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@ -83,6 +83,18 @@
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\newcommand{\GW}{GW}
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\newcommand{\XC}{\text{xc}}
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% orbitals, gaps, etc
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\newcommand{\eps}{\varepsilon}
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\newcommand{\IP}{I}
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\newcommand{\EA}{A}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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\newcommand{\Eg}{E_\text{gap}}
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\newcommand{\EgFun}{\Eg^\text{fund}}
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\newcommand{\EgOpt}{\Eg^\text{opt}}
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\newcommand{\EB}{E_B}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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@ -164,12 +176,12 @@
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\let\oldmaketitle\maketitle
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\let\maketitle\relax
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\title{A Chemist Guide to the Bethe-Salpeter Formalism}
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\title{A Chemist Guide to the Bethe-Salpeter Equation Formalism}
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\date{\today}
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\begin{tocentry}
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\centering
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\includegraphics[width=0.8\textwidth]{../TOC/TOC}
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\includegraphics[width=\textwidth]{../TOC/TOC}
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\end{tocentry}
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@ -219,22 +231,53 @@ The study of neutral electronic excitations in condensed matter systems, from mo
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{History}
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%\section{History}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Originally developed in the framework of nuclear physics, \cite{Salpeter_1951}
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the use of the BSE formalism in condensed-matter physics emerged in the 1960's at the semi-empirical tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000}
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Three decades latter, the first \textit{ab initio} implementations, starting with small clusters \cite{Onida_1995,Rohlfing_1998} and extended semiconductors and wide-gap insulators, \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999}
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paved the way to the popularization in the solid-state physics community of the BSE formalism.
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Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} the BSE formalism has emerged in condensed-matter physics around the 1960's at the semi-empirical tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000}
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Three decades later, the first \textit{ab initio} implementations, starting with small clusters \cite{Onida_1995,Rohlfing_1998} extended semiconductors and wide-gap insulators, \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999} paved the way to the popularization in the solid-state physics community of the BSE formalism.
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Following early applications to periodic polymers and molecules, [REFS] the BSE formalism gained much momentum in the quantum chemistry community with in particular several benchmarks \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same running parameters (geometries, basis sets) than the available reference higher-level calculations such as CC3. Such comparisons were grounded in the development of codes replacing the planewave solid-state physics paradigm by well documented correlation-consistent Gaussian basis sets, together with adequate auxiliary bases when resolution-of-the-identity techniques were used. [REFS]
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Following early applications to periodic polymers and molecules, [REFS] BSE gained much momentum in the quantum chemistry community with, in particular, several benchmarks \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same parameters (geometries, basis sets, etc) than the available higher-level reference calculations, \cite{Schreiber_2008} such as CC3. \cite{Christiansen_1995}
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Such comparisons were grounded in the development of codes replacing the plane-wave paradigm of solid-state physics by well-documented correlation-consistent Gaussian basis sets, \cite{Dunning_1989} together with adequate auxiliary bases when resolution-of-the-identity (RI) techniques were used. [REFS]
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An important conclusion drawn from these calculations was that the quality of the BSE excitation energies are strongly correlated to the deviation of the preceding $GW$ HOMO-LUMO gap with the experimental (IP-AE) photoemission gap. Standard $G_0W_0$ calculations starting with Kohn-Sham (KS) eigenstates generated with (semi)local functionals yield much larger HOMO-LUMO gaps than the input KS one, but still too small as compared to the experimental (AE-IP) value. Such an underestimation of the (IP-AE) gap leads to a similar underestimation of the lowest optical excitation energies.
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An important conclusion drawn from these calculations was that the quality of the BSE excitation energies is strongly correlated to the deviation of the preceding $GW$ HOMO-LUMO gap
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\begin{equation}
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\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW}
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\end{equation}
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with the experimental (photoemission) fundamental gap \cite{Bredas_2014}
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\begin{equation}
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\EgFun = I - A,
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\end{equation}
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where $I = E_0^{N-1} - E_0^N$ and $A = E_0^{N+1} - E_0^N$ are the ionization potential and the electron affinity of the $N$-electron system, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ corresponds to the $N$-electron ground-state energy.
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Such a residual HOMO-LUMO gap problem can be significantly improved by adopting exchange-correlation (xc) functionals with a tuned amount of exact exchange that yield a much improved KS HOMO-LUMO gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016} Alternatively, self-consistent schemes, 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} As a result, BSE excitation singlet energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations such as CC3. For sake of illustration, an average 0.2 eV error was found for the well-known Thiel set comprising more than a hundred representative singlet excitations from a large variety of representative molecules.
