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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Denis Jacquemin at 2020-06-08 11:44:16 +0200
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%% Created for Pierre-Francois Loos at 2020-06-08 22:10:57 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Packer_1996,
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Author = {Packer, M. K. and Dalskov, E. K. and Enevoldsen, T. and Jensen, H. J. and Oddershede, J.},
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Date-Added = {2020-06-08 21:57:16 +0200},
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Date-Modified = {2020-06-08 22:10:55 +0200},
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Doi = {10.1063/1.472430},
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Journal = {J. Chem. Phys.},
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Pages = {5886--5900},
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Title = {A New Implementation of the Second-Order Polarization Propagator Approximation (SOPPA): The Excitation Spectra of Benzene and Naphthalene},
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Volume = {105},
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Year = {1996}}
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@article{Wu_2019,
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Author = {Xin‐Ping Wu and Indrani Choudhuri and Donald G. Truhlar},
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Date-Added = {2020-06-05 20:35:01 +0200},
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@ -228,7 +228,7 @@ Future directions of developments and improvements are also discussed.
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In its press release announcing the attribution of the 2013 Nobel prize in Chemistry to Karplus, Levitt, and Warshel, the Royal Swedish Academy of Sciences concluded by stating \textit{``Today the computer is just as important a tool for chemists as the test tube.
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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 for jewellery, \cite{Prandini_2019} to the understanding, \eg, of the basic principles behind organic photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. \cite{Kippelen_2009,Improta_2016,Wu_2019}
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The study of optical excitations (also known as neutral 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 for jewellery, \cite{Prandini_2019} to the understanding, \eg, of the basic principles behind organic photovoltaics, photocatalysis or DNA damage under irradiation. \cite{Kippelen_2009,Improta_2016,Wu_2019}
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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} including computational scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known TD-DFT issues. \cite{Blase_2018}
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\\
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@ -236,21 +236,20 @@ The present \textit{Perspective} aims at describing the current status and upcom
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\section{Theory}
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%%%%%%%%%%%%%%%%%%%%%%
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The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Onida_2002,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} \hl{je parlerais aussi de SOPPA ici ? A citer au moins une fois ?}
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% originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018}
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The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Onida_2002,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques \cite{Dreuw_2015} or the polarization propagator approaches (like SOPPA\cite{Packer_1996}) in quantum chemistry.
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While the one-body density stands as the basic variable in density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
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\begin{equation}
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G(\bx t,\bx't') = -i \mel{\Nel}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{\Nel},
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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 (\ie, higher in energy than the highest-occupied energy level, also known as Fermi level), an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for $t < t'$ the propagation of an electron hole (often simply called a hole) is monitored.
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\\
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%===================================
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\subsection{Charged excitations}
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%===================================
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A central property of the one-body Green's function is that its frequency-dependent (\ie, dynamical) spectral representation has poles at the charged excitation energies of the system
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A central property of the one-body Green's function is that its frequency-dependent (\ie, dynamical) spectral representation has poles at the charged excitation energies (\ie, the ionization potentials and electron affinities) of the system
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\begin{equation}\label{eq:spectralG}
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G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) },
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\end{equation}
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@ -265,16 +264,15 @@ Using the equation-of-motion formalism for the creation/destruction operators, i
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\qty[ \pdv{}{t_1} - h(\br_1) ] G(1,2) - \int d3 \, \Sigma(1,3) G(3,2)
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= \delta(1,2),
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\end{equation}
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where we introduce the composite index, \eg, $1 \equiv (\bx_1 t_1)$.
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where we introduce the composite index, \eg, $1 \equiv (\bx_1 t_1)$.
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Here, $\delta$ is Dirac's delta function, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
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Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}],
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dropping spin variables for simplicity, one gets the familiar eigenvalue equation, \ie,
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\begin{equation}
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h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
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\end{equation}
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which formally resembles 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{The spin variable has disappear. How do we deal with this?}
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which formally resembles 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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\\
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%===================================
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@ -287,23 +285,25 @@ Developed by Lars Hedin in 1965 with application to the interacting homogeneous
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\cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Reining_2017,Golze_2019} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}).
