From 94299192a9a7bc436e5cc4b60345199e5a0dad28 Mon Sep 17 00:00:00 2001 From: Antoine MARIE Date: Tue, 13 Dec 2022 16:14:00 +0100 Subject: [PATCH] some more refs --- Manuscript/SRGGW.bib | 79 ++++++++++++++++++++++++++++++++++++++++++++ Manuscript/SRGGW.tex | 6 ++-- 2 files changed, 82 insertions(+), 3 deletions(-) diff --git a/Manuscript/SRGGW.bib b/Manuscript/SRGGW.bib index 04cc462..79f6d95 100644 --- a/Manuscript/SRGGW.bib +++ b/Manuscript/SRGGW.bib @@ -4965,6 +4965,39 @@ year = {2010}, bdsk-url-1 = {https://link.aps.org/doi/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, author = {Godby, R. W. and Schl\"uter, M. and Sham, L. J.}, @@ -6146,6 +6179,17 @@ 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}} +@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, author = {Strinati, G.}, date-added = {2020-05-18 21:40:28 +0200}, @@ -15981,6 +16025,41 @@ 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, title = {Long-{{Range Corrected DFT Meets GW}}: {{Vibrationally Resolved Photoelectron Spectra}} from {{First Principles}}}, author = {Gallandi, Lukas and K{\"o}rzd{\"o}rfer, Thomas}, diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 807f5a4..52e0c5d 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -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} 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. -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 \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. 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} -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. 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 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 $\bSig^{\qs}$.