saving work: up to GW
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@ -206,37 +210,53 @@ Future directions of developments and improvements are also discussed.
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%%% INTRODUCTION %%%
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%%%%%%%%%%%%%%%%%%%%
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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 \titou{as} a tool for chemists as the test tube. Simulations are so realistic that they predict the outcome of traditional experiments.''} Martin Karplus Nobel lecture moderated this bold 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 the scientist to develop \textit{``approximate practical methods''}. This is where the methodology community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems.
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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 bold 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 the scientist to develop \textit{``approximate practical methods''}. This is where the methodology 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. The present Perspective aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism that, while sharing many features with time-dependent density functional theory (TD-DFT), including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known difficulties.
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The 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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The present 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 cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
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The Bethe-Salpeter equation 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,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with e.g. the algebraic diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads:
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The 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,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015}
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While the density \titou{and density matrix} stand as the basic variables in DFT \titou{and Hartree-Fock}, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads
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\begin{equation}
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G(xt,x't') = -i \langle N | T \left[ {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't') \right] | N \rangle
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G(\bx t,\bx't') = -i \mel{N}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{N},
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\end{equation}
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where $| N \rangle $ is the N-electron ground-state wavefunction. The operators ${\hat \psi}(xt)$ and ${\hat \psi}^{\dagger}(x't')$ remove/add an electron in space-spin-time positions (xt) and (x't'), while $T$ is the time-ordering operator. For (t>t') the one-body Green's function provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in (xt) that was previously introduced in (x't'), while for (t<t') it is the propagation of a hole that is monitored. \\
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where $\ket{N}$ is the $N$-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 \titou{$T$ is the time-ordering operator}.
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For ($t > t'$) the one-body Green's function 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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\noindent{\textbf{Charged excitations.}} A central property of the one-body Green's function is that its spectral representation presents poles at the charged excitation energies of the system :
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{Charged excitations.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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A central property of the one-body Green's function is that its spectral representation presents poles at the charged excitation energies 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 \, \text{sgn}(\varepsilon_s - \mu ) },
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\end{equation}
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where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s(N+1) - E_0(N)$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0(N) - E_s(N-1)$ for $\varepsilon_s < \mu$.
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The quantities $E_s(N+1)$ and $E_s(N-1)$ are the total energy of the $s$th excited state of the $(N+1)$ and $(N-1)$-electron systems, while $E_0(N)$ is the $N$-electron ground-state energy.
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\titou{The $f_s$'s are the so-called Lehmann amplitudes that reduce to one-body orbitals in the case of single-determinant many-body wave functions [more ??].}
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Unlike Kohn-Sham eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are thus the proper \titou{charging} energies of the $N$-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbitals.
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Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies
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\begin{equation}\label{eq:Gmotion}
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\qty[ \pdv{}{t_1} - h({\bf r}_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 usual composite index, \eg, $1 \equiv (\bx_1,t_1)$.
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Here, $h$ is the \titou{one-body Hartree Hamiltonian} and $\Sigma$ the so-called exchange-correlation self-energy operator.
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Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one obtains the familiar eigenvalue equation, \ie,
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\begin{equation}
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G(x,x'; \omega ) = \sum_n \frac{ f_s(x) f^*_s(x') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) } \label{spectralG}
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h({\bf r}) f_s({\bf r}) + \int d{\bf r}' \; \Sigma({\bf r},{\bf r}'; \varepsilon_s ) f_s({\bf r}) = \varepsilon_s f_s({\bf r})
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\end{equation}
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where $\varepsilon_s = E_s(N+1) - E_0(N)$ for $\varepsilon_s > \mu$ ($\mu$ chemical potential, $\eta$ small positive infinitesimal) and $\varepsilon_s = E_0(N) - E_s(N-1)$ for $\varepsilon_s < \mu$. The quantities $E_s(N+1)$ and $E_s(N-1)$ are the total energy of the $s$-th excited state of the $(N+1)$ and $(N-1)$-electron systems, while $E_0(N)$ is the $N$-electron ground-state energy. Contrary to the Kohn-Sham eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are thus the proper charging energies of the N-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbitals. The $\lbrace f_s \rbrace$ are called the Lehmann amplitudes that reduce to one-body orbitals in the case of single-determinent many-body wave functions [more ??].
