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Manuscript/BSE_JPCL.tex
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\begin{abstract}
The many-body Green's function Bethe-Salpeter equation (BSE) 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, able 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.
With a similar computational cost than time-dependent density-functional theory (TD-DFT), BSE@$GW$ is then able to provide an accuracy on par with the most accurate global hybrid functionals without the unsettling choice of the exchange-correlation functional.
With a similar computational cost than time-dependent density-functional theory (TD-DFT), the BSE formalism is then able to provide an accuracy on par with the most accurate global \xavier{and range-separated} hybrid functionals without the unsettling choice of the exchange-correlation functional, \xavier{resolving further know issues (e.g. charge-transfer excitations) and offering a well-defined path to dynamical kernels.}
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.
Future directions of developments and improvements are also discussed.
@ -363,27 +363,27 @@ However, remaining a low order perturbative approach starting with single-determ
\subsection{Neutral excitations}
%===================================
While TD-DFT starts with the variation of the charge density $\rho(1)$ with respect to an external local perturbation $U(1)$, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the two-body Green's function $G(1,2)$ with respect to a non-local external perturbation $U(1,2)$:
While TD-DFT starts with the variation of the charge density $\rho(1)$ with respect to an external local perturbation $U(1)$, the BSE formalism considers a generalized 4-points susceptibility, or two-particle correlation function, that monitors the variation of the one-body Green's function $G(1,1')$ with respect to a non-local external perturbation $U(2,2')$:
\begin{equation}
\chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
\quad \rightarrow \quad
L(12,34) \stackrel{\BSE}{=} -i \pdv{G(1,2)}{U(3,4)},
L(1, 2;1',2' ) \stackrel{\BSE}{=} \pdv{G(1,1')}{U(2',2)},
\end{equation}
with the formal relations $\chi(1,2) = L(11,22)$ and $\rho(1) = -iG(1,1^{+})$ as a bridge between the TD-DFT and BSE worlds.
where we follow the notations by Strinati.\cite{Strinati_1988} The formal relation $\chi(1,2) = -i L(1,2;1^+,2^+)$ with $\rho(1) = -iG(1,1^{+})$ offers a direct bridge between the TD-DFT and the BSE worlds.
The equation of motion for $G$ [see Eq.~\eqref{eq:Gmotion}] can be reformulated in the form of a Dyson equation
\begin{equation}
G = G_0 + G_0 \Sigma G,
G = G_0 + G_0 ( v_H + U + \Sigma ) G,
\end{equation}
that relates the full (interacting) Green's function, $G$, to its Hartree version, $G_0$, obtained by replacing the $\lbrace \varepsilon_p, f_p \rbrace$ by the Hartree eigenvalues and eigenfunctions.
The derivative with respect to $U$ of the Dyson equation yields
that relates the full (interacting) Green's function, $G$, to its non-interacting version, $G_0$, with $v_H$ and $U$ the Hartree and external potential, respectively, and $\Sigma$ the xc self-energy.
The derivative with respect to $U$ of this Dyson equation yields
\begin{multline}\label{eq:DysonL}
L(12,34) = L_0(12,34)
L(1,2;1',2') = L_0(1,2;1',2') +
\\
\int d5678 \, L_0(12,56) \Xi^{\BSE}(5,6,7,8) L(78,34),
\int d3456 \, L_0(1,4;1',3) \Xi^{\BSE}(3,5;4,6) L(6,2;5,2'),
\end{multline}
where $L_0 = \partial G_0 / \partial U$ is the Hartree 4-point susceptibility and
where $L_0(1,2;1',2') = G(1,2')G(2,1')$ is the non-interacting 4-point susceptibility and
\begin{equation}
\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) + \pdv{\Sigma(5,6)}{G(7,8)}
i\Xi^{\BSE}(3,5;4,6) = v(3,6) \delta(34) \delta(56) + i \pdv{\Sigma(3,4)}{G(6,5)}
\end{equation}
is the so-called BSE kernel.
This equation can be compared to its TD-DFT analog
@ -397,7 +397,7 @@ where
is the TD-DFT kernel.
Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel
\begin{equation}\label{eq:BSEkernel}
\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) -W(5,6) \delta(57) \delta(68 ),
i \Xi^{\BSE}(3,5;4,6) = v(3,6) \delta(34) \delta(56) -W(3^+,4) \delta(36) \delta(45 ),
\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: