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%% This BibTeX bibliography file was created using BibDesk.
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%% Created for Pierre-Francois Loos at 2020-06-05 21:33:09 +0200
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%% Created for Denis Jacquemin at 2020-06-08 11:44:16 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@ -17,7 +18,8 @@
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Pages = {251--263},
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Title = {Computational Studies of Photocatalysis with Metal--Organic Frameworks},
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Volume = {2},
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Year = {2019}}
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Year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1002/eem2.12051}}
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@phdthesis{Rebolini_PhD,
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Author = {E. Rebolini},
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@ -44,11 +46,11 @@
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@article{Ankudinov_2003,
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Date-Added = {2020-05-25 11:42:58 +0200},
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Date-Modified = {2020-05-25 11:43:50 +0200},
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Date-Modified = {2020-06-08 11:43:50 +0200},
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Doi = {10.1103/PhysRevB.67.115120},
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Title = {Dynamic Screening Effects in X-Ray Absorption Spectra},
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Volume = {67},
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Year = {2003},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.67.115120}}
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@ -56,7 +58,7 @@
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@article{Ma_2009a,
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Author = {Ma, Yuchen and Rohlfing, Michael and Molteni, Carla},
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Date-Added = {2020-05-25 08:51:27 +0200},
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Date-Modified = {2020-05-25 08:51:27 +0200},
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@ -64,7 +66,7 @@
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Numpages = {4},
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Pages = {241405},
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Publisher = {American Physical Society},
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Title = {Excited states of biological chromophores studied using many-body perturbation theory: Effects of resonant-antiresonant coupling and dynamical screening},
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Title = {Excited States of Biological Chromophores Studied Using Many-Body Perturbation Theory: Effects of Resonant-Antiresonant Coupling and Dynamical Screening},
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@ -156,11 +158,10 @@
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@article{Azarias_2017,
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Author = {Azarias, Clo{\'e} and Habert, Chlo\'{e} and Budz\'{a}k, \check{S}imon and Blase, Xavier and Duchemin, Ivan and Jacquemin, Denis},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-06-08 11:23:53 +0200},
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Doi = {10.1021/acs.jpca.7b05222},
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Eprint = {https://doi.org/10.1021/acs.jpca.7b05222},
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Journal = {J. Phys. Chem. A},
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Note = {PMID: 28738157},
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Pages = {6122-6134},
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Title = {Calculations of n→π* Transition Energies: Comparisons Between TD-DFT, ADC, CC, CASPT2, and BSE/GW Descriptions},
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@ -190,13 +191,13 @@
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@article{Bauernschmitt_1996,
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Author = {Bauernschmitt,R{\"u}diger and Ahlrichs,Reinhart},
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Pages = {9047-9052},
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Title = {Stability analysis for solutions of the closed shell Kohn--Sham equation},
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Title = {Stability Analysis for Solutions of the Closed Shell Kohn--Sham Equation},
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@ -356,13 +357,13 @@
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@article{Blase_2018,
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Author = {Blase, Xavier and Duchemin, Ivan and Jacquemin, Denis},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-05-18 21:40:28 +0200},
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Doi = {10.1039/C7CS00049A},
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Journal = {Chem. Soc. Rev.},
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Pages = {1022-1043},
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Publisher = {The Royal Society of Chemistry},
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Title = {The Bethe--Salpeter equation in chemistry: relations with TD-DFT{,} applications and challenges},
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Title = {The Bethe--Salpeter Equation in Chemistry: Relations with TD-DFT{,} Applications and Challenges},
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Url = {http://dx.doi.org/10.1039/C7CS00049A},
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Volume = {47},
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Year = {2018},
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@ -423,13 +424,13 @@
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@article{Bruneval_2015,
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Author = {Bruneval,Fabien and Hamed,Samia M. and Neaton,Jeffrey B.},
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Eprint = {https://doi.org/10.1063/1.4922489},
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Journal = {J. Chem. Phys.},
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Number = {24},
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Pages = {244101},
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Title = {A systematic benchmark of the ab initio Bethe-Salpeter equation approach for low-lying optical excitations of small organic molecules},
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Title = {A Systematic Benchmark of the ab initio Bethe-Salpeter Equation Approach for Low-Lying Optical Excitations of Small Organic Molecules},
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Url = {https://doi.org/10.1063/1.4922489},
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Volume = {142},
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Year = {2015},
