Done with theory and history for now

Manuscript/BSE_JPCL.tex

@@ -235,8 +235,8 @@ The present \textit{Perspective} aims at describing the current status and upcom
 \section{Theory}
 %%%%%%%%%%%%%%%%%%%%%%
 
-The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of  Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} 
+The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of  Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques , originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018} in quantum chemistry. \cite{Dreuw_2015} 
-While the density stands as the basic variable in density-functional theory DFT, \cite{Hohenberg_1964,Kohn_1965,ParrBook} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
+While the one-body density stands as the basic variable in density-functional theory DFT, \cite{Hohenberg_1964,Kohn_1965,ParrBook} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
 \begin{equation}
     G(\bx t,\bx't') = -i \mel{N}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{N},
 \end{equation}
@@ -262,13 +262,13 @@ Using the equation-of-motion formalism for the creation/destruction operators, i
     = \delta(1,2),
 \end{equation}
 where we introduce the usual composite index, \eg, $1 \equiv (\bx_1,t_1)$. 
-Here, $h$ is the \titou{one-body Hartree Hamiltonian} and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
+Here, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
-Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one obtains the familiar eigenvalue equation, \ie,
+Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one gets the familiar eigenvalue equation, \ie,
 \begin{equation}
 	h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
 \end{equation}
-which resembles formally the KS equation with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.  
+which resembles formally the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.  
-The knowledge of $\Sigma$ allows to obtain the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.  
+The knowledge of $\Sigma$ allows to access the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.  
 \titou{[INTRODUCE QUASIPARTICLES and OTHER solutions ??]}
 \titou{The spin variable has disappear. How do we deal with this?}
 \\
@@ -281,7 +281,7 @@ While the equations reported above are formally exact, it remains to provide an
 This is where Green's function practical theories differ. 
 Developed by Lars Hedin in 1965 with application to the interacting homogeneous electron gas, \cite{Hedin_1965} the $GW$ approximation
 \cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Golze_2019rev} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}). 
-The obtained equation, when compared with the equation for the time-evolution of $G$ [see Eq.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy
+The resulting equation, when compared with the equation for the time-evolution of $G$ [see Eq.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy
 \begin{equation}\label{eq:Sig}
     \Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3),
 \end{equation}
@@ -298,30 +298,45 @@ that can be regarded as the lowest-order perturbation in terms of the screened C
 \end{gather}
 where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulomb potential. 
 
-\titou{In practice, the input $G$ and $\chi_0$ needed to build $\Sigma^{\GW}$ are taken to be the ``best'' Green's function and susceptibility that can be easily calculated, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.}
+%%% FIG 1 %%%
+\begin{figure}
+	\includegraphics[width=0.7\linewidth]{fig1/fig1}
+	\caption{
+	Hedin's pentagon: its connects the Green's function $G$, its non-interacting analog $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb potential $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965}
+	The path made of back arrow shows the $GW$ process which bypasses the computation of $\Gamma$ (gray arrows).
+	As input, one must provide KS (or HF) orbitals and their corresponding senergies.
+	Depending on the level of self-consistency of the $GW$ calculation, only the orbital energies or both the orbitals and their energies are corrected.
+	As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$ which can then be used to compute the BSE neutral excitations.
+	\label{fig:pentagon}}
+\end{figure} 
+%%% %%% %%%
+
+In practice, the input $G$ and $\chi_0$ required to initially build $\Sigma^{\GW}$ are taken as the ``best'' Green's function and susceptibility that can be easily computed, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.
 Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained as follows:
 \begin{equation}
     \varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
     \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\GW}) - V^{\XC}}{\phi_p^{\KS}}.
 \end{equation}
-\titou{T2: Shall we introduce the renormalization factor?}
+\titou{T2: Shall we introduce the renormalization factor or its non-linear version?
+Here, I would prefer to introduce the frequency-dependent quasiparticle equation and talks about quasiparticle and satellites, and the way of solving this equation in practice (linearization, etc).}
 Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique. 
 This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988} 
 surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
 In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the LDA KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV [REFS] with an $\mathcal{O}(N^4)$ computational scaling (see below). 
 
