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@ -187,7 +187,7 @@ 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) [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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The Bethe-Salpeter equation formalism [REFS] 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 \left[ {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't') \right] | N \rangle
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\end{equation}
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@ -197,20 +197,22 @@ where $| N \rangle $ is the N-electron ground-state wavefunction. The operators
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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 ) } \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 $\lbrace \varepsilon_s \rbrace$ 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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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 that reduce to one-body orbitals in the case of mono-determinental many-body wavefunctions [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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Using the equation of motion for the creation/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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where we use the notation $1 = (x_1,t_1)$. Here $h$ is the one-body Hartree Hamiltonian and $\Sigma$ the so-called exchange-correlation self-energy operator. 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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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. [INTRODUCE QUASIPARTICLES and OTHER solutions ??] \\
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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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\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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@ -219,10 +221,16 @@ where $W$ is the dynamically screened Coulomb potential and $\Gamma$ a ``vertex"
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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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with $\chi_0$ the well-known independent electron susceptibility and $v$ the bare Coulomb potential. In practice, the input $G$ and $\chi_0$ needed to buld $\Sigma$ are taken to be the best Green's function and susceptibility that can be easily calculated, namely the DFT (or HF) ones where the $\lbrace \varepsilon_s, f_s \rbrace$ of equation~\ref{spectralG} are taken to be DFT Kohn-Sham (or HF) eigenstates. Taking then $( \Sigma^{GW}-V^{XC} )$ as a correction to the $V^{XC}$ DFT exchange correlation potential, a first-order correction to the input Kohn-Sham $\lbrace \varepsilon_n^{KS} \rbrace$ energies is obtained as follows:
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\begin{equation}
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\varepsilon_n^{GW} = \varepsilon_n^{KS} +
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\langle \phi_n^{KS} | \Sigma^{GW}(\varepsilon_n^{GW}) -V^{XC} | \phi_n^{KS} \rangle
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\end{equation}
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Such an approach, where input Kohn-Sham energies are corrected to yield better electronic energy levels, is labeled the single-shot, or perturbative, $G_0W_0$ technique. This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, [REFS]
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surfaces [REFs], and 2D systems, [REFS] allowing to dramatically reduced the errors associated with Kohn-Sham eigenvalues in conjunction with the common LDA approximation.
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In particular, the well-known ``band gap" problem, [REFS] namely the underestimation of the occupied to unoccupied bands energy gap at the LDA Kohn-Sham level, was dramatically reduced, bringing the agreement with experiment to within a very few tenths of an eV [REFS] with an $\mathcal{O}(N^4)$ computational cost scaling (see below). As 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. [REFS] However, remaining a low order perturbative approach starting with mono-determinental mean-field solutions, it is not intended to explore strongly correlated systems. [REFS to Hubbard cluster and discuss bubbles in a Note ???] \\
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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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\noindent {\textbf{Neutral excitations.}} While TD-DFT starts with 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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@ -233,18 +241,18 @@ The equation of motion for $G$ (Eqn.~\ref{Gmotion}) can be reformulated in the f
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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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that relates the full (interacting) $G$ to the Hartree $G_0$ obtained by replacing the $\lbrace \varepsilon_s, f_s \rbrace$ by the Hartree eigenvalues and eigenfunctions.
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The derivation by $U$ 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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L(12,34) &= L^0(12,34) + \nonumber \\
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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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with $L_0 = \partial G_0 / \partial U$ the Hartree 4-point susceptibility 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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This equation can be compared to its TD-DFT analog:
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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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@ -252,11 +260,11 @@ 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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Plugging now the $GW$ self-energy, in a scheme that we label the BSE/$GW$ approach, leads to an approximation to the 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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where it is traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. Taking the static limit $W(\omega=0)$ for the screened Coulomb potential, that replaces thus the static DFT exchange-correlation kernel, and expressing equation~\ref{DysonL} in the standard product space $\lbrace \phi_i({\bf r}) \phi_a({\bf r}') \rbrace$ where (i,j) and (a,b) index occupied and virtual orbitals, 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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@ -278,13 +286,13 @@ 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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with electron-hole (e-h) eigenstates 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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\phi_i({r}_h) \phi_a({r}_e) + Y_{ia}^{ \lambda}
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\phi_i({r}_e) \phi_a({r}_h) \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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where $\lambda$ index the electronic excitations. The $\lbrace \phi_{i/a} \rbrace$ are the input (Kohn-Sham) eigenstates used to build the $GW$ self-energy. The resonant 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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@ -295,25 +303,40 @@ W_{ai,bj} = \int d{\bf r} d{\bf r}'
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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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STOP EQUATIONS \\\\
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$(ai|bj)$ bare Coulomb term. As compared to TD-DFT :
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\begin{itemize}
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\item the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{GW} \rbrace$ replace the Kohn-Sham eigenvalues
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\item the non-local screened Coulomb matrix elements replaces the DFT exchange-correlation kernel.
