diff --git a/Manuscript/BSE_JPCL.bib b/Manuscript/BSE_JPCL.bib index 269ded9..cd00b48 100644 --- a/Manuscript/BSE_JPCL.bib +++ b/Manuscript/BSE_JPCL.bib @@ -1,7 +1,7 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-06-04 22:08:23 +0200 +%% Created for Pierre-Francois Loos at 2020-06-05 12:56:19 +0200 %% Saved with string encoding Unicode (UTF-8) @@ -722,18 +722,18 @@ Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}} @article{Duchemin_2020, -author = {Duchemin, Ivan and Blase, Xavier}, -title = {Robust Analytic-Continuation Approach to Many-Body GW Calculations}, -journal = { J. Chem. Theory Comput. }, -volume = {16}, -number = {3}, -pages = {1742-1756}, -year = {2020}, -doi = {10.1021/acs.jctc.9b01235}, - note ={PMID: 32023052}, -URL = { https://doi.org/10.1021/acs.jctc.9b01235}, -eprint = { https://doi.org/10.1021/acs.jctc.9b01235} -} + Author = {Duchemin, Ivan and Blase, Xavier}, + Doi = {10.1021/acs.jctc.9b01235}, + Eprint = {https://doi.org/10.1021/acs.jctc.9b01235}, + Journal = {J. Chem. Theory Comput.}, + Note = {PMID: 32023052}, + Number = {3}, + Pages = {1742-1756}, + Title = {Robust Analytic-Continuation Approach to Many-Body GW Calculations}, + Url = {https://doi.org/10.1021/acs.jctc.9b01235}, + Volume = {16}, + Year = {2020}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b01235}} @article{Dunning_1989, Author = {T. H. {Dunning, Jr.}}, @@ -2380,7 +2380,7 @@ eprint = { https://doi.org/10.1021/acs.jctc.9b01235} Date-Modified = {2020-05-18 21:40:28 +0200}, Doi = {10.1021/acs.jpclett.7b02740}, Eprint = {https://doi.org/10.1021/acs.jpclett.7b02740}, - Journal = { J. Phys. Chem. Lett. }, + Journal = {J. Phys. Chem. Lett.}, Note = {PMID: 29280376}, Number = {2}, Pages = {306-312}, @@ -12745,21 +12745,6 @@ eprint = { https://doi.org/10.1021/acs.jctc.9b01235} Year = {2016}, Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4940139}} -@article{Boulanger_2014, - Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier}, - Doi = {10.1021/ct401101u}, - File = {/Users/loos/Zotero/storage/KTW3SS9F/Boulanger_2014.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Language = {en}, - Month = mar, - Number = {3}, - Pages = {1212--1218}, - Title = {Fast and {{Accurate Electronic Excitations}} in {{Cyanines}} with the {{Many}}-{{Body Bethe}}\textendash{}{{Salpeter Approach}}}, - Volume = {10}, - Year = {2014}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct401101u}} - @article{Bruneval_2009, Author = {Bruneval, Fabien}, Doi = {10.1103/PhysRevLett.103.176403}, @@ -14645,90 +14630,95 @@ eprint = { https://doi.org/10.1021/acs.jctc.9b01235} Bdsk-Url-2 = {https://doi.org/10.1088/1367-2630/14/5/053020}} @article{Nguyen_2019, - title = {Finite-Field Approach to Solving the Bethe-Salpeter Equation}, - author = {Nguyen, Ngoc Linh and Ma, He and Govoni, Marco and Gygi, Francois and Galli, Giulia}, - journal = {Phys. Rev. Lett.}, - volume = {122}, - issue = {23}, - pages = {237402}, - numpages = {6}, - year = {2019}, - month = {Jun}, - publisher = {American Physical Society}, - doi = {10.1103/PhysRevLett.122.237402}, - url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.237402} -} + Author = {Nguyen, Ngoc Linh and Ma, He and Govoni, Marco and Gygi, Francois and Galli, Giulia}, + Doi = {10.1103/PhysRevLett.122.237402}, + Issue = {23}, + Journal = {Phys. Rev. Lett.}, + Month = {Jun}, + Numpages = {6}, + Pages = {237402}, + Publisher = {American Physical Society}, + Title = {Finite-Field Approach to Solving the Bethe-Salpeter Equation}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.237402}, + Volume = {122}, + Year = {2019}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.122.237402}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.122.237402}} @article{Boulanger_2014, -author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier}, -title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe–Salpeter Approach}, -journal = {J. Chem. Theory Comput. }, -volume = {10}, -number = {3}, -pages = {1212-1218}, -year = {2014}, -doi = {10.1021/ct401101u}, - note ={PMID: 26580191}, -URL = { https://doi.org/10.1021/ct401101u}, -eprint = { https://doi.org/10.1021/ct401101u} -} + Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier}, + Doi = {10.1021/ct401101u}, + Eprint = {https://doi.org/10.1021/ct401101u}, + Journal = {J. Chem. Theory Comput.