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Manuscript/BSE_JPCL.tex
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@ -201,7 +201,8 @@
\begin{abstract}
The many-body Green's function Bethe-Salpeter formalism is steadily asserting itself as a new efficient and accurate tool in the armada of computational methods available to chemists in order to predict neutral electronic excitations in molecular systems.
In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable charged excitations and quasiparticle energies, and the Bethe-Salpeter formalism, allowing to catch excitonic effects, has shown to provide accurate excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
In this \textit{Perspective} article, we provide a historical overview of the Bethe-Salpeter formalism, with a particular focus on its condensed-matter roots, and we propose a critical review of its strengths and weaknesses for different chemical situations, such as \titou{bla bla bla}.
In this \textit{Perspective} article, we provide a historical overview of the Bethe-Salpeter formalism, with a particular focus on its condensed-matter roots.
We also propose a critical review of its strengths and weaknesses for different chemical situations, such as \titou{bla bla bla}.
Future directions of developments and improvements are also discussed.
\end{abstract}
@ -264,6 +265,7 @@ Such an underestimation of the fundamental gap leads to a similar underestimatio
\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.
Because of this, we have $\EgOpt < \EgFun$.
\titou{T2: I will include a figure here.}
Such a residual HOMO-LUMO gap problem can be significantly improved by adopting exchange-correlation (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}
@ -271,14 +273,6 @@ As a result, BSE excitation singlet energies starting from such improved quasipa
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.
A very remarkable success of the BSE formalism lies in the description of the charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT calculations adopting standard functionals. \cite{Dreuw_2004}
Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes, cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004}
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}
The analysis of the screened Coulomb potential matrix elements in the BSE kernel (see below) 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_2011b,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.\\
%We now leave the description of successes to discuss difficulties and Perspectives.\\
%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%
@ -387,7 +381,7 @@ where
\end{equation}
is the TD-DFT kernel.
Plugging now the $GW$ self-energy, in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel
\begin{align*}
\begin{align*}\label{eq:BSEkernel}
\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) -W(5,6) \delta(57) \delta(68 ),
\end{align*}
where it is traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$.
@ -441,9 +435,26 @@ As compared to TD-DFT,
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. \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Challenges}
\section{Successes \& Challenges}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%==========================================
\subsection{Charge-transfer excited states}
%==========================================
A very remarkable success of the BSE formalism lies in the description of the charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT calculations adopting standard functionals. \cite{Dreuw_2004}
Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes, cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004}
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}
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_2011b,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.\\
%==========================================
\subsection{Solvent effects}
%==========================================
\titou{T2: introduce discussion about coupling between BSE and solvent models.}
We now leave the description of successes to discuss difficulties and Perspectives.\\
%==========================================
\subsection{The computational challenge}
%==========================================
@ -465,14 +476,14 @@ While TD-DFT with RSH can benefit from tuning the range-separation parameter as
benchmarks \cite{Jacquemin_2017b,Rangel_2017}
a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$. \cite{Holzer_2018b}
a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$. \cite{Holzer_2018b}\\
%==========================================
\subsection{The challenge of analytic gradients}
%==========================================
An additional issue concerns the formalism taken to calculate the ground-state energy for a given atomic configuration. Since the BSE formalism presented so far concerns the calculation of the electronic excitations, namely the difference of energy between the GS and the ES, gradients of the ES absolute energy require
This points to another direction for the BSE foramlism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}
This points to another direction for the BSE foramlism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}\\
%==========================================
\subsection{The double excitation challenge}
@ -541,11 +552,12 @@ One of these approximations is the so-called \textit{static} approximation, whic
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.
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.\\
\noindent {\textbf{Core-level spectroscopy.}}. \\
XANES,
\cite{Olovsson_2009,Vinson_2011}
%==========================================
\subsection{Core-level spectroscopy}
%==========================================
XANES, \cite{Olovsson_2009,Vinson_2011}
diabatization and conical intersections \cite{Kaczmarski_2010}