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which resembles formally the Kohn-Sham equation with the difference that the self-energy $\Sigma$ is non-local, energy dependent and non-hermitian.
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The knowledge of $\Sigma$ allows thus to obtain the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation. [INTRODUCE QUASIPARTICLES and OTHER solutions ??] \\
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Using the equation of motion for the creation/destruction operators, it can be shown formally that $G$ verifies :
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\begin{equation}
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\qty[ \pdv{}{t_1} - h({\bf r}_1) ] G(1,2) - \int d3 \; \Sigma(1,3) G(3,2)
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= \delta(1,2) \label{Gmotion}
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\end{equation}
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where we introduce the usual composite index, \eg, $1 \equiv (x_1,t_1)$. Here, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ the so-called exchange-correlation self-energy operator. Using the spectral representation of $G$, one obtains a familiar eigenvalue equation:
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\begin{equation}
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h({\bf r}) f_s({\bf r}) + \int d{\bf r}' \; \Sigma({\bf r},{\bf r}'; \varepsilon_s ) f_s({\bf r}) = \varepsilon_s f_s({\bf r})
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\end{equation}
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which resembles formally the Kohn-Sham equation with the difference that the self-energy $\Sigma$ is non-local, energy dependent and non-hermitian. The knowledge of the self-energy operators allows thus to obtain the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation. [INTRODUCE QUASIPARTICLES and OTHER solutions ??] \\
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\noindent{\textbf{The $GW$ self-energy.}} While the presented equations are formally exact, it remains to provide an expression for the exchange-correlation self-energy operator $\Sigma$. This is where Green's function practical theories differ. Developed by Lars Hedin in 1965 with application to the interacting homogeneous electron gas, \cite{Hedin_1965} the $GW$ approximation
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\cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Golze_2019rev} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation. The obtained equation, when compared with the equation for the time-evolution of $G$ (Eqn.~\ref{Gmotion}), leads to a formal expression for the self-energy :
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{The $GW$ self-energy.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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While the presented equations are formally exact, it remains to provide an expression for the exchange-correlation self-energy operator $\Sigma$. This is where Green's function practical theories differ. Developed by Lars Hedin in 1965 with application to the interacting homogeneous electron gas, \cite{Hedin_1965} the $GW$ approximation
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\cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Golze_2019rev} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation. The obtained equation, when compared with the equation for the time-evolution of $G$ [Eqn.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy :
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\begin{equation}
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\Sigma(1,2) = i \int d34 \; G(1,4) W(3,1^{+}) \Gamma(42,3)
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\end{equation}
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@ -254,7 +274,10 @@ Such an approach, where input Kohn-Sham energies are corrected to yield better e
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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 Kohn-Sham eigenvalues in conjunction with common local or gradient-corrected approximations to the exchange-correlation 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 Kohn-Sham level, was dramatically reduced, bringing the agreement with experiment to within a very few tenths of an eV [REFS] with an $\mathcal{O}(N^4)$ computational cost scaling (see below). As another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite size and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999} However, remaining a low order perturbative approach starting with mono-determinental mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995} \\
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\noindent {\textbf{Neutral excitations.}} While TD-DFT starts with the variation of the charge density $\rho$ with respect to an external local perturbation, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{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, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation:
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\begin{equation}
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\chi(1,2) \stackrel{\DFT}{=} \frac{ \partial \rho(1) }{\partial U(2) }
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\;\; \rightarrow \;\;
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@ -359,7 +382,10 @@ In practice, the main bottleneck for standard BSE calculations as compared to TD
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The field of low-scaling $GW$ calculations is however witnessing significant advances. While the sparcity of ..., \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid 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}
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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 blooming in quantum chemistry withing the Interpolative Separable Density Fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in 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 comprising several hundred atoms on standard laboratory clusters. \\
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\noindent {\textbf{The Triplet Instability Challenge.}} The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{The Triplet Instability Challenge.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
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stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels.
