10.1063/1.4953039},
Journal = {J. Chem. Phys.},
Pages = {220901},
Title = {Fundamental aspects of time-dependent density functional theory},
Volume = {144},
Year = {2016},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4953039}}
@article{Christiansen_1995a,
Author = {Ove Christiansen and Henrik Koch and Poul J{\o}rgensen},
Date-Added = {2020-06-10 22:40:39 +0200},
@ -2321,13 +2333,8 @@
@article{Strinati_1988,
Author = {Strinati, G.},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Doi = {10.1007/BF02725962},
Issn = {1826-9850},
Date-Modified = {2020-06-19 14:11:57 +0200},
Journal = {Riv. Nuovo Cimento},
Language = {en},
Month = dec,
Number = {12},
Pages = {1--86},
Title = {Application of the {{Green}}'s Functions Method to the Study of the Optical Properties of Semiconductors},
Volume = {11},
@ -11270,14 +11277,10 @@
@article{Tozer_2000,
Author = {Tozer, David J. and Handy, Nicholas C.},
Date-Added = {2020-01-01 21:36:39 +0100},
Date-Modified = {2020-01-01 21:36:39 +0100},
Date-Modified = {2020-06-19 14:20:04 +0200},
Doi = {10.1039/a910321j},
File = {/Users/loos/Zotero/storage/TFJP3V8Z/Tozer and Handy - 2000 - On the determination of excitation energies using .pdf},
Issn = {14639076, 14639084},
Journal = {Phys. Chem. Chem. Phys.},
Language = {en},
Number = {10},
Pages = {2117-2121},
Pages = {2117--2121},
Title = {On the Determination of Excitation Energies Using Density Functional Theory},
Volume = {2},
Year = {2000},

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Notes/BSEdyn-notes.tex
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\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-2}
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-2}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem}
@ -13,144 +13,20 @@
]{hyperref}
\urlstyle{same}
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\SI}{\textcolor{blue}{supplementary material}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}}
% coordinates
\newcommand{\br}{\mathbf{r}}
\newcommand{\dbr}{d\br}
% methods
\newcommand{\evGW}{ev$GW$}
\newcommand{\qsGW}{qs$GW$}
\newcommand{\GOWO}{$G_0W_0$}
\newcommand{\Hxc}{\text{Hxc}}
\newcommand{\xc}{\text{xc}}
\newcommand{\Ha}{\text{H}}
\newcommand{\co}{\text{x}}
%
\newcommand{\Norb}{N_\text{orb}}
\newcommand{\Nocc}{O}
\newcommand{\Nvir}{V}
\newcommand{\IS}{\lambda}
% operators
\newcommand{\hH}{\Hat{H}}
% methods
\newcommand{\KS}{\text{KS}}
\newcommand{\HF}{\text{HF}}
\newcommand{\RPA}{\text{RPA}}
\newcommand{\RPAx}{\text{RPAx}}
\newcommand{\dRPAx}{\text{dRPAx}}
\newcommand{\BSE}{\text{BSE}}
\newcommand{\TDABSE}{\text{BSE(TDA)}}
\newcommand{\dBSE}{\text{dBSE}}
\newcommand{\TDAdBSE}{\text{dBSE(TDA)}}
\newcommand{\GW}{GW}
\newcommand{\GF}{\text{GF2}}
\newcommand{\stat}{\text{stat}}
\newcommand{\dyn}{\text{dyn}}
\newcommand{\TDA}{\text{TDA}}
% energies
\newcommand{\Enuc}{E^\text{nuc}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E^\text{HF}}
\newcommand{\EBSE}{E^\text{BSE}}
\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
% orbital energies
\newcommand{\e}[1]{\eps_{#1}}
\newcommand{\eHF}[1]{\eps^\text{HF}_{#1}}
\newcommand{\eKS}[1]{\eps^\text{KS}_{#1}}
\newcommand{\eQP}[1]{\eps^\text{QP}_{#1}}
\newcommand{\eGW}[1]{\eps^{GW}_{#1}}
\newcommand{\eGF}[1]{\eps^{\text{GF2}}_{#1}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
% Matrix elements
\newcommand{\Sig}[1]{\Sigma_{#1}}
\newcommand{\SigGW}[1]{\Sigma^{\GW}_{#1}}
\newcommand{\SigGF}[1]{\Sigma^{\GF}_{#1}}
\newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\sERI}[2]{[#1|#2]}
% excitation energies
\newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}}
\newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}}
\newcommand{\OmBSE}[1]{\Omega_{#1}^{\text{BSE}}}
\newcommand{\spinup}{\downarrow}
\newcommand{\spindw}{\uparrow}
\newcommand{\singlet}{\uparrow\downarrow}
\newcommand{\triplet}{\uparrow\uparrow}
% Matrices
\newcommand{\bO}{\mathbf{0}}
\newcommand{\bH}{\mathbf{H}}
\newcommand{\bV}{\mathbf{V}}
\newcommand{\bI}{\mathbf{1}}
\newcommand{\bb}{\mathbf{b}}
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bx}{\mathbf{x}}
% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\kcal}{kcal/mol}
\DeclareMathOperator*{\argmax}{argmax}
\DeclareMathOperator*{\argmin}{argmin}
% orbitals, gaps, etc
\newcommand{\updw}{\uparrow\downarrow}
\newcommand{\upup}{\uparrow\uparrow}
\newcommand{\eps}{\varepsilon}
\newcommand{\IP}{I}
\newcommand{\EA}{A}
\newcommand{\HOMO}{\text{HOMO}}
\newcommand{\LUMO}{\text{LUMO}}
\newcommand{\Eg}{E_\text{g}}
\newcommand{\EgFun}{\Eg^\text{fund}}
\newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\begin{document}
\title{Dynamical Kernels}
\title{A Tale of Three 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.
We discuss the physical properties and accuracy of three distinct dynamical (\ie, frequency-dependent) kernels for the computation of excitation energies within linear response theory:
i) an \textit{a priori} built kernel inspired by the dressed time-dependent density-functional theory (TD-DFT) kernel proposed by Maitra and coworkers [\href{https://doi.org/10.1063/1.1651060}{J.~Chem.~Phys.~120, 5932 (2004)}],
ii) the dynamical kernel stemming from the Bethe-Salpeter equation (BSE) formalism derived originally by Strinati [\href{https://doi.org/10.1007/BF02725962}{Riv.~Nuovo Cimento 11, 1--86 (1988)}], and
iii) the second-order BSE kernel derived by Yang and coworkers [\href{https://doi.org/10.1063/1.4824907}{J.~Chem.~Phys.~139, 154109 (2013)}].
In particular, using a simple two-level model, we analyze the appearance of spurious excitations, as first evidenced by Romaniello and collaborators [\href{https://doi.org/10.1063/1.3065669}{J.~Chem.~Phys.~130, 044108 (2009)}], due to the approximate nature of the kernels.
%\\
%\bigskip
%\begin{center}
@ -162,23 +38,58 @@ Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approxima
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\section{Linear response theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Linear response is a powerful approach that allows to directly access the optical excitations $\omega_s$ of a given electronic system (such as a molecule) and their corresponding oscillator strengths $f_s$ [extracted from their eigenvectors $\T{(\bX_s \bY_s)}$] via the response of the system to a weak electromagnetic field. \cite{Casida_1995}
From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995}
\begin{equation} \label{eq:LR}
\begin{pmatrix}
\bR(\omega_s) & \bC(\omega_s)
\\
-\bC(-\omega_s) & -\bR(-\omega_s)
\end{pmatrix}
\begin{pmatrix}
\bX_s
\\
\bY_s
\end{pmatrix}
=
\omega_s
\begin{pmatrix}
\bX_s
\\
\bY_s
\end{pmatrix}
\end{equation}
where the explicit expressions of the resonant and coupling blocks, $\bR(\omega)$ and $\bC(\omega)$, depend on the level of approximation that one employs.
Neglecting the coupling block between the resonant and anti-resonants parts, $\bR(\omega)$ and $-\bR(-\omega_s)$, is known as the Tamm-Dancoff approximation (TDA).
