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Manuscript/QUEST_WIREs.tex

@@ -299,7 +299,7 @@ to be extrapolated. This relation is valid in the regime of a sufficiently large
 correction largely dominates.
 However, in practice, due to the residual higher-order terms, the coefficient $\alpha^{(m)}$ deviates slightly from unity.
 
-Using Eq.(\ref{eqx}) the estimated error on the CIPSI energy is calculated as
+Using Eq.~\eqref{eqx} the estimated error on the CIPSI energy is calculated as
 \begin{equation}
   E_{\text{CIPSI}}^{(m)} - E_{\text{FCI}}^{(m)}
   = \qty(E_\text{var}^{(m)}+E_{\text{rPT2}}^{(m)}) - E_{\text{FCI}}^{(m)}
@@ -311,11 +311,11 @@ state is given by
   \Delta E_{\text{FCI}}^{(m)}
   = \qty[ E_\text{var}^{(m)} + E_{\text{rPT2}} + \qty(\alpha^{(m)}-1) E_{\text{rPT2}} ]
   - \qty[ E_\text{var}^{(0)} + E_{\text{rPT2}} + \qty(\alpha^{(0)}-1) E_{\text{rPT2}} ]
-  + O\qty[{E_{\text{rPT2}}^2 }]
+  + \mathcal{O}\qty[{E_{\text{rPT2}}^2 }],
 \end{equation}
-which evidences that the error in $\Delta E_{\text{FCI}}^{(m)}$ can be expressed as $\qty(\alpha^{(m)}-\alpha^{(0)}) E_{\text{rPT2}} + O\qty[{E_{\text{rPT2}}^2}]$.
+which evidences that the error in $\Delta E_{\text{FCI}}^{(m)}$ can be expressed as $\qty(\alpha^{(m)}-\alpha^{(0)}) E_{\text{rPT2}} + \mathcal{O}\qty[{E_{\text{rPT2}}^2}]$.
 
-Now, for the largest systems considered here, $\qty|{E_{\text{rPT2}}}|$ can be as large as 2~eV and, thus,
+Now, for the largest systems considered here, $\abs{E_{\text{rPT2}}}$ can be as large as 2~eV and, thus,
 the accuracy of the excitation energy estimates strongly depends on our ability to compensate the errors in the calculations.
 Here, we greatly enhance the compensation of errors by making use of
 our selection procedure ensuring that the PT2 values of both states
@@ -325,12 +325,12 @@ E_{\text{rPT2}}^{(0)} \approx E_{\text{rPT2}}^{(m)}$, and
 by using a common set of state-averaged natural orbitals with equal weights for the ground and excited states.
 This last feature tends to make the values of $\alpha^{(0)}$ and $\alpha^{(m)}$ very close to each other, such that the error on the energy difference
 is decreased.
-In the ideal case where we would be able to fully correlate the CIPSI calculations for the ground- and excited-states, the fluctuations of
+In the ideal case where we would be able to fully correlate the CIPSI calculations associated with the ground and excited states, the fluctuations of
 $\Delta E_\text{CIPSI}^{(m)}(n)$ as a function of $n$ would completely vanish and the exact excitation energy would be obtained from the first CIPSI iterations.
 Quite remarkably, in practice, numerical experience shows that the fluctuations with respect to the extrapolated value $\Delta E_\text{FCI}^{(m)}$ are small,
 zero-centered, almost independent of $n$ when not too close iteration
 numbers are considered, and display a Gaussian-like distribution.
-In addition, the fluctuations are found to be (very weakly) dependent on the iteration number $n$ (see, Fig.\ref{fig2}), so
+In addition, as stated just above, the fluctuations are found to be (very weakly) dependent on the iteration number $n$ (see Fig.~\ref{fig:histo}), so
 this dependence will not significantly alter our results and will not be considered here.
 We thus introduce the following random variable
 \begin{equation}
@@ -348,7 +348,7 @@ A natural choice for $\sigma^2(n)$, playing here the role of a variance, is
 \begin{equation}
 \sigma^2(n) \propto \qty[E_{\text{rPT2}}^{(m)}(n)]^2 + \qty[E_{\text{rPT2}}^{(0)}(n)]^2,
 \end{equation}
-which vanishes in the large-$n$ limit as it should be.
+which vanishes in the large-$n$ limit as it should.
 
