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@ -187,30 +187,119 @@ In its press release announcing the attribution of the 2013 Nobel prize in Chemi
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The study of neutral electronic excitations in condensed matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the prediction of the colour of precious metals and stones for jewellery, to the understanding e.g. of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. The present Perspective aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism that, while sharing many features with time-dependent density functional theory (TD-DFT), including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known difficulties.
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The Bethe-Salpeter equation formalism belongs to the family of Green's function many-body perturbation theories (MBPT). While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT take the one-body and two-body Green's function as central quantities. The (time-ordered) one-body Green's function reads:
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The Bethe-Salpeter equation formalism belongs to the family of Green's function many-body perturbation theories (MBPT) [REFS] to which belong as well the Algebraic Diagrammatic Construction (ADC) techniques in quantum chemistry. [REFS] While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads:
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\begin{equation}
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G(xt,x't') = -i \langle N | T {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't')} | N \rangle
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G(xt,x't') = -i \langle N | T \left[ {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't') \right] | N \rangle
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\end{equation}
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where $| N \rangle $ is the N-electron ground-state wavefunction. The operators ${\hat \psi}(xt)$ and ${\hat \psi}^{\dagger}(x't')}$ remove/add an electron in space-spin-time positions (xt) and (x't'), while $T$ is the time-ordering operator. For (t>t') the one-body Green's function provides the amplitude of probability of finding
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where $| N \rangle $ is the N-electron ground-state wavefunction. The operators ${\hat \psi}(xt)$ and ${\hat \psi}^{\dagger}(x't')$ remove/add an electron in space-spin-time positions (xt) and (x't'), while $T$ is the time-ordering operator. For (t>t') the one-body Green's function provides the amplitude of probability of finding bla bla \\
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A central property of the one-body Green's function is that its spectral representation presents poles at the charged excitation of the system :
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\noindent{\textbf{Charged excitations.}} A central property of the one-body Green's function is that its spectral representation presents poles at the charged excitation energies of the system :
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\begin{equation}
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G(x,x'; \omega ) = \sum_n \frac{ f_s(x) f^*_s(x') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu})
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G(x,x'; \omega ) = \sum_n \frac{ f_s(x) f^*_s(x') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) } \label{spectralG}
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\end{equation}
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where $\varepsilon_s = E_s(N+1) - E_0(N)$ for $\varepsilon_s > \mu$ ($\mu$ chemical potential, $\eta$ small positive infinitesimal) and $\varepsilon_s = E_0(N) - E_s(N-1)$ for $\varepsilon_s < \mu$. The quantities $E_s(N+1)$ and $E_s(N-1)$ are the total energy of the s-th excited state of the (N+1) and (N-1)-electron systems, while $E_0(N)$ is the N-electron ground-state energy. Contrary to the Kohn-Sham eigenvalues, the Green's function poles $\varepsilon_s$ are the proper charging energies of the N-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbtials. The $\labrace f_s \rbrace$ are called the Lehmann amplitudes.
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where $\varepsilon_s = E_s(N+1) - E_0(N)$ for $\varepsilon_s > \mu$ ($\mu$ chemical potential, $\eta$ small positive infinitesimal) and $\varepsilon_s = E_0(N) - E_s(N-1)$ for $\varepsilon_s < \mu$. The quantities $E_s(N+1)$ and $E_s(N-1)$ are the total energy of the s-th excited state of the (N+1) and (N-1)-electron systems, while $E_0(N)$ is the N-electron ground-state energy. Contrary to the Kohn-Sham eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are thus the proper charging energies of the N-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbitals. The $\lbrace f_s \rbrace$ are called the Lehmann amplitudes [more ??].
