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BSE-PES.tex
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@ -150,18 +150,18 @@
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\kcal}{kcal/mol}
\newcommand{\NEEL}{Univ. Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\CEA}{ Univ. Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France }
\newcommand{\CEA}{Universit\'e Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France}
\begin{document}
\title{Ground-State Potential Energy Surfaces Within the Bethe-Salpeter Formalism: Pros and Cons}
\author{Xavier \surname{Blase}}
\email{xavier.blase@neel.cnrs.fr }
\affiliation{\NEEL}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Ivan \surname{Duchemin}}
\email{ivan.duchemin@cea.fr}
\affiliation{\CEA}
@ -171,9 +171,9 @@
\author{Denis \surname{Jacquemin}}
\email{denis.jacquemin@univ-nantes.fr}
\affiliation{\CEISAM}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Xavier \surname{Blase}}
\email{xavier.blase@neel.cnrs.fr }
\affiliation{\NEEL}
\begin{abstract}
%\begin{wrapfigure}[12]{o}[-1.25cm]{0.4\linewidth}
@ -182,7 +182,7 @@
%\end{wrapfigure}
The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of molecular systems.
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 near their equilibrium distance.
Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium bond length.
Thanks to comparisons with 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 and equilibrium bond distances.
However, we sometimes observe unphysical irregularities on the ground-state PES in relation with the appearance of satellite resonances with a weight similar to that of the $GW$ quasiparticle peak.
\end{abstract}
@ -195,28 +195,28 @@ However, we sometimes observe unphysical irregularities on the ground-state PES
%%%%%%%%%%%%%%%%%%%%%%%%
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,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} is a valuable alternative that has gained momentum in the past few years for studying 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}
\cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} is a valuable alternative that has gained momentum in the past few years for studying 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}
It now stands 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 exchange-correlation 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}
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,Loos_2019}
A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytic nuclear gradients (\ie, the first derivatives of the energy with respect to the nuclear displacements) for both the ground and excited states, \cite{Furche_2002} preventing efficient applications to the study of chemoluminescence, fluorescence and other related processes \cite{Navizet_2011} associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients (\ie, the first derivatives of the energy with respect to the nuclear displacements) for both the ground and excited states, \cite{Furche_2002} preventing efficient applications to the study of chemoluminescence, fluorescence and other related processes \cite{Navizet_2011} associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
While calculations of the $GW$ quasiparticle energy ionic gradients is becoming popular,
\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} In this study devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the KS-DFT (LDA) forces as its ground-state analog.
\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} In this study devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the Kohn-Sham (KS) LDA forces as its ground-state analog.
Contrary to TD-DFT which relies on Kohn-Sham density-functional theory (KS-DFT) \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the BSE ground-state energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
Contrary to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the BSE ground-state energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
It then remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020}
As a matter of fact, in the largest recent available benchmark study \cite{Holzer_2018} of the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, \cite{Harding_2008} and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation (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}
As a matter of fact, in the largest recent available benchmark study \cite{Holzer_2018} of the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, \cite{Harding_2008} and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation theorem (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}
Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016}
With such an approximation, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either KS-DFT or $GW$ eigenvalues in the construction of the propagator becomes further relevant, increasing accordingly the number of possible definitions for the ground-state correlation energy.
Finally, renormalizing or not the Coulomb interaction by the interaction strength $\IS$ in the Dyson equation for the interacting polarizability leads to two different versions of the BSE correlation energy. \cite{Holzer_2018}
Finally, renormalizing or not the Coulomb interaction by the interaction strength $\IS$ in the Dyson equation for the interacting polarizability (see below) leads to two different versions of the BSE correlation energy. \cite{Holzer_2018}
Here, in analogy to the random-phase approximation (RPA)-type formalisms \cite{Furche_2008,Toulouse_2009,Toulouse_2010,Angyan_2011,Ren_2012} and similarly to Refs.~\onlinecite{Olsen_2014,Maggio_2016,Holzer_2018}, the ground-state BSE energy is calculated in the adiabatic connection framework.
Embracing this definition, the purpose of the present study is to investigate the quality of ground-state PES near equilibrium obtained within the BSE approach for several diatomic molecules.
The location of the minima on the ground-state PES is of particular interest.
This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
Thanks to comparison with both similar and state-of-art computational approaches, we show that the ACFDT@BSE@$GW$ approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods in terms of absolute energies and equilibrium distances.
Thanks to comparisons with both similar and state-of-art computational approaches, we show that the ACFDT@BSE@$GW$ approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods in terms of absolute energies and equilibrium distances.
However, we also observe that, 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. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
%The paper is organized as follows.
@ -299,12 +299,12 @@ In the absence of instabilities (\ie, $\bA{\IS} - \bB{\IS}$ is positive-definite
where the excitation amplitudes are
\begin{subequations}
\begin{align}
\bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS},
\bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{+1/2} \bZ{\IS},
\\
\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{+1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
\end{align}
\end{subequations}
With Mulliken's notation of the bare two-electron integrals
Introducing the so-called Mulliken notation for the bare two-electron integrals
\begin{equation}
\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
\end{equation}
@ -324,7 +324,7 @@ the BSE matrix elements read
\end{subequations}
where $\eGW{p}$ are the $GW$ quasiparticle energies.
%In the standard BSE implementation, the screened Coulomb potential $\W{}{\IS}$ is taken to be static $(\omega \rightarrow 0)$.
