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\begin{document}	

\title{Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies}

\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}
\author{Anthony \surname{Scemama}}
	\email{scemama@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}
\author{Ivan \surname{Duchemin}}
	\email{ivan.duchemin@cea.fr}
	\affiliation{\CEA}
\author{Denis \surname{Jacquemin}}
	\email{denis.jacquemin@univ-nantes.fr}
	\affiliation{\CEISAM}
\author{Xavier \surname{Blase}}
	\email{xavier.blase@neel.cnrs.fr }
	\affiliation{\NEEL}

\begin{abstract}
The combination of the many-body Green's function $GW$ approximation and the Bethe-Salpeter equation (BSE) formalism 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. 
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 topology 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 for the considered systems.  
However, we sometimes observe unphysical irregularities on the ground-state PES in relation with difficulties in the identification of a few $GW$ quasiparticle energies.
\\
\bigskip
\begin{center}
	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
\end{center}
\bigskip
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Introduction}
%\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%

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 
\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 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}
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-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}
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 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}
While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly 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 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.

In contrast to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state analog, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
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}
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} 
Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas (HEG). \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 (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.

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 minimum on the ground-state PES is of particular interest.
This study is a preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
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. 
%In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
%In particular, the general equations are reported in Subsec.~\ref{sec:BSE}, and the corresponding equations obtained in a finite basis in Subsec.~\ref{sec:BSE_basis}.
%Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study.
%Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
%Section \ref{sec:PES} reports  ground-state PES for various diatomic molecules. 
%Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Theory}
%\label{sec:theo}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{The Bethe-Salpeter equation}
%\label{sec:BSE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to compute the neutral (optical) excitations of the system and their associated oscillator strengths, the BSE expresses the two-body propagator \cite{Strinati_1988, Martin_2016} 
\begin{multline}
\label{eq:BSE}
	\LBSE{}(1,2,1',2') = \LBSE{0}(1,2,1',2') 
	\\
	+ \int d3 d4 d5  d6  \LBSE{0}(1,4,1',3) \XiBSE{}(3,5,4,6) \LBSE{}(6,2,5,2')
\end{multline}
as the linear response of the one-body Green's function $\G{}$ with respect to a general non-local external potential
\begin{equation}
	\XiBSE{}(3,5,4,6) = i \fdv{[\vc{\Ha}(3) \delta(3,4) + \Sig{\xc}(3,4)]}{\G{}(6,5)},
\end{equation}
which takes into account the self-consistent variation of the Hartree potential 
\begin{equation}
	\vc{\Ha}(1) = - i \int d2 \vc{}(2) \G{}(2,2^+),
\end{equation}
(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$.
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.
In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have
\begin{equation}
	\SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2),
\end{equation}
where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to 
\begin{equation}
	\XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4),
\end{equation}
where, as commonly done, we have neglected the term $\delta \W{}{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982}
Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{BSE in a finite basis}
%\label{sec:BSE_basis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For a closed-shell system in a finite basis, to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$), one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} 
\begin{equation}
\label{eq:LR}
	\begin{pmatrix}
		\bA{\IS}	&	\bB{\IS}	\\
		-\bB{\IS}	&	-\bA{\IS}	\\
	\end{pmatrix}
	\begin{pmatrix}
		\bX{\IS}_m	\\
		\bY{\IS}_m	\\
	\end{pmatrix}
	=
	\Om{m}{\IS}
	\begin{pmatrix}
		\bX{\IS}_m	\\
		\bY{\IS}_m	\\
	\end{pmatrix},
\end{equation}
where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interaction strength $\IS$, $\T{}$ is the matrix transpose, and we assume real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$.
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.
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.

