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BSEdyn.bib
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@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-22 20:38:15 +0200
%% Created for Pierre-Francois Loos at 2020-06-26 09:45:06 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Petersilka_1996,
Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross},
Date-Added = {2020-06-26 09:43:33 +0200},
Date-Modified = {2020-06-26 09:45:05 +0200},
Doi = {10.1103/PhysRevLett.76.1212},
Journal = {Phys. Rev. Lett.},
Pages = {1212},
Title = {Excitation Energies From Time-Dependent Density-Functional Theory},
Volume = {76},
Year = {1996}}
@article{Nielsen_1980,
Author = {Egon S. Nielsen and Poul Jorgensen},
Date-Added = {2020-06-22 20:37:32 +0200},
@ -17,7 +28,8 @@
Pages = {6238},
Title = {Transition moments and dynamic polarizabilities in a second order polarization propagator approach},
Volume = {73},
Year = {1980}}
Year = {1980},
Bdsk-Url-1 = {https://doi.org/10.1063/1.440119}}
@article{Oddershede_1977,
Author = {Jens Oddershede and Poul Jorgensen},
@ -28,7 +40,8 @@
Pages = {1541},
Title = {An order analysis of the particle--hole propagator},
Volume = {66},
Year = {1977}}
Year = {1977},
Bdsk-Url-1 = {https://doi.org/10.1063/1.434118}}
@phdthesis{Huix-Rotllant_PhD,
Author = {M. Huix-Rotllant},

111
BSEdyn.tex
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@ -209,6 +209,7 @@ Similar to the ubiquitous adiabatic approximation in time-dependent density-func
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 within the random phase approximation.
Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to assess the clear improvements brought by dynamical corrections.
%\\
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
@ -223,7 +224,7 @@ Our calculations are benchmarked against high-level (coupled-cluster) calculatio
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
The Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is to the $GW$ approximation \cite{Hedin_1965,Golze_2019} of many-body perturbation theory (MBPT) \cite{Martin_2016} what time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995} is to Kohn-Sham density-functional theory (KS-DFT), \cite{Hohenberg_1964,Kohn_1965} an affordable way of computing the neutral excitations of a given electronic system.
In recent years, it has been shown to be a valuable tool for computational chemists with a large number of systematic benchmark studies on large molecular systems appearing in the literature \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Krause_2017,Gui_2018} (see Ref.~\onlinecite{Blase_2018} for a recent review).
In recent years, it has been shown to be a valuable tool for computational chemists with a large number of systematic benchmark studies on large families of molecular systems appearing in the literature \cite{Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018} (see Ref.~\onlinecite{Blase_2018} for a recent review).
Taking the optical gap (\ie, the lowest optical excitation energy) as an example, BSE builds on top of a $GW$ calculation by adding up excitonic effects $\EB$ to the $GW$ HOMO-LUMO gap
\begin{equation}
@ -402,7 +403,7 @@ Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Om{s}{} )
\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Om{s}{} t^{34} }} { \Om{s}{} - ( \e{a} - \e{i} ) + i \eta }
\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \frac{\Om{s}{}}{2}) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \frac{\Om{s}{}}{2}) \tau_{34} } ].
\end{multline}
More details are provided in the Appendix. As a final step, we express the terms $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')] }{N,s}$ and $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ from Eq.~\eqref{eq:BSE_2} in the standard electron-hole product (or single-excitation) space.
with $t^{34} = (t_3 + t_4)/2$ and $\tau_{34} = t_3 -t_4$.More details are provided in the Appendix. As a final step, we express the terms $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')] }{N,s}$ and $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ from Eq.~\eqref{eq:BSE_2} in the standard electron-hole product (or single-excitation) space.
This is done by expanding the field operators over a complete orbital basis of creation/destruction operators.
For example, we have (see derivation in the Appendix)
\begin{multline} \label{eq:spectral65}
@ -413,7 +414,7 @@ For example, we have (see derivation in the Appendix)
\\
\times \qty[ \theta( \tau_{65} ) e^{- i ( \e{p} - \frac{\Om{s}{}}{2} ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i ( \e{q} + \frac{\Om{s}{}}{2}) \tau_{65} } ],
\end{multline}
with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$. The $\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}$ are the unknown particle-hole amplitudes in the molecular orbitals product basis.
%================================
@ -459,8 +460,8 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
= \iint d\br d\br' \, \MO{i}(\br) \MO{j}^*(\br) W(\br ,\br'; \omega) \MO{a}^*(\br') \MO{b}(\br').