In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is built within the direct RPA scheme:
In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme, \ie,
\begin{subequations}
\label{eq:wrpa}
\begin{align}
@ -335,7 +335,7 @@ In the standard BSE approach, the screened Coulomb potential $\W{}{\IS}$ is buil
& = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
\end{align}
\end{subequations}
with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual molecular orbitals product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals \footnote{In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$.}
with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual orbital product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals \footnote{In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $W$.}
\begin{multline}
\label{eq:W}
\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
@ -362,7 +362,7 @@ where $\eHF{p}$ are the HF orbital energies.
The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening
%namely setting $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$.
In this limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations:
In this limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock (HF) eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations:
\begin{subequations}
\begin{align}
\label{eq:LR_RPAx-A}
@ -437,7 +437,7 @@ Even for weakly correlated systems, triplet instabilities are much more common,
%\section{Computational details}
%\label{sec:comp_details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we have considered here.
All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) HF starting point, which is a very adequate choice in the case of the (small) systems that we have considered here.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting point to compute the BSE neutral excitations.
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}.
@ -520,7 +520,6 @@ The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold
The values in parenthesis have been obtained by fitting a Morse potential to the PES.
}
\label{tab:Req}
\begin{ruledtabular}
\begin{tabular}{llcccccccc}
& & \mc{8}{c}{Molecules} \\
@ -540,45 +539,17 @@ The values in parenthesis have been obtained by fitting a Morse potential to the
& cc-pVTZ & 1.393 & 3.004 & 2.968 & 2.405 & 2.095 & 2.144 & 2.383 & 2.636 \\
& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\
BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & (2.698) \\
& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 &2.068(2.074) & 2.116 & (2.389) & (2.647) \\
& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\
& cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)} &\gb{(2.408)} &\gb{(2.070)} &\gb{(2.130)} &\gb{(2.383)} &\rb{(2.640)}\\
RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & (2.629) \\
& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 &2.065(2.048) & 2.114 & (2.370) & (2.584) \\
& cc-pVQZ & 1.382 &3.013(2.998) & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
& cc-pVQZ & 1.382 & 2.997 & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\
& cc-pVTZ & 1.395 & 3.003 & 2.943 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
& cc-pVQZ & 1.394 & 3.011 &\gb{(2.943)} &\gb{(2.393)} &\gb{(2.041)} &\gb{(2.105)} &\gb{(2.367)} &\gb{(2.563)} \\
RPA@HF & cc-pVDZ & 1.431 & 3.021 & 2.999 & 2.424 & 2.083 & 2.134 & 2.416 & 2.623 \\
& cc-pVTZ & 1.388 & 2.978 & 2.939 & 2.396 & 2.045 & 2.110 & 2.362 & 2.579 \\
& cc-pVQZ & 1.386 & 2.994 &\gb{(2.946)} &\gb{(2.385)} &\gb{(2.042)} &\gb{(2.104)} &\gb{(2.365)} &\gb{(2.571)} \\
% FROZEN CORE VERSION
% Method & Basis & \ce{H2} & \ce{LiH}& \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl}\\
% \hline
% CC3 & cc-pVDZ & 1.438 & 3.052 & 3.014 & 2.115 & 2.167 & 2.447 & 2.741 & 2.438 \\
% & cc-pVTZ & 1.403 & 3.036 & 2.985 & 2.087 & 2.150 & 2.405 & 2.672 & 2.414 \\
% & cc-pVQZ & 1.402 & 3.037 & 2.985 & 2.080 & 2.142 & 2.398 & 2.667 & 2.413 \\
% CCSD & cc-pVDZ & 1.438 & 3.044 & 3.006 & 2.101 & 2.149 & 2.435 & 2.695 & 2.433 \\
% & cc-pVTZ & 1.403 & 3.012 & 2.954 & 2.064 & 2.126 & 2.382 & 2.629 & 2.409 \\
% & cc-pVQZ & 1.402 & 3.020 & 2.953 & 2.059 & 2.118 & 2.380 & 2.621 & 2.398 \\
% CC2 & cc-pVDZ & 1.426 & & & & & & & \\
% & cc-pVTZ & 1.393 & & & & & & & \\
% & cc-pVQZ & 1.391 & & & & & & & \\
% MP2 & cc-pVDZ & 1.426 & 3.049 & 3.012 & 2.134 & 2.167 & 2.433 & 2.681 & 2.429 \\
% & cc-pVTZ & 1.393 & 3.026 & 2.990 & 2.104 & 2.151 & 2.395 & 2.640 & 2.407 \\
% & cc-pVQZ & 1.391 & 3.026 & 2.990 & 2.098 & 2.144 & 2.389 & 2.638 & 2.405 \\
% BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & & & & & & & \\
% & cc-pVTZ & 1.404 & & & & & & & \\
% & cc-pVQZ & 1.399 & & & & & & & \\
% RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & & & & & & & \\
% & cc-pVTZ & 1.388 & & & & & & & \\
% & cc-pVQZ & 1.382 & & & & & & & \\
% RPAx@HF & cc-pVDZ & 1.428 & & & & & & & \\
% & cc-pVTZ & 1.395 & & & & & & & \\
% & cc-pVQZ & 1.394 & & & & & & & \\
% RPA@HF & cc-pVDZ & 1.431 & & & & & & & \\
% & cc-pVTZ & 1.388 & & & & & & & \\
% & cc-pVQZ & 1.386 & & & & & & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
@ -632,7 +603,8 @@ See {\SI} for additional potential energy curves with other basis sets and withi
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and XB is indebted to Valerio Olevano for numerous discussions.
%PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and
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.
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''.}