 %%% FIGURE 2 %%%
 \begin{figure}
@@ -357,19 +357,19 @@ which vanishes in the large-$n$ limit as it should be.
 \caption{Histogram of the random variable $X^{(m)}$ (see, text). About 200 values of the transition energies
 for the 13 five- and six-membered ring molecules, both for the singlet and triplet transitions and for a number of CIPSI iterations, are used.
 The number $M$ of iterations kept is chosen according to the statistical test presented in the text.}
-\label{fig2}
+\label{fig:histo}
 \end{figure}
 
 The histogram of $X^{(m)}$ resulting from the excitation energies
 obtained at different values of the CIPSI iterations $n$
 and for the 13 five- and six-membered ring molecules, both for the singlet and triplet transitions,
-is shown in Fig.\ref{fig2}. To avoid transient effects, only excitation energies at sufficiently large $n$ are retained in the data set.
+is shown in Fig.~\ref{fig:histo}. To avoid transient effects, only excitation energies at sufficiently large $n$ are retained in the data set.
 The criterion used to decide from which precise value of $n$ the data should be kept will be presented below. In our application, the total number
-of values employed to make the histogram is about 200. The dashed line of Fig.\ref{fig2} represents the best Gaussian fit
+of values employed to make the histogram is about 200. The dashed line of Fig.~\ref{fig:histo} represents the best Gaussian fit
 (in the sense of least-squares) reproducing the data.
 As seen, the distribution can be described by the Gaussian probability
 \begin{equation}
-P\qty[X^{(m)}] \propto e^{-\frac{{X^{(m)}}^2} {2{\sigma^{*}}^2}}
+P\qty[X^{(m)}] \propto \exp[-\frac{{X^{(m)}}^2} {2{\sigma^{*}}^2} ]
 \end{equation}
 where $\sigma^{*2}$ is some "universal" variance depending only
 on the way the correlated selection of both states is done, not on the molecule considered in our set.
@@ -380,7 +380,7 @@ $$\Delta E_\text{FCI}^{(m)} = \frac{ \sum_{n=1}^M  \frac{\Delta E_\text{CIPSI}^{
 $$
 where $M$ is the number of data kept.
 Now, regarding the estimate of the error on $\Delta E_\text{FCI}^{(m)}$ some caution is required since, although the distribution is globally Gaussian-like
-(see Fig.\ref{fig2}) there exists
+(see Fig.~\ref{fig:histo}) there exists
 some significant departure from it and we need to take this feature into account.
 
 More precisely, we search for a confidence interval $\mathcal{I}$ such that the true value of the excitation energy $\Delta E_{\text{FCI}}^{(m)}$ lies within one standard deviation of $\Delta E_\text{CIPSI}^{(m)}$, i.e., $P\qty( \Delta E_{\text{FCI}}^{(m)} \in \qty[ \Delta E_\text{CIPSI}^{(m)} \pm \sigma ] \; \Big| \; \mathcal{G}) = 0.6827$.
@@ -1238,9 +1238,10 @@ MAE			&				& 0.22	& 0.16	& 0.22	& 0.11	& 0.12	& 0.05	& 0.04	& 0.02	& 0.20	& 0.22
 \end{threeparttable}
 \end{sidewaystable}
 
+%%% FIGURE 5 %%%
 \begin{figure}
 	\centering
-	\includegraphics[width=0.9\textwidth]{histograms}
+	\includegraphics[width=0.9\textwidth]{fig5}
 	\caption{Distribution of the error (in eV) in excitation energies (with respect to the TBE/aug-cc-pVTZ values) for various methods for the entire QUEST database considering only closed-shell compounds.
 	Only the ``safe'' TBEs are considered (see Table \ref{tab:TBE}).
 	See Table \ref{tab:stat} for the values of the corresponding statistical quantities.