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@ -605,7 +606,7 @@
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@article{Cudazzo_2013,
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Author = {Cudazzo, Pierluigi and Gatti, Matteo and Rubio, Angel and Sottile, Francesco},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-05-18 21:40:28 +0200},
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Doi = {10.1103/PhysRevB.88.195152},
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Journal = {Phys. Rev. B},
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@ -613,7 +614,7 @@
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Numpages = {5},
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Pages = {195152},
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Publisher = {American Physical Society},
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Title = {Frenkel versus charge-transfer exciton dispersion in molecular crystals},
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Title = {Frenkel versus Charge-Transfer Exciton Dispersion in Molecular Crystals},
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Url = {https://link.aps.org/doi/10.1103/PhysRevB.88.195152},
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@ -734,10 +735,10 @@
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@article{Duchemin_2020,
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Author = {Duchemin, Ivan and Blase, Xavier},
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Date-Modified = {2020-06-08 11:24:10 +0200},
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Doi = {10.1021/acs.jctc.9b01235},
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Note = {PMID: 32023052},
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@ -761,11 +762,10 @@
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Author = {Elliott, Joshua D. and Colonna, Nicola and Marsili, Margherita and Marzari, Nicola and Umari, Paolo},
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Journal = {J. Chem. Theory Comput.},
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Note = {PMID: 30998361},
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Number = {6},
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Pages = {3710-3720},
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Title = {Koopmans Meets Bethe--Salpeter: Excitonic Optical Spectra without GW},
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@ -793,7 +793,7 @@
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@article{Faber_2011b,
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Author = {Faber, Carina and Janssen, Jonathan Laflamme and C\^ot\'e, Michel and Runge, E. and Blase, X.},
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Publisher = {American Physical Society},
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Title = {Electron-phonon coupling in the C${}_{60}$ fullerene within the many-body $GW$ approach},
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Title = {Electron-Phonon Coupling in the C${}_{60}$ Fullerene within the Many-body $GW$ Approach},
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Url = {https://link.aps.org/doi/10.1103/PhysRevB.84.155104},
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@ -831,13 +831,13 @@
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Author = {Foerster,D. and Koval,P. and S{\'a}nchez-Portal,D.},
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@ -978,11 +977,10 @@
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Author = {Gui, Xin and Holzer, Christof and Klopper, Wim},
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Title = {Accuracy Assessment of GW Starting Points for Calculating Molecular Excitation Energies Using the Bethe--Salpeter Formalism},
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@ -1006,13 +1004,13 @@
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Author = {H{\"{a}}ser,Marco and Alml{\"{o}}f,Jan},
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@ -1077,13 +1075,13 @@
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@article{Holzer_2018a,
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||||
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|
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@ -1271,11 +1269,10 @@
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@article{Jacquemin_2015b,
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Author = {Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
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||||
Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-05-18 21:40:28 +0200},
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@ -1287,11 +1284,10 @@
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@article{Jacquemin_2017,
|
||||
Author = {Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-05-18 21:40:28 +0200},
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@ -1303,11 +1299,10 @@
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@article{Jacquemin_2017b,
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||||
Author = {Jacquemin, Denis and Duchemin, Ivan and Blondel, Aymeric and Blase, Xavier},
|
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Date-Modified = {2020-05-18 21:40:28 +0200},
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@ -1337,11 +1332,10 @@
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@article{Kaltak_2014,
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Author = {Kaltak, Merzuk and Klime\v{s}, Ji\v{i}\'{i} and Kresse, Georg},
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@ -1353,11 +1347,10 @@
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@article{Korbel_2014,
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|
||||
@ -1563,7 +1556,7 @@
|
||||
@article{Ljungberg_2015,
|
||||
Author = {Ljungberg, M. P. and Koval, P. and Ferrari, F. and Foerster, D. and S\'anchez-Portal, D.},
|
||||
Date-Added = {2020-05-18 21:40:28 +0200},
|
||||
Date-Modified = {2020-05-18 21:40:28 +0200},
|
||||
Date-Modified = {2020-06-08 11:34:06 +0200},
|
||||
Doi = {10.1103/PhysRevB.92.075422},
|
||||
Issue = {7},
|
||||
Journal = {Phys. Rev. B},
|
||||
@ -1571,7 +1564,7 @@
|
||||
Numpages = {18},
|
||||
Pages = {075422},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Cubic-scaling iterative solution of the Bethe-Salpeter equation for finite systems},
|
||||
Title = {Cubic-Scaling Iterative Solution of the Bethe-Salpeter Equation for Finite Systems},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.92.075422},
|
||||
Volume = {92},
|
||||
Year = {2015},
|
||||
@ -1739,13 +1732,13 @@
|
||||
Abstract = {We describe state of the art methods for the calculation of electronic excitations in solids and molecules{,} based on many body perturbation theory{,} and we discuss some applications of these methods to the study of band edges and absorption processes in representative materials used as photoelectrodes for water splitting.},
|
||||
Author = {Ping, Yuan and Rocca, Dario and Galli, Giulia},
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
Pages = {2437-2469},
|
||||
