corrections Secs. II and III

Manuscript/Ec.tex

@@ -309,14 +309,14 @@ By introducing a Lagrange multiplier $\lambda$ to control the trust-region size,
 The addition of the level shift $\lambda \geq 0$ removes the negative eigenvalues and ensures the positive definiteness of the Hessian matrix by reducing the step size.
 By choosing the right value of $\lambda$, $\norm{\bk}$ is constrained into a hypersphere of radius $\Delta$ and is able to evolve from the Newton direction at $\lambda = 0$ to the steepest descent direction as $\lambda$ grows.
 The evolution of the trust radius during the optimization and the use of a condition to reject the step when the energy rises ensure the convergence of the algorithm.
-More detail can be found in Ref.~\onlinecite{Nocedal_1999}.
+More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 
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 \section{Computational details}
 \label{sec:compdet}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The geometries of the twelve systems considered in the present study were all obtained at the CC3/aug-cc-pVTZ level of theory and were extracted from a previous study. \cite{Loos_2020a}
-Note that, for the sake of consistency, the geometry of benzene considered here is different from one of Ref.~\onlinecite{Loos_2020e} which was obtained at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
+Note that, for the sake of consistency, the geometry of benzene considered here is different from the one of Ref.~\onlinecite{Loos_2020e} which was obtained at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
 The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations were performed with CFOUR, \cite{Matthews_2020} the CCSD(T) and CR-CC(2,3) calculations were made with GAMESS 2014R1, \cite{gamess} and MP5 calculations were computed with GAUSSIAN 09. \cite{g09}
 The CIPSI calculations were performed with QUANTUM PACKAGE. \cite{Garniron_2019}
 In the current implementation, the selection step and the PT2 correction are computed simultaneously via a hybrid semistochastic algorithm.\cite{Garniron_2017,Garniron_2019} %(which explains the statistical error associated with the PT2 correction in the following).
@@ -334,7 +334,7 @@ Using optimized orbitals has the undeniable advantage to produce, for a given va
 For the benzene molecule, we also explore the use of localized orbitals (LOs) which are produced with the Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals in several orbital windows in order to preserve a strict $\sigma$-$\pi$ separation in the planar systems considered here. \cite{Loos_2020e}
 Because they take advantage of the local character of electron correlation, localized orbitals have been shown to provide faster convergence towards the FCI limit compared to natural orbitals. \cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020,Loos_2020e}
 As we shall see below, employing optimized orbitals has the advantage to produce an even smoother and faster convergence of the SCI energy toward the FCI limit.
-Note both localized and optimized orbitals do break the spatial symmetry.
+Note that both localized and optimized orbitals do break the spatial symmetry.
 Unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Chilkuri_2021} the present wave functions do not fulfill this property as we aim for the lowest possible energy of a closed-shell singlet state.
 We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.