start ground state

BSE-PES.tex

@@ -1,4 +1,4 @@
-\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
+\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
 \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
 \usepackage[version=4]{mhchem}
 
@@ -157,7 +157,7 @@
 
 \begin{document}	
 
-\title{Ground- and excited-state potential energy surfaces within the Bethe-Salpeter equation formalism}
+\title{Ground-State Potential Energy Surfaces Within the Bethe-Salpeter Formalism: Pros and Cons}
 
 \author{Xavier \surname{Blase}}
 \email{xavier.blase@neel.cnrs.fr }
@@ -174,14 +174,12 @@
 %	\centering
 %  \includegraphics[width=\linewidth]{TOC}
 %\end{wrapfigure}
-The many-body Green's function 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 for medium and large molecular systems. 
-In particular, BSE@\textit{GW} faithfully treats charge-transfer states and avoids the delicate choice of the exchange-correlation functional from an ever-growing zoo of functionals.
-However, contrary to TD-DFT, there is no clear definition of the ground- and excited-state energies within BSE. 
-This makes the development of analytical nuclear gradients a particularly tricky task.
-Here, we provide an unambiguous definition of the ground-state BSE energy, and we calculate the excited-state BSE energy of a given state by adding the corresponding BSE excitation energy to the ground-state BSE energy which is calculated in the framework of the adiabatic-connection fluctuation-dissipation theorem.
-Embracing this definition which treats on equal footing ground and excited states at the BSE level, we study the topological features of the ground- and singlet excited-state potential energy surfaces (PES) for several diatomic molecules.
-Our aim is to know whether or not the BSE formalism is able to reproduce faithfully the main features of these PES near equilibrium, and, in particular, the location of the minima on the ground- and excited-state PES. 
-\titou{Thanks to comparison with both similar and state-of-art computational approaches, we show that the present approach is surprisingly accurate.}
+The many-body Green's function Bethe-Salpeter equation (BSE) formalism performed on top of a $GW$ calculation has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies for medium and large molecular systems. 
+Although less popular, the BSE formalism can also be employed to compute ground-state energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
+Here, we study the topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules.
+Our aim is to know whether or not the BSE formalism is able to reproduce faithfully the main features of these PES near equilibrium, and, in particular, the location of the minima on the ground-state potential energy surfaces. 
+Thanks to comparison with both similar and state-of-art computational approaches, we show that the present ACFDT-based approach is surprisingly accurate, and can even compete with coupled cluster methods.
+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.
 \end{abstract}
 
 \maketitle
@@ -222,23 +220,23 @@ A significant limitation of the BSE formalism as compared to TD-DFT lies in the
 \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 ground-state analogue.
 
 Contrary to TD-DFT, the ground-state correlation energy calculated at the BSE level remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020}
-As a matter of fact, in the largest recent available benchmark study of the 26 small molecules forming the HEAT test set, \cite{Holzer_2018} the BSE correlation energy, as evaluated within the adiabatic connection formulation (AC-BSE), was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018} 
+As a matter of fact, in the largest recent available benchmark study of the 26 small molecules forming the HEAT test set, \cite{Holzer_2018} the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation (ACFDT) framework, was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018} 
 Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016} 
 With such an approximation, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either KS-DFT or {\GW} eigenvalues in the construction of the propagator becomes further relevant, increasing accordingly the number of possible definitions for the ground-state correlation energy. 
-Finally, renormalizing or not the Coulomb interaction by the coupling parameter $\lambda$ in the Dyson equation for the interacting polarizability leads to two different versions of the AC-BSE correlation energy. 
+Finally, renormalizing or not the Coulomb interaction by the coupling parameter $\lambda$ in the Dyson equation for the interacting polarizability leads to two different versions of the BSE correlation energy. 
 
-Here, in analogy to the random-phase approximation (RPA) formalism,  \cite{Furche_2008} the ground-state BSE energy is calculated via the ``trace'' formula (see below). The excited-state BSE energy is then computed by adding the BSE excitation energy of the selected state to the ground-state BSE energy.
+\alert{Here, in analogy to the random-phase approximation (RPA) formalism,  \cite{Furche_2008} the ground-state BSE energy is calculated via the ``trace'' formula (see below). The excited-state BSE energy is then computed by adding the BSE excitation energy of the selected state to the ground-state BSE energy.
 This definition of the energy has the advantage of treating at the same level of theory the ground state and the excited states.
 Embracing this definition, the purpose of the present study is to investigate the quality of ground- and excited-state PES near equilibrium obtained within the BSE approach.
 The location of the minima on the ground- and (singlet and triplet) excited-state PES is of particular interest.
-This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@{\GW} formalism.
+This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@{\GW} formalism.}
 