-Further improvements may be obtained via partial self-consistency.
+Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature.
-There exist two main types of partially self-consistent $GW$ methods: 
+Further improvements may be obtained via self-consistency of the Hedin's equations (see Fig.~\ref{fig:pentagon}).
+There exists two main types of self-consistent $GW$ methods: 
 i) \textit{``eigenvalue-only quasiparticle''} $GW$ (ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011} 
-where the quasiparticle (QP) energies are updated at each iteration, and 
+where the quasiparticle energies are updated at each iteration, and 
 ii) \textit{``quasiparticle self-consistent''} $GW$ (qs$GW$), \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} 
-where one updates both the QP energies and the corresponding orbitals.
+where one updates both the quasiparticle energies and the corresponding orbitals.
 Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
 
 However, self-consistency does not always improve things, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
 Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006, Stan_2009, Rostgaard_2010, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Wilhelm_2018}
 In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.
-A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998, Holm_1999,Holm_2000, Garcia-Gonzalez_2001} 
+A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998, Holm_1999,Holm_2000,Garcia-Gonzalez_2001} 
 This was further evidenced in real extended systems by several authors. \cite{Schone_1998, Ku_2002, Kutepov_2016, Kutepov_2017}
 However, other approximations may have caused such deterioration, \eg, pseudo-potentials \cite{deGroot_1995} or finite-basis set effects. \cite{Friedrich_2006}
 These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
@@ -330,21 +345,11 @@ For finite systems such as atoms and molecules, the situation is less controvers
 Another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite size and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999} 
 However, remaining a low order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995}\\
 
-%%% FIG 1 %%%
-\begin{figure}
-	\includegraphics[width=0.7\linewidth]{fig1/fig1}
-	\caption{
-	Hedin's pentagon: its connects the Green's function $G$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965}
-	The path made of back arrow shows the $GW$ process which bypasses the computation of $\Gamma$.
-	\label{fig:pentagon}}
-\end{figure} 
-%%% %%% %%%
-
 %===================================
 \subsection{Neutral excitations}
 %===================================
 
-While TD-DFT starts with the  variation of the charge density $\rho$ with respect to an external local perturbation \titou{$U(1)$}, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation \titou{U($1,2$)}: 
+While TD-DFT starts with the  variation of the charge density $\rho$ with respect to an external local perturbation \titou{$U(1)$}, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation \titou{$U(1,2)$}: 
 \begin{equation}
     \chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
    \quad  \rightarrow \quad
@@ -432,17 +437,6 @@ As compared to TD-DFT,
 \titou{T2: would it be useful to say that there is 100\% exact exchange in BSE@$GW$?}   
 We emphasise that these equations can be solved at exactly the same cost as the standard TD-DFT equations once the quasiparticle energies and screened Coulomb potential $W$ are inherited from preceding $GW$ calculations. This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, emphasizing its pros and cons.  \\
 
-%===================================
-%\subsection{Practical considerations}
-%===================================
-
-%From a practical point of view, it is important to understand that, to compute the BSE neutral excitations, one must perform, beforehand, several calculations.
-%First, a KS-DFT (or HF) calculation has to be performed in order to get orbitals and their corresponding energies.
-%Then, these quantities are used as input variables for the $GW$ calculation, whose main purpose is to correct these quantities.
-%Depending on the level of self-consistency, only the orbital energies or both the orbitals and their energies are corrected.
-%The $GW$ calculation being performed, the BSE excitation energies can then be computed using as input the $GW$ quasiparticle energies and the screened Coulomb operator $W$. 
-%\\
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Historical overview}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -487,11 +481,11 @@ Such an underestimation of the fundamental gap leads to a similar underestimatio
 \begin{equation}
 	\EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB,
 \end{equation}
-where $\EB$ is the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind.
+where $\EB$ is the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind (see Fig.~\ref{fig:gaps}).
 Because of this, we have $\EgOpt < \EgFun$.
 
