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BSEdyn.bib
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@ -11,16 +11,12 @@
Date-Added = {2020-07-24 13:26:39 +0200},
Date-Modified = {2020-07-24 13:26:46 +0200},
Doi = {10.1021/acs.jpclett.8b02058},
Eprint = {https://doi.org/10.1021/acs.jpclett.8b02058},
Journal = {J. Phys. Chem. Lett.},
Number = {16},
Pages = {4646--4651},
Title = {Theoretical 0--0 Energies with Chemical Accuracy},
Url = {https://doi.org/10.1021/acs.jpclett.8b02058},
Volume = {9},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.8b02058}}
Year = {2018}}
@article{Loos_2020g,
Author = {P. F. Loos and D. Jacquemin},
Date-Added = {2020-07-22 22:42:52 +0200},

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BSEdyn.tex
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@ -206,7 +206,7 @@
\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.
Here, going beyond the static approximation, we compute the dynamical correction of 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 the dynamical correction.
%\\
@ -226,7 +226,7 @@ Our calculations are benchmarked against high-level (coupled-cluster) calculatio
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 (or optical) 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 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).
Qualitatively, 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 (\ie, the electron-hole binding energy) $\EB$ to the $GW$ HOMO-LUMO gap
Qualitatively, 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 (\ie, the electron-hole binding energy $\EB$) to the $GW$ HOMO-LUMO gap
\begin{equation}
\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \eps_{\HOMO}^{\GW},
\end{equation}
@ -242,7 +242,7 @@ where
\begin{equation} \label{eq:Egfun}
\EgFun = I^N - A^N
\end{equation}
is the the fundamental gap, \cite{Bredas_2014} $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N} - E_0^{N+1}$ being the ionization potential and the electron affinity of the $N$-electron system, respectively.
is the fundamental gap, $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N} - E_0^{N+1}$ being the ionization potential and the electron affinity of the $N$-electron system, respectively.
Here, $E_S^{N}$ is the total energy of the $S$th excited state of the $N$-electron system, and $E_0^N$ corresponds to its ground-state energy.
Because the excitonic effect corresponds physically to the stabilization implied by the attraction of the excited electron and its hole left behind, we have $\EgOpt < \EgFun$.
Due to the smaller amount of screening in molecules as compared to solids, a faithful description of excitonic effects is paramount in molecular systems.
@ -254,9 +254,9 @@ Although these double excitations are usually experimentally dark (which means t
They are particularly important in the faithful description of the ground state of open-shell molecules, \cite{Casida_2005,Romaniello_2009a,Huix-Rotllant_2011,Loos_2020c}
and they are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
Double excitations play also a significant role in the correct location of the excited states of polyenes that are closely related to rhodopsin, a biological pigment found in the rods of the retina and involved in the visual transduction. \cite{Olivucci_2010,Robb_2007,Manathunga_2016}
In butadiene, for example, while the bright $1 ^1B_u$ state has a clear $\HOMO \ra \LUMO$ single-excitation character, the dark $2 ^1A_g$ state includes a substantial fraction of doubly-excited character from the $\HOMO^2 \ra \LUMO^2$ double excitation (roughly $30\%$), yet dominant contributions from the $\HOMO-1 \ra \LUMO$ and $\HOMO \ra \LUMO+1$ single excitations. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
In butadiene, for example, while the bright $1 ^1B_u$ state has a clear $\HOMO \ra \LUMO$ single-excitation character, the dark $2 ^1A_g$ state includes a substantial fraction of doubly-excited character from the $\HOMO^2 \ra \LUMO^2$ double excitation (roughly $30\%$), yet with dominant contributions from the $\HOMO-1 \ra \LUMO$ and $\HOMO \ra \LUMO+1$ single excitations. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
Going beyond the static approximation is tricky and very few groups have dared to take the plunge. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
Going beyond the static approximation is difficult and very few groups have been addressing the problem. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
Nonetheless, it is worth mentioning the seminal work of Strinati on core excitons in semiconductors, \cite{Strinati_1982,Strinati_1984,Strinati_1988} in which the dynamical screening effects were taken into account through the dielectric matrix, and where he observed an increase of the binding energy over its value for static screening and a narrowing of the Auger width below its value for a core hole.
Following Strinati's footsteps, Rohlfing and coworkers have developed an efficient way of taking into account, thanks to first-order perturbation theory, the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation (TDA). \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
With such a scheme, they have been able to compute the excited states of biological chromophores, showing that taking into account the electron-hole dynamical screening is important for an accurate description of the lowest $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b}
@ -448,9 +448,9 @@ with $X_{jb,S} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,S}$ and $Y_{jb,S} = \mel{N}{\h
Neglecting the anti-resonant terms, $Y_{jb,S}$, in the dynamical BSE, which are (usually) much smaller than their resonant counterparts, $X_{jb,S}$, leads to the well-known TDA.