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Using the equation of motion of the creation and destruction operators, it can be shown formally that $G$ verifies :
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\begin{equation}
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\left( \frac{\partial }{\partial t_1} - h({\bf r}_1) \right) G(1,2) - \int d3 \; \Sigma(1,3) G(3,2)
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= \delta(1,2) \label{Gmotion}
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\end{equation}
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where $h$ is the one-body Hartree Hamiltonian and $\Sigma$ the so-called exchange-correlation self-energy operator. After Fourier transform and using the spectral representation of $G$, one obtains a familiar eigenvalue equation:
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\begin{equation}
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h({\bf r}) f_s({\bf r}) + \int d{\bf r}' \; \Sigma({\bf r},{\bf r}'; \varepsilon_s ) f_s({\bf r}) = \varepsilon_s f_s({\bf r})
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\end{equation}
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which resembles formally the Kohn-Sham equation with the difference that the self-energy $\Sigma$ is non-local, energy dependent and non-hermitian. The knowledge of the self-energy operators allows thus 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. \\
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\noindent{\textbf{The $GW$ self-energy.}} While the presented equations are formally exact, it remains to provide an expression for the exchange-correlation self-energy operator $\Sigma$. 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 follows the path of linear response by considering the variation of $G$ with respect to an external perturbation. The obtained equation, when compared with the equation for the time-evolution of $G$ (Eqn.~\ref{Gmotion}) leads to a formal expression for the self-energy :
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\begin{equation}
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\Sigma(1,2) = i \int d34 \; G(1,4) W(3,1^{+}) \Gamma(42,3)
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\end{equation}
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where $W$ is the dynamically screened Coulomb potential and $\Gamma$ a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \mathcal{O}(W)$ where $\mathcal{O}(W)$ means a corrective term with leading linear order in terms of $W$. The neglect of the vertex leads to the so-called $GW$ approximation for $\Sigma$ that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with :
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\begin{align}
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W(1,2) &= v(1,2) + \int d34 \; v(1,2) \chi_0(3,4) W(4,2) \\
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\chi_0(1,2) &= -i \int d34 \; G(2,3) G(4,2)
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\end{align}
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with $\chi_0$ the well-known independent electron susceptibility. In practice, $G$ and $\chi_0$ are taken to be the best Green's function and susceptibility that can be easily calculated, namely the DFT ones where the $\lbrace \varepsilon_s, f_s \rbrace$ of equation~\ref{spectralG} are taken to be DFT Kohn-Sham eigenstates. Such an approach, labeled e.g. $GW$@PBE0, if the starting Kohn-Sham eigenstates are generated with the PBE0 functional, bla bla \\
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\noindent {\textbf{Neutral excitations.}} The search for neutral (optical) excitations follow from then a path very similar to TD-DFT. While TD-DFT strives to calculate the variation of the charge density $\rho$ with respect to an external local perturbation, 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::
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\begin{equation}
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\chi(1,2) \stackrel{DFT}{=} \frac{ \partial \rho(1) }{\partial U(2) }
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\;\; \rightarrow \;\;
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L(12,34) \stackrel{BSE}{=} -i \frac{ \partial G(1,2) } { \partial U(3,4) }
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\end{equation}
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%with the relation $\chi(1,2) = L(11,22)$ since $\rho(1) = -iG(1,1^{+})$, as a first bridge between the TD-DFT and BSE worlds.
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The equation of motion for $G$ (Eqn.~\ref{Gmotion}) can be reformulated in the form of a Dyson equation:
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\begin{equation}
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G = G_0 + G_0 \Sigma G
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\end{equation}
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that relates the full (interacting) $G$ to the Hartree $G_0$ that can be obtained by replacing the $\lbrace \varepsilon_s, f_s \rbrace$ by the Hartree eogenvalues and eigenfunctions.