}, + Note = {PMID: 26580191}, + Number = {3}, + Pages = {1212-1218}, + Title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe--Salpeter Approach}, + Url = {https://doi.org/10.1021/ct401101u}, + Volume = {10}, + Year = {2014}, + Bdsk-Url-1 = {https://doi.org/10.1021/ct401101u}} @article{Spataru_2013, - title = {Electronic and optical gap renormalization in carbon nanotubes near a metallic surface}, - author = {Spataru, Catalin D.}, - journal = {Phys. Rev. B}, - volume = {88}, - issue = {12}, - pages = {125412}, - numpages = {8}, - year = {2013}, - month = {Sep}, - publisher = {American Physical Society}, - doi = {10.1103/PhysRevB.88.125412}, - url = {https://link.aps.org/doi/10.1103/PhysRevB.88.125412} -} + Author = {Spataru, Catalin D.}, + Doi = {10.1103/PhysRevB.88.125412}, + Issue = {12}, + Journal = {Phys. Rev. B}, + Month = {Sep}, + Numpages = {8}, + Pages = {125412}, + Publisher = {American Physical Society}, + Title = {Electronic and optical gap renormalization in carbon nanotubes near a metallic surface}, + Url = {https://link.aps.org/doi/10.1103/PhysRevB.88.125412}, + Volume = {88}, + Year = {2013}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.88.125412}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.88.125412}} @article{Rohlfing_2012, - title = {Redshift of Excitons in Carbon Nanotubes Caused by the Environment Polarizability}, - author = {Rohlfing, Michael}, - journal = {Phys. Rev. Lett.}, - volume = {108}, - issue = {8}, - pages = {087402}, - numpages = {5}, - year = {2012}, - month = {Feb}, - publisher = {American Physical Society}, - doi = {10.1103/PhysRevLett.108.087402}, - url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402} -} + Author = {Rohlfing, Michael}, + Doi = {10.1103/PhysRevLett.108.087402}, + Issue = {8}, + Journal = {Phys. Rev. Lett.}, + Month = {Feb}, + Numpages = {5}, + Pages = {087402}, + Publisher = {American Physical Society}, + Title = {Redshift of Excitons in Carbon Nanotubes Caused by the Environment Polarizability}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402}, + Volume = {108}, + Year = {2012}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.108.087402}} @article{Yin_2014, - title = {Charge-Transfer Excited States in Aqueous DNA: Insights from Many-Body Green's Function Theory}, - author = {Yin, Huabing and Ma, Yuchen and Mu, Jinglin and Liu, Chengbu and Rohlfing, Michael}, - journal = {Phys. Rev. Lett.}, - volume = {112}, - issue = {22}, - pages = {228301}, - numpages = {5}, - year = {2014}, - month = {Jun}, - publisher = {American Physical Society}, - doi = {10.1103/PhysRevLett.112.228301}, - url = {https://link.aps.org/doi/10.1103/PhysRevLett.112.228301} -} + Author = {Yin, Huabing and Ma, Yuchen and Mu, Jinglin and Liu, Chengbu and Rohlfing, Michael}, + Doi = {10.1103/PhysRevLett.112.228301}, + Issue = {22}, + Journal = {Phys. Rev. Lett.}, + Month = {Jun}, + Numpages = {5}, + Pages = {228301}, + Publisher = {American Physical Society}, + Title = {Charge-Transfer Excited States in Aqueous DNA: Insights from Many-Body Green's Function Theory}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.112.228301}, + Volume = {112}, + Year = {2014}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.112.228301}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.112.228301}} @article{Li_2017b, - title = {Correlated electron-hole mechanism for molecular doping in organic semiconductors}, - author = {Li, Jing and D'Avino, Gabriele and Pershin, Anton and Jacquemin, Denis and Duchemin, Ivan and Beljonne, David and Blase, Xavier}, - journal = {Phys. Rev. Materials}, - volume = {1}, - issue = {2}, - pages = {025602}, - numpages = {9}, - year = {2017}, - month = {Jul}, - publisher = {American Physical Society}, - doi = {10.1103/PhysRevMaterials.1.025602}, - url = {https://link.aps.org/doi/10.1103/PhysRevMaterials.1.025602} -} + Author = {Li, Jing and D'Avino, Gabriele and Pershin, Anton and Jacquemin, Denis and Duchemin, Ivan and Beljonne, David and Blase, Xavier}, + Doi = {10.1103/PhysRevMaterials.1.025602}, + Issue = {2}, + Journal = {Phys. Rev. Materials}, + Month = {Jul}, + Numpages = {9}, + Pages = {025602}, + Publisher = {American Physical Society}, + Title = {Correlated electron-hole mechanism for molecular doping in organic semiconductors}, + Url = {https://link.aps.org/doi/10.1103/PhysRevMaterials.1.025602}, + Volume = {1}, + Year = {2017}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevMaterials.1.025602}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevMaterials.1.025602}} diff --git a/Manuscript/BSE_JPCL.tex b/Manuscript/BSE_JPCL.tex index 79e4581..c2ed4c9 100644 --- a/Manuscript/BSE_JPCL.tex +++ b/Manuscript/BSE_JPCL.tex @@ -199,7 +199,7 @@ \begin{abstract} The many-body Green's function Bethe-Salpeter equation (BSE) formalism is steadily asserting itself as a new efficient and accurate tool in the armada of computational methods available to chemists in order to predict neutral electronic excitations in molecular systems. -In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable charged excitations and quasiparticle energies, and the Bethe-Salpeter formalism, able to catch excitonic effects, has shown to provide accurate \xavier{singlet ?} excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV. +In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable charged excitations and quasiparticle energies, and the Bethe-Salpeter formalism, able to catch excitonic effects, has shown to provide accurate singlet excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV. With a similar computational cost as time-dependent density-functional theory (TD-DFT), the BSE formalism is then able to provide an accuracy on par with the most accurate global and range-separated hybrid functionals without the unsettling choice of the exchange-correlation functional, resolving further known issues (\textit{e.g.}, charge-transfer excitations) and offering a well-defined path to dynamical kernels. In this \textit{Perspective} article, we provide a historical overview of the BSE formalism, with a particular focus on its condensed-matter roots. We also propose a critical review of its strengths and weaknesses for different chemical situations. @@ -304,13 +304,13 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo %%% FIG 1 %%% \begin{figure} - \includegraphics[width=0.7\linewidth]{fig1/fig1} + \includegraphics[width=0.55\linewidth]{fig1/fig1} \caption{ 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} The path made of back arrow shows the $GW$ process which bypasses the computation of $\Gamma$ (gray arrows). As input, one must provide KS (or HF) orbitals and their corresponding energies. Depending on the level of self-consistency of the $GW$ calculation, only the orbital energies or both the orbitals and their energies are corrected. - As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$, which can then be used to compute the BSE neutral excitations. + As output, $GW$ provides corrected quantities, \ie, quasiparticle energies and $W$, which can then be used to compute the BSE neutral excitations of the system of interest. \label{fig:pentagon}} \end{figure} %%% %%% %%% @@ -429,7 +429,9 @@ with electron-hole ($eh$) eigenstates written as + Y_{ia}^{m} \phi_i(\br_e) \phi_a(\br_h) ], \end{equation} where $m$ indexes the electronic excitations. -The $\lbrace \phi_{i/a} \rbrace$ are, in the case of $G_0W_0$ and ev$GW$, the input (KS) eigenstates used to build the $GW$ self-energy. They are here taken to be real in the case of finite size systems. +The $\lbrace \phi_{i/a} \rbrace$ are, in the case of $G_0W_0$ and ev$GW$, the input (KS) eigenstates used to build the $GW$ self-energy. +They are here taken to be real in the case of finite-size systems. +(In the case of complex orbitals, we refer the reader to Ref.~\citenum{Holzer_2019} for a correct use of complex conjugation.) The resonant and coupling parts of the BSE Hamiltonian read \begin{gather} R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \kappa (ia|jb) - W_{ij,ab}, @@ -451,7 +453,7 @@ $(ia|jb)$ bare Coulomb term defined as \end{equation} Neglecting the coupling term $C$ between the resonant term $R$ and anti-resonant term $-R^*$ in Eq.