Publisher = {The Royal Society of Chemistry},
|
||||
Title = {Electronic excitations in light absorbers for photoelectrochemical energy conversion: first principles calculations based on many body perturbation theory},
|
||||
Title = {Electronic Excitations in Light Absorbers for Photoelectrochemical Energy Conversion: First Principles Calculations Based on Many Body Perturbation Theory},
|
||||
Url = {http://dx.doi.org/10.1039/C3CS00007A},
|
||||
Volume = {42},
|
||||
Year = {2013},
|
||||
@ -1784,13 +1777,13 @@
|
||||
@article{Rebolini_2016,
|
||||
Author = {Rebolini,Elisa and Toulouse,Julien},
|
||||
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|
||||
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|
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|
||||
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||||
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|
||||
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|
||||
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|
||||
Title = {Range-separated time-dependent density-functional theory with a frequency-dependent second-order Bethe-Salpeter correlation kernel},
|
||||
Title = {Range-Separated Time-Dependent Density-Functional Theory with a Frequency-Dependent Second-Order Bethe-Salpeter Correlation Kernel},
|
||||
Url = {https://doi.org/10.1063/1.4943003},
|
||||
Volume = {144},
|
||||
Year = {2016},
|
||||
@ -1887,7 +1880,7 @@
|
||||
@article{Rohlfing_2000,
|
||||
Author = {Rohlfing, Michael and Louie, Steven G.},
|
||||
Date-Added = {2020-05-18 21:40:28 +0200},
|
||||
Date-Modified = {2020-05-18 21:40:28 +0200},
|
||||
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|
||||
Doi = {10.1103/PhysRevB.62.4927},
|
||||
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|
||||
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|
||||
@ -1895,7 +1888,7 @@
|
||||
Numpages = {0},
|
||||
Pages = {4927--4944},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Electron-hole excitations and optical spectra from first principles},
|
||||
Title = {Electron-hole Excitations and Optical Spectra from First Principles},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.62.4927},
|
||||
Volume = {62},
|
||||
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|
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@ -1923,13 +1916,13 @@
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||||
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||||
Author = {Romaniello,P. and Sangalli,D. and Berger,J. A. and Sottile,F. and Molinari,L. G. and Reining,L. and Onida,G.},
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||||
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||||
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||||
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|
||||
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|
||||
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||||
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|
||||
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||||
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||||
Year = {2009},
|
||||
@ -2059,13 +2052,13 @@
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||||
@article{Sears_2011,
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||||
Author = {Sears,John S. and Koerzdoerfer,Thomas and Zhang,Cai-Rong and Br{\'{e}}das,Jean-Luc},
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||||
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||||
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||||
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||||
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|
||||
Title = {Orbital Instabilities and Triplet States fFrom Time-Dependent Density Functional Theory and Long-Range Corrected Functionals},
|
||||
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||||
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||||
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@ -2074,13 +2067,13 @@
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||||
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||||
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||||
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||||
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||||
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|
||||
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||||
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||||
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||||
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@ -2186,11 +2179,10 @@
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||||
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||||
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||||
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||||
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|
||||
@ -2388,11 +2380,10 @@
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||||
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||||
Author = {Wilhelm, Jan and Golze, Dorothea and Talirz, Leopold and Hutter, J{\"u}rg and Pignedoli, Carlo A.},
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||||
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||||
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||||
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||||
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||||
@ -2510,11 +2501,10 @@
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||||
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|
||||
Author = {Parrish, Robert M. and Burns, Lori A. and Smith, Daniel G. A. and Simmonett, Andrew C. and DePrince, A. Eugene and Hohenstein, Edward G. and Bozkaya, U{\u g}ur and Sokolov, Alexander Yu. and Di Remigio, Roberto and Richard, Ryan M. and Gonthier, J{\'e}r{\^o}me F. and James, Andrew M. and McAlexander, Harley R. and Kumar, Ashutosh and Saitow, Masaaki and Wang, Xiao and Pritchard, Benjamin P. and Verma, Prakash and Schaefer, Henry F. and Patkowski, Konrad and King, Rollin A. and Valeev, Edward F. and Evangelista, Francesco A. and Turney, Justin M. and Crawford, T. Daniel and Sherrill, C. David},
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||||
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||||
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||||
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||||
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||||
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||||
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Author = {E. Rebolini and J. Toulouse and A. Savin},
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||||
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||||
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||||
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||||
Editor = {S. K. Ghosh and P. K. Chattaraj},
|
||||
Note = {preprint at http://arxiv.org/abs/1304.1314},
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||||
Pages = {367-390},
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||||
Publisher = {CRC Press},
|
||||