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\cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017} This is equivalent to the best TD-DFT results obtained by scanning a large variety of global hybrid functionals with varying fraction of exact exchange.
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Standard $G_0W_0$ calculations starting with Kohn-Sham (KS) eigenstates generated with (semi)local functionals yield much larger HOMO-LUMO gaps than the input KS gap
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\begin{equation}
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\Eg^{\KS} = \eps_{\LUMO}^{\KS} - \varepsilon_{\HOMO}^{\KS},
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\end{equation}
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but still too small as compared to the experimental value, \ie,
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\begin{equation}
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\Eg^{\KS} \ll \Eg^{G_0W_0} < \EgFun.
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\end{equation}
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Such an underestimation of the fundamental gap leads to a similar underestimation of the optical gap $\EgOpt$, \ie, the lowest optical excitation energy.
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\begin{equation}
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\EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB,
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\end{equation}
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where $\EB$ is the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind.
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Because of this, we have $\EgOpt < \EgFun$.
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A very remarkable success of the BSE formalism lies in the description of the charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT calculations adopting standard functionals. \cite{Dreuw_2004} Similar difficulties emerge as well in solid-state physics for semiconductors where extended Wannier excitons are characterized by weakly overlapping electrons and holes, causing a dramatic deficit of spectral weight at low energy. \cite{Botti_2004} These difficulties can be ascribed to the lack of long-range electron-hole interaction with local XC functionals that can be cured through an exact exchange contribution, a solution that explains in particular the success of range-separated hybrids for the description of CT excitations. \cite{Stein_2009} The analysis of the screened Coulomb potential matrix elements in the BSE kernel (see Eqn.~\ref{Wmatel}) reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc.) where 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{Blase_2011b,Baumeier_2012,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems. We now leave the description of successes to discuss difficulties and Perspectives.\\
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Such a residual HOMO-LUMO gap problem can be significantly improved by adopting exchange-correlation (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, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011} where corrected eigenvalues, and possibly orbitals, \cite{Faleev_2004, vanSchilfgaarde_2006, Kotani_2007, Ke_2011} 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 excitation singlet 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 more than hundred 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 global hybrid functionals with various amounts of exact exchange.
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A very remarkable success of the BSE formalism lies in the description of the charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT calculations adopting standard functionals. \cite{Dreuw_2004}
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Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes, 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 in particular 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 below) reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where screening reduces the long-range electron-hole interactions.
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The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011b,Baumeier_2012,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems.\\
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%We now leave the description of successes to discuss difficulties and Perspectives.\\
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%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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@ -256,8 +299,7 @@ A central property of the one-body Green's function is that its spectral represe
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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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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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\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 KS 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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@ -511,12 +553,12 @@ diabatization and conical intersections \cite{Kaczmarski_2010}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion.}
