Rewrote benzene

Manuscript/stochastic_triples.tex

@@ -217,28 +217,30 @@ accelerators.\cite{ma_2011,haidar_2015,dinapoli_2014,springer_2018}
 
 In this section we illustrate the convergence of the statistical error of the perturbative triples correction as a function of the computational cost.
 The benzene molecule serves as our reference system for conducting frozen-core CCSD(T) calculations with the cc-pVTZ and cc-pVQZ basis sets.
-Essentially, this involves the correlation of 30 electrons across either 258 or 503 molecular orbitals.
+Essentially, this involves the correlation of 30 electrons using either 258 or 503 molecular orbitals.
 
 \begin{figure}
 \includegraphics[width=\columnwidth]{benzene_tz.pdf}
 \includegraphics[width=\columnwidth]{benzene_qz.pdf}
-	\caption{\label{fig:benzene} Convergence of the energy of benzene as a function of the execution time of the program. The top curve corresponds to the cc-pVTZ basis set and the bottom curve to cc-pVQZ. The blue line represents the exact CCSD(T) energy.}
+	\caption{\label{fig:benzene} Energy convergence of benzene plotted against the program execution time, showing comparisons between the cc-pVTZ (upper curve) and cc-pVQZ (lower curve) basis sets. The blue lines indicate the exact CCSD(T) energies.}
 \end{figure}
 
 \begin{figure}
 \includegraphics[width=\columnwidth]{benzene_err.pdf}
-	\caption{\label{fig:benzene_err} Convergence of the statistical error of the perturbative triples contribution in benzene as a function of the percentage of computed contributions, in the cc-pVTZ and cc-pVQZ basis sets.}
+	\caption{\label{fig:benzene_err} Convergence of the statistical error of the perturbative triples contribution in benzene as a function of the percentage of computed contributions, for both cc-pVTZ and cc-pVQZ basis sets.}
 \end{figure}
 
-Figure~\ref{fig:benzene} shows the convergence of the CCSD(T) energy as a function of the execution time of the program with both basis sets. We observe that the exact CCSD(T) energy is alwasy within $2\sigma$, showing that the statistical error is reliable. 
+Figure~\ref{fig:benzene} shows the convergence of the CCSD(T) energy as a function of the program execution time using the two basis sets.
+Notably, the exact CCSD(T) energy always falls within $2\sigma$, affirming the reliability of the statistical error.
 Figure~\ref{fig:benzene_err} displays the statistical error as a function of the percentage of computed contributions.
-In both figures, the discontinuities in the curves are due to changes in the splitting between deterministic and stochastic components, leading to a change in the estimated value, and a reduction of the statistical error.
-For the both basis sets, the chemical accuracy (\SI{1.6}{\milli \hartree}) is reached using less than 1\% of the contributions.
-A precision of (\SI{0.1}{\milli \hartree}) is obtained using respectively 32\%  and 15\% for the cc-pVTZ and cc-pVQZ basis sets.
-The faster convergence for the largest basis set was expected: using a larger basis set increases the number of tiny contributions, and keeps the number of large contributions rather constant. Hence, this curve shows that the proposed algorithm is better adapted to fewer electrons in large basis sets than many electrons in small basis sets.
+Noteworthy in both figures are the curve discontinuities, attributable to readjustments in the separation between the deterministic and stochastic components of the calculation.
+These updates lead to revised estimates and a diminution in statistical error. 
+
+Achieving chemical accuracy, defined as \SI{1.6}{\milli\hartree}, necessitates less than 1\% of the total contributions in both basis sets.
+Attaining a \SI{0.1}{\milli\hartree} precision level requires computation of 32\% and 15\% of the contributions for cc-pVTZ and cc-pVQZ, respectively.
+The more rapid convergence observed with the larger basis set aligns with expectations, as expanding the basis set tends to increase the proportion of minor contributions while maintaining a relatively steady count of significant contributions.
+This trend underscores the algorithm's enhanced suitability for systems with fewer electrons and extensive basis sets, as opposed to larger electron counts in smaller basis sets.
 
-% Discuter les cassures dans les courbes
-% Ajouter une courbe de convergence de l'erreur
 
 \subsection{Vibrational frequency of copper chloride}