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BSEdyn.bib
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@ -1,13 +1,25 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-19 21:54:18 +0200
%% Created for Pierre-Francois Loos at 2020-06-21 22:26:07 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2020d,
Author = {P. F. Loos and E. Fromager},
Date-Added = {2020-06-21 21:41:42 +0200},
Date-Modified = {2020-06-21 21:42:24 +0200},
Doi = {10.1063/5.0007388},
Journal = {J. Chem. Phys.},
Pages = {214101},
Title = {A weight-dependent local correlation density-functional approximation for ensembles},
Volume = {152},
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1063/5.0007388}}
@article{Maitra_2016,
Author = {N. T. Maitra},
Date-Added = {2020-06-19 14:18:29 +0200},
@ -237,23 +249,6 @@
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2012.689872}}
@article{Ary98,
Author = {F Aryasetiawan and O Gunnarsson},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Doi = {10.1088/0034-4885/61/3/002},
Journal = {Rep. Prog. Phys.},
Month = {mar},
Number = {3},
Pages = {237--312},
Publisher = {{IOP} Publishing},
Title = {{The GW method}},
Url = {https://doi.org/10.1088%2F0034-4885%2F61%2F3%2F002},
Volume = {61},
Year = 1998,
Bdsk-Url-1 = {https://doi.org/10.1088%2F0034-4885%2F61%2F3%2F002},
Bdsk-Url-2 = {https://doi.org/10.1088/0034-4885/61/3/002}}
@article{Azarias_2017,
Author = {Azarias, Clo{\'e} and Habert, Chlo\'{e} and Budz\'{a}k, \check{S}imon and Blase, Xavier and Duchemin, Ivan and Jacquemin, Denis},
Date-Added = {2020-05-18 21:40:28 +0200},
@ -800,14 +795,9 @@
@article{Dreuw_2005,
Author = {Dreuw, Andreas and Head-Gordon, Martin},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-06-20 13:38:30 +0200},
Doi = {10.1021/cr0505627},
File = {/Users/loos/Zotero/storage/WKGXAHGE/Dreuw_2005.pdf},
Issn = {0009-2665, 1520-6890},
Journal = {Chem. Rev.},
Language = {en},
Month = nov,
Number = {11},
Pages = {4009--4037},
Title = {Single-{{Reference}} Ab {{Initio Methods}} for the {{Calculation}} of {{Excited States}} of {{Large Molecules}}},
Volume = {105},
@ -1068,21 +1058,6 @@
Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevB.37.10159},
Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.37.10159}}
@article{Golze_2019rev,
Author = {Golze, Dorothea and Dvorak, Marc and Rinke, Patrick},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Doi = {10.3389/fchem.2019.00377},
Issn = {2296-2646},
Journal = {Front. Chem.},
Pages = {377},
Title = {The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy},
Url = {https://www.frontiersin.org/article/10.3389/fchem.2019.00377},
Volume = {7},
Year = {2019},
Bdsk-Url-1 = {https://www.frontiersin.org/article/10.3389/fchem.2019.00377},
Bdsk-Url-2 = {https://doi.org/10.3389/fchem.2019.00377}}
@article{Gui_2018,
Author = {Gui, Xin and Holzer, Christof and Klopper, Wim},
Date-Added = {2020-05-18 21:40:28 +0200},
@ -1903,16 +1878,12 @@
@article{Reining_2017,
Author = {Reining, Lucia},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-06-20 13:48:16 +0200},
Doi = {10.1002/wcms.1344},
File = {/Users/loos/Zotero/storage/VDXYCLGF/Reining_2017.pdf},
Issn = {17590876},
Journal = {Wiley Interdiscip. Rev. Comput. Mol. Sci.},
Language = {en},
Month = dec,
Pages = {e1344},
Shorttitle = {The {{GW}} Approximation},
Title = {The {{GW}} Approximation: Content, Successes and Limitations: {{The GW}} Approximation},
Volume = {8},
Year = {2017},
Bdsk-Url-1 = {https://dx.doi.org/10.1002/wcms.1344}}
@ -2547,13 +2518,11 @@
@article{Golze_2019,