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The resulting equation, when compared with the equation for the time-evolution of $G$ [see Eq.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy
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\begin{equation}\label{eq:Sig}
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\Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3),
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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 $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$. \hl{vs ne vlz pas simplement dire que c'est des corrections de + grand ordre ?}
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The neglect of the vertex leads to the so-called $GW$ approximation of the self-energy
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where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is the so-called ``vertex" function.
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%where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$.
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The neglect of the vertex, \ie, $\Gamma(42,3) = \delta(23) \delta(24)$, leads to the so-called $GW$ approximation of the self-energy
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\begin{equation}\label{eq:SigGW}
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\Sigma^{\GW}(1,2) = i \, G(1,2) W(2,1^{+}),
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\Sigma^{\GW}(1,2) = i \, G(1,2) W(2,1^{+}),
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\end{equation}
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that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with
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\begin{gather}
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W(1,2) = v(1,2) + \int d34 \, v(1,2) \chi_0(3,4) W(4,2), \label{eq:defW}
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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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\label{eq:defW}
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\\
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\chi_0(1,2) = -i \int d34 \, G(2,3) G(4,2),
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\chi_0(1,2) = -i \int d34 \, G(2,3) G(4,2),
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\end{gather}
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where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulomb potential.
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%%% FIG 1 %%%
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\begin{figure}[h]
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\begin{figure}[ht]
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\includegraphics[width=0.55\linewidth]{fig1/fig1}
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\caption{
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Hedin's pentagon connects the Green's function $G$, its non-interacting analog $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb potential $W$, and the self-energy $\Sigma$ through a set of five integro-differential equations known as Hedin's equations. \cite{Hedin_1965}
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@ -318,8 +318,8 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
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In practice, the input $G$ and $\chi_0$ required to initially build $\Sigma^{\GW}$ are taken as the ``best'' Green's function and susceptibility that can be easily computed, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.
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Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained by solving the so-called quasiparticle equation
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\begin{equation} \label{eq:QP-eq}
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\omega = \varepsilon_p^{\KS} +
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\mel{\phi_p^{\KS}}{\Sigma^{\GW}(\omega) - V^{\XC}}{\phi_p^{\KS}}.
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\omega = \varepsilon_p^{\KS} +
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\mel{\phi_p^{\KS}}{\Sigma^{\GW}(\omega) - V^{\XC}}{\phi_p^{\KS}}.
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\end{equation}
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As a non-linear equation, the quasiparticle equation \eqref{eq:QP-eq} has various solutions $\varepsilon_{p,s}^{\GW}$ associated with spectral weights $Z(\varepsilon_{p,s}^{\GW})$, where
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\begin{equation}
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@ -330,14 +330,14 @@ In addition to the principal quasiparticle solution $\varepsilon_{p}^{\GW} \equi
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Because, one is usually interested only by the quasiparticle solution, in practice, Eq.~\eqref{eq:QP-eq} is often linearized around $\omega = \varepsilon_p^{\KS}$ as follows:
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\begin{equation}
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\varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
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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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\varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
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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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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} and
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surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_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 local-density approximation (LDA) KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV with a computational cost scaling quartically with the system size (see below). A compilation of data for $G_0W_0$ applied to extended inorganic semiconductors can be found in Ref.~\citenum{Shishkin_2007}.
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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} and
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surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_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 local-density approximation (LDA) KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV with a computational cost scaling quartically with the system size (see below). A compilation of data for $G_0W_0$ applied to extended inorganic semiconductors can be found in Ref.~\citenum{Shishkin_2007}.
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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 Hedin's equations (see Fig.~\ref{fig:pentagon}).
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@ -363,13 +363,14 @@ However, remaining a low-order perturbative approach starting with single-determ
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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 $\rho(1)$ 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 $U(2,2')$:
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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, 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 $U(2,2')$: \cite{Strinati_1988}
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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(1, 2;1',2' ) \stackrel{\BSE}{=} \pdv{G(1,1')}{U(2',2)},
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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(1, 2;1',2' ) \stackrel{\BSE}{=} \pdv{G(1,1')}{U(2',2)}.
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\end{equation}
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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.