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
\begin{equation}
\label{eq:small-LR}
	(\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{m}{\IS} = (\Om{m}{\IS})^2 \bZ{m}{\IS},
\end{equation}
where the excitation amplitudes are
\begin{subequations}
\begin{align}
	(\bX{\IS} + \bY{\IS})_m = (\Om{m}{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{+1/2} \bZ{m}{\IS},
	\\
	(\bX{\IS} - \bY{\IS})_m = (\Om{m}{\IS})^{+1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{m}{\IS}.
\end{align}
\end{subequations}
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}
and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\IS$ 
\begin{equation}
	\W{pq,rs}{\IS} = \iint  \MO{p}(\br{}) \MO{q}(\br{}) \W{}{\IS}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}')  \dbr{} \dbr{}',
\end{equation}
the BSE matrix elements read
\begin{subequations}
\begin{align}
	\label{eq:LR_BSE-A}
	\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb}   -  \W{ij,ab}{\IS} ],
	\\ 
	\label{eq:LR_BSE-B}
	\BBSE{ia,jb}{\IS} & =  \IS \qty[ 2 \ERI{ia}{bj} -   \W{ib,aj}{\IS} ],
\end{align}
\end{subequations}
where $\eGW{p}$ are the $GW$ quasiparticle energies. 
In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme, \ie,
\begin{subequations}
\label{eq:wrpa}
\begin{align}
	\W{}{\IS}(\br{},\br{}') 
	& = \int \frac{\epsilon_{\IS}^{-1}(\br{},\br{}''; \omega=0)}{\abs*{\br{}' - \br{}''}} \dbr{}'' , 
	\\ 
	\epsilon_{\IS}(\br{},\br{}'; \omega) 
	& = \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 orbital product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals
\begin{multline}
\label{eq:W}
	\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} +  2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
	\\
	\times  \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}),
\end{multline}
where the spectral weights at coupling strength $\IS$ read
\begin{equation}
	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
\end{equation}
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 $\W{}{}$.

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
\begin{subequations}
\begin{align}
	\label{eq:LR_RPA-A}
	\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) +   2 \IS \ERI{ia}{jb},
	\\ 
	\label{eq:LR_RPA-B}
	\BRPA{ia,jb}{\IS} & =   2 \IS \ERI{ia}{bj},
\end{align}
\end{subequations}
where $\eHF{p}$ are the Hartree-Fock (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 so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$. 
In this limit, the $GW$ quasiparticle energies reduce to the 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}
	\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \qty[ 2 \ERI{ia}{jb} -    \ERI{ij}{ab} ],
	\\ 
	\label{eq:LR_RPAx-B}
	\BRPAx{ia,jb}{\IS} & =  \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ].
\end{align}
\end{subequations}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Ground-state BSE energy}
%\label{sec:BSE_energy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The key quantity to define in the present context is the total BSE ground-state energy $\EBSE$.
Although this choice is not unique, \cite{Holzer_2018} we propose here to define it as
\begin{equation}
\label{eq:EtotBSE}
	\EBSE = \Enuc + \EHF + \EcBSE,
\end{equation}
where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and electronic ground-state HF energy (respectively), and
\begin{equation}
\label{eq:EcBSE}
	\EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP{\IS}) d\IS
\end{equation}
is the ground-state BSE correlation energy computed in the adiabatic connection framework, where 
\begin{equation}
\label{eq:K}
	\bK = 
	\begin{pmatrix}
		\btA{\IS=1}	&	\bB{\IS=1}	\\
		\bB{\IS=1}	&	\btA{\IS=1}	\\
	\end{pmatrix}
\end{equation}
is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS} = \IS \ERI{ia}{bj}$],
\begin{equation}
\label{eq:2DM}
	\bP{\IS} = 
	\begin{pmatrix}
		\bY{\IS} \T{(\bY{\IS})}		&	\bY{\IS} \T{(\bX{\IS})}	\\
		\bX{\IS} \T{(\bY{\IS})}		&	\bX{\IS} \T{(\bX{\IS})}	\\
	\end{pmatrix}
	-
	\begin{pmatrix}
		\bO		&	\bO	\\
		\bO		&	\bI	\\
	\end{pmatrix}
\end{equation} 
is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace.
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}
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.
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.
However, as commonly done within RPA and RPAx, \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. 