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The derivation of the Dyson equation yields :
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\begin{align}
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L(12,34) &= L^0(12,34) + \\
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& \int d5678 \; L^0(12,34) \Xi(5,6,7,8) L(78,34)
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\label{DysonL}
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\end{align}
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with $L_0 = \partial G_0 / \partial U$ the Hartree propagator and:
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\begin{align*}
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\Xi(5,6,7,8) = v(5,7) \delta(56) \delta(78) + \frac{ \partial \Sigma(5,6) }{ \partial G(7,8) }
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\end{align*}
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with $v$ the bare Coulomb potential. This equation resembles very much that relating the full susceptibility $\chi$ with the independent-electron one $\chi_0$ within TD-DFT, namely :
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\begin{equation}
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\chi(1,2) = \chi_0(1,2) + \int d34 \; \chi_0(1,3) \Xi^{DFT}(3,4) \chi(4,2)
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\end{equation}
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with $\Xi^{DFT}$ the TD-DFT kernel :
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\begin{equation}
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\Xi^{DFT}(3,4) = v(3,4) + \frac{ \partial V^{XC}(3) }{ \partial \rho(4) }
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\end{equation}
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Plugging now the $GW$ self-energy, in a scheme that we we label the BSE/$GW$ approach, leads to a simplfied BSE kernel
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\begin{align*}
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\Xi(5,6,7,8) = v(5,7) \delta(56) \delta(78) -W(5,6) \delta(57) \delta(68 )
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\end{align*}
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where it is traditional to neglect the derivative $\partial W \partial G$ that introduces higher orders in $W$. Taking the static limit $W(\omega=0$ for the screened Coulomb potential, that replaces the static DFT exchange-correlation kernel, and expressing equation~\ref{DysonL} in the standard propduct space $\lbrace \phi_i({\bf r}) \phi_a({\bf r}') \rbrace$ where (i,j, ..) and (a,b,..) indexe occupied and virtual one-body eigenstates, respectively, leads to an eigenvalue problem similar to the so-called Casida's equations in TD-DFT :
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\begin{equation}
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\left(
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\begin{matrix}
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R & C \\
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-C^* & R^{*}
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\end{matrix}
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\right) \cdot
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\left(
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\begin{matrix}
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X^{\lambda} \\
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Y^{\lambda}
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\end{matrix}
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\right) =
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\Omega_{\lambda}
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\left(
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\begin{matrix}
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X^{\lambda} \\
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Y^{\lambda}
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\end{matrix}
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\right)
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\end{equation}
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with electron-hole eigensolutions written:
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\begin{equation}
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\psi_{\lambda}^{eh}(r_e,r_h) = \sum_{ia} \left( X_{ia}^{ \lambda}
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\phi_i({\bf r}) \phi_a({\bf r}') + Y_{ia}^{ \lambda}
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\phi_i({\bf r}') \phi_a({\bf r}) \right)
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\end{equation}
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where $\lambda$ index the electronic excitations. The rsonnant part of the BSE Hamiltonian reads:
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\begin{align*}
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R_{ai,bj} = \left( \varepsilon_a^{GW} - \varepsilon_i^{GW} \right) \delta_{ij} \delta_{ab} + \eta (ai|bj) - W_{ai,bj}
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\end{align*}
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with $\eta=2,0$ for singlets/triplets and:
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\begin{equation}
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W_{ai,bj} = \int d{\bf r} d{\bf r}'
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\phi_i({\bf r}) \phi_j({\bf r}) W({\bf r},{\bf r}'; \omega=0)
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\phi_a({\bf r}') \phi_b({\bf r}')
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\end{equation}
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where we notice that the 2 occupied (virtual) eigenstates are taken at the same space position, in contrast with the
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$(ai|bj)$ bare Coulomb term. As compared to TD-DFT, the $GW$ quasiparticle energies $\labrace \varepsilon_{i/a}^{GW} \rbrace$ replace the Kohn-Sham eigenvalues and the non-local screened Coulomb matrix elements replaces the DFT exchange-correlation kernel. \\\\
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%% GW historical
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\cite{Hedin_1965}
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STOP EQUATIONS \\\\
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%% BSE historical
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