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BSE-PES.bib
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BSE-PES.bib
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%% Created for Pierre-Francois Loos at 2020-02-05 21:06:47 +0100
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%% Created for Pierre-Francois Loos at 2020-02-08 14:52:20 +0100
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@incollection{Hattig_2005c,
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Author = {Christof H{\"a}ttig},
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Booktitle = {Response Theory and Molecular Properties (A Tribute to Jan Linderberg and Poul J{\o}rgensen)},
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Date-Added = {2020-02-08 14:52:12 +0100},
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Date-Modified = {2020-02-08 14:52:20 +0100},
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Doi = {http://dx.doi.org/10.1016/S0065-3276(05)50003-0},
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Editor = {H.J. \AA\ Jensen},
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Issn = {0065-3276},
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Pages = {37--60},
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Publisher = {Academic Press},
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Series = {Advances in Quantum Chemistry},
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Title = {Structure Optimizations for Excited States with Correlated Second-Order Methods: CC2 and ADC(2)},
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Url = {http://www.sciencedirect.com/science/article/pii/S0065327605500030},
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Volume = {50},
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Year = {2005},
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Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0065327605500030},
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Bdsk-Url-2 = {http://dx.doi.org/10.1016/S0065-3276(05)50003-0}}
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@article{Psi4,
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Author = {Parrish, Robert M. and Burns, Lori A. and Smith, Daniel G. A. and Simmonett, Andrew C. and DePrince, A. Eugene and Hohenstein, Edward G. and Bozkaya, U{\u g}ur and Sokolov, Alexander Yu. and Di Remigio, Roberto and Richard, Ryan M. and Gonthier, J{\'e}r{\^o}me F. and James, Andrew M. and McAlexander, Harley R. and Kumar, Ashutosh and Saitow, Masaaki and Wang, Xiao and Pritchard, Benjamin P. and Verma, Prakash and Schaefer, Henry F. and Patkowski, Konrad and King, Rollin A. and Valeev, Edward F. and Evangelista, Francesco A. and Turney, Justin M. and Crawford, T. Daniel and Sherrill, C. David},
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Date-Added = {2020-02-08 14:50:26 +0100},
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Date-Modified = {2020-02-08 14:50:26 +0100},
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Doi = {10.1021/acs.jctc.7b00174},
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Eprint = {https://doi.org/10.1021/acs.jctc.7b00174},
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Journal = {J. Chem. Theory Comput.},
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Note = {PMID: 28489372},
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Number = {7},
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Pages = {3185--3197},
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Title = {Psi4 1.1: An Open-Source Electronic Structure Program Emphasizing Automation, Advanced Libraries, and Interoperability},
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Url = {https://doi.org/10.1021/acs.jctc.7b00174},
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Volume = {13},
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Year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00174}}
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@article{Harding_2008,
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Author = {M. E. Harding and J. Vazquez and B. Ruscic and A. K. Wilson and J. Gauss and J. F. Stanton},
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Date-Added = {2020-01-26 20:25:15 +0100},
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110
BSE-PES.tex
110
BSE-PES.tex
@ -186,10 +186,10 @@
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% \centering
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% \includegraphics[width=\linewidth]{TOC}
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%\end{wrapfigure}
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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.
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The combination of the many-body Green's function $GW$ and the Bethe-Salpeter equation (BSE) formalisms has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) for computing vertical transition energies and oscillator strengths in molecular systems.
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The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
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Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium bond length.
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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.
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Here, we study the topology of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium bond length.
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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 for the systems considered here.
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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.
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\end{abstract}
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@ -200,28 +200,28 @@ However, we sometimes observe unphysical irregularities on the ground-state PES
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%\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%
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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
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With a similar computational scaling as time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida} the many-body Green's function Bethe-Salpeter equation (BSE) formalism
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\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}
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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,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018}
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It now stands as a cost-effective computational method that can model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmarks. \cite{Jacquemin_2015,Bruneval_2015,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018}
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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}
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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_2015,Gui_2018}
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Moreover, when performed on top of a (partially) self-consistent {\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_2015,Gui_2018}
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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}
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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}
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While calculations of the $GW$ quasiparticle energy ionic gradients is becoming popular,
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\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.
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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 studies of excited-state processes (\eg, chemoluminescence and fluorescence) associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Bernardi_1996,Olivucci_2010,Navizet_2011,Robb_2007}
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While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
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\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 seminar work 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 contribution.
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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.
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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}
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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}
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In contrast to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
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Consequently, the BSE ground-state formalism remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020}
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In the largest available benchmark study \cite{Holzer_2018} encompassing 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}
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Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016}
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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.
