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BSEdyn.bib
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@ -14426,3 +14426,19 @@
Year = {2016},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.235113},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}}
@article{Boulanger_2014,
author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe–Salpeter Approach},
journal = {J. Chem. Theory Comput. },
volume = {10},
number = {3},
pages = {1212-1218},
year = {2014},
doi = {10.1021/ct401101u},
note ={PMID: 26580191},
URL = { https://doi.org/10.1021/ct401101u},
eprint = { https://doi.org/10.1021/ct401101u}
}

44
BSEdyn.tex
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@ -225,7 +225,7 @@ 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}
@ -404,7 +404,7 @@ 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}
@ -415,7 +415,7 @@ 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.
%================================
@ -523,6 +523,8 @@ 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 $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$.
@ -1082,10 +1084,10 @@ The data that support the findings of this study are available within the articl
\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.
@ -1094,26 +1096,29 @@ $(t_{1'} = t_1^{+})$
\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)
[L_0](x_1,3;x_{1'},4 & \;| \; \omega_1 ) =
\int \frac{ d\omega }{ 2i\pi } \; G(x_1,x_3;\omega) \; \times \\ & \times \; 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} ) \;
[L_0](x_1,3;x_{1'},4 &\; | \; \omega_1 ) = e^{ i \omega_1 t^{34} }
\int \frac{ d\omega }{ 2i\pi } \; G(x_1,x_3;\omega+ \frac{\omega_1}{2} ) \times \nonumber \\ & \times G(x_4,x_{1'};\omega-\frac{\omega_1}{2} ) \;
e^{ i \omega \tau_{34} }
\end{align}
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 }
\int & \frac{ d \omega }{2i\pi} \; G(x_1,x_3; \omega + \homu ) G(x_4,x_{1'}; \omega - \homu ) e^{ i \omega \tau }
= \sum_{bj} \Bigg\{ \\
& \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 }
\left[ \theta(\tau) e^{i ( \varepsilon_j + \homu ) \tau } + \theta(-\tau) e^{i ( \varepsilon_b - \homu ) \tau } \right] \\
- & \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 }
\left[ \theta(\tau) e^{i ( \varepsilon_j - \homu ) \tau } + \theta(-\tau) e^{i ( \varepsilon_b + \homu ) \tau } \right] \\
& \Bigg\} + \sum_{ab} \text{ pp terms } + \sum_{ij} \text{ hh terms }
\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}
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 }
@ -1151,7 +1156,14 @@ We now act on the N-electron ground-state with
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{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 {\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{align*}
leading to Eq.~\ref{eq:spectral65} with $\Omega_s = (E^N_s - E^N_0)$, $t_6 = \tau_{65}/2 + t^{65}$ and $t_5 = - \tau_{65}/2 + t^{65}$. \\
\center{ \rule{3cm}{1} }