Updates from Overleaf

BSE-PES.tex

@@ -226,7 +226,7 @@ This study is a first preliminary step towards the development of analytical nuc
 There are very few studies about the ground-state BSE energy for atomic and molecular systems \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020} and, to the best of our knowledge, the present study is the first to investigate the topology of the excited-state PES at the BSE level. 
 
 \xavier{ALTERNATIVE PARAGRAPH: 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, was 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{Holzer_2018} Such a modified BSE polarization propagator was inspired by a previous  study of the homogeneous interacting electron gaz.} \cite{Maggio_2016} 
+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}  With such an approximation, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either DFT Kohn-Sham 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, the renormalization or not of the Coulomb interaction by the coupling parameter $\lambda$ in the Dyson equation for the interacting polarizability leads to two different version of the AC-BSE correlation energy. }  
 
 \xavier{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.