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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}
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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} emphasizing further the lack of general agreement around the definition of the ground-state BSE energy.
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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.
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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.
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The location of the minima on the ground-state PES is of particular interest.
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This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
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The location of the minimum on the ground-state PES is of particular interest.
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This study is a preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
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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.
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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}
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@ -297,7 +297,7 @@ where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS
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The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively.
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In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
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In the absence of instabilities (\ie, $\bA{\IS} - \bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension
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In the absence of instabilities (\ie, when $\bA{\IS} - \bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension
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\begin{equation}
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\label{eq:small-LR}
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(\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS} = (\bOm{\IS})^2 \bZ{\IS},
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@ -351,7 +351,7 @@ where the spectral weights at coupling strength $\IS$ read
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\begin{equation}
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\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
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\end{equation}
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In the case of complex orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $\W{}{}$.
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In the case of complex orbitals, we refer the reader to Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $\W{}{}$.
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In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements
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\begin{subequations}
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@ -387,7 +387,7 @@ Although this choice is not unique, \cite{Holzer_2018} we propose here to define
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\label{eq:EtotBSE}
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\EBSE = \Enuc + \EHF + \EcBSE,
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\end{equation}
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where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), and
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where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and electronic ground-state HF energy (respectively), and
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\begin{equation}
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\label{eq:EcBSE}
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\EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP{\IS}) d\IS
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@ -416,43 +416,44 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS}
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\end{pmatrix}
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\end{equation}
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is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace.
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Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has been named ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018}
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Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF in the following, has been named ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018}
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Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintained as $\IS$ varies.
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Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be added.
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Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be, in principle, added.
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However, as commonly done within RPA and RPAx, \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study.
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Equation \eqref{eq:EcBSE} can also be straightforwardly applied to RPA and RPAx, the only difference being the expressions of $\bA{\IS}$ and $\bB{\IS}$ used to obtain the eigenvectors $\bX{\IS}$ and $\bY{\IS}$ entering the definition of $\bP{\IS}$ [see Eq.~\eqref{eq:2DM}].
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Equation \eqref{eq:EcBSE} can also be straightforwardly applied to RPA and RPAx, the only difference being the expressions of $\bA{\IS}$ and $\bB{\IS}$ used to obtain the eigenvectors $\bX{\IS}$ and $\bY{\IS}$ entering in the definition of $\bP{\IS}$ [see Eq.~\eqref{eq:2DM}].
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For RPA, these expressions have been provided in Eqs.~\eqref{eq:LR_RPA-A} and \eqref{eq:LR_RPA-B}, and their RPAx analogs in Eqs.~\eqref{eq:LR_RPAx-A} and \eqref{eq:LR_RPAx-B}.
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In the following, we will refer to these two types of calculations as RPA@HF and RPAx@HF, respectively.
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Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacing the HF orbital energies in Eq.~\eqref{eq:LR_RPA-A} by the $GW$ quasiparticles energies.
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Before going any further, several important comments are in order.
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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 classified as weakly correlated.
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However, singlet instabilities may appear in the presence of strong correlation, \eg, when the bond is stretched, hampering in particular the calculation of atomization energies. \cite{Holzer_2018}
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Note that, for spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), one rarely encounters singlet instabilities as these systems can be classified as weakly correlated.
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However, singlet instabilities may appear in the presence of strong correlation, \eg, when the bonds are stretched, hampering in particular the calculation of atomization energies. \cite{Holzer_2018}
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Even for weakly correlated systems, triplet instabilities are much more common, but triplet excitations do not contribute to the correlation energy in the ACFDT formulation. \cite{Toulouse_2009, Toulouse_2010, Angyan_2011}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Computational details}
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%\label{sec:comp_details}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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.
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All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are 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.
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Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting points to compute the BSE neutral excitations.
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In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the frequency-dependent quasiparticle equation.
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In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation.
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Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
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Finally, the infinitesimal $\eta$ has been set to zero for all calculations.
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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.
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Finally, the infinitesimal $\eta$ is set to zero for all calculations.
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The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] is performed with a 21-point Gauss-Legendre quadrature.
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Comparison with the so-called plasmon (or trace) formula \cite{Furche_2008} at the RPA level has confirmed the excellent accuracy of this quadrature scheme over $\IS$.
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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}
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For comparison purposes, we have also computed the PES at the second-order M{\o}ller-Plesset (MP2), as well as with various increasingly accurate CC methods, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3. \cite{Christiansen_1995b}
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These calculations have been performed with DALTON \cite{dalton} and PSI4. \cite{Psi4}
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The computational cost of these methods, in their usual implementation, scale as $\order*{N^5}$, $\order*{N^5}$, $\order*{N^6}$, and $\order*{N^7}$, respectively.
