done with revision

benzene.tex

@@ -112,7 +112,7 @@ However, performing SCI calculations rapidly becomes extremely tedious when one
 From a historical point of view, CIPSI is probably one of the oldest SCI algorithm. 
 It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see also Ref.~\onlinecite{Evangelisti_1983}).
 Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by our group enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019}
-In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
+In particular, we were able to compute highly-accurate ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
 CIPSI is also frequently used to provide accurate trial wave function for QMC calculations. \cite{Caffarel_2014,Caffarel_2016a,Caffarel_2016b,Giner_2013,Giner_2015,Scemama_2015,Scemama_2016,Scemama_2018,Scemama_2018b,Scemama_2019,Dash_2018,Dash_2019} 
 The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
 \alert{Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \cite{Garniron_2019} 
@@ -122,12 +122,13 @@ We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can
 Being late to the party, we obviously cannot report blindly our CIPSI results.
 However, following the philosophy of Eriksen \textit{et al.} \cite{Eriksen_2020} and Lee \textit{et al.}, \cite{Lee_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind.
 We then follow our usual ``protocol'' \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
-Natural orbitals are then computed based on this wave function, and a new, larger SCI calculation is performed with this new set of orbitals. 
+Natural orbitals are then computed based on this wave function, and a new, larger SCI calculation is performed with this new natural set of orbitals. 
 This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
 The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{(r)PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element. The magnitude of $E_\text{(r)PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
 As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems.
 We then linearly extrapolate the total SCI energy to $E_\text{(r)PT2} = 0$ (which effectively corresponds to the FCI limit). 
-Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Applencourt_2018} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a single state. We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero ($\sim 5 \times 10^{-3}$ a.u.).
+Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Applencourt_2018} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a singlet state. 
+We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero ($\sim 5 \times 10^{-3}$ a.u.).
 The corresponding energies are reported in Table \ref{tab:NOvsLO} as functions of the number of determinants in the variational space $N_\text{det}$.
 
 A second run has been performed with localized orbitals. 
@@ -146,7 +147,7 @@ As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of
 We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations. 
 The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of the correlation energy, $\Delta E_\text{var.}$ and $\Delta E_\text{var.} + E_\text{(r)PT2}$, as a function of $N_\text{det}$ (left panel).
 The right panel of Fig.~\ref{fig:CIPSI} is more instructive as it shows $\Delta E_\text{var.}$ as a function of $E_\text{(r)PT2}$, and their corresponding four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy. 
-\alert{In other words, the four largest variational wave functions are considered to perform the linear extrapolation.}
+\alert{(In other words, the four largest variational wave functions are considered to perform the linear extrapolation.)}
 From this figure, one clearly sees that the rPT2-based correction behaves more linearly than its corresponding PT2 version, and is thus systematically employed in the following.
 
 % Results