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}].
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}.
In the following, we will refer to these two types of calculations as RPA@HF and RPAx@HF, respectively.
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.

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.
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}
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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Computational details}
%\label{sec:comp_details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting points to compute the BSE neutral excitations.
In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
Finally, the infinitesimal $\eta$ is set to zero for all calculations.
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. 
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 $\IS$. 

For comparison purposes, we have also computed the PES at the second-order M{\o}ller-Plesset perturbation theory (MP2), as well as with various increasingly accurate CC methods, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3. \cite{Christiansen_1995b} 
These calculations have been performed with DALTON \cite{dalton} and PSI4. \cite{Psi4}
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.
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.
All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018}
As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian Gaussian functions. 
Unless otherwise stated, the frozen-core approximation is not applied in order to provide a fair comparison between methods.
We have, however, found that the conclusions drawn in the present study hold within the frozen-core approximation (see the {\SI} for additional information).

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.
This step is, by far, the computational bottleneck in our current implementation.
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*}
	\includegraphics[width=0.49\linewidth]{H2_GS_VQZ}
	\includegraphics[width=0.49\linewidth]{LiH_GS_VQZ}
\caption{
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-pVQZ basis set. 
%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
\label{fig:PES-H2-LiH}
}
\end{figure*}
%%% %%% %%%

%%% FIG 2 %%%
\begin{figure*}
	\includegraphics[height=0.35\linewidth]{LiF_GS_VQZ}
	\includegraphics[height=0.35\linewidth]{HCl_GS_VQZ}
\caption{
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-pVQZ basis set. 
%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
\label{fig:PES-LiF-HCl}
}
\end{figure*}
%%% %%% %%%

%%% FIG 3 %%%
\begin{figure*}
	\includegraphics[height=0.26\linewidth]{N2_GS_VQZ}
	\includegraphics[height=0.26\linewidth]{CO_GS_VQZ}
	\includegraphics[height=0.26\linewidth]{BF_GS_VQZ}
\caption{
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-pVQZ basis set. 
%Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
\label{fig:PES-N2-CO-BF}
}
\end{figure*}
%%% %%% %%%

