update MO labels and var 160 M

benzene.tex

@@ -132,15 +132,16 @@ Note that, unlike excited-state calculations where it is important to enforce th
 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. 
-Starting from the same natural orbitals, a Boys-Foster localization procedure \cite{Boys_1960} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals. Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene.
+Starting from the same natural orbitals, a Boys-Foster localization procedure \cite{Boys_1960} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals. 
-% MO Indices:
+\footnote{MO indices for Boys-Foster localization procedure:
-%[1-6]  # Core
+core [1--6];
-%[7,8,9,10,11,12,13,14,15,16,17,18] # Sigma occ
+$\sigma$ [7--18];
-%[19,20,21] # Pi occ
+$\pi$ [19--21]; 
-%[22,23,24] # Pi virt 1
+$\pi^*$ [22--24];
-%[25,26,27,28,29,30,31,32,33,34,35,36] # Sigma virt 1
+$\sigma^*$ [25--36];
-%[39,41,42,43,46,49,50,53,54,55,56,57,71,72,73,74,82,83,84,85,87,92,93,98] # Pi virt 2
+higher-lying $\pi$ [39,41--43,46,49,50,53--57,71--74,82--85,87,92,93,98];
-%[37,38,40,44,45,47,48,51,52,58,59,60,61,62,63,64,65,66,67,68,69,70,75,76,77,78,79,80,81,86,88,89,90,91,94,95,96,97,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114] # Sigma virt 2
+higher-lying $\sigma$ [37,38,40,44,45,47,48,51,52,58--70,75--81,86,88--91,94--97,99--114].}
+Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene.
 As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of $N_\text{det}$, the variational energy as well as the PT2-corrected energies are much lower with localized orbitals than with natural orbitals. 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 $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ as a function of $N_\text{det}$ (left panel).
 The right panel of Fig.~\ref{fig:CIPSI} shows $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$, and their corresponding \titou{two}-point linear extrapolation curves that we have used to get our final estimate of the correlation energy.
@@ -209,7 +210,7 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
     20\,971\,520 & $-231.474\,019$  & $-231.561\,315(430)$  & $-231.560\,063(424)$    &   $-231.508\,714$  & $-231.564\,707(275)$   & $-231.564\,223(273)$  \\
     41\,943\,040 & $-231.487\,978$  & $-231.564\,529(382)$  & $-231.563\,593(377)$    &   $-231.519\,122$  & $-231.567\,419(240)$   & $-231.567\,069(238)$  \\
     83\,886\,080 & $-231.501\,334$  & $-231.566\,994(317)$  & $-231.566\,325(314)$    &   $-231.528\,568$  & $-231.570\,084(199)$   & $-231.569\,832(198)$  \\
-   167\,772\,160 & \\
+   167\,772\,160 & $-231.514\,009$  & & & $-231.536\,655$\\
   \end{tabular}  
 \end{ruledtabular}  
 \end{table*}