Author = {Golze, Dorothea and Dvorak, Marc and Rinke, Patrick},
Date-Added = {2020-05-18 21:37:56 +0200},
Date-Modified = {2020-05-18 21:38:00 +0200},
Date-Modified = {2020-06-20 13:49:12 +0200},
Doi = {10.3389/fchem.2019.00377},
Issn = {2296-2646},
Journal = {Front. Chem.},
Pages = {377},
Title = {The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy},
Url = {https://www.frontiersin.org/article/10.3389/fchem.2019.00377},
Volume = {7},
Year = {2019},
Bdsk-Url-1 = {https://www.frontiersin.org/article/10.3389/fchem.2019.00377},
@ -4027,14 +3996,10 @@
Abstract = {The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequencydependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequencydependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the groundstate exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. Many exact results, limits, and expansions about those limits are given in the Appendices.},
Author = {Carrascal, Diego J. and Ferrer, Jaime and Maitra, Neepa and Burke, Kieron},
Date-Added = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-06-21 21:38:08 +0200},
Doi = {10.1140/epjb/e2018-90114-9},
File = {/Users/loos/Zotero/storage/YFNPCZLK/Carrascal et al. - 2018 - Linear response time-dependent density functional .pdf},
Issn = {1434-6028, 1434-6036},
Journal = {Eur. Phys. J. B},
Language = {en},
Month = jul,
Number = {7},
Pages = {142},
Title = {Linear Response Time-Dependent Density Functional Theory of the {{Hubbard}} Dimer},
Volume = {91},
Year = {2018},

129
Notes/BSEdyn-notes.rty Normal file
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@ -0,0 +1,129 @@
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\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{\Hx}{\text{Hx}}
\newcommand{\xc}{\text{xc}}
\newcommand{\Ha}{\text{H}}
\newcommand{\co}{\text{c}}
\newcommand{\ex}{\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{\bR}{\mathbf{R}}
\newcommand{\bS}{\mathbf{S}}
\newcommand{\bX}{\mathbf{X}}
\newcommand{\bY}{\mathbf{Y}}
\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{\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}
% orbitals, gaps, etc
\newcommand{\updw}{\uparrow\downarrow}
\newcommand{\upup}{\uparrow\uparrow}
\newcommand{\eps}{\epsilon}
\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}

464
Notes/BSEdyn-notes.tex
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@ -45,53 +45,59 @@ Linear response theory is a powerful approach that allows to directly access the
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)
\bR^{\sigma}(\omega_s) & \bC^{\sigma}(\omega_s)
\\
-\bC(-\omega_s) & -\bR(-\omega_s)
-\bC^{\sigma}(-\omega_s) & -\bR^{\sigma}(-\omega_s)
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX_s
\bX_s^{\sigma}
\\
\bY_s
\bY_s^{\sigma}
\end{pmatrix}
=
\omega_s
\begin{pmatrix}
\bX_s
\bX_s^{\sigma}
\\
\bY_s
\bY_s^{\sigma}
\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 $-\bR(-\omega)$, 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.
where the explicit expressions of the resonant and coupling blocks, $\bR^{\sigma}(\omega)$ and $\bC^{\sigma}(\omega)$, depend on the spin manifold ($\sigma =$ $\updw$ for singlets and $\sigma =$ $\upup$ for triplets) and the level of approximation that one employs.
Neglecting the coupling block [\ie, $\bC^{\sigma}(\omega) = 0$] between the resonant and anti-resonants parts, $\bR(\omega)$ and $-\bR(-\omega)$, 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 $\omega \to -\omega$.
Therefore, without loss of generality, we will restrict our analysis to positive frequencies.