Series = {Concepts and Methods in Modern Theoretical Chemistry Vol. 1},
|
||||
Title = {Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H$_2$ molecule},
|
||||
Title = {Electronic Excitation Energies of Molecular Systems from the Bethe-Salpeter Equation: Example of the H$_2$ Molecule},
|
||||
Year = {2013}}
|
||||
|
||||
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||||
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||||
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||||
Author = {Guareschi, Riccardo and Zulfikri, Habiburrahman and Daday, Csaba and Floris, Franca Maria and Amovilli, Claudio and Mennucci, Benedetta and Filippi, Claudia},
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||||
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||||
@ -6495,9 +6484,8 @@
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||||
Isbn = {978-1-56081-588-4 978-0-471-18576-5},
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||||
Note = {OCLC: 27226211},
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||||
Publisher = {{VCH}},
|
||||
Title = {Excited States and Photochemistry of Organic Molecules},
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||||
Year = {1995}}
|
||||
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||||
Address = {Amsterdam; Boston (Mass.); Paris},
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||||
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||||
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||||
Isbn = {978-0-08-045519-8},
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||||
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||||
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||||
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@ -11046,7 +11033,7 @@
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||||
@article{Zhang_1996,
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||||
Author = {Zhang, Nan and Hayase, Takasuke and Kawamata, Hiroshi and Nakao, Kojiro and Nakajima, Atsushi and Kaya, Koji},
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||||
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|
||||
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||||
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||||
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||||
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||||
@book{March_1999,
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||||
Address = {London},
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||||
Date-Modified = {2020-06-08 11:36:57 +0200},
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Editor = {March, Norman H.},
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||||
File = {/Users/loos/Zotero/storage/7VMQ86HQ/Farid_1999.pdf},
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||||
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Language = {eng},
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||||
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||||
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||||
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||||
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||||
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|
||||
Title = {Correlation energy for the homogeneous electron gas: Exact Bethe-Salpeter solution and an approximate evaluation},
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||||
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||||
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||||
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||||
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Title = {Electronic Excitations in Push--Pull Oligomers and Their Complexes with Fullerene from Many-Body Green's Functions Theory with Polarizable Embedding},
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@ -14426,12 +14413,13 @@
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
Doi = {10.1063/1.1867373},
|
||||
Eprint = {https://doi.org/10.1063/1.186737},
|
||||
Journal = {The Journal of Chemical Physics},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Number = {10},
|
||||
Pages = {104513},
|
||||
Title = {Electronic excitation energies of molecules in solution: State specific and linear response methods for nonequilibrium continuum solvation models},
|
||||
Title = {Electronic Excitation Energies of Molecules in Solution: State Specific and Linear Response Methods for Nonequilibrium Continuum Solvation Models},
|
||||
Url = {https://doi.org/10.1063/1.1867373},
|
||||
Volume = {122},
|
||||
Year = {2005},
|
||||
@ -14533,10 +14524,10 @@
|
||||
|
||||
@article{Improta_2016,
|
||||
Author = {Improta, Roberto and Santoro, Fabrizio and Blancafort, Llu{\'\i}s},
|
||||
Date-Modified = {2020-06-08 11:23:44 +0200},
|
||||
Doi = {10.1021/acs.chemrev.5b00444},
|
||||
Eprint = {https://doi.org/10.1021/acs.chemrev.5b00444},
|
||||
Journal = {Chem. Rev.},
|
||||
Note = {PMID: 26928320},
|
||||
Number = {6},
|
||||
Pages = {3540-3593},
|
||||
Title = {Quantum Mechanical Studies on the Photophysics and the Photochemistry of Nucleic Acids and Nucleobases},
|
||||
@ -14626,14 +14617,14 @@
|
||||
|
||||
@article{Ren_2012b,
|
||||
Author = {Xinguo Ren and Patrick Rinke and Volker Blum and J{\"u}rgen Wieferink and Alexandre Tkatchenko and Andrea Sanfilippo and Karsten Reuter and Matthias Scheffler},
|
||||
Date-Modified = {2020-06-04 20:21:26 +0200},
|
||||
Date-Modified = {2020-06-08 11:29:40 +0200},
|
||||
Doi = {10.1088/1367-2630/14/5/053020},
|
||||
Journal = {New J. Phys.},
|
||||
Month = {may},
|
||||
Number = {5},
|
||||
Pages = {053020},
|
||||
Publisher = {{IOP} Publishing},
|
||||
Title = {Resolution-of-identity Approach to Hartree{\textendash}Fock, hybrid density functionals, {RPA}, {MP}2 and {GW} with numeric atom-centered orbital basis functions},
|
||||
Title = {Resolution-of-identity Approach to Hartree{\textendash}Fock, Hybrid Fensity Functionals, {RPA}, {MP}2 and {GW} with Numeric Atom-Centered Orbital Basis Functions},
|
||||
Url = {https://doi.org/10.1088%2F1367-2630%2F14%2F5%2F053020},
|
||||
Volume = {14},
|
||||
Year = 2012,
|
||||
@ -14658,10 +14649,10 @@
|
||||
|
||||
@article{Boulanger_2014,
|
||||
Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
|
||||
Date-Modified = {2020-06-08 11:25:05 +0200},
|
||||
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},
|
||||
@ -14672,6 +14663,7 @@
|
||||
|
||||
@article{Spataru_2013,
|
||||
Author = {Spataru, Catalin D.},
|
||||
Date-Modified = {2020-06-08 11:31:48 +0200},
|
||||
Doi = {10.1103/PhysRevB.88.125412},
|
||||
Issue = {12},
|
||||
Journal = {Phys. Rev. B},
|
||||
@ -14679,7 +14671,7 @@
|
||||
Numpages = {8},
|
||||
Pages = {125412},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Electronic and optical gap renormalization in carbon nanotubes near a metallic surface},
|
||||
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},
|
||||
|
||||
@ -199,11 +199,11 @@
|
||||
%%%%%%%%%%%%%%%%
|
||||
|
||||
\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 singlet excitation energies in many chemical scenarios with a typical error of $0.1$--$0.3$ eV.
