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@ -230,9 +230,9 @@ In its press release announcing the attribution of the 2013 Nobel prize in Chemi
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Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003}
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Martin Karplus' Nobel lecture moderated this statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are \textit{``much too complicated to be soluble''}, urging scientists to develop \textit{``approximate practical methods''}. This is where the electronic structure community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems.
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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, \eg, of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology.
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% T2: shall we add a few references here?
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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, \eg, of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. \titou{[REFS]}
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The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Dreuw_2005} including computational scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
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\\
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%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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@ -245,7 +245,8 @@ While the one-body density stands as the basic variable in density-functional th
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\end{equation}
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where $\ket{\Nel}$ is the $\Nel$-electron ground-state wave function.
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The operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.
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For $t > t'$, $G$ provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for $t < t'$ the propagation of a hole is monitored.\\
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For $t > t'$, $G$ provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for $t < t'$ the propagation of a hole is monitored.
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\\
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%===================================
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\subsection{Charged excitations}
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@ -272,7 +273,6 @@ Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one get
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\end{equation}
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which resembles formally the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.
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The knowledge of $\Sigma$ allows to access 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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%\titou{[INTRODUCE QUASIPARTICLES and OTHER solutions ??]}
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\titou{The spin variable has disappear. How do we deal with this?}
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\\
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@ -333,14 +333,10 @@ Because, one is usually interested only by the quasiparticle solution, in practi
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Z_p(\varepsilon_p^{\KS}) \mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\KS}) - V^{\XC}}{\phi_p^{\KS}}.
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\end{equation}
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%\titou{T2: Shall we introduce the renormalization factor or its non-linear version?
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%Here, I would prefer to introduce the frequency-dependent quasiparticle equation and talks about quasiparticle and satellites, and the way of solving this equation in practice (linearization, etc).}
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Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique.
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This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988}
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surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
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In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the LDA KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV [REFS] with a computational cost scaling quartically with the number of basis functions (see below).
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In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the LDA KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV \titou{[REFS]} with a computational cost scaling quartically with the number of basis functions (see below).
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Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature.
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Further improvements may be obtained via self-consistency of the Hedin's equations (see Fig.~\ref{fig:pentagon}).
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@ -353,28 +349,27 @@ Note that a starting point dependence remains in ev$GW$ as the orbitals are not
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However, self-consistency does not always improve things, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
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Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006, Stan_2009, Rostgaard_2010, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Wilhelm_2018}
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In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.
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A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998, Holm_1999,Holm_2000,Garcia-Gonzalez_2001}
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This was further evidenced in real extended systems by several authors. \cite{Schone_1998, Ku_2002, Kutepov_2016, Kutepov_2017}
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However, other approximations may have caused such deterioration, \eg, pseudo-potentials \cite{deGroot_1995} or finite-basis set effects. \cite{Friedrich_2006}
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In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.\cite{deGroot_1995,Schone_1998,Ku_2002,Friedrich_2006,Kutepov_2016,Kutepov_2017}
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A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998,Holm_1999,Holm_2000,Garcia-Gonzalez_2001}
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These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
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For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011,Blase_2011,Faber_2011,Caruso_2012,Caruso_2013,Caruso_2013a,Caruso_2013b,Koval_2014,Hung_2016,Blase_2018,Jacquemin_2017}
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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. \cite{Campillo_1999}
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However, remaining a low order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995}\\
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However, remaining a low order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995}
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\\
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%===================================
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\subsection{Neutral excitations}
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%===================================
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While TD-DFT starts with the variation of the charge density $\rho$ with respect to an external local perturbation $U(1)$, 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 $U(1,2)$:
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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 $U(1,2)$:
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\begin{equation}
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\chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
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\quad \rightarrow \quad
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L(12,34) \stackrel{\BSE}{=} -i \pdv{G(1,2)}{U(3,4)}.
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L(12,34) \stackrel{\BSE}{=} -i \pdv{G(1,2)}{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 $G$ [see Eq.~\ref{eq:Gmotion}] can be reformulated in the form of a Dyson equation
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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.
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The equation of motion for $G$ [see Eq.~\eqref{eq: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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@ -404,7 +399,7 @@ Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that w
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\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) -W(5,6) \delta(57) \delta(68 ),
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\end{equation}
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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}
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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's equations in TD-DFT:
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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:
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\begin{equation}
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\begin{pmatrix}
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R & C
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@ -431,29 +426,34 @@ with electron-hole ($eh$) eigenstates written as
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+ Y_{ia}^{m} \phi_i(\br_e) \phi_a(\br_h) ],
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\end{equation}
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where $m$ indexes the electronic excitations.
