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Pierre-Francois Loos 2020-01-27 12:48:18 +01:00
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BSE-PES.bib
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@ -9505,8 +9505,8 @@
Author = {Ivan Duchemin and Xavier Blase},
Date-Added = {2019-10-23 10:00:45 +0200},
Date-Modified = {2020-01-26 11:17:42 +0100},
Journal = {J. Chem. Theory Comput.},
Pages = {submitted},
Journal = {arXiv:1912.06459},
Pages = {},
Title = {Robust Analytic Continuation Approach to Many-Body GW Calculations},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}}

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BSE-PES.tex
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@ -184,7 +184,9 @@ However, we also observe, in some cases, unphysical irregularities on the ground
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With a similar computational cost to time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida} the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is a valuable alternative with early \textit{ab initio} calculations in condensed matter physics dated back to the end of the 90's. \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999}
In the past few years, BSE has gained momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Tiago_2008,Sai_2008,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} and is now a serious candidate as a computationally inexpensive method that can effectively model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmark calculations. \cite{Jacquemin_2015,Bruneval_2015,Blase_2016,Jacquemin_2016,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018}
In the past few years, BSE has gained momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017}
\textcolor{red}{[Removed Tiago2008 and Sai2008]}
and is now a serious candidate as a computationally inexpensive method that can effectively model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmark calculations. \cite{Jacquemin_2015,Bruneval_2015,Blase_2016,Jacquemin_2016,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018}
One of the main advantages of BSE compared to TD-DFT is that it allows a faithful description of charge-transfer states. \cite{Lastra_2011,Blase_2011b,Baumeier_2012,Duchemin_2012,Cudazzo_2013,Ziaei_2016}
Moreover, when performed on top of a (partially) self-consistently {\evGW} calculation, \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} BSE@{\evGW} has been shown to be weakly dependent on its starting point (\ie, on the xc functional selected for the underlying DFT calculation). \cite{Jacquemin_2016,Gui_2018}
However, similar to adiabatic TD-DFT, \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} the static version of BSE cannot describe multiple excitations. \cite{Romaniello_2009a,Sangalli_2011}
@ -387,13 +389,14 @@ Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacin
Several important comments are in order here.
For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities as these systems can be labeled as weakly correlated.
However, singlet instabilities may appear in the presence of strong correlation (\eg, when the bond is stretched).
In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}.
Even for weakly correlated systems, triplet instabilities are much more common.
However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Sawada_1957b, Rowe_1968} the triplet excitations do not contribute in the ACFDT formulation, which is an indisputable advantage of this approach.
However, singlet instabilities may appear in the presence of strong correlation, \eg, when the bond is stretched, \textcolor{red}{ hampering in particular the calculation of atomisation energies}.
%% In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}.
Even for weakly correlated systems, triplet instabilities are much more common \textcolor{red}{but triplet excitations do not contribute to the correlation energy in the ACFDT formulation.}
\textcolor{red}{ IS THIS NEEDED NOW ? However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Sawada_1957b, Rowe_1968} the triplet excitations do not contribute in the ACFDT formulation, which is an indisputable advantage of this approach.
Indeed, although the plasmon and adiabatic connection formulations are equivalent for RPA, \cite{Sawada_1957b, Gell-Mann_1957, Fukuta_1964, Furche_2008} this is not the case at the BSE and RPAx levels. \cite{Toulouse_2009, Toulouse_2010, Angyan_2011, Li_2020}
For RPAx, an alternative plasmon expression (equivalent to its ACFDT analog) can be found if exchange is also added to the interaction kernel [see Eq.~\eqref{eq:K}]. \cite{Angyan_2011}
However, to the best of our knowledge, such alternative plasmon expression does not exist for BSE.
However, to the best of our knowledge, such alternative plasmon expression does not exist for BSE. }
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%\section{Computational details}
@ -404,11 +407,12 @@ Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculatio
In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
Finally, the infinitesimal $\eta$ has been set to zero for all calculations.
The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] has been performed with a 21-point Gauss-Legendre quadrature.
The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] has been performed with a 21-point Gauss-Legendre quadrature. \textcolor{red}{ Comparison with the so-called plasmon (or Trace) formula~\cite{Furche_2008} at the RPA level confirmed the excellent convergency of such a $\lambda$-sampling scheme. }
For comparison purposes, we have also computed the PES at the MP2, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} levels of theory using DALTON. \cite{dalton}
All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018}
As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions.
Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fairer comparison between methods.
Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fair comparison between methods.
However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI}).
Because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum for each quadrature point, we perform several complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
@ -582,9 +586,10 @@ See {\SI} for additional potential energy curves with other basis sets and withi
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\begin{acknowledgements}
PFL would like to thank Anthony Scemama for technical assistance.
\textcolor{red}{Authors are indebted to Valerio Olevano for numerous discussions.}
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''. }
\end{acknowledgements}
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