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Manuscript/BSE_JPCL.bib
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@ -1,13 +1,78 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-05-18 21:44:02 +0200 %% Created for Pierre-Francois Loos at 2020-05-25 11:57:31 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@phdthesis{Rebolini_PhD,
Author = {E. Rebolini},
Date-Added = {2020-05-25 11:57:30 +0200},
Date-Modified = {2020-05-25 11:57:30 +0200},
School = {Universit{\'e} Pierre et Marie Curie --- Paris VI},
Title = {Range-Separated Density-Functional Theory for Molecular Excitation Energies},
Url = {https://tel.archives-ouvertes.fr/tel-01027522},
Year = {2014},
Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
@article{Baumeier_2012b,
Author = {Baumeier, Bj\"{o}rn and Andrienko, Denis and Ma, Yuchen and Rohlfing, Michael},
Date-Added = {2020-05-25 11:56:57 +0200},
Date-Modified = {2020-05-25 11:56:57 +0200},
Doi = {10.1021/ct2008999},
Journal = {J. Chem. Theory Comput.},
Pages = {997--1002},
Title = {Excited States of Dicyanovinyl-Substituted Oligothiophenes from Many-Body Green's Functions Theory},
Volume = {8},
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1021/ct300311x}}
@article{Ankudinov_2003,
Author = {A. L. Ankudinov and A. I. Nesvizhskii and J. J. Rehr},
Date-Added = {2020-05-25 11:42:58 +0200},
Date-Modified = {2020-05-25 11:43:50 +0200},
Doi = {10.1103/PhysRevB.67.115120},
Journal = {Phys. Rev. B},
Pages = {115120},
Title = {Dynamic screening effects in x-ray absorption spectra},
Volume = {67},
Year = {2003},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.67.115120}}
@article{Ma_2009a,
Author = {Ma, Yuchen and Rohlfing, Michael and Molteni, Carla},
Date-Added = {2020-05-25 08:51:27 +0200},
Date-Modified = {2020-05-25 08:51:27 +0200},
Doi = {10.1103/PhysRevB.80.241405},
Issue = {24},
Journal = {Phys. Rev. B},
Month = {Dec},
Numpages = {4},
Pages = {241405},
Publisher = {American Physical Society},
Title = {Excited states of biological chromophores studied using many-body perturbation theory: Effects of resonant-antiresonant coupling and dynamical screening},
Url = {https://link.aps.org/doi/10.1103/PhysRevB.80.241405},
Volume = {80},
Year = {2009},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.80.241405},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.80.241405}}
@article{Ma_2009b,
Author = {Ma, Yuchen and Rohlfing, Michael and Molteni, Carla},
Date-Added = {2020-05-25 08:51:27 +0200},
Date-Modified = {2020-05-25 08:51:27 +0200},
Doi = {10.1021/ct900528h},
Journal = {J. Chem. Theory. Comput.},
Pages = {257--265},
Title = {Modeling the Excited States of Biological Chromophores within Many-Body Green's Function Theory},
Volume = {6},
Year = {2009},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.80.241405},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.80.241405}}
@article{Albrecht_1997, @article{Albrecht_1997,
Author = {Albrecht, Stefan and Onida, Giovanni and Reining, Lucia}, Author = {Albrecht, Stefan and Onida, Giovanni and Reining, Lucia},
Date-Added = {2020-05-18 21:40:28 +0200}, Date-Added = {2020-05-18 21:40:28 +0200},
@ -1557,24 +1622,6 @@
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S002199911730671X}, Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S002199911730671X},
Bdsk-Url-2 = {https://doi.org/10.1016/j.jcp.2017.09.012}} Bdsk-Url-2 = {https://doi.org/10.1016/j.jcp.2017.09.012}}
@article{Ma_2009,
Author = {Ma, Yuchen and Rohlfing, Michael and Molteni, Carla},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Doi = {10.1103/PhysRevB.80.241405},
Issue = {24},
Journal = {Phys. Rev. B},
Month = {Dec},
Numpages = {4},
Pages = {241405},
Publisher = {American Physical Society},
Title = {Excited states of biological chromophores studied using many-body perturbation theory: Effects of resonant-antiresonant coupling and dynamical screening},
Url = {https://link.aps.org/doi/10.1103/PhysRevB.80.241405},
Volume = {80},
Year = {2009},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.80.241405},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.80.241405}}
@article{Martin_1959, @article{Martin_1959,
Author = {Martin, Paul C. and Schwinger, Julian}, Author = {Martin, Paul C. and Schwinger, Julian},
Date-Added = {2020-05-18 21:40:28 +0200}, Date-Added = {2020-05-18 21:40:28 +0200},
@ -14282,9 +14329,9 @@
Year = {2011}, Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3655352}} Bdsk-Url-1 = {https://doi.org/10.1063/1.3655352}}
@article{Baumeier_2012, @article{Baumeier_2012a,
Author = {Baumeier, Bj\"{o}rn and Andrienko, Denis and Rohlfing, Michael}, Author = {Baumeier, Bj\"{o}rn and Andrienko, Denis and Rohlfing, Michael},
Date-Modified = {2020-02-05 20:52:41 +0100}, Date-Modified = {2020-05-25 11:57:00 +0200},
Doi = {10.1021/ct300311x}, Doi = {10.1021/ct300311x},
Journal = {J. Chem. Theory Comput.}, Journal = {J. Chem. Theory Comput.},
Number = {8}, Number = {8},

152
Manuscript/BSE_JPCL.tex
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@ -506,7 +506,7 @@ Similar difficulties emerge in solid-state physics for semiconductors where exte
These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals. These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals.