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contaminating as well TD-DFT calculations with popular range-separated hybrids (RSH) that generally contains a large fraction of exact exchange in the long-range. \cite{Sears_2011}
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While TD-DFT with RSH can benefit from tuning the range-separation parameter 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.
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@ -368,14 +394,17 @@ benchmarks \cite{Jacquemin_2017b,Rangel_2017}
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a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$. \cite{Holzer_2018b}
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\noindent {\textbf{The challenge of Analytic gradients.}} \\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{The challenge of analytic gradients.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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An additional issue concerns the formalism taken to calculate the ground-state energy for a given atomic configuration. Since the BSE formalism presented so far concerns the calculation of the electronic excitations, namely the difference of energy between the GS and the ES, gradients of the ES absolute energy require
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This points to another direction for the BSE foramlism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}
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\noindent {\textbf{Dynamical kernels and multiple excitations.}} \\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{Dynamical kernels and multiple excitations.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\cite{Zhang_2013}
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@ -387,7 +416,9 @@ XANES,
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diabatization and conical intersections \cite{Kaczmarski_2010}
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\noindent {\textbf{The Concept of dynamical properties.}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{The Concept of dynamical properties.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
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Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009,Sangalli_2011,ReiningBook}
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To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
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@ -454,10 +485,9 @@ In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its
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This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$.
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Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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%%%%%%%%%%%%%%%%%%
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\noindent {\textbf{Conclusion.}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\paragraph{Conclusion.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here goes the conclusion.
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -1,13 +1,140 @@
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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 Pierre-Francois Loos at 2020-04-17 20:05:26 +0200
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%% Created for Pierre-Francois Loos at 2020-05-13 10:12:24 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@inbook{Casida_1995,
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Author = {M. E. Casida},
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Date-Added = {2020-05-13 10:10:00 +0200},
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Date-Modified = {2020-05-13 10:10:00 +0200},
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Doi = {10.1142/9789812830586_0005},