-Such a residual HOMO-LUMO gap problem can be significantly improved by adopting xc functionals with a tuned amount of exact exchange \cite{Stein_2009,Kronik_2012} that yield a much improved KS gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016} 
+Such a residual gap problem can be significantly improved by adopting xc functionals with a tuned amount of exact exchange \cite{Stein_2009,Kronik_2012} that yield a much improved KS gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016} 
-Alternatively, self-consistent schemes, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011} where corrected eigenvalues, and possibly orbitals, \cite{Faleev_2004, vanSchilfgaarde_2006, Kotani_2007, Ke_2011} are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rangel_2016,Kaplan_2016,Caruso_2016} 
+Alternatively, self-consistent schemes such as ev$GW$ and qs$GW$, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011} where corrected eigenvalues, and possibly orbitals, are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rangel_2016,Kaplan_2016,Caruso_2016} 
 As a result, BSE excitation singlet energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations.
 For sake of illustration,  an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering more than hundred representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017} 
 This is equivalent to the best TD-DFT results obtained by scanning a large variety of global hybrid functionals with various amounts of exact exchange. 
@@ -579,7 +573,7 @@ However, we also observe that, in some cases, unphysical irregularities on the g
 \subsection{Unphysical discontinuities} 
 %==========================================
 The GW approximation of many-body perturbation theory has been highly successful at predicting the electronic properties of solids and molecules. \cite{Onida_2002, Aryasetiawan_1998, Reining_2017}
-However, it is also known to be inadequate to model strongly correlated systems. \cite{Romaniello_2009, Romaniello_2012, DiSabatino_2015, DiSabatino_2016, Tarantino_2017}
+However, it is also known to be inadequate to model strongly correlated systems. \cite{Romaniello_2009a,Romaniello_2012,DiSabatino_2015,DiSabatino_2016,Tarantino_2017,DiSabatino_2019}
 Here, we have found severe shortcomings of two widely-used variants of $GW$ in the weakly correlated regime.
 
 We report unphysical irregularities and discontinuities in some key experimentally-measurable quantities computed within the $GW$ approximation 
@@ -603,7 +597,7 @@ In a recent article, \cite{Loos_2018} while studying a model two-electron system
 \subsection{The double excitation challenge}
 %==========================================
 As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
-Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009,Sangalli_2011,Zhang_2013,ReiningBook}
+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}
 To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
 In most cases, this can be done by solving a set of linear equations of the form
 \begin{equation}
@@ -652,7 +646,7 @@ with
 	\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_2009,Sangalli_2011,ReiningBook}
+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.
@@ -668,7 +662,7 @@ In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its
 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.\\
 
-Beyond the static approximation \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009,Romaniello_2009,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
+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}
 
 %==========================================
 \subsection{Core-level spectroscopy}