It can be cured through an exact exchange contribution, a solution that explains in particular the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012} It can be cured through an exact exchange contribution, a solution that explains in particular the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012}
The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where screening reduces the long-range electron-hole interactions. The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where screening reduces the long-range electron-hole interactions.
The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011,Baumeier_2012,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems.\\ The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems.\\
%========================================== %==========================================
\subsection{Solvent effects} \subsection{Solvent effects}
@ -606,75 +606,95 @@ This shortcoming has been thoroughly described in several previous studies.\cite
%In a recent article, \cite{Loos_2018} while studying a model two-electron system, we have observed that, within partially self-consistent $GW$ (such as ev$GW$ and qs$GW$), one can observe, in the weakly correlated regime, (unphysical) discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, HOMO-LUMO gap, total and correlation energies, as well as vertical excitation energies). %In a recent article, \cite{Loos_2018} while studying a model two-electron system, we have observed that, within partially self-consistent $GW$ (such as ev$GW$ and qs$GW$), one can observe, in the weakly correlated regime, (unphysical) discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, HOMO-LUMO gap, total and correlation energies, as well as vertical excitation energies).
%========================================== %==========================================
\subsection{The double excitation challenge} \subsection{Beyond the static approximation}
%========================================== %==========================================
As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness. Going beyond the static approximation is a difficult challenge which has been, nonetheless, embraced by several groups around the world.\cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009b,Sangalli_2011,Zhang_2013,ReiningBook} As mentioned earlier in this \textit{Perspective}, most of BSE calculations are performed within the so-called static approximation, which substitutes the dynamically-screened (\ie, frequency-dependent) Coulomb potential $W(\omega)$ by its static limit $W(\omega = 0)$.
To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system. It is important to mention that diagonalizing the BSE Hamiltonian in the static approximation corresponds to solving a \textit{linear} eigenvalue problem in the space of single excitations, while it is, in its dynamical form, a non-linear eigenvalue problem (in the same space) which is much harder to solve from a numerical point of view.
In most cases, this can be done by solving a set of linear equations of the form In complete analogy with the ubiquitous adiabatic approximation in TD-DFT, one key consequence of the static approximation is that double (and higher) excitations are completely absent from the BSE spectrum, which obviously hampers the applicability of BSE as double excitation may play, indirectly, a key role in photochemistry mechanisms.
\begin{equation} Higher excitations would be explicitly present in the BSE Hamiltonian by ``unfolding'' the dynamical BSE kernel, and one would recover a linear eigenvalue problem with, nonetheless, a much larger dimension.
\label{eq:lin_sys} Corrections to take into account the dynamical nature of the screening may or may not recover these multiple excitations.
\bA \bx = \omega \bx, However, dynamical corrections permit, in any case, to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations (and, in particular, non-interacting double excitations).
\end{equation}
where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
If we assume that the operator $\bA$ has a matrix representation of size $K \times K$, this \textit{linear} set of equations yields $K$ excitation energies.
However, in practice, $K$ might be very large, and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
\begin{equation}
\label{eq:lin_sys_split}
\begin{pmatrix}
\bA_1 & \tr{\bb} \\
\bb & \bA_2 \\
\end{pmatrix}
\begin{pmatrix}
\bx_1 \\
\bx_2 \\
\end{pmatrix}
= \omega
\begin{pmatrix}
\bx_1 \\
\bx_2 \\
\end{pmatrix},
\end{equation}
where the blocks $\bA_1$ and $\bA_2$, of sizes $K_1 \times K_1$ and $K_2 \times K_2$ (with $K_1 + K_2 = K$), can be associated with, for example, the single and double excitations of the system.
Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
Solving separately each row of the system \eqref{eq:lin_sys_split} yields From a more practical point of view, dynamical effects have been found to affect the positions and widths of core-exciton resonances in semiconductors, \cite{Strinati_1982,Strinati_1984} rare gas solids, and transition metals. \cite{Ankudinov_2003}
\begin{subequations} Thanks to first-order perturbation theory, Rohlfing and coworkers have developed an efficient way of taking into account the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
\begin{gather} With such as 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}
\label{eq:row1} Studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that \textit{``the influence of dynamical screening on the excitation energies is about $0.1$ eV for the lowest $\pi \ra \pi^*$ transitions, but for the lowest $n \ra \pi^*$ transitions the influence is larger, up to $0.25$ eV.''} \cite{Ma_2009b}
\bA_1 \bx_1 + \tr{\bb} \bx_2 = \omega \bx_1, Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Salpeter kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016,Rebolini_PhD}
In these two latter studies, they also followed a (non-self-consistent) perturbative approach within the Tamm-Dancoff approximation with a renormalization of the first-order perturbative correction.
%Finally, let us also mentioned the work of Romaniello and coworkers, \cite{Romaniello_2009b,Sangalli_2011} in which the authors genuinely accessed additional excitations by solving the non-linear, frequency-dependent eigenvalue problem.
%However, it is based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations.
\\ \\
\label{eq:row2}
\bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1.
\end{gather}
\end{subequations}
Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
\begin{equation}
\label{eq:non_lin_sys}
\Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1
\end{equation}
with
\begin{equation}
\Tilde{\bA}_1(\omega) = \bA_1 + \tr{\bb} (\omega \bI - \bA_2)^{-1} \bb
\end{equation}
which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{Romaniello_2009b,Sangalli_2011,ReiningBook}
How have we been able to reduce the dimension of the problem while keeping the same number of solutions? %==========================================
To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent. %\subsection{The double excitation challenge}
In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension. %==========================================
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Gatti_2007,Garniron_2018} %As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension. %Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009b,Sangalli_2011,Zhang_2013,ReiningBook}
However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analogue given by Eq.~\eqref{eq:lin_sys}. %To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently. %In most cases, this can be done by solving a set of linear equations of the form
%\begin{equation}
One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value. % \label{eq:lin_sys}
For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$. % \bA \bx = \omega \bx,
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature. %\end{equation}
This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$. %where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.\\ %If we assume that the operator $\bA$ has a matrix representation of size $K \times K$, this \textit{linear} set of equations yields $K$ excitation energies.
%However, in practice, $K$ might be very large, and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
Beyond the static approximation \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019} %\begin{equation}
% \label{eq:lin_sys_split}
% \begin{pmatrix}
% \bA_1 & \tr{\bb} \\
% \bb & \bA_2 \\
% \end{pmatrix}
% \begin{pmatrix}
% \bx_1 \\
% \bx_2 \\
% \end{pmatrix}
% = \omega
% \begin{pmatrix}
% \bx_1 \\
% \bx_2 \\
% \end{pmatrix},
%\end{equation}
%where the blocks $\bA_1$ and $\bA_2$, of sizes $K_1 \times K_1$ and $K_2 \times K_2$ (with $K_1 + K_2 = K$), can be associated with, for example, the single and double excitations of the system.
%Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
%
%Solving separately each row of the system \eqref{eq:lin_sys_split} yields
%\begin{subequations}
%\begin{gather}
% \label{eq:row1}
% \bA_1 \bx_1 + \tr{\bb} \bx_2 = \omega \bx_1,
% \\
% \label{eq:row2}
% \bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1.
%\end{gather}
%\end{subequations}
%Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
%\begin{equation}
% \label{eq:non_lin_sys}
% \Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1
%\end{equation}
%with
%\begin{equation}
% \Tilde{\bA}_1(\omega) = \bA_1 + \tr{\bb} (\omega \bI - \bA_2)^{-1} \bb
%\end{equation}
%which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
%For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{Romaniello_2009b,Sangalli_2011,ReiningBook}
%
%How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
%To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
%In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension.
%This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Gatti_2007,Garniron_2018}
%Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension.
%However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analogue given by Eq.~\eqref{eq:lin_sys}.
%Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.
%
%One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value.
%For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
%In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
%This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$.
%Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.\\
%========================================== %==========================================
\subsection{Core-level spectroscopy} \subsection{Core-level spectroscopy}