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%where we follow the notations by Strinati.\cite{Strinati_1988}
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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 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 ( v_H + U + \Sigma ) G,
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@ -383,8 +384,8 @@ The derivative with respect to $U$ of this Dyson equation yields the self-consis
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\end{multline}
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where $L_0(1,2;1',2') = G(1,2')G(2,1')$ is the non-interacting 4-point susceptibility and
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\begin{equation}
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i\,\Xi^{\BSE}(3,5;4,6) = v(3,6) \delta(34) \delta(56) + i \pdv{\Sigma(3,4)}{G(6,5)}
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\end{equation}
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i\,\Xi^{\BSE}(3,5;4,6) = v(3,6) \delta(34) \delta(56) + i \pdv{\Sigma(3,4)}{G(6,5)}
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\end{equation}
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is the so-called BSE kernel.
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This equation can be compared to its TD-DFT analog
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\begin{equation}
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@ -402,7 +403,8 @@ Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that w
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= v(3,6) \delta(34) \delta(56) -W(3^+,4) \delta(36) \delta(45 ),
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\end{multline}
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where it is customary 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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At that stage, the BSE kernel is fully dynamical. 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 spatial orbitals and $(a,b)$ are unoccupied spatial orbitals), leads to an eigenvalue problem similar to the so-called Casida equations in TD-DFT: \cite{Casida_1995}
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At that stage, the BSE kernel is fully dynamical, \ie, it explicitly depends on the frequency $\omega$.
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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 spatial orbitals and $(a,b)$ are unoccupied spatial orbitals], leads to an eigenvalue problem similar to the so-called Casida equations in TD-DFT: \cite{Casida_1995}
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\begin{equation} \label{eq:BSE-eigen}
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\begin{pmatrix}
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R & C
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@ -444,8 +446,7 @@ with $\kappa=2,0$ for singlets/triplets and
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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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$(ia|jb)$ bare Coulomb term defined as
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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 $(ia|jb)$ 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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@ -462,7 +463,7 @@ This defines the standard (static) BSE@$GW$ scheme that we discuss in this \text
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\section{Historical overview}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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 tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000}
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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 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_1999b} 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, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in the quantum chemistry community with, in particular, several benchmarks \cite{Boulanger_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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@ -479,7 +480,7 @@ with the experimental (photoemission) fundamental gap
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where $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system (see Fig.~\ref{fig:gaps}).
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%%% FIG 2 %%%
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\begin{figure*}[h]
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\begin{figure*}[ht]
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\includegraphics[width=0.7\linewidth]{gaps}
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\caption{
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Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
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@ -508,7 +509,7 @@ where $\EB$ is the excitonic effect, that is, the stabilization implied by the a
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Such a residual gap problem can be significantly improved by adopting 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}
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Alternatively, self-consistent schemes such as ev$GW$ and qs$GW$, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011} where corrected eigenvalues, and possibly orbitals, 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}
|
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As a result, BSE singlet excitation energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations.
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For sake of illustration, an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering ca. 200 representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017}
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For sake of illustration, an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering roughly ca.~200 representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017}
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This is equivalent to the best TD-DFT results obtained by scanning a large variety of hybrid functionals with various amounts of exact exchange.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
@ -522,12 +523,12 @@ This is equivalent to the best TD-DFT results obtained by scanning a large varie
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A very remarkable success of the BSE formalism lies in the description of charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT adopting standard (semi-)local functionals. \cite{Dreuw_2004}
|
||||
Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes (Fig.~\ref{fig:CTfig}), cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004}
|
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These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals.
|
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It can be cured through an exact exchange contribution, a solution that explains the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012}
|
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It can be cured through an exact exchange contribution, a solution that explains the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012}
|
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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 the 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{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to the modeling of key applications such as doping, \cite{Li_2017b} photovoltaics or photocatalysis in organic systems.\\
|
||||
The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to the modeling of key applications such as doping, \cite{Li_2017b} photovoltaics or photocatalysis in organic systems.\\
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure}[h]
|
||||
\begin{figure}[ht]
|
||||
\includegraphics[width=0.6\linewidth]{CTfig}
|
||||
\caption{
|
||||
Symbolic representation of extended Wannier exciton with large electron-hole average distance (top), and Frenkel (local) and charge-transfer (CT) excitations at a donor-acceptor interface (bottom).