|
||||
The many-body Green's function Bethe-Salpeter equation (BSE) formalism is steadily asserting itself as a new efficient and accurate tool in the ensemble of computational methods available to chemists in order to predict optical excitations in molecular systems.
|
||||
In particular, the combination of the so-called $GW$ approximation of many-body perturbation theory, giving access to reliable ionization energies and electron affinities, 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.
|
||||
We also propose a critical review of its strengths and weaknesses in different chemical situations.
|
||||
Future directions of developments and improvements are also discussed.
|
||||
\end{abstract}
|
||||
|
||||
@ -228,16 +228,15 @@ Future directions of developments and improvements are also discussed.
|
||||
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.
|
||||
Simulations are so realistic that they predict the outcome of traditional experiments.''} \cite{Nobel_2003}
|
||||
Martin Karplus' Nobel lecture moderated this statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are \textit{``much too complicated to be soluble''}, urging scientists to develop \textit{``approximate practical methods''}. This is where the electronic structure community stands, attempting to develop robust approximations to study with increasing accuracy the properties of ever more complex systems.
|
||||
|
||||
The study of neutral electronic excitations in condensed-matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the prediction of the colour of precious metals for jewellery, \cite{Prandini_2019} to the understanding, \eg, of the basic principles behind organic photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. \cite{Kippelen_2009,Improta_2016,Wu_2019}
|
||||
The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984} including computational scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known issues. \cite{Blase_2018}
|
||||
The present \textit{Perspective} aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} that, while sharing many features with time-dependent density-functional theory (TD-DFT), \cite{Runge_1984} including computational scaling with system size, relies on a very different formalism, with specific difficulties but also potential solutions to known TD-DFT issues. \cite{Blase_2018}
|
||||
\\
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Theory}
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Onida_2002,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015}
|
||||
The BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Onida_2002,ReiningBook} together with, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} \hl{je parlerais aussi de SOPPA ici ? A citer au moins une fois ?}
|
||||
% originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018}
|
||||
While the one-body density stands as the basic variable in density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
|
||||
\begin{equation}
|
||||
@ -256,7 +255,7 @@ A central property of the one-body Green's function is that its frequency-depend
|
||||
G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu ) },
|
||||
\end{equation}
|
||||
where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{\Nel+1} - E_0^{\Nel}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{\Nel} - E_s^{\Nel-1}$ for $\varepsilon_s < \mu$.
|
||||
Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
|
||||
Here, $E_s^{\Nel}$ is the total energy of the $s$\textsuperscript{th} excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy.
|
||||
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 (see below).
|
||||
Unlike Kohn-Sham (KS) eigenvalues, the poles of the Green's function $\lbrace \varepsilon_s \rbrace$ are proper addition/removal energies of the $\Nel$-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.
|
||||
@ -266,14 +265,14 @@ Using the equation-of-motion formalism for the creation/destruction operators, i
|
||||
\qty[ \pdv{}{t_1} - h(\br_1) ] G(1,2) - \int d3 \, \Sigma(1,3) G(3,2)
|
||||
= \delta(1,2),
|
||||
\end{equation}
|
||||
where we introduce the usual composite index, \eg, $1 \equiv (\bx_1 t_1)$.
|
||||
where we introduce the composite index, \eg, $1 \equiv (\bx_1 t_1)$.
|
||||
Here, $\delta$ is Dirac's delta function, $h$ is the one-body Hartree Hamiltonian and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
|
||||
Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}],
|
||||
dropping spin variables for simplicity, one gets the familiar eigenvalue equation, \ie,
|
||||
\begin{equation}
|
||||
h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
|
||||
\end{equation}
|
||||
which resembles formally the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.