%%% FIG 4 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{F2_GS_VQZ}
\caption{
Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVQZ 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}
%%% %%% %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Potential energy surfaces}
%\label{sec:PES}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 %%%
\begin{squeezetable}
\begin{table*}
\caption{
Equilibrium bond length $\Req$ (in bohr) and correlation energy $\Ec$ (in millihartree) for the ground state of diatomic molecules obtained with the cc-pVQZ basis set at various levels of theory. 
For each system and each method, the correlation energy is computed at its respective equilibrium bond length (\ie, $R = \Req$).
When irregularities appear in the PES, the $\Req$ values are reported in parenthesis and they have been obtained by fitting a Morse potential to the PES.
The error (in \%) compared to the reference CC3 values are reported in square brackets.
}
\label{tab:Req}
	\begin{ruledtabular}
	\begin{tabular}{lllllllll}
					&	\mc{8}{c}{Equilibrium bond length $\Req$ (bohr) }	\\
									\cline{2-9}
	Method			&	\tabc{\ce{H2}}		&	\tabc{\ce{LiH}}		&	\tabc{\ce{LiF}}		&	\tabc{\ce{HCl}}		&	\tabc{\ce{N2}}		&	\tabc{\ce{CO}}		&	\tabc{\ce{BF}}		&	\tabc{\ce{F2}}				\\
	\hline
	CC3				&	1.402				&	3.019				&	2.963				&	2.403				&	2.075				&	2.136				&	2.390				&	2.663				\\
	CCSD			&	1.402[$+0.0\%$]	&	3.020[$+0.0\%$]	&	2.953[$-0.3\%$]		&	2.398[$-0.2\%$]	&	2.059[$-0.8\%$]	&	2.118[$-0.8\%$]	&	2.380[$-0.4\%$]	&	2.621[$-1.6\%$]	\\
	CC2				&	1.391[$-0.8\%$]	&	2.989[$-1.0\%$]	&	2.982[$+0.6\%$]		&	2.396[$-0.3\%$]	&	2.106[$+1.5\%$]	&	2.156[$+0.9\%$]	&	2.393[$+0.1\%$]	&	2.665[$+0.1\%$]	\\
	MP2				&	1.391[$-0.8\%$]	&	3.008[$-0.4\%$]	&	2.970[$+0.2\%$]		&	2.395[$-0.3\%$]	&	2.091[$+0.8\%$]	&	2.137[$+0.1\%$]	&	2.382[$-0.3\%$]	&	2.634[$-1.1\%$]	\\
	BSE@{\GOWO}@HF	&	1.399[$-0.2\%$]	&	3.017[$-0.1\%$]	&	(2.974)[$+0.4\%$]	&	2.400[$-0.1\%$]	&	2.065[$-0.5\%$]	&	2.134[$-0.1\%$]	&	2.385[$-0.2\%$]	&	(2.640)[$-0.9\%$]	\\
	RPA@{\GOWO}@HF	&	1.382[$-1.4\%$]	&	2.997[$-0.7\%$]	&	(2.965)[$+0.1\%$]	&	2.370[$-1.5\%$]	&	2.043[$-1.5\%$]	&	2.132[$-0.2\%$]	&	2.365[$-1.1\%$]	&	(2.571)[$-3.5\%$]	\\
	RPAx@HF			&	1.394[$-0.6\%$]	&	3.011[$-0.3\%$]	&	2.944[$-0.6\%$]		&	2.391[$-0.5\%$]	&	2.041[$-1.6\%$]	&	2.104[$-1.5\%$]	&	2.366[$-1.0\%$]	&	2.565[$-3.7\%$]	\\
	RPA@HF			&	1.386[$-1.1\%$]	&	2.994[$-0.8\%$]	&	2.946[$-0.6\%$]		&	2.382[$-0.9\%$]	&	2.042[$-1.6\%$]	&	2.103[$-1.5\%$]	&	2.364[$-1.1\%$]	&	2.573[$-3.4\%$]	\\
\hline
					&	\mc{8}{c}{Correlation energy $-\Ec$ (millihartree)}	\\
									\cline{2-9}
	Method			&	\tabc{\ce{H2}}		&	\tabc{\ce{LiH}}		&	\tabc{\ce{LiF}}		&	\tabc{\ce{HCl}}		&	\tabc{\ce{N2}}		&	\tabc{\ce{CO}}		&	\tabc{\ce{BF}}		&	\tabc{\ce{F2}}				\\
	\hline
	CC3				&	40.4			&	70.0			&	383.7				&	382.2			&	494.4			&	477.6			&	447.5				&	668.9				\\
	CCSD			&	40.4[$+0.0\%$]	&	69.8[$-0.2\%$]	&	372.6[$-2.9\%$]		&	370.8[$-3.0\%$]	&	470.6[$-4.8\%$]	&	455.2[$-4.7\%$]	&	432.9[$-3.3\%$]	&	644.0[$-3.7\%$]	\\
	CC2				&	33.3[$-17.6\%$]	&	57.3[$-18.1\%$]	&	376.7[$-1.8\%$]		&	356.9[$-6.6\%$]	&	488.0[$-1.3\%$]	&	465.5[$-2.5\%$]	&	427.3[$-4.5\%$]	&	654.9[$-2.1\%$]	\\
	MP2				&	33.2[$-17.9\%$]	&	57.9[$-17.2\%$]	&	373.0[$-2.8\%$]		&	355.7[$-6.9\%$]	&	478.0[$-3.3\%$]	&	455.0[$-4.7\%$]	&	421.6[$-5.8\%$]	&	644.3[$-3.7\%$]	\\
	BSE@{\GOWO}@HF	&	46.5[$+15.2\%$]	&	78.1[$+11.6\%$]	&	388.9[$+1.4\%$]		&	385.9[$+1.0\%$]	&	499.1[$+1.0\%$]	&	481.2[$+0.8\%$]	&	453.1[$+1.3\%$]	&	675.7[$+1.0\%$]	\\
	RPA@{\GOWO}@HF	&	57.6[$+42.6\%$]	&	101.1[$+44.5\%$]&	473.1[$+23.3\%$]	&	451.2[$+18.1\%$]&	580.3[$+17.4\%$]&	566.5[$+18.6\%$]&	545.5[$+21.9\%$]	&	794.3[$+18.8\%$]	\\
	RPAx@HF			&	37.9[$-6.2\%$]	&	65.2[$-6.8\%$]	&	343.6[$-10.5\%$]	&	344.2[$-9.9\%$]	&	427.2[$-13.6\%$]&	416.3[$-12.8\%$]&	399.1[$-10.8\%$]	&	586.1[$-12.4\%$]	\\
	RPA@HF			&	57.3[$+42.0\%$]	&	100.2[$+43.2\%$]&	465.9[$+21.4\%$]	&	442.7[$+15.8\%$]&	569.4[$+15.2\%$]&	555.9[$+16.4\%$]&	537.7[$+20.2\%$]	&	781.3[$+16.8\%$]	\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}
\end{squeezetable}

Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}.
Their 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 both significantly overestimate the FCI correlation 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.

Despite the shallow nature of its PES, the scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
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-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 is considered herein. 
Moreover, this is consistent with the study by Maggio and Kresse on the HEG showing that BSE slightly overestimates the correlation energy as compared to QMC reference data. \cite{Maggio_2016} 
Similarly, the much larger overestimation of the correlation energy that we observe at the RPA@$GW$ level was also observed for the HEG. 
Care must be taken however in drawing comparisons since the HEG study of Ref.~\onlinecite{Maggio_2016} was performed starting with LDA eigenstates.

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 as it underestimates the bond lengths by a few hundredths of bohr only.
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 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. 
We note that these irregularities are much smaller than the differences between the BSE and the other RPA-like techniques (RPA, RPAx, RPA@$GW$) leaving BSE unambiguously more accurate than these approaches.

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 molecules. 
In particular, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, the performance of BSE@{\GOWO}@HF is outstanding with an error of the order of $1\%$ on the correlation energy.
Importantly, it 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 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 $2.065$, $2.134$, and $2.385$ 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 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 error on the correlation energy of $1\%$ and 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. 
This is an important outcome of the present study as the difficulties encountered at large interatomic distances (\ie, close to the dissociation limit) do not prevent the BSE approach to be potentially useful and accurate in the vicinity of equilibrium distances. 
Furthermore, preliminary calculations could not detect any singlet instabilities in the vicinity of the lowest singlet excited-state minima.

As a final remark, we would like to mention that although we have considered here only a limited set of compounds, our correlation energy mean absolute error (MAE) with BSE@{\GOWO}@HF of $5.5$ mHa (as compared to CC3) is significantly smaller than the one obtained with MP2, CC2, and CCSD ($18.2$, $13.1$ and $13.5$ mHa respectively). 
For comparison, the RPA-related formalisms return larger MAEs of $75.6$, $43.1$, and $68.2$ mHa for BSE@{\GOWO}@HF, RPAx@HF, and RPA@HF, respectively.

%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Conclusion}
%\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 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).
For the larger systems considered here, we have observed that BSE@{\GOWO} recovers $99\%$ of the CC3 correlation energy.
Moreover, because triplet states do not contribute to the ACFDT correlation energy and singlet instabilities do not appear for weakly-correlated systems around their equilibrium structure, the present scheme does not suffer from singlet nor triplet instabilities.
However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies. 
Such an unphysical behaviour stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found.
This shortcoming 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 method.
%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.

%%%%%%%%%%%%%%%%%%%%%%%%
\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.~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 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}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for additional potential energy curves computed with other basis sets and within the frozen-core approximation, as well as tables gathering equilibrium distances for smaller basis sets (cc-pVDZ and cc-pVTZ).

\bibliography{BSE-PES,BSE-PES-control}

\end{document}