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\end{itemize}
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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 Perspective, emphasizing its pros and cons. \\
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%% BSE historical
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\cite{Salpeter_1951}
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The use of the BSE formalism in condensed-matter physics emerged in the early 50s at the semi-empirical tight-binding level with the study of the optical properties of excitonic \cite{Sham_1966,Strinati_1984,Delerue_2000}
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A decade latter, the first \textit{ab initio} implementations , starting with small clusters \cite{Onida_1995,Rohlfing_1998} before addressing the case of extended solids such as semmiconductors and wide-gap ionic insulators (Li$_2$O, LiF, MgO), \cite{Albrecht_1997,Benedict_1998} and simple surfaces, \cite{Rohlfing_1999}
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paved the way to the popularization in the solid-state physics community of the Bethe-Salpeter equation formalism.
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Originally developed in the framework of nuclear physics, \cite{Salpeter_1951}
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the use of the BSE formalism in condensed-matter physics emerged in the 60s at the semi-empirical tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000}
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Three decades latter, the first \textit{ab initio} implementations, starting with small clusters \cite{Onida_1995,Rohlfing_1998} and extended semiconductors and wide-gap insulators, \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999}
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paved the way to the popularization in the solid-state physics community of the BSE formalism.
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Following early applications to periodic polymers and molecules, [REFS] the BSE formalism gained much momentum in the quantum chemistry community with in particular several benchmarks \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same running parameters (geometries, basis sets) than the available reference higher-level calculations such as CC3. [REFS] Such comparisons were grounded in the development of codes replacing the planewave solid-state physics paradigm by well documented correlation-consistent Gaussian basis sets, together with adequate auxiliary bases when resolution-of-the-identity techniques were used. [REFS]
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An important conclusion drawn from these calculations was that the quality of the BSE excitation energies are strongly correlated to the deviation of the preceding $GW$ HOMO-LUMO gap with the experimental (IP-AE) photoemission gap. Standard $G_0W_0$ calculations starting with Kohn-Sham eigenstates generated with (semi)local functionals yield much larger HOMO-LUMO gaps than the input Kohn-Sham one, but still too small as compared to the experimental (AE-IP) value. Such an underestimation of the (IP-AE) gap leads to a similar underestimation of the lowest optical excitation energies.
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Such a residual HOMO-LUMO gap problem can be significantly improved by adopting exchange-correlation functionals that yield a much improved HOMO-LUMO gap as a starting point for the $GW$ correction. Obviously, optimally tuned functionals such that the Kohn-Sham HOMO-LUMO gap matches the $\Delta$SCF (AE-IP) value, yields excellent $GW$ gaps and much improved resulting BSE excitations. Alternatively, self-consistent schemes, where ...
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too small by up to an eV in the case of small
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\vskip 5cm
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interest from the quantum chemistry community BSE formalism
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chemistry oriented reviews with, e.g., the language of localized basis and resolution-of-the-identity techniques, \cite{Ren_2012} or applications related to organic molecular systems, photoelectrochemistry, etc. \cite{Ping_2013,Leng_2016,Blase_2018}
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large molecular benchmarks with comparisons to TD-DFT and higher level wavefunctions techniques such as CC3 \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018}
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large molecular benchmarks with comparisons to TD-DFT and higher level wavefunctions techniques such as CC3
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Reinjecting the corrected eigenvalues into the construction of $G$ and $W$ leads to a partially self-consistent scheme where eigenvalues are updated, while keeping input one-body orbitals frozen to their Kohn-Sham ansatz. Such a simple self-consistent scheme is labelled ev$GW$, [REFS] as a simple alternative to more involved self-consistent schemes where both eigenvalues and eigenstates are updated. [REFS] \\
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charge transfer
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classification into local-, Rydberg-, or charge transfer-type
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\cite{Hirose_2017} ad developed extensively in to the TD-DFT community.
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