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As shown in Refs.~\onlinecite{Hattig_2005c,Budzak_2017}, CC3 provides extremely accurate ground-state (and excited-state) geometries, and will be taken as reference in the present study.
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All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018}
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As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions.
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Unless otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fair comparison between methods.
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However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI} for additional information).
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As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian Gaussian functions.
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Unless otherwise stated, the frozen-core approximation is not applied in order to provide a fair comparison between methods.
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We have, however, found that the conclusions drawn in the present study hold within the frozen-core approximation (see the {\SI} for additional information).
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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.
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This step is, by far, the computational bottleneck in our current implementation.
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However, we are currently pursuing different avenues to lower this cost by computing the two-electron density matrix of Eq.~\eqref{eq:2DM} via a quadrature in frequency space. \cite{Duchemin_2019,Duchemin_2020}
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However, we are currently pursuing different avenues to lower the cost of this step by computing the two-electron density matrix of Eq.~\eqref{eq:2DM} via a quadrature in frequency space. \cite{Duchemin_2019,Duchemin_2020}
|
||||
|
||||
%%% FIG 1 %%%
|
||||
\begin{figure*}
|
||||
@ -506,9 +507,9 @@ Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various
|
||||
%\section{Potential energy surfaces}
|
||||
%\label{sec:PES}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed the ground-state PES of several closed-shell diatomic molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{HCl}, \ce{N2}, \ce{CO}, \ce{BF}, and \ce{F2}.
|
||||
The PES of these molecules for various methods are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances and correlation energies are gathered in Table \ref{tab:Req}.
|
||||
Both of these properties have been computed with Dunning's cc-pVQZ basis set.
|
||||
In order to illustrate the performance of the BSE-based adiabatic connection formulation, we compute the ground-state PES of several closed-shell diatomic molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{HCl}, \ce{N2}, \ce{CO}, \ce{BF}, and \ce{F2}.
|
||||
The PES of these molecules are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances and correlation energies are gathered in Table \ref{tab:Req}.
|
||||
Both of these properties are computed with Dunning's cc-pVQZ basis set.
|
||||
Graphs and tables for additional basis sets can be found in the {\SI}.
|
||||
|
||||
%%% TABLE I %%%
|
||||
@ -539,8 +540,8 @@ The error (in \%) compared to the reference CC3 values are reported in square br
|
||||
& $\Ec$ & 46.498[$+15.15\%$] & 78.075[$+11.58\%$] & 388.907[$+1.36\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] \\
|
||||
RPA@{\GOWO}@HF & $\Req$ & 1.382[$-1.43\%$] & 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & \gb{(2.043)} & \gb{(2.110)} & \gb{(2.367)} & (2.571)[$-3.45\%$] \\
|
||||
& $\Ec$ & 57.567[$+42.56\%$] & 101.092[$+44.47\%$] & 473.053[$+23.29\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] \\
|
||||
RPAx@HF & $\Req$ & 1.394[$-0.57\%$] & 3.011[$-0.26\%$] & 2.944[$-0.64\%$] & 2.391[$-0.50\%$] & 2.041[$-1.64\%$] & 2.104[$-1.50\%$] & 2.366[$-1.00\%$] & \gb{(2.563)} \\
|
||||
& $\Ec$ & 37.886[$-6.18\%$] & 65.203[$-6.82\%$] & 343.604[$-10.45\%$] & 344.249[$-9.93\%$] & 427.170[$-13.60\%$] & 416.315[$-12.83\%$] & 399.060[$-10.82\%$] & xxx.xxx[$+0.00\%$] \\
|
||||
RPAx@HF & $\Req$ & 1.394[$-0.57\%$] & 3.011[$-0.26\%$] & 2.944[$-0.64\%$] & 2.391[$-0.50\%$] & 2.041[$-1.64\%$] & 2.104[$-1.50\%$] & 2.366[$-1.00\%$] & 2.565[$-3.68\%$] \\
|
||||
& $\Ec$ & 37.886[$-6.18\%$] & 65.203[$-6.82\%$] & 343.604[$-10.45\%$] & 344.249[$-9.93\%$] & 427.170[$-13.60\%$] & 416.315[$-12.83\%$] & 399.060[$-10.82\%$] & 586.090[$-12.38\%$] \\
|
||||
RPA@HF & $\Req$ & 1.386[$-1.14\%$] & 2.994[$-0.83\%$] & 2.946[$-0.57\%$] & 2.382[$-0.87\%$] & 2.042[$-1.59\%$] & 2.103[$-1.54\%$] & 2.364[$-1.09\%$] & \gb{(2.571)} \\
|
||||
& $\Ec$ & 57.332[$+41.98\%$] & 100.164[$+43.15\%$] & 465.905[$+21.43\%$] & 442.675[$+15.83\%$] & 569.384[$+15.17\%$] & 555.857[$+16.39\%$] & 537.685[$+20.16\%$] & xxx.xxx[$+0.00\%$] \\
|
||||
\end{tabular}
|
||||
@ -548,11 +549,11 @@ The error (in \%) compared to the reference CC3 values are reported in square br
|
||||
\end{table*}
|
||||
\end{squeezetable}
|
||||
|
||||
Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds.