\end{equation}
%\xavier{\sout{ A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting $\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s}$ and $L_0(\bx_1,4;\bx_{1'},3; \Om{s}{})$ onto $\MO{i}^*(\bx_1) \MO{a}(\bx_{1'})$ instead of
%$\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$. } }
\xavier{\sout{ A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting $\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s}$ and $L_0(\bx_1,4;\bx_{1'},3; \Om{s}{})$ onto $\MO{i}^*(\bx_1) \MO{a}(\bx_{1'})$ instead of
$\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$. } }
%================================
@ -521,6 +522,8 @@ The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
\\
\times \qty[ \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ].
\end{multline}
\xavier{One can verify that in the static limit, that can be obtained with
$\Om{m}{\RPA} \rightarrow +\infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to the static ${W}_{ij,ab}$ ones, and the dBSE formalism recovers the form of the standard BSE formalism.}
Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{s} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{s}{})$ has no resonances.
Furthermore, $\Om{ib}{s}$ and $\Om{ja}{s}$ are necessarily negative quantities for in-gap low-lying BSE excitations.
Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$.
@ -1074,16 +1077,17 @@ This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 i
%%%%%%%%%%%%%%%%%%%%%%%%
The data that support the findings of this study are available within the article and its {\SI}.
\begin{widetext}
\appendix
\section{$L_0(1,3; 1',4)$ $(t_1)$-time Fourier transform}
\label{appendixA}
We derive in this Appendix Eqs.~\ref{eq:iL0} to ~\ref{eq:iL0bis}.
Defining the $t_1$-time Fourier transform of $iL_0(1,3;4,1')$ with
Defining the $t_1$-time Fourier transform of $L_0(1,3;4,1')$ with
$(t_{1'} = t_1^{+})$
\begin{align}
[iL_0](x_1,3;x_{1'},4 \; | \; \omega_1 ) =
[L_0](x_1,3;x_{1'},4 \; | \; \omega_1 ) = -i
\int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1')
\end{align}
we plug-in the Fourier expansion of the Green's function, e.g.
@ -1091,65 +1095,82 @@ $(t_{1'} = t_1^{+})$
G(1,3) = \int \frac{ d\omega }{ 2\pi } G(x_1,x_3;\omega) e^{-i \omega \tau_{13} }
\end{align*}
with $\tau_{13} = (t_1-t_3)$ to obtain:
\begin{align}
[iL_0](x_1,3;x_{1'},4 & \;| \; \omega_1 ) =
\int \frac{ d\omega }{ 2\pi } \; G(x_1,x_3;\omega) \; \times \\ & \times \; G(x_4,x_{1'};\omega-\omega_1)
\begin{equation}
[L_0](x_1,3;x_{1'},4 \;| \; \omega_1 ) =
\int \frac{ d\omega }{ 2i\pi } \; G(x_1,x_3;\omega) \; G(x_4,x_{1'};\omega-\omega_1)
e^{ i \omega t_3 } e^{-i (\omega-\omega_1) t_4 } \nonumber
\end{align}
With the change of variable $\omega \rightarrow \omega + {\omega_1}/2$ one obtains readily
\begin{align}
[iL_0](x_1,3;x_{1'},4 &\; | \; \omega_1 ) = e^{ i \omega_1 t^{34} }
\int \frac{ d\omega }{ 2\pi } \; G(x_1,x_3;\omega+ \frac{\omega_1}{2} ) \times \nonumber \\ & \times G(x_4,x_{1'};\omega-\frac{\omega_1}{2} ) \;
\end{equation}
With the change of variable $\omega \\to \omega + {\omega_1}/2$ one obtains readily
\begin{equation}
[L_0](x_1,3;x_{1'},4 \; | \; \omega_1 ) = e^{ i \omega_1 t^{34} }
\int \frac{ d\omega }{ 2i\pi } \; G\qty(x_1,x_3;\omega+ \frac{\omega_1}{2} ) G\qty(x_4,x_{1'};\omega-\frac{\omega_1}{2} ) \;
e^{ i \omega \tau_{34} }
\end{align}
\end{equation}
with $\tau_{34} = ( t_3 - t_4 )$ and $t^{34}= (t_3+t_4)/2$.
