From c5814e73bc93cd0d7d9820265576fb3334716f93 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 29 Jun 2020 11:53:29 +0200 Subject: [PATCH] modif xavier --- BSEdyn.bib | 19 +++++-- BSEdyn.tex | 113 ++++++++++++++++++++++++----------------- Notes/BSEdyn-notes.tex | 11 ++-- 3 files changed, 89 insertions(+), 54 deletions(-) diff --git a/BSEdyn.bib b/BSEdyn.bib index 0ff0482..4698db5 100644 --- a/BSEdyn.bib +++ b/BSEdyn.bib @@ -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}, diff --git a/BSEdyn.tex b/BSEdyn.tex index ce04dbd..09be96b 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -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. @@ -1091,66 +1095,83 @@ $(t_{1'} = t_1^{+})$ 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} \end{document} diff --git a/Notes/BSEdyn-notes.tex b/Notes/BSEdyn-notes.tex index 4cefb1a..10972e0 100644 --- a/Notes/BSEdyn-notes.tex +++ b/Notes/BSEdyn-notes.tex @@ -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