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here goes the conclusion.
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%% ACKNOWLEDGEMENTS %%%
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\section*{Acknowledgments}
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%%%%%%%%%%%%%%%%%%%%%%%%
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DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support.
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@ -1,13 +1,354 @@
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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 2020-05-13 11:30:11 +0200
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%% Created for Pierre-Francois Loos at 2020-05-14 13:56:12 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Ke_2011,
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Doi = {10.1103/PhysRevB.76.165106},
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Journal = {J. Chem. Theory Comput.},
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Pages = {877--883},
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Title = {Assessing {{{\emph{GW}}}} {{Approaches}} for {{Predicting Core Level Binding Energies}}},
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Journal = {J. Chem. Theory Comput.},
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Language = {en},
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Month = dec,
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Number = {12},
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Pages = {5665--5687},
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Shorttitle = {{{{\emph{GW}}}} 100},
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Title = {{{{\emph{GW}}}} 100: {{Benchmarking}} {{{\emph{G}}}}{\textsubscript{0}}{{{\emph{W}}}}{\textsubscript{0}} for {{Molecular Systems}}},
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Volume = {11},
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Year = {2015},
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Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.5b00453}}
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Doi = {10.1021/ct300648t},
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File = {/Users/loos/Zotero/storage/SLTIFNCK/vanSetten_2013.pdf},
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Issn = {1549-9618, 1549-9626},
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Journal = {J. Chem. Theory Comput.},
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Language = {en},
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Month = jan,
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||||
Number = {1},
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||||
Pages = {232--246},
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Shorttitle = {The {{{\emph{GW}}}} -{{Method}} for {{Quantum Chemistry Applications}}},
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Doi = {10.1103/PhysRevLett.93.126406},
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Language = {en},
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@article{Shishkin_2007,
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Author = {Shishkin, M. and Kresse, G.},
|
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Date-Added = {2020-05-14 13:39:36 +0200},
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Date-Modified = {2020-05-14 13:39:36 +0200},
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|
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||||
@article{Faber_2011,
|
||||
Author = {Faber, Carina and Attaccalite, Claudio and Olevano, V. and Runge, E. and Blase, X.},
|
||||
Date-Added = {2020-05-14 13:39:24 +0200},
|
||||
Date-Modified = {2020-05-14 13:39:24 +0200},
|
||||
Doi = {10.1103/PhysRevB.83.115123},
|
||||
Issn = {1098-0121, 1550-235X},
|
||||
Journal = {Phys. Rev. B},
|
||||
Language = {en},
|
||||
Month = mar,
|
||||
Number = {11},
|
||||
Pages = {115123},
|
||||
Title = {First-Principles {{GW}} Calculations for {{DNA}} and {{RNA}} Nucleobases},
|
||||
Volume = {83},
|
||||
Year = {2011},
|
||||
Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.83.115123}}
|
||||
|
||||
@article{Faber_2011b,
|
||||
Author = {Faber, Carina and Janssen, Jonathan Laflamme and C\^ot\'e, Michel and Runge, E. and Blase, X.},
|
||||
Date-Added = {2020-05-14 13:39:24 +0200},
|
||||
Date-Modified = {2020-05-14 13:39:24 +0200},
|
||||
Doi = {10.1103/PhysRevB.84.155104},
|
||||
Issue = {15},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Oct},
|
||||
Numpages = {5},
|
||||
Pages = {155104},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Electron-phonon coupling in the C${}_{60}$ fullerene within the many-body $GW$ approach},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.84.155104},
|
||||
Volume = {84},
|
||||
Year = {2011},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.84.155104},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.84.155104}}
|
||||
|
||||
@phdthesis{Faber_2014,
|
||||
Author = {Faber, Carina},