Using now the Lehman representation of the Green's functions (Eq.~\ref{eq:G-Lehman}), and picking up the poles associated with the occupied (virtual) states in the upper (lower) half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains using the residue theorem (with $\tau = \tau_{34})$
\begin{align*}
\int \frac{ d \omega }{2i\pi} & \; G(x_1,x_3; \omega + \homu ) G(x_4,x_{1'}; \omega - \homu ) e^{ i \omega \tau } = \\
& \theta( \tau ) \sum_{nj} \frac{ \phi_n(x_1) \phi_n^*(x_3) \phi_j(x_4) \phi_j^*(x_{1'}) } { \varepsilon_j + \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_j + \homu ) \tau } \\
+ & \theta( \tau_ ) \sum_{jn} \frac{ \phi_j(x_1) \phi_j^*(x_3) \phi_n(x_4) \phi_n^*(x_{1'}) } { \varepsilon_j - \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_j - \homu ) \tau } \\
- & \theta(- \tau ) \sum_{nb} \frac{ \phi_n(x_1) \phi_n^*(x_3) \phi_b(x_4) \phi_b^*(x_{1'}) } { \varepsilon_b + \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_b + \homu ) \tau } \\
- & \theta(- \tau ) \sum_{bn} \frac{ \phi_b(x_1) \phi_b^*(x_3) \phi_n(x_4) \phi_n^*(x_{1'}) } { \varepsilon_b - \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_b - \homu ) \tau }
\end{align*}
Projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$ selects the first and fourth lines of the right-hand side, leading to Eq.~\ref{eq:iL0bis}
\begin{equation}
\begin{split}
\int \frac{ d \omega }{2i\pi} \; G\qty(x_1,x_3; \omega + \homu ) G\qty(x_4,x_{1'}; \omega - \homu ) e^{ i \omega \tau }
& = \sum_{bj}
\frac{ \phi_b(x_1) \phi_b^*(x_3) \phi_j(x_4) \phi_j^*(x_{1'})} { \omega_1 - (\varepsilon_b - \varepsilon_j) + i\eta }
\qty[ \theta(\tau) e^{i ( \varepsilon_j + \homu ) \tau } + \theta(-\tau) e^{i ( \varepsilon_b - \homu ) \tau } ]
\\
& - \sum_{bj} \frac{ \phi_j(x_1) \phi_j^*(x_3) \phi_b(x_4) \phi_b^*(x_{1'})} { \omega_1 + (\varepsilon_b - \varepsilon_j ) -i\eta }
\qty[ \theta(\tau) e^{i ( \varepsilon_j - \homu ) \tau } + \theta(-\tau) e^{i ( \varepsilon_b + \homu ) \tau } ]
\\
& + \sum_{ab} \text{ pp terms } + \sum_{ij} \text{ hh terms }
\end{split}
\end{equation}
where (pp) and (hh) labels particle-particle and hole-hole channels neglected here.
Projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$ selects the first line of the RHS, leading to Eq.~\ref{eq:iL0bis}
with $ (\omega_1 \rightarrow \Omega_s )$.
\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ in the electron/hole product basis }
We now derive in some more details Eq.~\ref{eq:spectral65}.
Starting with:
\begin{align*}
\begin{equation}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
& = \theta(\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5) }{N,s} \\
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(5) \hpsi(6) }{N,s}
\end{align*}
we use the relation between operators in their Eisenberg and Schr\"{o}dinger representations (Eq.~\ref{Eisenberg}) to obtain:
\begin{align*}
\langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle = \\
& + \theta(\tau_{65}) \mel{N}{ \hpsi(x_6) e^{-i{\hat H} \tau_{65}} \hpsi^{\dagger}(x_5) }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }\\
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(x_5) e^{ i{\hat H} \tau_{65}} \hpsi(x_6) }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 }
\end{align*}
= \theta(\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5) }{N,s}
- \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(5) \hpsi(6) }{N,s}
\end{equation}
we use the relation between operators in their HeEisenberg and Schr\"{o}dinger representations (Eq.~\ref{Eisenberg}) to obtain:
\begin{equation}
\langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle = \\
+ \theta(\tau_{65}) \mel{N}{ \hpsi(x_6) e^{-i{\hat H} \tau_{65}} \hpsi^{\dagger}(x_5) }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }
- \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(x_5) e^{ i{\hat H} \tau_{65}} \hpsi(x_6) }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 }
\end{equation}
with $E^N_0$ the N-electron ground-state energy and $E^N_s$ the enrgy of the s-th excited state $| N,s \rangle$. Expanding now the field operators with creation/destruction operators in the MO basis
\begin{align*}
\hpsi(x_6) = \sum_p \phi_p(x_6) {\hat a}_p \;\;\; \text{and} \;\;\;
\hpsi^{\dagger}(x_5) = \sum_q \phi_q^{*}(x_5)
\hpsi(x_6) & = \sum_p \phi_p(x_6) {\hat a}_p
& \;\;\; \text{and} \;\;\;
\hpsi^{\dagger}(x_5) & = \sum_q \phi_q^{*}(x_5)
{\hat a}^{\dagger}_q
\end{align*}
one obtains
\begin{align*}
\langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle =
\sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5) \; \times\\
& \big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p e^{-i{\hat H} \tau_{65}} {\hat a}^{\dagger}_q }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 } \\
& - \theta(-\tau_{65}) \mel{N}{ {\hat a}^{\dagger}_q e^{ i{\hat H} \tau_{65}} {\hat a}_p }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 } \; \big]
\end{align*}
We now act on the N-electron ground-state with
\begin{equation}
\langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle =
\sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5) \;
\big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p e^{-i{\hat H} \tau_{65}} {\hat a}^{\dagger}_q }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 } \\
- \theta(-\tau_{65}) \mel{N}{ {\hat a}^{\dagger}_q e^{ i{\hat H} \tau_{65}} {\hat a}_p }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 } \; \big]
\end{equation}
We now act on the $N$-electron ground-state with
\begin{align*}
e^{i{\hat H} \tau_{65} } {\hat a}^{\dagger}_p | N \rangle &=
e^{i ( E^N_0 + \varepsilon_p ) \tau_{65} } | N \rangle \\
e^{i ( E^N_0 + \varepsilon_p ) \tau_{65} } | N \rangle &\\
e^{ -i{\hat H} \tau_{65} } {\hat a}_q | N \rangle &=
e^{-i ( E^N_0 - \varepsilon_q ) \tau_{65} } | N \rangle
\end{align*}
where $\lbrace \varepsilon_{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. Taking the associated bras that we plug into the MOs product basis expansion of $\langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle $ one obtains Eq.~\ref{eq:spectral65}. \\
where $\lbrace \varepsilon_{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. Taking the associated bras that we plug into the MOs product basis expansion of $\langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle $ one obtains:
\begin{equation}
\langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle =
\sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5) \;
\big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p {\hat a}^{\dagger}_q }{N,s} e^{ -i \varepsilon_p \tau_{65} } e^{ - i \Omega_s t_5 }
- \theta(-\tau_{65}) \mel{N}{ {\hat a}^{\dagger}_q {\hat a}_p }{N,s} e^{ -i \varepsilon_q \tau_{65} } e^{ - i \Omega_s t_6 } \; \big]
\end{equation}
leading to Eq.~\ref{eq:spectral65} with $\Omega_s = (E^N_s - E^N_0)$, $t_6 = \tau_{65}/2 + t^{65}$ and $t_5 = - \tau_{65}/2 + t^{65}$. \\
\end{widetext}
\bibliography{BSEdyn}

11
Notes/BSEdyn-notes.tex
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@ -41,13 +41,13 @@ In particular, using a simple two-level model, we analyze, for each kernel, the
\section{Linear response theory}
\label{sec:LR}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Linear response theory 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}
Linear response theory 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{Oddershede_1977,Casida_1995,Petersilka_1996}
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^{\sigma}(\omega_s) & \bC^{\sigma}(\omega_s)
\\
-\bC^{\sigma}(-\omega_s) & -\bR^{\sigma}(-\omega_s)
-\bC^{\sigma}(-\omega_s)^* & -\bR^{\sigma}(-\omega_s)^*
\end{pmatrix}
\cdot
\begin{pmatrix}
@ -64,8 +64,8 @@ From a practical point of view, these quantities are obtained by solving non-lin
\end{pmatrix}
\end{equation}
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 $\omega \to -\omega$.
Neglecting the coupling block [\ie, $\bC^{\sigma}(\omega) = 0$] between the resonant and anti-resonants parts, $\bR^{\sigma}(\omega)$ and $-\bR^{\sigma}(-\omega)^*$, is known as the Tamm-Dancoff approximation (TDA).
In the absence of symmetry breaking, \cite{Dreuw_2005} 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.
In the one-electron basis of (real) spatial orbitals $\lbrace \MO{p} \rbrace$, we will assume that the elements of the matrices defined in Eq.~\eqref{eq:LR} have the following generic forms: \cite{Dreuw_2005}
@ -434,6 +434,7 @@ As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appears d
Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} 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.
\titou{T2: add comment on how one can detect fake solutions?}
%%% TABLE I %%%
\begin{table*}
@ -723,7 +724,7 @@ What have we learned 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.
He also thanks Xavier Blase for numerous insightful discussions on dynamical kernels.}
He also thanks Xavier Blase and Juliette Authier for numerous insightful discussions on dynamical kernels.}
%%%%%%%%%%%%%%%%%%%%%%%%
% BIBLIOGRAPHY