The central point here is that, thanks to their non-linear nature stemming from their frequency dependence, dynamical kernels potentially generate 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} have the following generic form:
In the one-electron basis of (real) spatial orbitals $\lbrace \MO{p} \rbrace$, the elements of the matrices defined in Eq.~\eqref{eq:LR} have the following generic forms: \cite{Dreuw_2005}
\begin{subequations}
\begin{gather}
\begin{split}
R_{ia,jb}(\omega)
& = \iint \MO{i}(\br) \MO{a}(\br) \bR(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
\\
& = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + 2 \sigma \ERI{ia}{jb} - \ERI{ib}{ja} + f_{ia,jb}^\sigma(\omega)
\end{split}
R_{ia,jb}^{\sigma}(\omega) = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + f_{ia,jb}^{\Hxc,\sigma}(\omega)
\\
\begin{split}
C_{ia,jb}(\omega)
& = \iint \MO{i}(\br) \MO{a}(\br) \bC(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
\\
& = 2 \sigma \ERI{ia}{bj} - \ERI{ij}{ba} + f_{ia,bj}^\sigma(\omega)
\end{split}
C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\Hxc,\sigma}(\omega)
\end{gather}
\end{subequations}
where $\sigma = 1 $ or $0$ for singlet ($\updw$) and triplet ($\upup$) excited states (respectively), and
where $\delta_{pq}$ is the Kronecker delta, $\e{p}$ is the one-electron energy associated with $\MO{p}$, and
\begin{equation} \label{eq:kernel}
f_{ia,jb}^{\Hxc,\sigma}(\omega)
= \iint \MO{i}(\br) \MO{a}(\br) f^{\Hxc,\sigma}(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
\end{equation}
In Eq.~\eqref{eq:kernel},
\begin{equation} \label{eq:kernel-Hxc}
f^{\Hxc,\sigma}(\omega) = f^{\Hx,\sigma} + f^{\co,\sigma}(\omega)
\end{equation}
is the (spin-resolved) Hartree-exchange-correlation (Hxc) dynamical kernel.
Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $p$ and $q$ indicate arbitrary orbitals.
As readily seen from Eq.~\eqref{eq:kernel-Hxc}, only the correlation (c) part of the kernel is frequency dependent and, in a wave function context, the static Hartree-exchange (Hx) matrix elements read
\begin{equation}
f_{ia,jb}^{\Hx,\sigma} = \sigma \ERI{ia}{jb} - \ERI{ib}{ja}
\end{equation}
where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively), and
\begin{equation}
\ERI{ia}{jb} = \iint \MO{i}(\br) \MO{a}(\br) \frac{1}{\abs{\br - \br'}} \MO{j}(\br') \MO{b}(\br') d\br d\br'
\end{equation}
are the usual (bare) two-electron integrals.
Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $f^{\sigma}(\omega)$ 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 $\sigma$.
In the case of a spin-independent kernel, we will drop the superscript $\sigma$.
Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -104,7 +110,7 @@ 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 \bc = \omega \bc
\bA \cdot \bc = \omega \, \bc
\end{equation}
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.
@ -115,6 +121,7 @@ However, in practice, $N$ might be (very) large (\eg, equal to the total number
\bA_1 & \T{\bb} \\
\bb & \bA_2 \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bc_1 \\
\bc_2 \\
@ -132,22 +139,22 @@ Solving separately each row of the system \eqref{eq:lin_sys_split} and assuming
\begin{subequations}
\begin{gather}
\label{eq:row1}
\bA_1 \bc_1 + \T{\bb} \bc_2 = \omega \bc_1
\bA_1 \cdot \bc_1 + \T{\bb} \cdot \bc_2 = \omega \, \bc_1
\\
\label{eq:row2}
\bc_2 = (\omega \bI - \bA_2)^{-1} \bb \bc_1
\bc_2 = (\omega \, \bI - \bA_2)^{-1} \cdot \bb \cdot \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) \bc_1 = \omega \bc_1
\Tilde{\bA}_1(\omega) \cdot \bc_1 = \omega \, \bc_1
\end{equation}
with
\begin{equation}
\Tilde{\bA}_1(\omega) = \bA_1 + \T{\bb} (\omega \bI - \bA_2)^{-1} \bb
\Tilde{\bA}_1(\omega) = \bA_1 + \T{\bb} \cdot (\omega \, \bI - \bA_2)^{-1} \cdot \bb
\end{equation}
which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
which has, by construction, exactly the same solutions as the linear system \eqref{eq:lin_sys} but a smaller dimension.
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the single-excitation basis 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?
@ -240,66 +247,104 @@ This system contains two orbitals and the numerical values of the various quanti
\end{align}
\end{subequations}
This yields the following exact singlet and triplet excitation energies
\begin{align} \label{sec:exact}
\begin{align} \label{eq:exact}
\omega_{1}^{\updw} & = 1.92145
&
\omega_{3}^{\updw} & = 3.47880
&
\omega_{1}^{\upup} & = 1.47085
\end{align}
that we are going to use a reference for the remaining of this study.
where $\omega_{1}^{\updw}$ and $\omega_{3}^{\updw}$ are the singlet single and double excitations (respectively), and $\omega_{1}^{\upup}$ is the triplet single excitation.