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Editor = {D. P. Chong},
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Pages = {155--192},
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Publisher = {World Scientific, Singapore},
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Series = {Recent Advances in Density Functional Methods},
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Shorttitle = {Asymptotic Correction Approach to Improving Approximate Exchange\textendash{}Correlation Potentials},
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Title = {Asymptotic Correction Approach to Improving Approximate Exchange\textendash{}Correlation Potentials: {{Time}}-Dependent Density-Functional Theory Calculations of Molecular Excitation Spectra},
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Author = {Casida, M.E. and Huix-Rotllant, M.},
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Date-Added = {2020-05-13 10:10:00 +0200},
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Date-Modified = {2020-05-13 10:10:00 +0200},
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Journal = {Annu. Rev. Phys. Chem.},
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Title = {Progress in Time-Dependent Density-Functional Theory},
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@article{Huix-Rotllant_2010,
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Author = {{Huix-Rotllant}, Miquel and Natarajan, Bhaarathi and Ipatov, Andrei and Muhavini Wawire, C. and Deutsch, Thierry and Casida, Mark E.},
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Date-Added = {2020-05-13 10:10:00 +0200},
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Bdsk-Url-1 = {https://doi.org/10.1039/c0cp00273a}}
|
||||
|
||||
@article{Huix-Rotllant_2011,
|
||||
Author = {{Huix-Rotllant}, Miquel and Ipatov, Andrei and Rubio, Angel and Casida, Mark E.},
|
||||
Date-Added = {2020-05-13 10:10:00 +0200},
|
||||
Date-Modified = {2020-05-13 10:10:00 +0200},
|
||||
Doi = {10.1016/j.chemphys.2011.03.019},
|
||||
File = {/Users/loos/Zotero/storage/A4JUV4M4/Huix-Rotllant et al. - 2011 - Assessment of dressed time-dependent density-funct.pdf},
|
||||
Issn = {03010104},
|
||||
Journal = {Chem. Phys.},
|
||||
Language = {en},
|
||||
Month = nov,
|
||||
Number = {1},
|
||||
Pages = {120-129},
|
||||
Title = {Assessment of Dressed Time-Dependent Density-Functional Theory for the Low-Lying Valence States of 28 Organic Chromophores},
|
||||
Volume = {391},
|
||||
Year = {2011},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1016/j.chemphys.2011.03.019}}
|
||||
|
||||
@article{Runge_1984,
|
||||
Author = {Runge, E. and Gross, E. K. U.},
|
||||
Date-Added = {2020-05-13 10:09:39 +0200},
|
||||
Date-Modified = {2020-05-13 10:09:39 +0200},
|
||||
Doi = {10.1103/PhysRevLett.52.997},
|
||||
Journal = PRL,
|
||||
Pages = {997--1000},
|
||||
Title = {Density-Functional Theory for Time-Dependent Systems},
|
||||
Volume = 52,
|
||||
Year = 1984,
|
||||
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.52.997}}
|
||||
|
||||
@article{Dreuw_2005,
|
||||
Author = {Dreuw, Andreas and Head-Gordon, Martin},
|
||||
Date-Added = {2020-05-13 10:09:21 +0200},
|
||||
Date-Modified = {2020-05-13 10:09:21 +0200},
|
||||
Doi = {10.1021/cr0505627},
|
||||
File = {/Users/loos/Zotero/storage/WKGXAHGE/Dreuw_2005.pdf},
|
||||
Issn = {0009-2665, 1520-6890},
|
||||
Journal = {Chem. Rev.},
|
||||
Language = {en},
|
||||
Month = nov,
|
||||
Number = {11},
|
||||
Pages = {4009--4037},
|
||||
Title = {Single-{{Reference}} Ab {{Initio Methods}} for the {{Calculation}} of {{Excited States}} of {{Large Molecules}}},
|
||||
Volume = {105},
|
||||
Year = {2005},
|
||||
Bdsk-Url-1 = {https://dx.doi.org/10.1021/cr0505627}}
|
||||
|
||||
@misc{Nobel_2003,
|
||||
Author = {{{The Royal Swedish Academy of Sciences}}},