|
||||
which formally resembles the KS equation \cite{Kohn_1965} with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.
|
||||
The knowledge of $\Sigma$ allows to access the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.
|
||||
%% \titou{The spin variable has disappear. How do we deal with this?}
|
||||
\\
|
||||
@ -290,7 +289,7 @@ The resulting equation, when compared with the equation for the time-evolution o
|
||||
\begin{equation}\label{eq:Sig}
|
||||
\Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3),
|
||||
\end{equation}
|
||||
where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$.
|
||||
where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$. \hl{vs ne vlz pas simplement dire que c'est des corrections de + grand ordre ?}
|
||||
The neglect of the vertex leads to the so-called $GW$ approximation of the self-energy
|
||||
\begin{equation}\label{eq:SigGW}
|
||||
\Sigma^{\GW}(1,2) = i \, G(1,2) W(2,1^{+}),
|
||||
@ -310,8 +309,8 @@ where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulo
|
||||
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 of the system of interest.
|
||||
Depending on the level of self-consistency in 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 optical excitations of the system of interest.
|
||||
\label{fig:pentagon}}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
@ -432,7 +431,7 @@ with electron-hole ($eh$) eigenstates written as
|
||||
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.
|
||||
(In the case of complex orbitals, we refer the reader to Ref.~\citenum{Holzer_2019} for a correct use of complex conjugation.)
|
||||
%(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},
|
||||
@ -452,22 +451,22 @@ $(ia|jb)$ bare Coulomb term defined as
|
||||
\phi_i(\br) \phi_a(\br) v(\br-\br')
|
||||
\phi_j(\br') \phi_b(\br').
|
||||
\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.
|
||||
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 (TDA).
|
||||
|
||||
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.
|
||||
This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, highlighting its \emph{pros} and \emph{cons}.
|
||||
\\
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Historical overview}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
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}
|
||||
Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} the BSE formalism has emerged in condensed-matter physics around the 1960's at the tight-binding level with the study of the optical properties of simple semiconductors. \cite{Sham_1966,Strinati_1984,Delerue_2000}
|
||||
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}
|
||||
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}
|
||||
Such comparisons were grounded in the development of codes replacing the plane-wave paradigm of solid-state physics by Gaussian basis sets, 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}
|
||||
@ -509,8 +508,8 @@ 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 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.
|
||||
For sake of illustration, an average error of $0.2$ eV was found for the well-known Thiel set \cite{Schreiber_2008} gathering ca. 200 representative singlet excitations from a large variety of representative molecules. \cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017}
|
||||
This is equivalent to the best TD-DFT results obtained by scanning a large variety of hybrid functionals with various amounts of exact exchange.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Successes \& Challenges}
|
||||
@ -523,15 +522,15 @@ This is equivalent to the best TD-DFT results obtained by scanning a large varie
|
||||
A very remarkable success of the BSE formalism lies in the description of charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT adopting standard (semi-)local functionals. \cite{Dreuw_2004}
|
||||
Similar difficulties emerge in solid-state physics for semiconductors where extended Wannier excitons, characterized by weakly overlapping electrons and holes (Fig.~\ref{fig:CTfig}), cause a dramatic deficit of spectral weight at low energy. \cite{Botti_2004}
|
||||
These difficulties can be ascribed to the lack of long-range electron-hole interaction with local xc functionals.
|
||||
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}
|
||||
The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where screening reduces the long-range electron-hole interactions.
|
||||
The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to important applications such as doping, \cite{Li_2017b} photovoltaics or photocatalysis in organic systems.\\
|
||||
It can be cured through an exact exchange contribution, a solution that explains the success of optimally-tuned range-separated hybrids for the description of CT excitations. \cite{Stein_2009,Kronik_2012}
|
||||
The analysis of the screened Coulomb potential matrix elements in the BSE kernel [see Eq.~\eqref{eq:BSEkernel}] reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc) where the screening reduces the long-range electron-hole interactions.
|
||||
The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Rocca_2010,Cudazzo_2010,Lastra_2011,Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2013,Yin_2014} opening the way to the modeling of key applications such as doping, \cite{Li_2017b} photovoltaics or photocatalysis in organic systems.\\
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure}[h]
|
||||
\includegraphics[width=0.6\linewidth]{CTfig}
|
||||
\caption{
|
||||
Symbolic representation of extended Wannier exciton with large electron-hole average distance (top), and Frenkel and charge-transfer (CT) excitations at a donor-acceptor interface (bottom).
|
||||
Symbolic representation of extended Wannier exciton with large electron-hole average distance (top), and Frenkel (local) and charge-transfer (CT) excitations at a donor-acceptor interface (bottom).
