successes

Manuscript/BSE_JPCL.tex

@@ -305,7 +305,7 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
 \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}
+	Hedin's pentagon 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.
@@ -400,7 +400,7 @@ Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that w
 \end{equation}
 where it is  traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. \cite{Hanke_1980,Strinati_1982,Strinati_1984}
 Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied orbitals and $(a,b)$ are unoccupied orbitals), leads to an eigenvalue problem similar to the so-called Casida equations in TD-DFT:
-\begin{equation}
+\begin{equation} \label{eq:BSE-eigen}
     \begin{pmatrix}
 		R & C  
 		\\
@@ -531,25 +531,29 @@ The success of the BSE formalism to treat CT excitations has been demonstrated i
 Recent attempts to merge the $GW$ and BSE formalisms with model polarizable environments at the PCM or QM/MM levels
 \cite{Baumeier_2014,Duchemin_2016,Li_2016,Varsano_2016,Duchemin_2018,Li_2019,Tirimbo_2020} paved the way not only to interesting applications but also to a better understanding of the merits of these approaches relying on the use of the  screened Coulomb potential designed to capture  polarization effects at all spatial ranges. As a matter of fact,
 dressing the bare Coulomb potential with the reaction field matrix 
-$$
-v({\bf r},{\bf r}') \longrightarrow v({\bf r},{\bf r}') + v^{\text{reac}}({\bf r},{\bf r}'; \omega)
-$$
-in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility [Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc.) with the same complexity as in the gas phase. The reaction field matrix $v^{\text{reac}}({\bf r},{\bf r}'; \omega)$ describes the potential generated in ${\bf r}'$ by the charge rearrangements in the polarizable environment induced by a source charge located in ${\bf r}$, with $\bf r$ and ${\bf r}'$ in the quantum mechanical (QM) subsystem of interest. The reaction field is dynamical since the dielectric properties of the environment, such as the macroscopic dielectric constant $\epsilon_M(\omega)$, are in principle frequency dependent.   Once the reaction field matrix known, with  typically   $\mathcal{O}(N^3)$ operations, the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
+\begin{equation}
+	v(\br,\br') \longrightarrow v(\br,\br') + v^{\text{reac}}(\br,\br'; \omega)
+\end{equation}
+\titou{in the relation between} the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc) with the same complexity as in the gas phase. 
+The reaction field matrix $v^{\text{reac}}(\br,\br'; \omega)$ describes the potential generated in $\br'$ by the charge rearrangements in the polarizable environment induced by a source charge located in $\br$, where $\br$ and $\br'$ lie in the quantum mechanical subsystem of interest. 
+The reaction field is dynamical since the dielectric properties of the environment, such as the macroscopic dielectric constant $\epsilon_M(\omega)$, are in principle frequency dependent.   
+Once the reaction field matrix is known, with typically $\order*{N^3}$ operations (where $\Norb$ is the number of orbitals), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
 
-A remarkable property \cite{Duchemin_2018} of the BSE formalism combined with a polarizable environment is that the scheme described here above, with electron-electron and electron-hole interactions renormalized by the reaction field, allows to capture both linear-response (LR)  and state-specific (SS) contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing Frenkel and CT excitations. This is an important advantage as compared e.g. to TD-DFT calculations where LR and SS effects have to be explored with different formalisms. 
+A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field allows to capture both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing Frenkel and CT excitations. 
+This is an important advantage as compared to, \eg, TD-DFT where linear-response and state-specific effects have to be explored with different formalisms. 
 
-To date, the effect of the environment on fast electronic excitations is only included by considering the low-frequency optical response of the polarizable medium (e.g. considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant for water in the optical range), neglecting the variations with frequency of the dielectric constant in the optical range. Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, namely in situations where neither the adiabatic nor antiadiabatic limits are expected to be valid (for a recent discussion, see Ref.
-~\citenum{Huu_2020}). \\
+To date, environmental effects on fast electronic excitations are only included by considering the low-frequency optical response of the polarizable medium (\eg,  considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant for water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range. 
+Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, in situations where neither the adiabatic nor antiadiabatic limits are expected to be valid (for a recent discussion, see Ref.
+~\citenum{Huu_2020}).
 
-
-
- We now leave the description of successes to discuss difficulties and Perspectives.\\
+We now leave the description of successes to discuss difficulties and future directions of developments and improvements.
+\\
 
 %==========================================
 \subsection{The computational challenge} 
 %==========================================
-As emphasized above, the BSE eigenvalue equation in the occupied-to-virtual product space is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
-Searching iteratively for the lowest eigenstates presents the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT (where $\Norb$ is the number of orbitals). 
+As emphasized above, the BSE eigenvalue equation in the single-excitation space [see Eq.~\eqref{eq:BSE-eigen}] is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
+Searching iteratively for the lowest eigenstates exhibits the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT. 
 Concerning the construction of the BSE Hamiltonian, it is no more expensive than building the TD-DFT one with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques. 
 At the price of sacrificing the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\order*{\Norb^3}$ operations using iterative techniques. \cite{Ljungberg_2015} 
 With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015}
@@ -726,12 +730,12 @@ In these two latter studies, they also followed a (non-self-consistent) perturba
 %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}
 %==========================================
-XANES, \cite{Olovsson_2009,Vinson_2011}
+%XANES, \cite{Olovsson_2009,Vinson_2011}
 
 
-diabatization and conical intersections \cite{Kaczmarski_2010}
+%diabatization and conical intersections \cite{Kaczmarski_2010}