In Eq.~\eqref{eq:BSE-final},
\begin{equation}
\ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'),
\ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}(\br) \MO{q}(\br) v(\br -\br') \MO{r}(\br') \MO{s}(\br'),
\end{equation}
are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and
are the bare two-electron integrals in the (real-valued) spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and
\begin{multline} \label{eq:wtilde}
\widetilde{W}_{pq,rs}(\Om{S}{})
= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{pq,rs}(\omega)
@ -460,7 +460,7 @@ are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}
is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{S} = \Om{S}{} - ( \eGW{q} - \eGW{p} )$ and
\begin{equation}
W_{pq,rs}({\omega})
= \iint d\br d\br' \, \MO{p}(\br) \MO{q}^*(\br) W(\br ,\br'; \omega) \MO{r}^*(\br') \MO{s}(\br').
= \iint d\br d\br' \, \MO{p}(\br) \MO{q}(\br) W(\br ,\br'; \omega) \MO{r}(\br') \MO{s}(\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 $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$. } }
@ -1141,7 +1141,7 @@ The small correction on the $^1A_g$ state might be explained by its rather diffu
\section{Conclusion}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
The BSE formalism is quickly gaining momentum in the electronic structure community thanks to its attractive computational scaling with system size and its overall accuracy for modeling single excitations of various natures.
The BSE formalism is quickly gaining momentum in the electronic structure community thanks to its attractive computational scaling with system size and its overall accuracy for modeling single excitations of various natures in large molecular systems.
It now stands as a genuine cost-effective excited-state method and is regarded as a valuable alternative to the popular TD-DFT method.
However, the vast majority of the BSE calculations are performed within the static approximation in which, in complete analogy with the ubiquitous adiabatic approximation in TD-DFT, the dynamical BSE kernel is replaced by its static limit.
One key consequence of this static approximation is the absence of higher excitations from the BSE optical spectrum.
@ -1183,7 +1183,7 @@ Combining the Fourier transform (with respect to $t_1$) of $L_0(1,4;1',3)$
[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 )
= -i \int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1'),
\end{align}
(where $t_{1'} = t_1^{+}$) with the Fourier expansion of the Green's function, \eg,
(where $t_{1'} = t_1^{+}$) with the \trashPFL{Fourier expansion} \titou{inverse Fourier transform} of the Green's function, \eg,
\begin{align}
G(1,3) = \int \frac{ d\omega }{ 2\pi } G(\bx_1,\bx_3;\omega) e^{-i \omega \tau_{13} },
\end{align}
@ -1239,9 +1239,9 @@ we employ the relationship between operators in their Heisenberg and Schr\"{o}di
\begin{equation}
\begin{split}
& \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} = \\
& + \theta(\tau_{65}) \mel{N}{ \hpsi(\bx_6) e^{-i\hH \tau_{65}} \hpsi^{\dagger}(\bx_5) }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 }
& + \theta(+\tau_{65}) \mel{N}{ \hpsi(\bx_6) e^{-i\hH \tau_{65}} \hpsi^{\dagger}(\bx_5) }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 }
\\
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(\bx_5) e^{ i\hH \tau_{65}} \hpsi(\bx_6) }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 }.
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(\bx_5) e^{+ i\hH \tau_{65}} \hpsi(\bx_6) }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 }.
\end{split}
\end{equation}
%with $E^N_0$ the $N$-electron ground-state energy and $E^N_S$ the energy of the $S$th excited state $\ket{N,S}$.
@ -1259,11 +1259,11 @@ one gets
= \sum_{pq} \phi_p(\bx_6) \phi_q^{*}(\bx_5)
[ & \theta(+\tau_{65}) \mel{N}{ \ha_p e^{-i \hH \tau_{65}} \ha^{\dagger}_q }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 }
\\
- & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } ].
- & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q e^{+ i \hH \tau_{65}} \ha_p }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } ].
\end{split}
\end{equation}
%We now act on the $N$-electron ground-state wave function with
\titou{Assuming now that the $\e{a}$'s and $\e{i}$'s are proper addition and removal energies (respectively)}, such as the $GW$ quasiparticle energies, one can use the following relations
\titou{Assuming now that the $\e{a}$'s and $\e{i}$'s are proper addition and removal energies (respectively)}, such as the $GW$ quasiparticle energies, one can use the following relationships
\begin{subequations}
\begin{align}
e^{+i\hH \tau_{65} } \ha^{\dagger}_p \ket{N} &=