We are going to use these as reference for the remaining of this study.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maitra's dynamical kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The kernel proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} in the context of dressed TDDFT (D-TDDFT) corresponds to an \textit{ad hoc} many-body theory correction to TDDFT.
More specifically, D-TDDFT adds manually to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian \eqref{eq:H-exact}.
The very same idea was taking further by Huix-Rotllant, Casida and coworkers. \cite{Huix-Rotllant_2011}
For the singlet states, we have
More specifically, D-TDDFT adds to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian: a single and double excitations, assumed to be strongly coupled, are isolated from among the spectrum and added manually to the static kernel.
The very same idea was taking further by Huix-Rotllant, Casida and coworkers, \cite{Huix-Rotllant_2011} and tested on a large set of molecules.
Here, we start instead from a HF reference.
The static problem corresponds then to the TDHF Hamiltonian, while in the TDA, it reduces to CIS.
For the two-level model, the reverse-engineering process of the exact Hamiltonian \eqref{eq:H-exact} yields
\begin{equation} \label{eq:f-Maitra}
f_M^{\updw}(\omega) = \frac{\abs*{\mel{S}{\hH}{D}}^2}{\omega - (\mel{D}{\hH}{D} - \mel{0}{\hH}{0}) }
f_M^{\co,\updw}(\omega) = \frac{\abs*{\mel{S}{\hH}{D}}^2}{\omega - (\mel{D}{\hH}{D} - \mel{0}{\hH}{0}) }
\end{equation}
while $f_M^{\upup}(\omega) = 0$.
The expression \eqref{eq:f-Maitra} can be easily obtained by folding the double excitation onto the single excitation starting from the Hamiltonian \eqref{eq:H-exact}, as explained in Sec.~\ref{sec:dyn}.
while $f_M^{\co,\upup}(\omega) = 0$.
The expression \eqref{eq:f-Maitra} can be easily obtained by folding the double excitation onto the single excitation, as explained in Sec.~\ref{sec:dyn}.
It is clear that one must know \textit{a priori} the structure of the Hamiltonian to construct such dynamical kernel, and this obviously hampers its applicability to realistic photochemical systems where it is sometimes hard to get a clear picture of the interplay between excited states. \cite{Boggio-Pasqua_2007}
For the two-level model, the non-linear equations defined in Eq.~\eqref{eq:LR} provides the following effective Hamiltonian
\begin{equation} \label{eq:H-M}
\bH_{M}(\omega) =
\bH_\text{D-TDHF}^{\sigma}(\omega) =
\begin{pmatrix}
R_M(\omega) & C_M(\omega)
R_M^{\sigma}(\omega) & C_M^{\sigma}(\omega)
\\
-C_M(-\omega) & -R_M(-\omega)
-C_M^{\sigma}(-\omega) & -R_M^{\sigma}(-\omega)
\end{pmatrix}
\end{equation}
with
\begin{subequations}
\begin{gather}
R_M(\omega) = \Delta\e{} + 2 \sigma \ERI{vc}{vc} - \ERI{vc}{vc} + f_M^{\sigma}(\omega)
\label{eq:R_M}
R_M^{\sigma}(\omega) = \Delta\e{} + 2 \sigma \ERI{vc}{vc} - \ERI{vc}{vc} + f_M^{\co,\sigma}(\omega)
\\
C_M(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} + f_M^{\sigma}(\omega)
\label{eq:C_M}
C_M^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} + f_M^{\co,\sigma}(\omega)
\end{gather}
\end{subequations}
which provides the following excitation energies when diagonalized:
\begin{align} \label{sec:M}
\omega_{1}^{\updw} & = 1.89314
&
\omega_{3}^{\updw} & = 3.44865
&
\omega_{1}^{\upup} & = 1.43794
\end{align}
Although not particularly accurate, this kernel provides exactly the right number of solutions (2 singlets and 1 triplet).