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The $\lbrace \phi_{i/a} \rbrace$ are the input (KS) eigenstates used to build the $GW$ self-energy.
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\titou{T2: this is only true in the case of $G_0W_0$.}
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The $\lbrace \phi_{i/a} \rbrace$ are, in the case of $G_0W_0$ and ev$GW$, the input (KS) eigenstates used to build the $GW$ self-energy.
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The resonant and anti-resonant parts of the BSE Hamiltonian read
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\begin{gather}
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R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \eta (ai|bj) - W_{ai,bj},
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R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \kappa (ia|jb) - W_{ij,ab},
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\\
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C_{ai,bj} = \eta (ai|jb) - W_{ai,jb},
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C_{ai,bj} = \kappa (ia|bj) - W_{ib,aj},
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\end{gather}
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with $\eta=2,0$ for singlets/triplets and
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with $\kappa=2,0$ for singlets/triplets and
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\begin{equation}\label{eq:Wmatel}
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W_{ai,bj} = \iint d\br d\br'
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W_{ij,ab} = \iint d\br d\br'
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\phi_i(\br) \phi_j(\br) W(\br,\br'; \omega=0)
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\phi_a(\br') \phi_b(\br'),
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\end{equation}
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where we notice that the two occupied (virtual) eigenstates are taken at the same position of space, in contrast with the
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$(ai|bj)$ bare Coulomb term.
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$(ia|jb)$ bare Coulomb term defined as
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\begin{equation}
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(ai|bj) = \iint d\br d\br'
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\phi_i(\br) \phi_a(\br) v(\br-\br')
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\phi_j(\br') \phi_b(\br'),
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\end{equation}
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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 KS eigenvalues
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\item the non-local screened Coulomb matrix elements replaces the DFT xc kernel.
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\end{itemize}
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\titou{T2: would it be useful to say that there is 100\% exact exchange in BSE@$GW$?}
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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 \textit{Perspective}, emphasizing its pros and cons. \\
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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.
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This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, emphasizing its pros and cons.
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\\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Historical overview}
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@ -462,8 +462,8 @@ We emphasise that these equations can be solved at exactly the same cost as the
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Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} the BSE formalism has emerged in condensed-matter physics around the 1960's 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 later, the first \textit{ab initio} implementations, starting with small clusters \cite{Onida_1995,Rohlfing_1998} extended semiconductors and wide-gap insulators, \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999} 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] BSE 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 parameters (geometries, basis sets, etc) than the available higher-level reference calculations, \cite{Schreiber_2008} such as CC3. \cite{Christiansen_1995}
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Such comparisons were grounded in the development of codes replacing the plane-wave paradigm of solid-state physics by well-documented correlation-consistent Gaussian basis sets, \cite{Dunning_1989} together with adequate auxiliary bases when resolution-of-the-identity (RI) techniques were used. [REFS]
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Following early applications to periodic polymers and molecules, \titou{[REFS]} BSE 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 parameters (geometries, basis sets, etc) than the available higher-level reference calculations, \cite{Schreiber_2008} such as CC3. \cite{Christiansen_1995}
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Such comparisons were grounded in the development of codes replacing the plane-wave paradigm of solid-state physics by well-documented correlation-consistent Gaussian basis sets, \cite{Dunning_1989} together with adequate auxiliary bases when resolution-of-the-identity (RI) techniques were used. \titou{[REFS]}
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An important conclusion drawn from these calculations was that the quality of the BSE excitation energies is strongly correlated to the deviation of the preceding $GW$ HOMO-LUMO gap
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\begin{equation}
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@ -495,7 +495,7 @@ but still too small as compared to the experimental value, \ie,
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\begin{equation}
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\Eg^{\KS} \ll \Eg^{G_0W_0} < \EgFun.
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\end{equation}
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Such an underestimation of the fundamental gap leads to a similar underestimation of the optical gap $\EgOpt$, \ie, the lowest optical excitation energy.
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Such an underestimation of the fundamental gap leads to a similar underestimation of the optical gap $\EgOpt$, \ie, the lowest optical excitation energy:
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\begin{equation}
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\EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB,
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\end{equation}
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@ -534,7 +534,7 @@ dressing the bare Coulomb potential with the reaction field matrix
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$$
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v({\bf r},{\bf r}') \longrightarrow v({\bf r},{\bf r}') + v^{\text{reac}}({\bf r},{\bf r}'; \omega)
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$$
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in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility (equation \ref{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.
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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.
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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.
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