|
||||
Date-Added = {2020-05-14 13:39:24 +0200},
|
||||
Date-Modified = {2020-05-14 13:39:24 +0200},
|
||||
File = {/Users/loos/Zotero/storage/ZPJDQPGS/Faber_PhD.pdf},
|
||||
School = {Universit{\'e} de Grenoble},
|
||||
Shorttitle = {Electronic, Excitonic and Polaronic Properties of Organic Systems within the Many-Body {{GW}} and {{Bethe}}-{{Salpeter}} Formalisms},
|
||||
Title = {Electronic, Excitonic and Polaronic Properties of Organic Systems within the Many-Body {{GW}} and {{Bethe}}-{{Salpeter}} Formalisms: Towards Organic Photovoltaics},
|
||||
Type = {{{PhD Thesis}}},
|
||||
Year = {2014}}
|
||||
|
||||
@article{Faber_2015,
|
||||
Author = {Faber, C. and Boulanger, P. and Attaccalite, C. and Cannuccia, E. and Duchemin, I. and Deutsch, T. and Blase, X.},
|
||||
Date-Added = {2020-05-14 13:39:24 +0200},
|
||||
Date-Modified = {2020-05-14 13:39:24 +0200},
|
||||
Doi = {10.1103/PhysRevB.91.155109},
|
||||
Issue = {15},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Apr},
|
||||
Numpages = {9},
|
||||
Pages = {155109},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Exploring Approximations to the $GW$ Self-Energy Ionic Gradients},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.91.155109},
|
||||
Volume = {91},
|
||||
Year = {2015},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.91.155109},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.91.155109}}
|
||||
|
||||
@article{Kronik_2012,
|
||||
Author = {Leeor Kronik and Tamar Stein and Sivan {Refaely-Abramson} and Roi Baer},
|
||||
Date-Added = {2020-05-14 13:35:26 +0200},
|
||||
Date-Modified = {2020-05-14 13:37:05 +0200},
|
||||
Doi = {10.1021/ct2009363},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Pages = {1515--1531},
|
||||
Title = {Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals},
|
||||
Volume = {8},
|
||||
Year = {2012},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/ct2009363}}
|
||||
|
||||
@article{Kraisler_2013,
|
||||
Author = {Kraisler, Eli and Kronik, Leeor},
|
||||
Date-Added = {2020-05-14 13:33:01 +0200},
|
||||
Date-Modified = {2020-05-14 13:33:01 +0200},
|
||||
Doi = {10.1103/PhysRevLett.110.126403},
|
||||
File = {/Users/loos/Zotero/storage/ELUWJQ66/Kraisler and Kronik - 2013 - Piecewise Linearity of Approximate Density Functio.pdf},
|
||||
Issn = {0031-9007, 1079-7114},
|
||||
Journal = {Phys. Rev. Lett.},
|
||||
Language = {en},
|
||||
Month = mar,
|
||||
Number = {12},
|
||||
Pages = {126403},
|
||||
Shorttitle = {Piecewise {{Linearity}} of {{Approximate Density Functionals Revisited}}},
|
||||
Title = {Piecewise {{Linearity}} of {{Approximate Density Functionals Revisited}}: {{Implications}} for {{Frontier Orbital Energies}}},
|
||||
Volume = {110},
|
||||
Year = {2013},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.110.126403}}
|
||||
|
||||
@article{Kraisler_2014,
|
||||
Author = {Kraisler, Eli and Kronik, Leeor},
|
||||
Date-Added = {2020-05-14 13:33:01 +0200},
|
||||
Date-Modified = {2020-05-14 13:33:01 +0200},
|
||||
Doi = {10.1063/1.4871462},
|
||||
File = {/Users/loos/Zotero/storage/SLYF4MBS/Kraisler and Kronik - 2014 - Fundamental gaps with approximate density function.pdf},
|
||||
Issn = {0021-9606, 1089-7690},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Language = {en},
|
||||
Month = may,
|
||||
Number = {18},
|
||||
Pages = {18A540},
|
||||
Shorttitle = {Fundamental Gaps with Approximate Density Functionals},
|
||||
Title = {Fundamental Gaps with Approximate Density Functionals: {{The}} Derivative Discontinuity Revealed from Ensemble Considerations},
|
||||
Volume = {140},
|
||||
Year = {2014},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.4871462}}
|
||||
|
||||
@article{Refaely-Abramson_2012,
|
||||
Author = {Sivan Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik},
|
||||
Date-Added = {2020-05-14 13:33:01 +0200},
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Date-Modified = {2020-05-14 13:33:01 +0200},
|
||||
Doi = {10.1103/PhysRevLett.109.226405},
|
||||
Journal = {Phys. Rev. X},
|
||||
Pages = {226405},
|
||||
Title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional},
|
||||
Volume = {109},
|
||||
Year = {2012},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.109.226405}}
|
||||
|
||||
@article{Stein_2010,
|
||||
Author = {Tamar Stein and Helen Eisenberg and Leeor Kronik and Roi Baer},
|
||||
Date-Added = {2020-05-14 13:33:01 +0200},
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||||
Date-Modified = {2020-05-14 13:33:01 +0200},
|
||||
Doi = {10.1103/PhysRevLett.105.266802},
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||||
Journal = {Phys. Rev. Lett.},
|
||||
Pages = {266802},
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Title = {Fundamental Gaps in Finite Systems from Eigenvalues of a Generalized Kohn-Sham Method},
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||||
Volume = {105},
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||||
Year = {2010},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.105.266802}}
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||||
|
||||
@article{Stein_2012,
|
||||
Author = {Tamar Stein and Jochen Autschbach and Niranjan Govind and Leeor Kronik and Roi Baer},
|
||||