 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 PES of the ground- and excited-states for various diatomic molecules. 
+Section \ref{sec:PES} reports  ground-state PES for various diatomic molecules. 
 Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -350,38 +348,27 @@ are the bare two-electron integrals, $\delta_{pq}$ is the Kronecker delta, and
 In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral direct (\ie, without exchange) dRPA excitation energies computed during the {\GW} calculation, and $\eta$ is a positive infimitesimal.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Ground- and excited-state BSE energy}
+\subsection{Ground-state BSE energy}
 \label{sec:BSE_energy}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The key quantity to define in the present context is the total BSE energy.
-Although not unique, we propose to define the BSE total energy of the $m$th state as
+The key quantity to define in the present context is the total ground-state BSE energy.
+Although not unique, we propose to define it as
 \begin{equation}
 \label{eq:EtotBSE}
-	\EBSE{m} = \Enuc + \EHF + \EcBSE + \OmBSE{m},
+	\EBSE{m} = \Enuc + \EHF + \EcBSE 
 \end{equation}
-where $\Enuc$ and $\EHF$ are the state-independent nuclear repulsion energy and ground-state HF energy (respectively),
+where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively),
 \begin{equation}
 \label{eq:EcBSE}
 	\EcBSE = \frac{1}{2} \qty[ {\sum_m}' \OmBSE{m} - \Tr(\bA) ]
 \end{equation}
-is the ground-state BSE correlation energy computed with the so-called trace formula, \cite{Schuck_Book, Rowe_1968, Sawada_1957b} and $\OmBSE{m}$ is the $m$th BSE excitation energy with the convention that, for the ground state ($m=0$), $\OmBSE{0} = 0$. 
-An elegant derivation of Eq.~\eqref{eq:EcBSE} has been recently proposed within the BSE formalism by Olevano and coworkers. \cite{Li_2020} 
-Note that, at the dRPA level, an alternative formulation does exist which consists in integrating along the adiabatic connection path. \cite{Gell-Mann_1957}
-These two dRPA formulations have been found to be equivalent in practice for both the uniform electron gas \cite{Sawada_1957b, Fukuta_1964, Furche_2008} and in molecules. \cite{Li_2020} 
-However, in the case of BSE, there is no guarantee that the two formalisms (trace \textit{vs} adiabatic connection) yields the same values.
-
-Equation \eqref{eq:EtotBSE} defines unambiguously the total BSE energy of the system for both ground and (singlet and triplet) excited states.
-It has also the indisputable advantage of treating on equal footing (\ie, at the same level of theory) the ground state and the excited states.
-Note that the prime in the sum of Eq.~\eqref{eq:EcBSE} means that we only consider in the summation the \textit{positive} excitation energies from the singlet and triplet manifolds.
-Indeed, in case of triplet instabilities, some of the triplet excitation energies are negative, and must be discarded as the resonant-only part of the BSE excitonic Hamiltonian has to be considered. 
-
-From a practical point of view, it is also convenient to split $\EcBSE = \EcsBSE + \EctBSE$ in its singlet ($\sigma \neq \sigma^{\prime}$) and triplet ($\sigma = \sigma^{\prime}$) contributions, \ie,
-\begin{equation}
-	\EcBSE 
-	= \underbrace{\frac{1}{2} \qty[ {\sum_m} \OmsBSE{m} - \Tr(\bAs) ]}_{\EcsBSE}
-	+ \underbrace{\frac{1}{2} \qty[ {\sum_m}'  \OmtBSE{m} - \Tr(\bAt) ]}_{\EctBSE}.
-\end{equation}
-As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily generalized to other theories (such as CIS, dRPA, or TDHF) by computing $\Ec$ and $\Om{m}$ at the these levels of theory.
+is the ground-state BSE correlation energy computed in the adiabatic connection framework 
+Note that the present formulation is different from the plasmon formulation. \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020}
+Note that, at the dRPA level, the plasmon and adiabatic connection formulations are equivalent. \cite{Sawada_1957b, Fukuta_1964, Furche_2008}
+Howewer, this is not the case at the BSE level. 
+ 
+One of the undisputable advantage of the adiabtic connection formulation is that the triplet does not contribute. 
+Therefore, the triplet instabilities, some of the triplet excitation energies are negative, and must be discarded as the resonant-only part of the BSE excitonic Hamiltonian has to be considered. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
@@ -396,8 +383,7 @@ For {\evGW}, the quasiparticle energies are obtained self-consistently and we ha
 Further details about our implementation of {\GOWO} and {\evGW} can be found in Refs.~\onlinecite{Loos_2018,Veril_2018}.
 Finally, the infinitesimal $\eta$ has been set to zero for all calculations.
 \titou{For sake of comparison, no frozen core approximation.
-The numerical integration required to compute the correlation energy along the adiabatic path has been computed with a 21-point Gauss-Legendre quadrature.
-This number of points is probably too big...}
+The numerical integration required to compute the correlation energy along the adiabatic path has been computed with a 21-point Gauss-Legendre quadrature.	}
 
 Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both singlet and triplet), we perform a complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3}$ computational cost.
 This step is, by far, the computational bottleneck in our current implementation.