Its accuracy could be certainly improved in a DFT context.
yielding the excitation energies reported in Table \ref{tab:Maitra} when diagonalized.
The TDHF Hamiltonian is obtained from Eq.~\eqref{eq:H-M} by setting $f_M^{\co,\sigma}(\omega) = 0$ in Eqs.~\eqref{eq:R_M} and \eqref{eq:C_M}.
In Fig.~\ref{fig:Maitra}, we plot $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (black and gray) and triplet (orange) manifolds.
The roots of $\det[\bH(\omega) - \omega \bI]$ indicate the excitation energies.
Because, there is nothing to dress for the triplet state, only the static TDHF excitation energy is reported.
%%% TABLE I %%%
\begin{table}
\caption{Singlet and triplet excitation energies at various levels of theory.
\label{tab:Maitra}
}
\begin{ruledtabular}
\begin{tabular}{|c|cccc|c|}
Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
\hline
$\omega_1^{\updw}$ & 1.91119 & 1.89758 & 1.90636 & 1.89314 & 1.92145 \\
$\omega_3^{\updw}$ & & & & 3.44865 & 3.47880 \\
\hline
Triplets & & & & & Exact \\
\hline
$\omega_1^{\upup}$ & 1.45585 & 1.43794 & 1.45585 & 1.43794 & 1.47085 \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
%%% FIGURE 1 %%%
\begin{figure}
\includegraphics[width=\linewidth]{Maitra}
\caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (black and gray) and triplet (orange) manifolds.
The static TDHF Hamiltonian (dashed) and dynamic D-TDHF Hamiltonian (solid) are considered.
\label{fig:Maitra}
}
\end{figure}
%%% %%% %%% %%%
Although not particularly accurate for the single excitations, Maitra's dynamical kernel allows to access the double excitation with good accuracy and provides exactly the right number of solutions (two singlets and one triplet).
Note that the correlation kernel is known to work best in the weak correlation regime (which is the case here) where the true excitations have a clear single and double excitation character, \cite{Loos_2019,Loos_2020d} but it is not intended to explore strongly correlated systems. \cite{Carrascal_2018}
Its accuracy for the single excitations could be certainly improved in a DFT context.
However, this is not the point of the present investigation.
Because $f_M^{\upup}(\omega) = 0$, the triplet excitation energy is equivalent to the TDHF excitation energy.
In the static approximation where $f_M^{\updw}(\omega) = 0$, the singlet excitations are also TDHF excitation energies.
Table \ref{tab:Maitra} also reports the slightly-improves (thanks to error compensation) CIS and D-CIS excitation energies.
Graphically, the curves obtained for CIS and D-CIS are extremely similar to the ones of TDHF and D-TDHF depicted in Fig.~\ref{fig:Maitra}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dynamical BSE kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Within MBPT, one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
Assuming that the dynamically-screened Coulomb potential has been calculated at the random-phase approximation (RPA) level and within the Tamm-Dancoff approximation (TDA), the expression of the $\GW$ quasiparticle energy is
\titou{Titou will add a bit more background information about the BSE dynamical kernel.}
Within the so-called $GW$ approximation of MBPT, \cite{Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
Assuming that the dynamically-screened Coulomb potential $W$ has been calculated at the random-phase approximation (RPA) level and within the TDA, the expression of the $\GW$ quasiparticle energy is
\begin{equation}
\e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
\end{equation}
@ -308,97 +353,13 @@ 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 wither the valence of conduction orbitals,
is the correlation part of the self-energy $\Sig{}$, and
\begin{equation}
Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
\end{equation}
is the renormalization factor.
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}
\bH^{\dBSE}(\omega) =
\begin{pmatrix}
R(\omega) & C(\omega)
\\
-C(-\omega) & -R(-\omega)
\end{pmatrix}
\end{equation}
with
\begin{subequations}
\begin{gather}
R(\omega) = \Delta\eGW{} + 2 \sigma \ERI{vc}{cv} - W_R(\omega)
\\
C(\omega) = 2 \sigma \ERI{vc}{cv} - W_C(\omega)
\end{gather}
\end{subequations}
($\sigma = 1$ for singlets and $\sigma = 0$ for triplets) and
\begin{subequations}
\begin{gather}
W_R(\omega) = \ERI{vv}{cc} + \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
\\
W_C(\omega) = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
\end{gather}
\end{subequations}
are the elements of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian.
It can be easily shown that solving the equation
\begin{equation}
\det[\bH^{\dBSE}(\omega) - \omega \bI] = 0
\end{equation}
yields 6 solutions (per spin manifold): 3 pairs of frequencies opposite in sign, which corresponds to the 3 resonant states and the 3 anti-resonant states.
As mentioned in Ref.~\cite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel.
Indeed, diagonalizing the exact Hamiltonian would produce two singlet solutions corresponding to the singly- and doubly-excited states, while there is only one triplet state (see discussion earlier in the section).
Therefore, there is one spurious solution for the singlet manifold and two spurious solution for the triplet manifold.
Within the static approximation, the BSE Hamiltonian is
\begin{equation}
\bH^{\BSE} =
\begin{pmatrix}
R^{\stat} & C^{\stat}
\\
-C^{\stat} & -R^{\stat}
\end{pmatrix}
\end{equation}
with
\begin{subequations}
\begin{gather}
R^{\stat} = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\eGW{})
\\
C^{\stat} = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0)
\end{gather}
\end{subequations}
In the static approximation, only one pair of solutions (per spin manifold) is obtained by diagonalizing $\bH^{\BSE}$.
There are, like in the dynamical case, opposite in sign.
Therefore, the static BSE Hamiltonian does not produce spurious excitations but misses the (singlet) double excitation.
Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the BSE Hamiltonian, \ie, $C(\omega) = 0$, allows to remove some of these spurious excitations.
In this case, the excitation energies are obtained by solving the simple equation $R(\omega) - \omega = 0$, which yields two solutions for each spin manifold.
There is thus only one spurious excitation in the triplet manifold, the two solutions of the singlet manifold corresponding to the single and double excitations.
Another way to access dynamical effects while staying in the static framework is to use perturbation theory.
To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
\begin{equation}
\bH^{\dBSE}(\omega) = \underbrace{\bH^{\BSE}}_{\bH^{(0)}} + \underbrace{\qty[ \bH^{\dBSE}(\omega) - \bH^{\BSE} ]}_{\bH^{(1)}}
\end{equation}
Thanks to (renormalized) first-order perturbation theory, one gets
\begin{equation}
\omega_{1,\sigma}^{\BSE1} = \omega_{1,\sigma}^{\BSE} + Z_{1} \T{\bV} \cdot \qty[ \bH^{\dBSE}(\omega = \omega_{1,\sigma}^{\BSE}) - \bH^{\BSE} ] \cdot \bV
\end{equation}
where
\begin{equation}
\bV =
\begin{pmatrix}
X \\ Y
\end{pmatrix}
\end{equation}
are the eigenvectors of $\bH^{\BSE}$, and
\begin{equation}
Z_{1} = \qty{ 1 - \T{\bV} \cdot \left. \pdv{\bH^{\dBSE}(\omega)}{\omega} \right|_{\omega = \omega_{1,\sigma}^{\BSE}} \cdot \bV }^{-1}
\end{equation}
This corresponds to a dynamical correction to the static excitations, and the TDA can be applied to the dynamical correction, a scheme we label as dTDA in the following.
which yields
Numerically, we get
\begin{align}
\Omega & = 2.769\,327
&
@ -407,7 +368,70 @@ which yields
\eGW{c} & = +1.373\,640
\end{align}
%%% FIGURE 1 %%%
One can now build the dynamical BSE (dBSE) Hamiltonian \cite{Strinati_1988}
\begin{equation} \label{eq:HBSE}
\bH_{\dBSE}^{\sigma}(\omega) =
\begin{pmatrix}
R_{\dBSE}^{\sigma}(\omega) & C_{\dBSE}^{\sigma}(\omega)
\\
-C_{\dBSE}^{\sigma}(-\omega) & -R_{\dBSE}^{\sigma}(-\omega)
\end{pmatrix}
\end{equation}
with
\begin{subequations}
\begin{gather}
R_{\dBSE}^{\sigma}(\omega) = \Delta\eGW{} + 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} - W^{\co}_R(\omega)
\\
C_{\dBSE}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vc}{cv} - W^{\co}_C(\omega)
\end{gather}
\end{subequations}
and
\begin{subequations}
\begin{gather}
W^{\co}_R(\omega) = \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
\\
W^{\co}_C(\omega) = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
\end{gather}
\end{subequations}
are the elements of the correlation part of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian.
