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22
BSEdyn.bib
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@ -12783,21 +12783,6 @@
Year = {2016},
Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4940139}}
@article{Boulanger_2014,
Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
Doi = {10.1021/ct401101u},
File = {/Users/loos/Zotero/storage/KTW3SS9F/Boulanger_2014.pdf},
Issn = {1549-9618, 1549-9626},
Journal = {J. Chem. Theory Comput.},
Language = {en},
Month = mar,
Number = {3},
Pages = {1212--1218},
Title = {Fast and {{Accurate Electronic Excitations}} in {{Cyanines}} with the {{Many}}-{{Body Bethe}}\textendash{}{{Salpeter Approach}}},
Volume = {10},
Year = {2014},
Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct401101u}}
@article{Bruneval_2009,
Author = {Bruneval, Fabien},
Doi = {10.1103/PhysRevLett.103.176403},
@ -14429,16 +14414,13 @@
@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},
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},
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}
}

104
BSEdyn.tex
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@ -207,9 +207,8 @@
\begin{abstract}
Similar to the ubiquitous adiabatic approximation in time-dependent density-functional theory, the static approximation, which substitutes a dynamical (\ie, frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the Bethe-Salpeter equation (BSE) formalism.
Here, going beyond the static approximation, we compute the dynamical correction in the electron-hole screening for molecular excitation energies thanks to a renormalized first-order perturbative correction to the static BSE excitation energies.
The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random phase approximation. \xavier{
\sout{Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approximation by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.}
Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to assess the clear improvements induced by dynamical corrections. }
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}
@ -1078,6 +1077,7 @@ 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}
@ -1095,28 +1095,32 @@ $(t_{1'} = t_1^{+})$
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}
[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)
\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}
[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} ) \;
\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 }
= \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*}
\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 )$.
@ -1125,47 +1129,47 @@ with $ (\omega_1 \rightarrow \Omega_s )$.
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 Heisenberg 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)
{\hat a}^{\dagger}_q
\hpsi(x_6) & = \sum_p \phi_p(x_6) {\hat a}_p
&
\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:
\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*}
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 $t_5 = - \tau_{65}/2 + t^{65}$. \\
\center{ \rule{3cm}{1} }
\end{widetext}
\bibliography{BSEdyn}