starting results and adding stuff in theory
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BSE-PES.tex
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BSE-PES.tex
@ -41,9 +41,8 @@
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\newcommand{\dbr}[1]{d\br{#1}}
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% methods
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\newcommand{\GW}{$GW$}
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\newcommand{\evGW}{ev\GW}
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\newcommand{\qsGW}{qs\GW}
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\xc}{\text{xc}}
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@ -64,10 +63,9 @@
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E^\text{HF}}
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\newcommand{\EBSE}{E^\text{BSE}}
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\newcommand{\EcRPA}{E_\text{c}^\text{dRPA}}
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\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
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\newcommand{\EcRPAx}{E_\text{c}^\text{RPAx}}
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\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
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\newcommand{\EcsBSE}{{}^1\EcBSE}
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\newcommand{\EctBSE}{{}^3\EcBSE}
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\newcommand{\IP}{\text{IP}}
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\newcommand{\EA}{\text{EA}}
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@ -77,7 +75,7 @@
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^{GW}_{#1}}
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\newcommand{\eevGW}[1]{\epsilon^\text{\evGW}_{#1}}
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\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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@ -101,7 +99,7 @@
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\Sig}[1]{\Sigma_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\ERI}[2]{(#1|#2)}
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@ -124,9 +122,9 @@
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\newcommand{\bSig}{\mathbf{\Sigma}}
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\newcommand{\bSigX}{\mathbf{\Sigma}^\text{x}}
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\newcommand{\bSigC}{\mathbf{\Sigma}^\text{c}}
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\newcommand{\bSigGW}{\mathbf{\Sigma}^\text{\GW}}
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\newcommand{\bSigGW}{\mathbf{\Sigma}^{GW}}
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\newcommand{\be}{\mathbf{\epsilon}}
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\newcommand{\beGW}{\mathbf{\epsilon}^\text{\GW}}
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\newcommand{\beGW}{\mathbf{\epsilon}^{GW}}
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\newcommand{\beGnWn}[1]{\mathbf{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bde}{\mathbf{\Delta\epsilon}}
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\newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}}
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@ -192,22 +190,22 @@ Moreover, when performed on top of a (partially) self-consistently {\evGW} calcu
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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}
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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}
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While calculations of the {\GW} quasiparticle energies ionic gradients is becoming popular,
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While calculations of the $GW$ quasiparticle energies 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 KS-DFT (LDA) forces as its ground-state analog.
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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.
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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 H?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}
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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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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 leads to two different versions of the BSE correlation energy. \cite{Holzer_2018}
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Here, in analogy to 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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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.
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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. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
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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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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.
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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. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
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%The paper is organized as follows.
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%In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
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@ -243,7 +241,7 @@ which takes into account the self-consistent variation of the Hartree potential
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\end{equation}
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(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$.
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In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2') = - i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables.
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In the {\GW} approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have
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In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have
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\begin{equation}
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\SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2),
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\end{equation}
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@ -300,10 +298,10 @@ In the case of BSE, the specific expression of the matrix elements are
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\label{eq:LR_BSE}
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\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \W{ia,bj}{\IS}(\omega = 0) - \lambda \ERI{ia}{jb},
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\\
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\BBSE{ia,jb}{\IS} & = \W{ia,jb}{\IS}(\omega = 0) - \lambda \ERI{ib}{ja} ,
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\BBSE{ia,jb}{\IS} & = \W{ia,jb}{\IS}(\omega = 0) - \lambda \ERI{ia}{bj} ,
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\end{align}
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\end{subequations}
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where $\eGW{p}$ are the {\GW} quasiparticle energies,
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where $\eGW{p}$ are the $GW$ quasiparticle energies,
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\begin{multline}
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\label{eq:W}
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\W{ia,jb}{\IS}(\omega) = 2 \ERI{ia}{jb}
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@ -323,9 +321,9 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA}
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\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{bj},
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\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{bj},
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\\
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\BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{jb},
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\BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{jb},
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\end{align}
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\end{subequations}
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where $\eHF{p}$ are the HF orbital energies.
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@ -369,11 +367,24 @@ 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}] has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer 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 here, has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer 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 maintain with respect to $\IS$.
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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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However, as commonly done within RPA and RPAx (\ie, RPA with exchange), \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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For RPA, these expressions have been provided in Eq.~\eqref{eq:LR_RPA}, and their RPAx analogs read
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPAx}
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\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{bj} - \IS \ERI{ia}{jb},
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\\
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\BRPAx{ia,jb}{\IS} & = 2 \IS \ERI{ia}{jb} - \IS \ERI{ia}{bj}.
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\end{align}
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\end{subequations}
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These two types of calculations will be refer to as RPA@HF and RPAx@HF respectively in the following.
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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} by the $GW$ quasiparticles energies.
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Several important comments are in order here.
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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 labeled as weakly correlated.
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However, singlet instabilities may appear in the presence of strong correlation (\eg, when the bond is stretched).
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@ -388,15 +399,14 @@ However, to the best of our knowledge, such alternative plasmon expression does
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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) Hartree-Fock (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 point to compute the BSE neutral excitations.
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These will be labeled as BSE@{\GOWO}.
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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) Hartree-Fock (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 point to compute the BSE neutral excitations.
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In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, 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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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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All the other calculations have been performed with our locally developed {\GW} software. \cite{Loos_2018,Veril_2018}
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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 fairer 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}).
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@ -410,27 +420,9 @@ However, we are currently pursuing different avenues to lower this cost by compu
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%\label{sec:PES}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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{N2}, \ce{CO}, \ce{BF}, \ce{F2}, and \ce{HCl}.