Manuscript/xbbib.bib

@@ -1,13 +1,252 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-05-15 23:27:42 +0200 
+%% Created for Pierre-Francois Loos at 2020-05-16 09:03:07 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
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+	Author = {S. {Di Sabatino} and J. A. Berger and P. Romaniello},
+	Date-Added = {2020-05-16 09:03:04 +0200},
+	Date-Modified = {2020-05-16 09:03:04 +0200},
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+	Year = {2019},
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+	Date-Added = {2020-05-16 08:58:04 +0200},
+	Date-Modified = {2020-05-16 08:58:04 +0200},
+	Doi = {10.1103/PhysRevB.85.155131},
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+	Date-Modified = {2020-05-16 08:57:51 +0200},
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+	Author = {Di Sabatino,S. and Berger,J. A. and Reining,L. and Romaniello,P.},
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+	Date-Added = {2020-05-16 08:57:34 +0200},
+	Date-Modified = {2020-05-16 08:57:34 +0200},
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+@article{Tarantino_2017,
+	Author = {Tarantino, Walter and Romaniello, Pina and Berger, J. A. and Reining, Lucia},
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+	Date-Modified = {2020-05-16 08:57:25 +0200},
+	Doi = {10.1103/PhysRevB.96.045124},
+	File = {/Users/loos/Zotero/storage/PE5YGU7F/Romaniello_2017.pdf},
+	Issn = {2469-9950, 2469-9969},
+	Journal = {Phys. Rev. B},
+	Language = {en},
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+	Pages = {045124},
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+@article{Tarantino_2018,
+	Author = {Walter Tarantino and Bernardo S Mendoza and Pina Romaniello and J A Berger and Lucia Reining},
+	Date-Added = {2020-05-16 08:57:25 +0200},
+	Date-Modified = {2020-05-16 08:57:25 +0200},
+	Journal = {J. Phys.: Condensed Matter},
+	Number = {13},
+	Pages = {135602},
+	Title = {Many-body perturbation theory and non-perturbative approaches: screened interaction as the key ingredient},
+	Volume = {30},
+	Year = {2018}}
+
+@book{Schirmer_2018,
+	Author = {Jochen Schirmer},
+	Date-Added = {2020-05-16 08:22:42 +0200},
+	Date-Modified = {2020-05-16 08:22:58 +0200},
+	Publisher = {Springer},
+	Title = {Many-Body Methods for Atoms, Molecules and Clusters},
+	Year = {2018}}
+
+@article{Schirmer_2004d,
+	Author = {Schirmer, J. and Trofimov, A. B.},
+	Date-Added = {2020-05-16 08:22:21 +0200},
+	Date-Modified = {2020-05-16 08:22:42 +0200},
+	Doi = {http://dx.doi.org/10.1063/1.1752875},
+	Journal = {J. Chem. Phys.},
+	Pages = {11449-11464},
+	Title = {Intermediate State Representation Approach to Physical Properties of Electronically Excited Molecules},
+	Url = {http://scitation.aip.org/content/aip/journal/jcp/120/24/10.1063/1.1752875},
+	Volume = {120},
+	Year = {2004},
+	Bdsk-Url-1 = {http://scitation.aip.org/content/aip/journal/jcp/120/24/10.1063/1.1752875},
+	Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.1752875}}
+
+@article{Trofimov_1997b,
+	Abstract = {The electronic spectrum of furan is investigated theoretically beyond the previous vertical-electronic description. A polarization propagator method referred to as second-order algebraic-diagrammatic construction (ADC(2)) has been used in the electronic structure calculations. The vibrational excitation accompanying the electronic transitions is described with the aid of a linear electron-vibrational coupling model. The spectral information thereby obtained permits extensive comparison with experiment. The average accuracy of the present method, estimated by comparing adiabatic transition energies, is better than 0.4 eV. Only for the lowest π-π∗ valance transition, V′(1A1) and V′(1B2), and for the Rydberg excitations agree The results for the other π-π∗ valence transitions, V(1B2), and for the Rydberg excitations agree well with findings of previous experimental and theoretical work. A (multistate) vibronic coupling effect involving the V′(1A1) and V(1B2) valence transitions and the 3s(1A2 Rydberg excitation is suggested as the reason for the highly diffuse character of the 5.7--6.7 eV photoabsorption band.},
+	Author = {A.B. Trofimov and J. Schirmer},
+	Date-Added = {2020-05-16 08:21:53 +0200},
+	Date-Modified = {2020-05-16 08:22:02 +0200},
+	Doi = {https://doi.org/10.1016/S0301-0104(97)00256-5},
+	Issn = {0301-0104},
+	Journal = {Chem. Phys.},
+	Number = {2},
+	Pages = {175--190},
+	Title = {Polarization Propagator Study of Electronic Excitation in key Heterocyclic Molecules II. Furan},
+	Url = {http://www.sciencedirect.com/science/article/pii/S0301010497002565},
+	Volume = {224},
+	Year = {1997},
+	Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0301010497002565},
+	Bdsk-Url-2 = {https://doi.org/10.1016/S0301-0104(97)00256-5}}
+
+@article{Trofimov_1997,