|
||||
Date-Added = {2020-05-13 10:00:08 +0200},
|
||||
Date-Modified = {2020-05-13 10:02:28 +0200},
|
||||
Howpublished = {Press release},
|
||||
Month = {Oct},
|
||||
Title = {The Nobel Prize in Chemistry 2013},
|
||||
Year = {2013}}
|
||||
|
||||
@article{Gatti_2007,
|
||||
Author = {M. Gatti and V. Olevano and L. Reining,and I. V. Tokatly},
|
||||
Date-Added = {2020-04-17 10:10:16 +0200},
|
||||
|
|
@ -1412,178 +1539,176 @@
|
|||
Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}}
|
||||
|
||||
@article{Botti_2004,
|
||||
title = {Long-Range Contribution to the Exchange-Correlation Kernel of Time-Dependent Density Functional Theory},
|
||||
author = {Botti, Silvana and Sottile, Francesco and Vast, Nathalie and Olevano, Valerio and Reining, Lucia and Weissker, Hans-Christian and Rubio, Angel and Onida, Giovanni and Del Sole, Rodolfo and Godby, R. W.},
|
||||
journal = {Phys. Rev. B},
|
||||
volume = {69},
|
||||
issue = {15},
|
||||
pages = {155112},
|
||||
numpages = {14},
|
||||
year = {2004},
|
||||
month = {Apr},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevB.69.155112},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevB.69.155112}
|
||||
}
|
||||
|
||||
Author = {Botti, Silvana and Sottile, Francesco and Vast, Nathalie and Olevano, Valerio and Reining, Lucia and Weissker, Hans-Christian and Rubio, Angel and Onida, Giovanni and Del Sole, Rodolfo and Godby, R. W.},
|
||||
Doi = {10.1103/PhysRevB.69.155112},
|
||||
Issue = {15},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Apr},
|
||||
Numpages = {14},
|
||||
Pages = {155112},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Long-Range Contribution to the Exchange-Correlation Kernel of Time-Dependent Density Functional Theory},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.69.155112},
|
||||
Volume = {69},
|
||||
Year = {2004},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.69.155112},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.69.155112}}
|
||||
|
||||
@article{Kaltak_2014,
|
||||
author = {Kaltak, Merzuk and Klime\v{s}, Ji\v{i}\'{i} and Kresse, Georg},
|
||||
title = {Low Scaling Algorithms for the Random Phase Approximation: Imaginary Time and Laplace Transformations},
|
||||
journal = {Journal of Chemical Theory and Computation},
|
||||
volume = {10},
|
||||
number = {6},
|
||||
pages = {2498-2507},
|
||||
year = {2014},
|
||||
doi = {10.1021/ct5001268},
|
||||
note ={PMID: 26580770},
|
||||
URL = { https://doi.org/10.1021/ct5001268},
|
||||
eprint = { https://doi.org/10.1021/ct5001268}
|
||||
}
|
||||
|
||||
Author = {Kaltak, Merzuk and Klime\v{s}, Ji\v{i}\'{i} and Kresse, Georg},
|
||||
Doi = {10.1021/ct5001268},
|
||||
Eprint = {https://doi.org/10.1021/ct5001268},
|
||||
Journal = {Journal of Chemical Theory and Computation},
|
||||
Note = {PMID: 26580770},
|
||||
Number = {6},
|
||||
Pages = {2498-2507},
|
||||
Title = {Low Scaling Algorithms for the Random Phase Approximation: Imaginary Time and Laplace Transformations},
|
||||
Url = {https://doi.org/10.1021/ct5001268},
|
||||
Volume = {10},
|
||||
Year = {2014},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/ct5001268}}
|
||||
|
||||
@article{Liu_2016,
|
||||
title = {Cubic Scaling $GW$: Towards Fast Quasiparticle Calculations},
|
||||
author = {Liu, Peitao and Kaltak, Merzuk and Klime\ifmmode \check{s}\else \v{s}\fi{}, Ji\ifmmode \check{r}\else \v{r}\fi{}\'{\i} and Kresse, Georg},
|
||||
journal = {Phys. Rev. B},
|
||||
volume = {94},
|
||||
issue = {16},
|
||||
pages = {165109},
|
||||
numpages = {13},
|
||||
year = {2016},
|
||||
month = {Oct},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevB.94.165109},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevB.94.165109}
|
||||
}
|
||||
|
||||
Author = {Liu, Peitao and Kaltak, Merzuk and Klime\ifmmode \check{s}\else \v{s}\fi{}, Ji\ifmmode \check{r}\else \v{r}\fi{}\'{\i} and Kresse, Georg},