|
||||
|
|
@ -549,16 +550,16 @@ As a matter of fact, dressing the bare Coulomb potential with the reaction field
|
|||
$[
|
||||
v(\br,\br') \longrightarrow v(\br,\br') + v^{\text{reac}}(\br,\br'; \omega)
|
||||
]$
|
||||
in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment with the same complexity as in the gas phase.
|
||||
in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment with the same complexity as in the gas phase.
|
||||
The reaction field matrix $v^{\text{reac}}(\br,\br'; \omega)$ describes the potential generated in $\br'$ by the charge rearrangements in the polarizable environment induced by a source charge located in $\br$, where $\br$ and $\br'$ lie in the quantum mechanical 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 is known, with typically $\order*{\Norb N_\text{MM}^2}$ operations (where $\Norb$ is the number of orbitals and $N_\text{MM}$ the number of polarizable atoms in the environment), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
|
||||
Once the reaction field matrix is known, with typically $\order*{\Norb N_\text{MM}^2}$ operations (where $\Norb$ is the number of orbitals and $N_\text{MM}$ the number of polarizable atoms in the environment), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
|
||||
|
||||
A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field captures both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing local (Frenkel) and CT excitations.
|
||||
A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field captures both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing local (Frenkel) and CT excitations.
|
||||
This is an important advantage as compared to, \eg, TD-DFT where linear-response and state-specific effects have to be explored with different formalisms.
|
||||
|
||||
To date, environmental effects on fast electronic excitations are only included by considering the low-frequency optical response of the polarizable medium (\eg, considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant of water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range.
|
||||
Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, situations where neither the adiabatic limit nor the antiadiabatic limit are expected to be valid (for a recent discussion, see Ref.
|
||||
To date, environmental effects on fast electronic excitations are only included by considering the low-frequency optical response of the polarizable medium (\eg, considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant of water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range.
|
||||
Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, situations where neither the adiabatic limit nor the antiadiabatic limit are expected to be valid (for a recent discussion, see Ref.
|
||||
~\citenum{Huu_2020}).
|
||||
|
||||
We now leave the description of successes to discuss difficulties and future directions of developments and improvements.
|
||||
|
|
@ -580,7 +581,7 @@ In practice, the main bottleneck for standard BSE calculations as compared to TD
|
|||
The field of low-scaling $GW$ calculations is however witnessing significant advances.
|
||||
While the sparsity of, for example, the overlap matrix in the atomic orbital basis allows to reduce the scaling in the large size limit, \cite{Foerster_2011,Wilhelm_2018} efficient real-space grids and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
|
||||
Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017}
|
||||
The separability of occupied and virtual states summations lying at the heart of these approaches are now spreading fast in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied for calculating with cubic scaling the susceptibility needed in random-phase approximation (RPA) and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020}
|
||||
The separability of occupied and virtual states summations lying at the heart of these approaches are now spreading fast in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied for calculating with cubic scaling the susceptibility needed in random-phase approximation (RPA) and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020}
|
||||
These ongoing developments pave the way to applying the $GW$@BSE formalism to systems containing several hundred atoms on standard laboratory clusters.
|
||||
\\
|
||||
|
||||
|
|
@ -589,7 +590,7 @@ These ongoing developments pave the way to applying the $GW$@BSE formalism to sy
|
|||
%==========================================
|
||||
\hl{Je ne pige pas la premiere phrase qui semble melanger differents concepts. A divisier ?}
|
||||
The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
|
||||
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels, contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range.
|
||||
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels, contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range.
|
||||
While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter(s) as a mean to act on the triplet instability, \cite{Sears_2011} BSE calculations do not offer this pragmatic way-out since the screened Coulomb potential that builds the kernel does not offer any parameter to tune.
|
||||
|
||||
Benchmark calculations \cite{Jacquemin_2017b,Rangel_2017} clearly concluded that triplets are notably too low in energy within BSE and that the use of the Tamm-Dancoff approximation was able to partly reduce this error.
|
||||
|
|
|
|||
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Reference in New Issue