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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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@ -274,7 +273,6 @@ dropping spin variables for simplicity, one gets the familiar eigenvalue equatio
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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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\\
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%===================================
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@ -289,21 +287,23 @@ The resulting equation, when compared with the equation for the time-evolution o
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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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\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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\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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\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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@ -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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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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@ -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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@ -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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@ -527,7 +528,7 @@ The analysis of the screened Coulomb potential matrix elements in the BSE kernel
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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.\\
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%%% FIG 3 %%%
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\begin{figure}[h]
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\begin{figure}[ht]
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\includegraphics[width=0.6\linewidth]{CTfig}
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\caption{
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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).
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