|
||||
Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}.
|
||||
Their corresponding PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
|
||||
For \ce{H2}, we take as reference the full configuration interaction (FCI) energies \cite{QP2} and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
|
||||
RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over- and under-shoot the FCI energy, respectively, RPAx@HF being the best match to FCI in the case of \ce{H2}.
|
||||
Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
|
||||
RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and both significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over- and undershoot the FCI energy, respectively, RPAx@HF being the best match to FCI in the case of \ce{H2}.
|
||||
Interestingly, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set (see Table in the {\SI}).
|
||||
For example, BSE@{\GOWO}@HF/cc-pVQZ is only off by $0.003$ bohr as compared to FCI/cc-pVQZ, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
|
||||
The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
|
||||
This is a general trend that is magnified in larger systems as the ones discussed below.
|
||||
@ -561,18 +562,18 @@ Despite the shallow nature of its PES, the scenario is almost identical for \ce{
|
||||
In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, theses surfaces running almost perfectly parallel to one another.
|
||||
Here again, the BSE@{\GOWO}@HF/cc-pVQZ equilibrium bond length is extremely accurate ($3.017$ bohr) as compared to CC3/cc-pVQZ ($3.019$ bohr).
|
||||
|
||||
The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are chemically interesting as they correspond to strongly polarized bonds towards the halogen atoms which are much more electronegative than the first row elements.
|
||||
For these ionic bonds, the performance of BSE@{\GOWO}@HF is terrific with an almost perfect match to the CC3 curve.
|
||||
The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are chemically interesting as they correspond to strongly polarized bonds towards the halogen atoms which are much more electronegative than the first-column elements.
|
||||
For these partially ionic bonds, the performance of BSE@{\GOWO}@HF is terrific with an almost perfect match to the CC3 curve.
|
||||
Maybe surprisingly, BSE@{\GOWO}@HF is on par with both CC2 and CCSD, and outperforms RPAx@HF by a big margin for these two molecules exhibiting charge transfer, the latter fact being also observed for the other diatomics discussed below.
|
||||
Interestingly, while CCSD and CC2 systematically underestimates the total energy, the BSE@{\GOWO}@HF energy is always lower than the reference CC3 energy.
|
||||
This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that we have studied.
|
||||
This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that is considered herein.
|
||||
|
||||
For \ce{HCl}, the data reported in Table \ref{tab:Req} show that the BSE@{\GOWO}@HF equilibrium bond lengths are again in very good agreement with the CC3 reference values.
|
||||
Contrary to CCSD which is known to provide slightly too short bond lengths, ACFDT@BSE underestimates the bond lengths by a few hundredths of bohr.
|
||||
However, in the case of \ce{LiF}, the attentive reader would have observed a small ``glitch'' in the $GW$-based curves very close to their minimum.
|
||||
For \ce{HCl}, the data reported in Table \ref{tab:Req} show that the BSE@{\GOWO}@HF equilibrium bond length is again in very good agreement with its CC3 counterpart.
|
||||
In contrast to CCSD which is known to provide slightly shorter bond lengths, ACFDT@BSE underestimates the bond lengths by a few hundredths of bohr.
|
||||
However, in the case of \ce{LiF}, the attentive reader can observe a small ``glitch'' in the $GW$-based curves very close to their minimum.
|
||||
As observed in Refs.~\onlinecite{vanSetten_2015,Maggio_2017,Loos_2018} and explained in details in Refs.~\onlinecite{Veril_2018,Duchemin_2020}, these irregularities, which makes particularly tricky the location of the minima, are due to ``jumps'' between distinct solutions of the $GW$ quasiparticle equation.
|
||||
Including a broadening via the increasing the value of $\eta$ in the $GW$ self-energy and the screened Coulomb operator soften the problem, but does not remove it completely.