Date-Added = {2020-05-14 13:33:01 +0200},
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Date-Modified = {2020-05-14 13:33:01 +0200},
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||||
Doi = {10.1021/jz3015937},
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Journal = {J. Phys. Chem. Lett.},
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||||
Pages = {3740},
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Title = {Curvature and Frontier Orbital Energies in Density Functional Theory},
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||||
Volume = {3},
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||||
Year = {2012},
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||||
Bdsk-Url-1 = {https://doi.org/10.1021/jz3015937}}
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||||
|
||||
@article{Bredas_2014,
|
||||
Author = {Bredas, Jean-Luc},
|
||||
Date-Added = {2020-05-14 13:30:22 +0200},
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Date-Modified = {2020-05-14 13:30:22 +0200},
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||||
Doi = {10.1039/C3MH00098B},
|
||||
File = {/Users/loos/Zotero/storage/QZ92N6TD/Bredas - 2014 - Mind the gap!.pdf},
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||||
Issn = {2051-6347, 2051-6355},
|
||||
Journal = {Mater Horiz},
|
||||
Language = {en},
|
||||
Number = {1},
|
||||
Pages = {17-19},
|
||||
Title = {Mind the Gap!},
|
||||
Volume = {1},
|
||||
Year = {2014},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1039/C3MH00098B}}
|
||||
|
||||
@article{Dunning_1989,
|
||||
Author = {T. H. {Dunning, Jr.}},
|
||||
Date-Added = {2020-05-14 13:26:08 +0200},
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Date-Modified = {2020-05-14 13:26:08 +0200},
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||||
Doi = {10.1063/1.456153},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Pages = {1007},
|
||||
Title = {Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen},
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||||
Volume = {90},
|
||||
Year = {1989},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.456153}}
|
||||
|
||||
@article{Christiansen_1995,
|
||||
Author = {Christiansen, Ove and Koch, Henrik and J{\o}rgensen, Poul},
|
||||
Date-Added = {2020-05-14 13:25:26 +0200},
|
||||
Date-Modified = {2020-05-14 13:25:32 +0200},
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Doi = {http://dx.doi.org/10.1063/1.470315},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Number = {17},
|
||||
Pages = {7429-7441},
|
||||
Title = {Response Functions in the CC3 Iterative Triple Excitation Model},
|
||||
Url = {http://scitation.aip.org/content/aip/journal/jcp/103/17/10.1063/1.470315},
|
||||
Volume = {103},
|
||||
Year = {1995},
|
||||
Bdsk-Url-1 = {http://scitation.aip.org/content/aip/journal/jcp/103/17/10.1063/1.470315},
|
||||
Bdsk-Url-2 = {http://dx.doi.org/10.1063/1.470315}}
|
||||
|
||||
@article{Schreiber_2008,
|
||||
Author = {Schreiber, M. and Silva-Junior, M. R. and Sauer, S. P. A. and Thiel, W.},
|
||||
Date-Added = {2020-05-14 13:25:05 +0200},
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||||
Date-Modified = {2020-05-14 13:56:09 +0200},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Pages = {134110},
|
||||
Title = {Benchmarks for Electronically Excited States: CASPT2, CC2, CCSD and CC3},
|
||||
Volume = {128},
|
||||
Year = {2008}}
|
||||
|
||||
@inbook{Casida_1995,
|
||||
Author = {M. E. Casida},
|
||||
Date-Added = {2020-05-13 10:10:00 +0200},
|
||||
|
|
@ -102,7 +443,7 @@
|
|||
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,
|
||||
Journal = {Phys. Rev. Lett.},
|
||||
Pages = {997--1000},
|
||||
Title = {Density-Functional Theory for Time-Dependent Systems},
|
||||
Volume = 52,
|
||||
|
|
|
|||
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Binary file not shown.
12
TOC/TOC.tex
12
TOC/TOC.tex
|
|
@ -39,10 +39,10 @@
|
|||
R & C \\
|
||||
-C^* & R^{*}
|
||||
\end{pmatrix}
|
||||
\begin{matrix}
|
||||
\begin{pmatrix}
|
||||
X_m \\
|
||||
Y_m
|
||||
\end{matrix}
|
||||
\end{pmatrix}
|
||||
=
|
||||
\Omega_{m}
|
||||
\begin{pmatrix}
|
||||
|
|
@ -53,18 +53,18 @@
|
|||
};
|
||||
|
||||
|
||||
\node [comp2, align=center] (phys) [below=of BSE, xshift=-4cm]
|
||||
\node [comp2, align=center] (phys) [left=of BSE, xshift=-2cm]
|
||||
{\LARGE Molecules};
|
||||
|
||||
\node [comp2, align=center] (chem) [below=of BSE, xshift=0cm]
|
||||
{\LARGE Materials};
|
||||
|
||||
\node [comp2, align=center] (bio) [below=of BSE, xshift=4cm]
|
||||
\node [comp2, align=center] (bio) [right=of BSE, xshift=2cm]
|
||||
{\LARGE Clusters};
|
||||
|
||||
\path
|
||||
(KS) edge [->,color=black] node [right,black] {Fundamental gap} (GW)
|
||||
(GW) edge [->,color=black] node [right,black] {Excitonic effect} (BSE)
|
||||
(KS) edge [->,color=black] node [right,black] {\LARGE Fundamental gap} (GW)
|
||||
(GW) edge [->,color=black] node [right,black] {\LARGE Excitonic effect} (BSE)
|
||||
(BSE) edge [->,color=black] node [above,black] {} (phys)
|
||||
(BSE) edge [->,color=black] node [above,black] {} (chem)
|
||||
(BSE) edge [->,color=black] node [above,black] {} (bio)
|
||||
|
|
|
|||
Loading…
Reference in New Issue