some more refs

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Antoine Marie 2022-12-13 16:14:00 +01:00
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2 changed files with 82 additions and 3 deletions

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@ -4966,6 +4966,39 @@
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.81.115105}, bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.81.115105},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.81.115105}} bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.81.115105}}
@article{Godby_1986,
title = {Accurate {{Exchange-Correlation Potential}} for {{Silicon}} and {{Its Discontinuity}} on {{Addition}} of an {{Electron}}},
author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.},
year = {1986},
journal = {Phys. Rev. Lett.},
volume = {56},
number = {22},
pages = {2415--2418},
doi = {10.1103/PhysRevLett.56.2415}
}
@article{Godby_1987,
title = {Trends in Self-Energy Operators and Their Corresponding Exchange-Correlation Potentials},
author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.},
year = {1987},
journal = {Phys. Rev. B},
volume = {36},
number = {12},
pages = {6497--6500},
doi = {10.1103/PhysRevB.36.6497}
}
@article{Godby_1987a,
title = {Quasiparticle Energies in {{GaAs}} and {{AlAs}}},
author = {Godby, R. W. and Schl{\"u}ter, M. and Sham, L. J.},
year = {1987},
journal = {Phys. Rev. B},
volume = {35},
number = {8},
pages = {4170--4171},
doi = {10.1103/PhysRevB.35.4170}
}
@article{Godby_1988, @article{Godby_1988,
author = {Godby, R. W. and Schl\"uter, M. and Sham, L. J.}, author = {Godby, R. W. and Schl\"uter, M. and Sham, L. J.},
date-added = {2020-05-18 21:40:28 +0200}, date-added = {2020-05-18 21:40:28 +0200},
@ -6146,6 +6179,17 @@
bdsk-url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.45.290}, bdsk-url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.45.290},
bdsk-url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}} bdsk-url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}}
@article{Strinati_1982,
title = {Dynamical Aspects of Correlation Corrections in a Covalent Crystal},
author = {Strinati, G. and Mattausch, H. J. and Hanke, W.},
year = {1982},
journal = {Physical Review B},
volume = {25},
number = {4},
pages = {2867--2888},
doi = {10.1103/PhysRevB.25.2867}
}
@article{Strinati_1988, @article{Strinati_1988,
author = {Strinati, G.}, author = {Strinati, G.},
date-added = {2020-05-18 21:40:28 +0200}, date-added = {2020-05-18 21:40:28 +0200},
@ -15981,6 +16025,41 @@
doi = {10.1021/acs.jctc.6b00774} doi = {10.1021/acs.jctc.6b00774}
} }
@article{Blase_2011,
title = {First-Principles \$\textbackslash mathit\{\vphantom\}{{GW}}\vphantom\{\}\$ Calculations for Fullerenes, Porphyrins, Phtalocyanine, and Other Molecules of Interest for Organic Photovoltaic Applications},
author = {Blase, X. and Attaccalite, C. and Olevano, V.},
year = {2011},
journal = {Physical Review B},
volume = {83},
number = {11},
pages = {115103},
doi = {10.1103/PhysRevB.83.115103}
}
@article{Shishkin_2007,
title = {Self-Consistent \${{GW}}\$ Calculations for Semiconductors and Insulators},
author = {Shishkin, M. and Kresse, G.},
year = {2007},
journal = {Physical Review B},
volume = {75},
number = {23},
pages = {235102},
doi = {10.1103/PhysRevB.75.235102}
}
@article{Wilhelm_2016,
title = {{{GW}} in the {{Gaussian}} and {{Plane Waves Scheme}} with {{Application}} to {{Linear Acenes}}},
author = {Wilhelm, Jan and Del Ben, Mauro and Hutter, J{\"u}rg},
year = {2016},
journal = {Journal of Chemical Theory and Computation},
volume = {12},
number = {8},
pages = {3623--3635},
issn = {1549-9618},
doi = {10.1021/acs.jctc.6b00380}
}
@article{Gallandi_2015, @article{Gallandi_2015,
title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}}, title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}},
author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas}, author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas},

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@ -52,7 +52,7 @@ Here comes the abstract.
One-body Green's functions provide a natural and elegant way to access charged excitations energies of a physical system. \cite{Martin_2016} One-body Green's functions provide a natural and elegant way to access charged excitations energies of a physical system. \cite{Martin_2016}
The one-body non-linear Hedin's equations give a recipe to obtain the exact interacting one-body Green's function and therefore the exact ionization potentials and electron affinities. \cite{Hedin_1965} The one-body non-linear Hedin's equations give a recipe to obtain the exact interacting one-body Green's function and therefore the exact ionization potentials and electron affinities. \cite{Hedin_1965}
Unfortunately, fully solving the Hedin's equations is out of reach and one must resort to approximations. Unfortunately, fully solving the Hedin's equations is out of reach and one must resort to approximations.
In particular, the GW approximation, \cite{Hedin_1965} which has first been mainly used in the context of solids \ant{ref?} and is now widely used for molecules as well \ant{ref?}, provides fairly accurate results for weakly correlated systems\cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021} In particular, the GW approximation, \cite{Hedin_1965} which has first been mainly used in the context of solids \cite{Strinati_1980,Strinati_1982,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely used for molecules as well \ant{ref?}, provides fairly accurate results for weakly correlated systems\cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021}
The $GW$ approximation is an approximation for the self-energy $\Sigma$ which role is to relate the exact interacting Green's function $G$ to a non-interacting reference one $G_0$ through the Dyson equation The $GW$ approximation is an approximation for the self-energy $\Sigma$ which role is to relate the exact interacting Green's function $G$ to a non-interacting reference one $G_0$ through the Dyson equation
\begin{equation} \begin{equation}
@ -131,10 +131,10 @@ In fact, these cases are related to the discontinuities and convergence problems
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence. One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example. Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
Therefore, one could try to optimise the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016} Therefore, one could try to optimise the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
Alternatively, one could solve this equation self-consistently leading to the eigenvalue only self-consistent scheme. \cite{Kaplan_2016} Alternatively, one could solve this equation self-consistently leading to the eigenvalue only self-consistent scheme. \cite{Shishkin_2006,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016}
To do so one use the energy of the quasi-particle solution of the previous iteration to build Eq.~(\ref{eq:G0W0}) and then solves for $\omega$ again until convergence is reached. To do so one use the energy of the quasi-particle solution of the previous iteration to build Eq.~(\ref{eq:G0W0}) and then solves for $\omega$ again until convergence is reached.
However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach. However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach.
Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. \cite{Marom_2012}
To update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~(\ref{eq:quasipart_eq}). To update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~(\ref{eq:quasipart_eq}).
To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$. To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.