|
||||
Doi = {10.1103/PhysRevB.94.165109},
|
||||
Issue = {16},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Oct},
|
||||
Numpages = {13},
|
||||
Pages = {165109},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Cubic Scaling $GW$: Towards Fast Quasiparticle Calculations},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.94.165109},
|
||||
Volume = {94},
|
||||
Year = {2016},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.94.165109},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.94.165109}}
|
||||
|
||||
@article{Duchemin_2019,
|
||||
author = {Duchemin,Ivan and Blase,Xavier },
|
||||
title = {Separable resolution-of-the-identity with all-electron Gaussian bases: Application to cubic-scaling RPA},
|
||||
journal = { J. Chem. Phys. },
|
||||
volume = {150},
|
||||
number = {17},
|
||||
pages = {174120},
|
||||
year = {2019},
|
||||
doi = {10.1063/1.5090605},
|
||||
URL = { https://doi.org/10.1063/1.5090605},
|
||||
eprint = { https://doi.org/10.1063/1.5090605}
|
||||
}
|
||||
Author = {Duchemin,Ivan and Blase,Xavier},
|
||||
Doi = {10.1063/1.5090605},
|
||||
Eprint = {https://doi.org/10.1063/1.5090605},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Number = {17},
|
||||
Pages = {174120},
|
||||
Title = {Separable resolution-of-the-identity with all-electron Gaussian bases: Application to cubic-scaling RPA},
|
||||
Url = {https://doi.org/10.1063/1.5090605},
|
||||
Volume = {150},
|
||||
Year = {2019},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}}
|
||||
|
||||
@article{Lu_2017,
|
||||
title = "Cubic Scaling Algorithms for RPA Correlation Using Interpolative Separable Density Fitting",
|
||||
journal = "J. Comput. Phys.",
|
||||
volume = "351",
|
||||
pages = "187 - 202",
|
||||
year = "2017",
|
||||
issn = "0021-9991",
|
||||
doi = "https://doi.org/10.1016/j.jcp.2017.09.012",
|
||||
url = "http://www.sciencedirect.com/science/article/pii/S002199911730671X",
|
||||
author = "Jianfeng Lu and Kyle Thicke",
|
||||
keywords = "Electronic structure theory, Density fitting, Random phase approximation, Fast algorithms, Contour integral",
|
||||
abstract = "We present a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in χ0 by use of Cauchy's integral formula. This introduces an additional integral to be carried out, for which we provide a geometrically convergent quadrature rule. Our scheme also uses the newly developed Interpolative Separable Density Fitting algorithm to further reduce the computational cost in a way analogous to that of the Resolution of Identity method."
|
||||
}
|
||||
Abstract = {We present a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in χ0 by use of Cauchy's integral formula. This introduces an additional integral to be carried out, for which we provide a geometrically convergent quadrature rule. Our scheme also uses the newly developed Interpolative Separable Density Fitting algorithm to further reduce the computational cost in a way analogous to that of the Resolution of Identity method.},
|
||||
Author = {Jianfeng Lu and Kyle Thicke},
|
||||
Doi = {https://doi.org/10.1016/j.jcp.2017.09.012},
|
||||
Issn = {0021-9991},
|
||||
Journal = {J. Comput. Phys.},
|
||||
Keywords = {Electronic structure theory, Density fitting, Random phase approximation, Fast algorithms, Contour integral},
|
||||
Pages = {187 - 202},
|
||||
Title = {Cubic Scaling Algorithms for RPA Correlation Using Interpolative Separable Density Fitting},
|
||||
Url = {http://www.sciencedirect.com/science/article/pii/S002199911730671X},
|
||||
Volume = {351},
|
||||
Year = {2017},
|
||||
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S002199911730671X},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1016/j.jcp.2017.09.012}}
|
||||
|
||||
@article{Yabana_1996,
|
||||
title = {Time-Dependent Local-Density Approximation in Real Time},
