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Pierre-Francois Loos 2020-01-27 14:50:07 +01:00
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BSE-PES.tex
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The many-body Green's function Bethe-Salpeter equation (BSE) formalism performed on top of a $GW$ calculation has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies for medium and large molecular systems. The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalisms have shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of molecular systems.
Although no clear consensus has been reached for the definition of the BSE ground-state energy, the BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT). \sout{Although no clear consensus has been reached for the definition of the BSE ground-state energy,} the BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules. \sout{Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules.
Our aim is to know whether or not the BSE formalism is able to reproduce faithfully the main features of these PES near equilibrium, and, in particular, the location of the minima on the ground-state PES. Our aim is to know whether or not the BSE formalism is able to reproduce faithfully the main features of these PES near equilibrium, and, in particular, the location of the minima on the ground-state PES.}
Thanks to comparisons with both similar and state-of-art computational approaches, we show that the present ACFDT-based approach is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies. Thanks to comparisons with \sout{both similar and} state-of-art computational approaches, we show that ACFDT-BSE is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies, \xavier{ concerning the description of ground-state PES near equilibrium°. }
However, we also observe, in some cases, unphysical irregularities on the ground-state PES, which are due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak. However, we also observe, in some cases, unphysical irregularities on the ground-state PES, which are due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak.
\xavier{[]184 words larger than 150]}
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -185,7 +186,7 @@ However, we also observe, in some cases, unphysical irregularities on the ground
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} 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,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} 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]} \xavier{ [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} 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} 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} 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}
@ -389,11 +390,11 @@ Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacin
Several important comments are in order here. 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. 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, \textcolor{red}{ hampering in particular the calculation of atomisation energies}. However, singlet instabilities may appear in the presence of strong correlation, \eg, when the bond is stretched, \xavier{ hampering in particular the calculation of atomisation energies}.
%% In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}. %% 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.} Even for weakly correlated systems, triplet instabilities are much more common \xavier{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. \xavier{ 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} 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} 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. }
@ -407,7 +408,7 @@ 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. 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}. 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. 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. \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. } 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. \xavier{ 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} 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} All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018}
@ -586,7 +587,7 @@ See {\SI} for additional potential energy curves with other basis sets and withi
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\begin{acknowledgements} \begin{acknowledgements}
PFL would like to thank Anthony Scemama for technical assistance. PFL would like to thank Anthony Scemama for technical assistance.
\textcolor{red}{Authors are indebted to Valerio Olevano for numerous discussions.} \xavier{XB is 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. 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. 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''. }