Note that, in this case, the correlation kernel is spin blind.
It can be easily shown that solving the equation
\begin{equation}
\det[\bH_{\dBSE}^{\sigma}(\omega) - \omega \bI] = 0
\end{equation}
yields 3 solutions per spin manifold (see Fig.~\ref{fig:dBSE}).
Their numerical values are reported in Table \ref{tab:dBSE} alongside other variants discussed below.
As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel.
Indeed, diagonalizing the exact Hamiltonian produces only two singlet solutions corresponding to the singly- and doubly-excited states, and one triplet state (see Sec.~\ref{sec:exact}).
Therefore, there is one spurious solution for the singlet manifold ($\omega_{2}^{\dBSE,\updw}$) and two spurious solutions for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$ and $\omega_{3}^{\dBSE,\upup}$).
It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\dBSE,\sigma}$ and $\omega_3^{\dBSE,\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
This evidences that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
%%% TABLE I %%%
\begin{table*}
\caption{BSE singlet and triplet excitation energies at various levels of theory.
\label{tab:dBSE}
}
\begin{ruledtabular}
\begin{tabular}{|c|ccccccc|c|}
Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
\hline
$\omega_1^{\updw}$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\
$\omega_2^{\updw}$ & & & & 2.78377 & & & & \\
$\omega_3^{\updw}$ & & & & 4.90134 & & & 4.90117 & 3.47880 \\
\hline
Triplets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
\hline
$\omega_1^{\upup}$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\
$\omega_2^{\upup}$ & & & & 2.76178 & & & & \\
$\omega_3^{\upup}$ & & & & 4.91545 & & & 4.91517 & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%%% %%% %%% %%%
%%% FIGURE 2 %%%
\begin{figure}
\includegraphics[width=\linewidth]{dBSE}
\caption{
@ -417,33 +441,29 @@ which yields
\end{figure}
%%% %%% %%% %%%
Figure \ref{fig:dBSE} shows the three resonant solutions (for the singlet and triplet spin manifold) of the dynamical BSE Hamiltonian $\bH(\omega)$ defined in Eq.~\eqref{eq:HBSE}, the curve being invariant with respect to the transformation $\omega \to - \omega$ (electron-hole symmetry).
Numerically, we find
\begin{align}
\omega_{1,\updw}^{\dBSE} & = 1.90527
&
\omega_{2,\updw}^{\dBSE} & = 2.78377
&
\omega_{3,\updw}^{\dBSE} & = 4.90134
\end{align}
for the singlet states, and
\begin{align}
\omega_{1,\upup}^{\dBSE} & = 1.46636
&
\omega_{2,\upup}^{\dBSE} & = 2.76178
&
\omega_{3,\upup}^{\dBSE} & = 4.91545
\end{align}
for the triplet states.
it is interesting to mention that, around $\omega = \omega_1^{\sigma}$ ($\sigma =$ $\updw$ or $\upup$), the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\sigma}$ and $\omega_3^{\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
Diagonalizing the static BSE Hamiltonian yields the following singlet and triplet excitation energies:
\begin{align}
\omega_{1,\updw}^{\BSE} & = 1.92778
&
\omega_{1,\upup}^{\BSE} & = 1.48821
\end{align}
which shows that the physical single excitation stemming from the dynamical BSE Hamiltonian is the lowest one for each spin manifold, \ie, $\omega_1^{\updw}$ and $\omega_1^{\upup}$.
Within the usual static approximation, the BSE Hamiltonian is
\begin{equation}
\bH_{\BSE}^{\sigma} =
\begin{pmatrix}
R_{\BSE}^{\sigma} & C_{\BSE}^{\sigma}
\\
-C_{\BSE}^{\sigma} & -R_{\BSE}^{\sigma}
\end{pmatrix}
\end{equation}
with
\begin{subequations}
\begin{gather}
R_{\BSE}^{\sigma} = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\eGW{})
\\
C_{\BSE}^{\sigma} = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0)
\end{gather}
\end{subequations}
In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:dBSE}).