|
||||
author = {Yabana, K. and Bertsch, G. F.},
|
||||
journal = {Phys. Rev. B},
|
||||
volume = {54},
|
||||
issue = {7},
|
||||
pages = {4484--4487},
|
||||
numpages = {0},
|
||||
year = {1996},
|
||||
month = {Aug},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevB.54.4484},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevB.54.4484}
|
||||
}
|
||||
Author = {Yabana, K. and Bertsch, G. F.},
|
||||
Doi = {10.1103/PhysRevB.54.4484},
|
||||
Issue = {7},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Aug},
|
||||
Numpages = {0},
|
||||
Pages = {4484--4487},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Time-Dependent Local-Density Approximation in Real Time},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.54.4484},
|
||||
Volume = {54},
|
||||
Year = {1996},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.54.4484},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.54.4484}}
|
||||
|
||||
@article{Rabani_2015,
|
||||
title = {Time-Dependent Stochastic Bethe-Salpeter Approach},
|
||||
author = {Rabani, Eran and Baer, Roi and Neuhauser, Daniel},
|
||||
journal = {Phys. Rev. B},
|
||||
volume = {91},
|
||||
issue = {23},
|
||||
pages = {235302},
|
||||
numpages = {10},
|
||||
year = {2015},
|
||||
month = {Jun},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevB.91.235302},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevB.91.235302}
|
||||
}
|
||||
|
||||
Author = {Rabani, Eran and Baer, Roi and Neuhauser, Daniel},
|
||||
Doi = {10.1103/PhysRevB.91.235302},
|
||||
Issue = {23},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Jun},
|
||||
Numpages = {10},
|
||||
Pages = {235302},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Time-Dependent Stochastic Bethe-Salpeter Approach},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.91.235302},
|
||||
Volume = {91},
|
||||
Year = {2015},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.91.235302},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.91.235302}}
|
||||
|
||||
@article{Elliott_2019,
|
||||
author = {Elliott, Joshua D. and Colonna, Nicola and Marsili, Margherita and Marzari, Nicola and Umari, Paolo},
|
||||
title = {Koopmans Meets Bethe–Salpeter: Excitonic Optical Spectra without GW},
|
||||
journal = {J. Chem. Theory Comput. },
|
||||
volume = {15},
|
||||
number = {6},
|
||||
pages = {3710-3720},
|
||||
year = {2019},
|
||||
doi = {10.1021/acs.jctc.8b01271},
|
||||
note ={PMID: 30998361},
|
||||
URL = { https://doi.org/10.1021/acs.jctc.8b01271},
|
||||
eprint = { https://doi.org/10.1021/acs.jctc.8b01271}
|
||||
}
|
||||
Author = {Elliott, Joshua D. and Colonna, Nicola and Marsili, Margherita and Marzari, Nicola and Umari, Paolo},
|
||||
Doi = {10.1021/acs.jctc.8b01271},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.8b01271},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 30998361},
|
||||
Number = {6},
|
||||
Pages = {3710-3720},
|
||||
Title = {Koopmans Meets Bethe--Salpeter: Excitonic Optical Spectra without GW},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.8b01271},
|
||||
Volume = {15},
|
||||
Year = {2019},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01271}}
|
||||
|
||||
@article{Vlcek_2017,
|
||||
author = {Vl\v{c}ek, Vojt\v{e}ch and Rabani, Eran and Neuhauser, Daniel and Baer, Roi},
|
||||
title = {Stochastic GW Calculations for Molecules},
|
||||
journal = {J. Chem. Theory Comput. },
|
||||
volume = {13},
|
||||
number = {10},
|
||||
pages = {4997-5003},
|
||||
year = {2017},
|
||||
doi = {10.1021/acs.jctc.7b00770},
|
||||
note ={PMID: 28876912},
|
||||
URL = { https://doi.org/10.1021/acs.jctc.7b00770},
|
||||
eprint = { https://doi.org/10.1021/acs.jctc.7b00770}
|
||||
}
|
||||
|
||||
Author = {Vl\v{c}ek, Vojt\v{e}ch and Rabani, Eran and Neuhauser, Daniel and Baer, Roi},
|
||||