The non-linear eigenvalue problem defined in Eq.~\eqref{eq:LR} has particle-hole symmetry which means that it is invariant via the transformation $\omega \to -\omega$, and, thanks to its non-linear nature stemming from its frequency dependence, it potentially generates more than just single excitations.
In a wave function context, introducing a spatial orbital basis $\lbrace \MO{p} \rbrace$, we assume here that the elements of the matrices defined in Eq.~\eqref{eq:LR} read
\begin{gather}
R_{ia,jb}(\omega) = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + 2 \sigma \ERI{ia}{jb} - \ERI{ib}{ja} + f_{ia,jb}^\sigma(\omega)
\\
C_{ia,jb}(\omega) = 2 \sigma \ERI{ia}{bj} - \ERI{ij}{ba} + f_{ia,bj}^\sigma(\omega)
\end{gather}
where $\sigma = 1 $ or $0$ for singlet ($\updw$) and triplet ($\upup$) excited states (respectively), $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $f(\omega)^{\sigma}$ is the correlation part of the spin-resolved kernel.
(Note that, usually, only the correlation part of the kernel is frequency dependent.)
In the case of a spin-independent kernel, we will drop the superscrit $\sigma$.
Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The concept of dynamical quantities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%s
As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes \cite{ReiningBook}.
To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009b,Sangalli_2011,ReiningBook}
To do so, let us consider the usual chemical scenario where one wants to get the optical excitations of a given system.
In most cases, this can be done by solving a set of linear equations of the form
\begin{equation}
\label{eq:lin_sys}
\bA \bc = \omega \bc
\end{equation}
where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bc$.
where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector .
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
However, in practice, $N$ might be very large (\eg, equal to the total number of single and double excitations generated from a reference Slater determinant), 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}
@ -198,7 +109,7 @@ However, in practice, $N$ might be very large, and it might therefore be practic
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
Solving separately each row of the system \eqref{eq:lin_sys_split} and assuming that $\omega \bI - \bA_2$ is invertible yields
\begin{subequations}
\begin{gather}
\label{eq:row1}
@ -218,33 +129,68 @@ with
\Tilde{\bA}_1(\omega) = \bA_1 + \T{\bb} (\omega \bI - \bA_2)^{-1} \bb
\end{equation}
which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2} \cite{ReiningBook}.
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{ReiningBook}
How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension.
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts.
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Garniron_2018,QP2}
Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension.
However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analogue given by Eq.~\eqref{eq:lin_sys}.
However, because there is usually no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analog given by Eq.~\eqref{eq:lin_sys}.
Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.
For example, assuming that $\bA_2$ is a diagonal matrix is of common practice (see, for example, Ref.~\onlinecite{Garniron_2018} and references therein).
One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value.
For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
Another of these approximations is the so-called \textit{static} approximation, which corresponds to fixing the frequency to a particular value.
For example, as commonly done within the Bethe-Salpeter equation (BSE) formalism, \cite{Strinati_1988} $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$.
Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.
A similar example in the context of time-dependent density-functional theory (TD-DFT) \cite{Runge_1984} is provided by the ubiquitous adiabatic approximation, \cite{Tozer_2000} which neglects all memory effects by making the exchange-correlation (xc) kernel static (\ie, frequency-independent). \cite{Maitra_2016}
These approximations come with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $N$ to $N_1$.
Coming back to our example, in the static (or adiabatic) approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $N_1$ excitation energies are associated with single excitations.
All additional solutions associated with higher excitations have been forever lost.
In the next section, we illustrate these concepts and the various levels of approximation that can be used to recover some of these dynamical effects.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A two-level model}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us consider a two-level quantum system made of two orbitals \cite{Romaniello_2009b}.
We will label these two orbitals as valence ($v$) and conduction ($c$) orbitals with respective one-electron energies $\e{v}$ and $\e{c}$.
Let us consider a two-level quantum system made of two orbitals \cite{Romaniello_2009b} in its singlet ground state (\ie, the lowest orbital is doubly occupied).