|
||||
Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
|
||||
Including a broadening via an increase of the $\eta$ value entering in the expression of the $GW$ self-energy and the screened Coulomb operator softens the problem, but does not remove it completely.
|
||||
%Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
|
||||
When irregularities are present in the PES, we have fitted a Morse potential of the form $M(R) = D_0\qty{1-\exp[-\alpha\qty(R-\Req)]}^2$ to the PES in order to provide an estimate of the equilibrium bond length.
|
||||
These values are reported in parenthesis in Table \ref{tab:Req}.
|
||||
For the smooth PES where one can obtain both the genuine minimum and the fitted minimum (\ie, based on the Morse curve), this procedure has been shown to be very accurate with an error of the order of $10^{-3}$ bohr in most cases.
|
||||
@ -580,11 +581,11 @@ For the smooth PES where one can obtain both the genuine minimum and the fitted
|
||||
Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple to single bond).
|
||||
The conclusions drawn for the previous systems also apply to these diatomic molecules.
|
||||
In particular, the performance of BSE@{\GOWO}@HF is outstanding, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, and systematically outperforms both CC2 and CCSD.
|
||||
One can notice some irregularities in the PES of \ce{BF} with the cc-pVDZ et cc-pVTZ basis sets (see {\SI}).
|
||||
One can notice some irregularities in the PES of \ce{BF} with the cc-pVDZ et cc-pVTZ basis sets (see the {\SI}).
|
||||
The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium bond lengths once again: at the BSE@{\GOWO}@HF/cc-pVQZ level of theory, we obtain \gb{$2.070$}, \gb{$2.130$}, and \gb{$2.383$} bohr for \ce{N2}, \ce{CO}, and \ce{BF}, respectively, which has to be compared with the CC3/cc-pVQZ values of $2.075$, $2.136$ and $2.390$ bohr, respectively.
|
||||
|
||||
As a final example, we consider the \ce{F2} molecule, a notoriously difficult case to treat due to the weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence its relatively long equilibrium bond length ($2.663$ bohr at the CC3/cc-pVQZ level).
|
||||
Similarly to what we have observed for \ce{LiF} and \ce{BF}, there are irregularities near the minimum of the {\GOWO}-based curves.
|
||||
Similarly to what is observed for \ce{LiF} and \ce{BF}, there are irregularities near the minimum of the {\GOWO}-based curves.
|
||||
However, BSE@{\GOWO}@HF is the closest to the CC3 curve, with an estimated bond length of $2.640$ bohr (via a Morse fit) at the BSE@{\GOWO}@HF/cc-pVQZ level.
|
||||
Note that, for this system, triplet (and then singlet) instabilities appear for quite short bond lengths.
|
||||
However, around the equilibrium structure, we have not encountered any instabilities.
|
||||
@ -594,7 +595,7 @@ However, around the equilibrium structure, we have not encountered any instabili
|
||||
%\label{sec:conclusion}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In this Letter, we hope to have illustrated that the ACFDT@BSE formalism is a promising methodology for the computation of accurate ground-state PES and their corresponding equilibrium structures.
|
||||
To do so, we have shown that calculating the BSE correlation energy computed within the ACFDT framework yield extremely accurate PES around equilibrium.
|
||||
To do so, we have shown that calculating the BSE correlation energy computed within the ACFDT framework yields extremely accurate PES around equilibrium.
|
||||
%(Their accuracy near the dissociation limit remains an open question.)
|
||||
We have illustrated this for 8 diatomic molecules for which we have also computed reference ground-state energies using coupled cluster methods (CC2, CCSD, and CC3).
|
||||
However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak.
|
||||
@ -607,10 +608,11 @@ We hope to be able to report on this in the near future.
|
||||
\section*{Acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and
|
||||
PFL and AS thank the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
|
||||
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.~2019-A0060801738), CALMIP (Toulouse) under allocation 2020-18005, and the CCIPL center installed in Nantes.
|
||||
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 also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Supporting Information}
|
||||
|
||||
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H2_GS_VQZ.pdf
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@ -214,7 +214,7 @@ When irregularities appear in the PES, the values are reported in parenthesis an
|
||||
& 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 & 2.944 & 2.391 & 2.041 & 2.104 & 2.366 &\gb{(2.563)} \\
|
||||
& cc-pVQZ & 1.394 & 3.011 & 2.944 & 2.391 & 2.041 & 2.104 & 2.366 & 2.565 \\
|
||||
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 & 2.946 & 2.382 & 2.042 & 2.103 & 2.364 &\gb{(2.571)} \\
|
||||
|
||||
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