+	Abstract = {The electronic excitation spectrum of pyrrole is studied using a polarization propagator method referred to as the second-order algebraic-diagrammatic construction (ADC(2)), along with a simple model for vibrational excitation accounting for all totally symmetric modes. The method describes the optical absorption profile of pyrrole with an expected accuracy of 0.2 -- 0.4 eV for the vertical excitation energies. The vibrational analysis provides for detailed additional spectroscopic information. In the singlet spectrum, besides the ns, np and nd (n = 3,4) Rydberg excitations, three π-π∗ valence transitions, V′(1A1), V(1B2) and V(1A1) can clearly be distinguished. No evidence is found for Rydberg-valence interaction near the equilibrium geometry. Substantial vibrational widths and distinct vibrational excitation patterns are predicted for the Rydberg series converging to the first and second ionization thresholds. Some new assignments of major spectral features are proposed. The long-wave absorption maximum in the 5.6 -- 6.6. eV region is explained exclusively by the presence of Rydberg transitions, while the most intense absorption in the short-wave band system (7.0 -- 8.3 ev) predominantly originates from the V(1B2) and V(1A1) valence transitions.},
+	Author = {A.B. Trofimov and J. Schirmer},
+	Date-Added = {2020-05-16 08:21:31 +0200},
+	Date-Modified = {2020-05-16 08:21:46 +0200},
+	Doi = {https://doi.org/10.1016/S0301-0104(96)00303-5},
+	Issn = {0301-0104},
+	Journal = {Chem. Phys.},
+	Number = {2},
+	Pages = {153--170},
+	Title = {Polarization Propagator Study of Electronic Excitation in key Heterocyclic Molecules I. Pyrrole},
+	Url = {http://www.sciencedirect.com/science/article/pii/S0301010496003035},
+	Volume = {214},
+	Year = {1997},
+	Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0301010496003035},
+	Bdsk-Url-2 = {https://doi.org/10.1016/S0301-0104(96)00303-5}}
+
+@article{Barth_1995,
+	Abstract = {A recently derived approximation scheme for the polarisation propagator has been applied in a study of discrete K-shell excitations in N2 and CO. The new scheme referred to as second-order algebraic diagrammatic construction (ADC(2)) provides a direct approach to excitation energies and transition moments and gives a consistent second-order and first-order treatment for transitions to singly and doubly excited states, respectively. The essential computational requisite is a Hermitean eigenvalue problem in the space of single and double excitations on the Hartree-Fock ground state. Spin-free decoupled ADC(2) working equations for the singlet-singlet and singlet-triplet transitions have been formulated and employed. As the only additional approximation, the mixing between configurations with a different number of excited core-level electrons has been neglected. The calculated excitation energies of both the core-valence and core-Rydberg transitions are in very good agreement with the experimental data and are distinctly improved with respect to previous theoretical work, including extended configuration interaction treatments. The authors emphasise the accuracy achieved for the oscillator strengths which yield a very satisfactory description for the intensity ratios of the dipole-allowed transitions. The absolute dipole oscillator strengths are in excellent accord with the experimental values of Kay et al. (1977).},
+	Author = {A. Barth and J. Schirmer},
+	Date-Added = {2020-05-16 08:20:57 +0200},
+	Date-Modified = {2020-05-16 08:21:13 +0200},
+	Doi = {10.1088/0022-3700/18/5/008},
+	Journal = {J. Phys. B: At. Mol. Phys.},
+	Month = {mar},
+	Number = {5},
+	Pages = {867--885},
+	Publisher = {{IOP} Publishing},
+	Title = {Theoretical Core-level Excitation Spectra of N$_2$ and CO by a new Polarisation Propagator Method},
+	Url = {https://doi.org/10.1088%2F0022-3700%2F18%2F5%2F008},
+	Volume = {18},
+	Year = 1995,
+	Bdsk-Url-1 = {https://doi.org/10.1088%2F0022-3700%2F18%2F5%2F008},
+	Bdsk-Url-2 = {https://doi.org/10.1088/0022-3700/18/5/008}}
+
+@article{Schirmer_1991,
+	Author = {Jochen Schirmer},
+	Date-Added = {2020-05-16 08:20:31 +0200},
+	Date-Modified = {2020-05-16 08:20:38 +0200},
+	Doi = {10.1103/PhysRevA.43.4647},
+	Journal = {Phys. Rev. A.},
+	Pages = {4647--4659},
+	Title = {Closed-Form Intermediate Representations of Many-Body Propagators and Resolvent Matrices},
+	Volume = {43},
+	Year = {1991},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.43.4647}}
+
+@article{Schirmer_1982,
+	Author = {Jochen Schirmer},
+	Date-Added = {2020-05-16 08:19:54 +0200},
+	Date-Modified = {2020-05-16 08:20:00 +0200},
+	Doi = {10.1103/PhysRevA.26.2395},
+	Journal = PRA,
+	Pages = {2395--2416},
+	Title = {Beyond the Random-Phase Approximation: a new Approximation Scheme for the Polarization Propagator},
+	Volume = 26,
+	Year = 1982,
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.26.2395}}
+
 @article{Duchemin_2020,
 	Author = {Ivan Duchemin and Xavier Blase},
 	Date-Added = {2020-05-15 23:27:41 +0200},
@@ -2420,8 +2659,9 @@
 	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.88.195152},
 	Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.88.195152}}
 
-@article{Romaniello_2009,
+@article{Romaniello_2009b,
 	Author = {Romaniello,P. and Sangalli,D. and Berger,J. A. and Sottile,F. and Molinari,L. G. and Reining,L. and Onida,G.},
+	Date-Modified = {2020-05-16 08:57:57 +0200},
 	Doi = {10.1063/1.3065669},
 	Eprint = {https://doi.org/10.1063/1.3065669},
 	Journal = {J. Chem. Phys.},