|
||||
Wannier and CT excitations require long-range electron-hole interaction accounting for the host dielectric constant.
|
||||
In the case of Wannier excitons, the binding energy $\EB$ can be well approximated by the standard hydrogenoid model where $\mu$ is the effective mass and $\epsilon$ is the dielectric constant.
|
||||
\label{fig:CTfig}}
|
||||
@ -542,7 +541,7 @@ The success of the BSE formalism to treat CT excitations has been demonstrated i
|
||||
\subsection{Combining BSE with PCM and QM/MM models}
|
||||
%==========================================
|
||||
|
||||
The ability to account for the effect on the excitation energies of an electrostatic and dielectric environment (an electrode, a solvent, a molecular interface, \ldots) is an important step towards the description of realistic systems.
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The ability to account for the effect on the excitation energies of an electrostatic and dielectric environment (an electrode, a solvent, a molecular interface... is an important step towards the description of realistic systems.
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Pioneering BSE studies demonstrated, for example, the large renormalization of charged and neutral excitations in molecular systems and nanotubes close to a metallic electrode or in bundles. \cite{Lastra_2011,Rohlfing_2012,Spataru_2013}
|
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Recent attempts to merge the $GW$ and BSE formalisms with model polarizable environments at the PCM or QM/MM levels
|
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\cite{Baumeier_2014,Duchemin_2016,Li_2016,Varsano_2016,Duchemin_2018,Li_2019,Tirimbo_2020} paved the way not only to interesting applications but also to a better understanding of the merits of these approaches relying on the use of the screened Coulomb potential designed to capture polarization effects at all spatial ranges.
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@ -550,16 +549,16 @@ As a matter of fact, dressing the bare Coulomb potential with the reaction field
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$[
|
||||
v(\br,\br') \longrightarrow v(\br,\br') + v^{\text{reac}}(\br,\br'; \omega)
|
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]$
|
||||
in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc) with the same complexity as in the gas phase.
|
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in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility [see Eq.~\eqref{eq:defW}] allows to perform $GW$ and BSE calculations in a polarizable environment with the same complexity as in the gas phase.
|
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The reaction field matrix $v^{\text{reac}}(\br,\br'; \omega)$ describes the potential generated in $\br'$ by the charge rearrangements in the polarizable environment induced by a source charge located in $\br$, where $\br$ and $\br'$ lie in the quantum mechanical subsystem of interest.
|
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The reaction field is dynamical since the dielectric properties of the environment, such as the macroscopic dielectric constant $\epsilon_M(\omega)$, are in principle frequency dependent.
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Once the reaction field matrix is known, with typically $\order*{\Norb N_\text{MM}^2}$ operations (where $\Norb$ is the number of orbitals and $N_\text{MM}$ the number of polarizable atoms in the environment), the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
|
||||
|
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A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field captures both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing Frenkel and CT excitations.
|
||||
A remarkable property \cite{Duchemin_2018} of the scheme described above, which combines the BSE formalism with a polarizable environment, is that the renormalization of the electron-electron and electron-hole interactions by the reaction field captures both linear-response and state-specific contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing local (Frenkel) and CT excitations.
|
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This is an important advantage as compared to, \eg, TD-DFT where linear-response and state-specific effects have to be explored with different formalisms.
|
||||
|
||||
To date, environmental effects on fast electronic excitations are only included by considering the low-frequency optical response of the polarizable medium (\eg, considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant of water in the optical range), neglecting the frequency dependence of the dielectric constant in the optical range.
|
||||
Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, in situations where neither the adiabatic limit nor the antiadiabatic limit are expected to be valid (for a recent discussion, see Ref.
|
||||
Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, \ie, situations where neither the adiabatic limit nor the antiadiabatic limit are expected to be valid (for a recent discussion, see Ref.
|
||||
~\citenum{Huu_2020}).
|
||||
|
||||
We now leave the description of successes to discuss difficulties and future directions of developments and improvements.
|
||||
@ -588,6 +587,7 @@ These ongoing developments pave the way to applying the $GW$@BSE formalism to sy
|
||||
%==========================================
|
||||
\subsection{The triplet instability challenge}
|
||||
%==========================================
|
||||
\hl{Je ne pige pas la premiere phrase qui semble melanger differents concepts. A divisier ?}
|
||||
The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
|
||||
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels, contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range.
|
||||
While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter(s) as a mean to act on the triplet instability, \cite{Sears_2011} BSE calculations do not offer this pragmatic way-out since the screened Coulomb potential that builds the kernel does not offer any parameter to tune.
|
||||
@ -611,7 +611,7 @@ From a theoretical point of view, the accurate prediction of excited electronic
|
||||
For the last two decades, TD-DFT has been the go-to method to compute absorption and emission spectra in large molecular systems.