Therefore, the static BSE Hamiltonian does not produce spurious excitations but misses the (singlet) double excitation, and it shows that the physical single excitation stemming from the dBSE Hamiltonian is the lowest one for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the dBSE Hamiltonian, allows to remove some of these spurious excitations.
There is thus only one spurious excitation in the triplet manifold ($\omega_{3}^{\BSE,\upup}$), the two solutions of the singlet manifold corresponding to the single and double excitations.
Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
%%% FIGURE 2 %%%
\begin{figure}
@ -455,56 +475,60 @@ which shows that the physical single excitation stemming from the dynamical BSE
\end{figure}
%%% %%% %%% %%%
Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
As one can see, the spurious solution $\omega_2^{\sigma}$ has disappeared, and two pairs of solutions remain for each spin manifold.
Numerically, we have
\begin{align}
\omega_{1,\updw}^{\TDAdBSE} & = 1.94005
&
\omega_{3,\updw}^{\TDAdBSE} & = 4.90117
\end{align}
for the singlet states, and
\begin{align}
\omega_{1,\upup}^{\TDAdBSE} & = 1.47070
&
\omega_{3,\upup}^{\TDAdBSE} & = 4.91517
\end{align}
while the static values are
\begin{align}
\omega_{1,\updw}^{\TDABSE} & = 1.95137
&
\omega_{1,\upup}^{\TDABSE} & = 1.49603
\end{align}
All these numerical results are gathered in Table \ref{tab:BSE}.
This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
A
Another way to access dynamical effects while staying in the static framework is to use perturbation theory.
To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
\begin{equation}
\bH_{\dBSE}^{\sigma}(\omega) = \underbrace{\bH_{\BSE}^{\sigma}}_{\bH^{(0)}} + \underbrace{\qty[ \bH_{\dBSE}^{\sigma}(\omega) - \bH_{\BSE}^{\sigma} ]}_{\bH^{(1)}}
\end{equation}
Thanks to (renormalized) first-order perturbation theory, one gets
\begin{equation}
\begin{split}
\omega_{1}^{\BSE1,\sigma}
& = \omega_{1}^{\BSE,\sigma}
\\
& + Z_{1}
\T{\begin{pmatrix}
X_1 \\ Y_1
\end{pmatrix}
}
\cdot \qty[ \bH^{\dBSE}(\omega = \omega_{1}^{\BSE,\sigma}) - \bH^{\BSE} ] \cdot
\begin{pmatrix}
X_1 \\ Y_1
\end{pmatrix}
\end{split}
\end{equation}
where
\begin{equation}
\bH^{\BSE}
\cdot
\begin{pmatrix}
X_1 \\ Y_1
\end{pmatrix}
= \omega_{1}^{\BSE,\sigma}
\begin{pmatrix}
X_1 \\ Y_1
\end{pmatrix}
\end{equation}
are the eigenvectors of $\bH^{\BSE}$, and
\begin{equation}
Z_{1} = \qty{ 1 -
\T{
\begin{pmatrix}
X_1 \\ Y_1
\end{pmatrix}
}
\cdot \left. \pdv{\bH^{\dBSE}(\omega)}{\omega} \right|_{\omega = \omega_{1}^{\BSE,\sigma}} \cdot
\begin{pmatrix}
X_1 \\ Y_1
\end{pmatrix}
}^{-1}
\end{equation}
This corresponds to a dynamical correction to the static excitations, and the TDA can be applied to the dynamical correction, a scheme we label as dTDA in the following.
The perturbatively-corrected values are also reported, which shows that this scheme is very efficient at reproducing the dynamical value.
The perturbatively-corrected values are also reported in Table \ref{tab:dBSE}, which shows that this scheme is very efficient at reproducing the dynamical value.
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*}
\caption{BSE singlet and triplet excitation energies at various levels of theory.
\label{tab:BSE}
}
\begin{ruledtabular}
\begin{tabular}{|c|ccccccc|c|}
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 \\
$\omega_2$ & & & & 2.78377 & & & & \\
$\omega_3$ & & & & 4.90134 & & & 4.90117 & 3.47880 \\
\hline
Triplets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
\hline
$\omega_1$ & 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 & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Second-order BSE kernel}
@ -617,7 +641,7 @@ 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.
In Ref.~\onlinecite{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

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