We will label these two orbitals, $\MO{v}$ and $\MO{c}$, as valence ($v$) and conduction ($c$) orbitals with respective one-electron Hartree-Fock (HF) energies $\e{v}$ and $\e{c}$.
In a more quantum chemical language, these correspond to the HOMO and LUMO orbitals (respectively).
The ground state has a one-electron configuration $v\bar{v}$, while the doubly-excited state has a configuration $c\bar{c}$.
There is then only one single excitation which corresponds to the transition $v \to c$.
As usual, this can produce a singlet singly-excited state of configuration $(v\bar{c} + c\bar{v})/\sqrt{2}$, and a triplet singly-excited state of configuration $(v\bar{c} - c\bar{v})/\sqrt{2}$ \cite{SzaboBook}.
The ground state $\ket{0}$ has a one-electron configuration $\ket{v\bar{v}}$, while the doubly-excited state $\ket{D}$ has a configuration $\ket{c\bar{c}}$.
There is then only one single excitation possible which corresponds to the transition $v \to c$ with different spin-flip configurations.
As usual, this can produce a singlet singly-excited state $\ket{S} = (\ket{v\bar{c}} + \ket{c\bar{v}})/\sqrt{2}$, and a triplet singly-excited state $\ket{T} = (\ket{v\bar{c}} - \ket{c\bar{v}})/\sqrt{2}$. \cite{SzaboBook}
For the singlet manifold, the exact Hamiltonian in the basis of the (spin-adapted) configuration state functions reads
\begin{equation}
\bH^{\updw} =
\begin{pmatrix}
\mel{0}{\hH}{0} & \mel{0}{\hH}{S} & \mel{0}{\hH}{D} \\
\mel{S}{\hH}{0} & \mel{S}{\hH}{S} & \mel{S}{\hH}{D} \\
\mel{D}{\hH}{0} & \mel{D}{\hH}{S} & \mel{D}{\hH}{D} \\
\end{pmatrix}
\end{equation}
with
\begin{gather}
\mel{0}{\hH}{0} \equiv \EHF = 2\e{v} - \ERI{vv}{vv}
\\
\mel{1}{\hH - \EHF}{1} = \Delta\e{} + \ERI{vc}{cv} - \ERI{vv}{cc}
\\
\mel{1}{\hH - \EHF}{1} = 2\Delta\e{} + \ERI{vv}{vv} + \ERI{cc}{cc} + 2\ERI{vc}{cv} - 4\ERI{vv}{cc}
\\
\mel{0}{\hH - \EHF}{1} = 0
\\
\mel{1}{\hH - \EHF}{2} = \sqrt{2}[\ERI{vc}{cc} - \ERI{cv}{vv}]
\\
\mel{0}{\hH - \EHF}{2} = \ERI{vc}{cv}
\end{gather}
and $\Delta\e{} = \e{c} - \e{v}$.
The energy of the only triplet state is simply $\mel{T}{\hH}{T} = \EHF + \Delta\e{} - \ERI{vv}{cc}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maitra's kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The kernel proposed by Maitra in the context of dressed TD-DFT corresponds to a static kernel to which
where a frequency-dependent kernel is build \textit{a priori} and manually for a particular excitation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dynamical BSE kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -570,7 +516,7 @@ This might not be the smartest way of decomposing the Hamiltonian though but it
\begin{tabular}{|c|ccccccc|c|}
Singlets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
\hline
$\omega_1$ & 1.84903 & 1.90941 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
$\omega_1$ & 1.84903 & 1.90940 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
$\omega_2$ & & & & & & & & \\
$\omega_3$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\
\hline
@ -597,6 +543,7 @@ This might not be the smartest way of decomposing the Hamiltonian though but it
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sangalli's kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\titou{This section is experimental...}
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.
@ -627,7 +574,6 @@ and
\end{gather}
\end{subequations}
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}
@ -639,7 +585,7 @@ For the two-level model, Sangalli's kernel reads
\begin{gather}
\Xi_H (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
\\
\Xi_C (\omega) = 0
\Xi_K (\omega) = 0
\end{gather}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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