~\eqref{eq:BSE-eigen}, leads to the well-known Tamm-Dancoff approximation. -As compared to TD-DFT, \; i) the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues, and ii) the non-local screened Coulomb matrix elements replaces the DFT xc kernel. +As compared to TD-DFT, i) the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues, and ii) the non-local screened Coulomb matrix elements replaces the DFT xc kernel. We emphasize that these equations can be solved at exactly the same cost as the standard TD-DFT equations once the quasiparticle energies and screened Coulomb potential $W$ are inherited from preceding $GW$ calculations. This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, highlighting its pros and cons. \\ @@ -463,14 +465,14 @@ This defines the standard (static) BSE@$GW$ scheme that we discuss in this \text 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} Three decades later, the first \textit{ab initio} implementations, starting with small clusters \cite{Onida_1995,Rohlfing_1998} extended semiconductors and wide-gap insulators, \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999b} paved the way to the popularization in the solid-state physics community of the BSE formalism. -Following early applications to periodic polymers and molecules, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in the quantum chemistry community with, in particular, several benchmarks \cite{Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same parameters (geometries, basis sets, etc) than the available higher-level reference calculations, \cite{Schreiber_2008} such as CC3. \cite{Christiansen_1995} +Following early applications to periodic polymers and molecules, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003} BSE gained much momentum in the quantum chemistry community with, in particular, several benchmarks \cite{Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same parameters (geometries, basis sets, etc) than the available higher-level reference calculations. \cite{Schreiber_2008} %such as CC3. \cite{Christiansen_1995} 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. \cite{Ren_2012b} 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 \begin{equation} \Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW}, \end{equation} -with the experimental (photoemission) fundamental gap \cite{Bredas_2014} +with the experimental (photoemission) fundamental gap \begin{equation} \EgFun = I^\Nel - A^\Nel, \end{equation} @@ -505,8 +507,8 @@ where $\EB$ is the excitonic effect, that is, the stabilization implied by the a 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} 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} -As a result, BSE excitation singlet energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations. -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} +As a result, BSE singlet excitation energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations. +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} 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. @@ -557,7 +559,7 @@ We now leave the description of successes to discuss difficulties and future dir As emphasized above, the BSE eigenvalue equation in the single-excitation space [see Eq.~\eqref{eq:BSE-eigen}] is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005} Searching iteratively for the lowest eigenstates exhibits the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT. Concerning the construction of the BSE Hamiltonian, it is no more expensive than building its TD-DFT analogue with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques. Explicit calculation of the full BSE Hamiltonian in transition space can be further avoided using density matrix perturbation theory, -\cite{Rocca_10,Nguyen_2019} not reducing though the $\order*{\Norb^4}$ scaling, sacrificing further the knowledge of the eigenvectors. +\cite{Rocca_2010,Nguyen_2019} not reducing though the $\order*{\Norb^4}$ scaling, sacrificing further the knowledge of the eigenvectors. Exploiting further the locality of localized atomic basis orbitals, the BSE absorption spectrum could be obtained with $\order*{\Norb^3}$ operations using such iterative techniques. \cite{Ljungberg_2015} With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015} diff --git a/Manuscript/fig1/fig1.pdf b/Manuscript/fig1/fig1.pdf index 347585c..3da5b82 100644 Binary files a/Manuscript/fig1/fig1.pdf and b/Manuscript/fig1/fig1.pdf differ diff --git a/Manuscript/fig1/fig1.tex b/Manuscript/fig1/fig1.tex index 182c73b..3338831 100644 --- a/Manuscript/fig1/fig1.tex +++ b/Manuscript/fig1/fig1.tex @@ -35,8 +35,8 @@ decoration={snake, \node [Op4, align=center] (Sigma) at (-3*0.587785, 3*0.809017) {$\Sigma$}; \node [Input, align=center] (In) [above=of G] {}; 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