|
||||
|
||||
In TD-DFT, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state.
|
||||
One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first- and second-order derivatives of the excited-state energy with respect to the nuclear displacements), which enables the exploration of excited-state PES. \cite{Furche_2002}
|
||||
One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first- and second-order derivatives of the excited-state energy with respect to atomic displacements), which enables the exploration of excited-state PES. \cite{Furche_2002}
|
||||
|
||||
A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients for both the ground and excited states, preventing efficient studies of excited-state processes.
|
||||
While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
|
||||
@ -628,11 +628,11 @@ Consequently, the BSE ground-state formalism remains in its infancy with very fe
|
||||
|
||||
A promising route, which closely follows RPA-type formalisms, \cite{Angyan_2011} is to calculated the ground-state BSE energy within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework. \cite{Furche_2005}
|
||||
Thanks to comparisons with both similar and state-of-art computational approaches, it was recently shown that the ACFDT@BSE@$GW$ approach yields extremely accurate PES around equilibrium, and can even compete with high-order coupled cluster methods in terms of absolute ground-state energies and equilibrium distances. \cite{Loos_2020}
|
||||
Their accuracy near the dissociation limit remains an open question. \cite{Caruso_2013,Olsen_2014,Colonna_2014,Hellgren_2015,Holzer_2018}
|
||||
However, their accuracy near the dissociation limit remains an open question. \cite{Caruso_2013,Olsen_2014,Colonna_2014,Hellgren_2015,Holzer_2018}
|
||||
Indeed, in the largest available benchmark study \cite{Holzer_2018} encompassing the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the ACFDT framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018}
|
||||
Moreover, it was also observed in Ref.~\citenum{Loos_2020} that, in some cases, unphysical irregularities on the ground-state PES show up due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies.
|
||||
Such an unphysical behavior stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found [see Eq.~\eqref{eq:QP-eq}].
|
||||
We refer the interested reader to Refs.~\citenum{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020} for a detailed discussion of this particular point.
|
||||
We refer the interested reader to Refs.~\citenum{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020} for detailed discussions.
|
||||
\\
|
||||
|
||||
%==========================================
|
||||
@ -669,12 +669,12 @@ Corrections to take into account the dynamical nature of the screening may or ma
|
||||
However, dynamical corrections permit, in any case, to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations (and, in particular, non-interacting double excitations).
|
||||
|
||||
From a more practical point of view, dynamical effects have been found to affect the positions and widths of core-exciton resonances in semiconductors, \cite{Strinati_1982,Strinati_1984} rare gas solids, and transition metals. \cite{Ankudinov_2003}
|
||||
Thanks to first-order perturbation theory, Rohlfing and coworkers have developed an efficient way of taking into account the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
|
||||
Thanks to first-order perturbation theory, Rohlfing and coworkers have developed an efficient way of taking into account the dynamical effects via a plasmon-pole approximation combined with TDA. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
|
||||
With such a scheme, they have been able to compute the excited states of biological chromophores, showing that taking into account the electron-hole dynamical screening is important for an accurate description of the lowest $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b}
|
||||
Studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that \textit{``the influence of dynamical screening on the excitation energies is about $0.1$ eV for the lowest $\pi \ra \pi^*$ transitions, but for the lowest $n \ra \pi^*$ transitions the influence is larger, up to $0.25$ eV.''} \cite{Ma_2009b}
|
||||
Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Salpeter kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
|
||||
Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016}
|
||||
In these two latter studies, they also followed a (non-self-consistent) perturbative approach within the Tamm-Dancoff approximation with a renormalization of the first-order perturbative correction.
|
||||
In these two latter studies, they also followed a (non-self-consistent) perturbative approach within TDA with a renormalization of the first-order perturbative correction.
|
||||
%Finally, let us also mentioned the work of Romaniello and coworkers, \cite{Romaniello_2009b,Sangalli_2011} in which the authors genuinely accessed additional excitations by solving the non-linear, frequency-dependent eigenvalue problem.
|
||||
%However, it is based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations.
|
||||
\\
|
||||
@ -759,7 +759,7 @@ In these two latter studies, they also followed a (non-self-consistent) perturba
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Conclusion}
|
||||
\section{Conclusions}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
Although far from being exhaustive, we hope to have provided, in the present \textit{Perspective}, a concise and fair assessment of the strengths and weaknesses of the BSE formalism of many-body perturbation theory.
|
||||
To do so, we have briefly reviewed the theoretical aspects behind BSE, and its intimate link with the underlying $GW$ calculation that one must perform to compute quasiparticle energies and the dynamically-screened Coulomb potential; two of the key input ingredients associated with the BSE formalism.
|
||||
|
||||
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