Doi = {10.1021/acs.jctc.7b00770},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.7b00770},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 28876912},
|
||||
Number = {10},
|
||||
Pages = {4997-5003},
|
||||
Title = {Stochastic GW Calculations for Molecules},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.7b00770},
|
||||
Volume = {13},
|
||||
Year = {2017},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00770}}
|
||||
|
||||
@article{Wilhelm_2018,
|
||||
author = {Wilhelm, Jan and Golze, Dorothea and Talirz, Leopold and Hutter, Jürg and Pignedoli, Carlo A.},
|
||||
title = {Toward GW Calculations on Thousands of Atoms},
|
||||
journal = {The Journal of Physical Chemistry Letters},
|
||||
volume = {9},
|
||||
number = {2},
|
||||
pages = {306-312},
|
||||
year = {2018},
|
||||
doi = {10.1021/acs.jpclett.7b02740},
|
||||
note ={PMID: 29280376},
|
||||
URL = { https://doi.org/10.1021/acs.jpclett.7b02740},
|
||||
eprint = { https://doi.org/10.1021/acs.jpclett.7b02740}
|
||||
}
|
||||
Author = {Wilhelm, Jan and Golze, Dorothea and Talirz, Leopold and Hutter, J{\"u}rg and Pignedoli, Carlo A.},
|
||||
Doi = {10.1021/acs.jpclett.7b02740},
|
||||
Eprint = {https://doi.org/10.1021/acs.jpclett.7b02740},
|
||||
Journal = {The Journal of Physical Chemistry Letters},
|
||||
Note = {PMID: 29280376},
|
||||
Number = {2},
|
||||
Pages = {306-312},
|
||||
Title = {Toward GW Calculations on Thousands of Atoms},
|
||||
Url = {https://doi.org/10.1021/acs.jpclett.7b02740},
|
||||
Volume = {9},
|
||||
Year = {2018},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.7b02740}}
|
||||
|
||||
@article{Foerster_2011,
|
||||
author = {Foerster,D. and Koval,P. and Sánchez-Portal,D. },
|
||||
title = {An O(N3) implementation of Hedin's GW approximation for molecules},
|
||||
journal = { J. Chem. Phys. },
|
||||
volume = {135},
|
||||
number = {7},
|
||||
pages = {074105},
|
||||
year = {2011},
|
||||
doi = {10.1063/1.3624731},
|
||||
URL = { https://doi.org/10.1063/1.3624731},
|
||||
eprint = { https://doi.org/10.1063/1.3624731}
|
||||
}
|
||||
Author = {Foerster,D. and Koval,P. and S{\'a}nchez-Portal,D.},
|
||||
Doi = {10.1063/1.3624731},
|
||||
Eprint = {https://doi.org/10.1063/1.3624731},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Number = {7},
|
||||
Pages = {074105},
|
||||
Title = {An O(N3) implementation of Hedin's GW approximation for molecules},
|
||||
Url = {https://doi.org/10.1063/1.3624731},
|
||||
Volume = {135},
|
||||
Year = {2011},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.3624731}}
|
||||
|
||||
@article{Gao_2020,
|
||||
author = {Gao, Weiwei and Chelikowsky, James R.},
|
||||
title = {Accelerating Time-Dependent Density Functional Theory and GW Calculations for Molecules and Nanoclusters with Symmetry Adapted Interpolative Separable Density Fitting},
|
||||
journal = {J. Chem. Theory Comput.},
|
||||
volume = {16},
|
||||
number = {4},
|
||||
pages = {2216-2223},
|
||||
year = {2020},
|
||||
doi = {10.1021/acs.jctc.9b01025},
|
||||
note ={PMID: 32074452},
|
||||
URL = { https://doi.org/10.1021/acs.jctc.9b01025},
|
||||
eprint = { https://doi.org/10.1021/acs.jctc.9b01025}
|
||||
}
|
||||
|
||||
|
||||
Author = {Gao, Weiwei and Chelikowsky, James R.},
|
||||
Doi = {10.1021/acs.jctc.9b01025},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.9b01025},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 32074452},
|
||||
Number = {4},
|
||||
Pages = {2216-2223},
|
||||
Title = {Accelerating Time-Dependent Density Functional Theory and GW Calculations for Molecules and Nanoclusters with Symmetry Adapted Interpolative Separable Density Fitting},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.9b01025},
|
||||
Volume = {16},
|
||||
Year = {2020},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b01025}}
|
||||
|
|
|
|||
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Reference in New Issue