final corrections

dynker.tex

@@ -28,8 +28,9 @@ We discuss the physical properties and accuracy of three distinct dynamical (\ie
 i) an \textit{a priori} built kernel inspired by the dressed time-dependent density-functional theory (TDDFT) kernel proposed by Maitra and coworkers [\href{https://doi.org/10.1063/1.1651060}{J.~Chem.~Phys.~120, 5932 (2004)}], 
 ii) the dynamical kernel stemming from the Bethe-Salpeter equation (BSE) formalism derived originally by Strinati [\href{https://doi.org/10.1007/BF02725962}{Riv.~Nuovo Cimento 11, 1--86 (1988)}], and
 iii) the second-order BSE kernel derived by Yang and coworkers [\href{https://doi.org/10.1063/1.4824907}{J.~Chem.~Phys.~139, 154109 (2013)}].
-In particular, using a simple two-level model, we analyze, for each kernel, the appearance of spurious excitations, as first evidenced by Romaniello and collaborators [\href{https://doi.org/10.1063/1.3065669}{J.~Chem.~Phys.~130, 044108 (2009)}], due to the approximate nature of the kernels.
-Prototypical examples of valence, charge-transfer, and Rydberg excited states are considered.
+The principal take-home message of the present paper is that dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated to higher-order excitations (such as the infamous double excitations), an unappreciated feature of dynamical quantities.
+We also analyze, for each kernel, the appearance of spurious excitations originating from the approximate nature of the kernels, as first evidenced by Romaniello \textit{et al.}~[\href{https://doi.org/10.1063/1.3065669}{J.~Chem.~Phys.~130, 044108 (2009)}].
+Using a simple two-level model, prototypical examples of valence, charge-transfer, and Rydberg excited states are considered.
 %\\
 %\bigskip
 %\begin{center}
@@ -509,8 +510,6 @@ This can be further verify by switching off gradually the electron-electron inte
 
 Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant parts of the dBSE Hamiltonian \eqref{eq:HBSE} does not change the situation in terms of spurious solutions: there is still one spurious excitation in the triplet manifold ($\omega_{2}^{\BSE,\upup}$), and the two solutions for the singlet manifold which corresponds to the single and double excitations.
 However, it does increase significantly the static excitations while the magnitude of the dynamical corrections is not altered by the TDA.
-%Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
-%The spin blindness of the dBSE kernel is probably to blame for the existence of this spurious triplet excitation.
 
 Another way to access dynamical effects while staying in the static framework is to use perturbation theory, \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e} a scheme we label as perturbative BSE (pBSE).
 To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
@@ -706,60 +705,11 @@ Again, the case of \ce{H2} is a bit peculiar as the perturbative treatment (pBSE
 Although frequency-independent, this additional term makes the singlet and triplet excitation energies very accurate.
 However, one cannot access the double excitation.
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%\subsection{The forgotten kernel: Sangalli's kernel}
-%\label{sec:Sangalli}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%\titou{This section is experimental...}
-%In Ref.~\onlinecite{Sangalli_2011}, Sangalli proposed a dynamical kernel (based on the second RPA) without (he claims) spurious excitations thanks to the design of a number-conserving approach which correctly describes particle indistinguishability and Pauli exclusion principle.
-%We will first start by writing down explicitly this kernel as it is given in obscure physicist notations in the original article.
-%
-%The Hamiltonian with Sangalli's kernel is (I think)
-%\begin{equation}
-%	\bH_\text{S}^{\sigma}(\omega) = 
-%	\begin{pmatrix}
-%		\bR_\text{S}^{\sigma}(\omega)	&	\bC_\text{S}^{\sigma}(\omega)
-%		\\
-%		-\bC_\text{S}^{\sigma}(-\omega)	&	-\bR_\text{S}^{\sigma}(-\omega)
-%	\end{pmatrix}
-%\end{equation}
-%with
-%\begin{subequations}
-%\begin{gather}
-%	R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
-%	\\
-%	C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
-%\end{gather}
-%\end{subequations}
-%and
-%\begin{subequations}
-%\begin{gather}
-%	f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
-%	\\
-%	c_{ia,mn}^{\sigma} = \sum_{jb,kc} \qty{ \qty[ \ERI{ij}{kc} \delta_{ab} + \ERI{kc}{ab} \delta_{ij} ] \qty[ R_{m,jc} R_{n,kb}
-%	+ R_{m,kb} R_{n,jc} ] }
-%\end{gather}
-%\end{subequations}
-%where $R_{m,ia}$ are the elements of the RPA eigenvectors.
-%
-%For the two-level model, Sangalli's kernel reads
-%\begin{align}
-%	R(\omega) & =  \Delta\eGW{} + f_R (\omega)
-%	\\
-%	C(\omega) & = f_C (\omega)
-%\end{align}
-%
-%\begin{gather}
-%	f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
-%	\\
-%	f_C (\omega) = 0
-%\end{gather}
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Take-home messages}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The take-home message of the present paper is that dynamical kernels have much more to give that one would think.
-In more scientific terms, dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated to higher-order excitations (such as the infamous double excitations).
+In more scientific terms, dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated to higher-order excitations (such as the infamous double excitations), an unappreciated feature of dynamical quantities.
 However, they sometimes give too much, and generate spurious excitations, \ie, excitation which does not corresponds to any physical excited state.
 The appearance of these factitious excitations is due to the approximate nature of the dynamical kernel.
 Moreover, because of the non-linear character of the linear response problem when one employs a dynamical kernel, it is computationally more involved to access these extra excitations.
@@ -773,9 +723,11 @@ We hope that the present contribution will foster new developments around dynami
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
 We would like to thank Xavier Blase, Elisa Rebolini, Pina Romaniello, Arjan Berger, Miquel Huix-Rotllant and Julien Toulouse for insightful discussions on dynamical kernels. 
-PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.%%%%%%%%%%%%%%%%%%%%%%%%
+PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
+%%%%%%%%%%%%%%%%%%%%%%%%
 
-% BIBLIOGRAPHY
+%%%%%%%%%%%%%%%%%%%%%%%%
 \bibliography{dynker}
+%%%%%%%%%%%%%%%%%%%%%%%%
 
 \end{document}