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The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ are represented in Fig.~\ref{fig:PES}, and the computed equilibrium distances are gathered in Table \ref{tab:Req}.
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The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ 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 are gathered in Table \ref{tab:Req}.
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Additional graphs for other basis sets can be found in the {\SI}.
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%%% FIG 1 %%%
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\begin{figure*}
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\includegraphics[width=0.45\linewidth]{H2_GS_VTZ}
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\includegraphics[width=0.45\linewidth]{LiH_GS_VTZ}
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\includegraphics[width=0.45\linewidth]{LiF_GS_VTZ}
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\includegraphics[width=0.45\linewidth]{N2_GS_VTZ}
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\includegraphics[width=0.45\linewidth]{CO_GS_VTZ}
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\includegraphics[width=0.45\linewidth]{BF_GS_VTZ}
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\includegraphics[width=0.45\linewidth]{F2_GS_VTZ}
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\includegraphics[width=0.45\linewidth]{HCl_GS_VTZ}
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\caption{
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Ground-state PES of diatomic molecules around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
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\label{fig:PES}
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}
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\end{figure*}
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%%% %%% %%%
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%%% TABLE I %%%
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\begin{table*}
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\caption{
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@ -499,20 +491,76 @@ Equilibrium distances (in bohr) of the ground state of diatomic molecules obtain
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\end{ruledtabular}
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\end{table*}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Hydrogen molecule}
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%\label{sec:H2}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Let us first start with the two smallest molecules, \ce{H2} and \ce{LiH} which are both linked by covalent bonds (see Fig.~\ref{fig:PES-H2-LiH}).
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For \ce{H2}, we take as reference the full configuration interaction (FCI) energies and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
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RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly overestimate (in absolute value) the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly underestimate and overestimate the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
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Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
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For example, with the cc-pVQZ basis set, BSE@{\GOWO}@HF is only off by $0.003$ bohr compared to FCI, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
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The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Lithium hydride and lithium fluoride}
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%\label{sec:LiX}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
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In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, and they are almost perfectly parallel.
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%%% FIG 1 %%%
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\begin{figure*}
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\includegraphics[width=0.49\linewidth]{H2_GS_VTZ}
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\includegraphics[width=0.49\linewidth]{LiH_GS_VTZ}
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\caption{
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Ground-state PES of \ce{H2} (left) and \ce{LiH} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
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\label{fig:PES-H2-LiH}
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}
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\end{figure*}
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%%% %%% %%%
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The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are interesting as they corresponds to a strongly polarized bond towards the halogen atoms which are much more electronegative than the first row elements.
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For these ionic bond, the performance of BSE@{\GOWO}@HF are terrific with an almost perfect match to the CC3 curve.
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For \ce{LiF}, the two curves starting to deviate a few tenths of bohr after the equilibrium geometry, but they remain tightly ... for much longer in the case of \ce{HCl}.
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Maybe surprisingly, BSE@{\GOWO}@HF outperforms both CC2 and CCSD, as well as RPAx@HF by a big margin for these two molecules exhibiting charge transfer.
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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.
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%%% FIG 2 %%%
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\begin{figure*}
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\includegraphics[width=0.49\linewidth]{LiF_GS_VTZ}
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\includegraphics[width=0.49\linewidth]{HCl_GS_VTZ}
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\caption{
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Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
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\label{fig:PES-LiF-HCl}
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}
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\end{figure*}
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%%% %%% %%%
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Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple bond to single bond).
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In that case again, the performance of BSE@{\GOWO}@HF are outstanding as shown in Fig.~\ref{fig:PES-N2-CO-BF}.
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%%% FIG 3 %%%
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\begin{figure*}
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\includegraphics[width=0.33\linewidth]{N2_GS_VTZ}
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\includegraphics[width=0.33\linewidth]{CO_GS_VTZ}
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\includegraphics[width=0.33\linewidth]{BF_GS_VTZ}
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\caption{
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Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
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Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
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\label{fig:PES-N2-CO-BF}
|
||||
}
|
||||
\end{figure*}
|
||||
%%% %%% %%%
|
||||
|
||||
The \ce{F2} molecule is a notoriously difficult case to treat due to the relative weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}).
|
||||
|
||||
%%% FIG 4 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{F2_GS_VTZ}
|
||||
\caption{
|
||||
Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set.
|
||||
Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
|
||||
\label{fig:PES-F2}
|
||||
}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\subsection{The isoelectronic sequence: \ce{N2}, \ce{CO}, and \ce{BF}}
|
||||
%\label{sec:isoN2}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\section{Conclusion}
|
||||
@ -520,10 +568,10 @@ Equilibrium distances (in bohr) of the ground state of diatomic molecules obtain
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In this Letter, we have shown that calculating the BSE correlation energy in the ACFDT framework yield extremely accurate PES around equilibrium.
|
||||
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.
|
||||
This shortcoming, which is entirely due to the quasiparticle nature of the underlying {\GW} calculation, has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
|
||||
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.
|
||||
This shortcoming, which is entirely due to the quasiparticle nature of the underlying $GW$ calculation, has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
|
||||
We believe that this central issue must be resolved if one wants to expand the applicability of the present methods.
|
||||
In the perspective of developing analytical nuclear gradients within the BSE@{\GW} formalism, we are currently investigating the accuracy of the ACFDT@BSE scheme for excited-state PES.
|
||||
In the perspective of developing analytical nuclear gradients within the BSE@$GW$ formalism, we are currently investigating the accuracy of the ACFDT@BSE scheme for excited-state PES.
|
||||
We hope to be able to report on this in the near future.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
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Reference in New Issue
Block a user