Typos

Manuscript/QUEST_WIREs.tex

@@ -290,7 +290,7 @@ The definition of the active space considered for each system as well as the num
 %------------------------------------------------
 In this section, we present our scheme to estimate the extrapolation error in SCI calculations.
 This new protocol is then applied to five- and six-membered ring molecules for which SCI calculations are particularly challenging even for small basis sets.
-Note that the present method does only applied to ``state-averaged'' SCI calculations where ground- and excited-state energies are produced during the same calculation with the same set of molecular orbitals, not to ``state-specific'' calculations where one computes solely the energy of a single state (like conventional ground-state calculations).
+Note that the present method does only apply to ``state-averaged'' SCI calculations where ground- and excited-state energies are produced during the same calculation with the same set of molecular orbitals, not to ``state-specific'' calculations where one computes solely the energy of a single state (like conventional ground-state calculations).
 
 For the $m$th excited state (where $m = 0$ corresponds to the ground state), we usually estimate its FCI energy $E_{\text{FCI}}^{(m)}$ by performing a linear extrapolation of its variational energy $E_\text{var}^{(m)}$ as a function of its rPT2 correction $E_{\text{rPT2}}^{(m)}$ as follows
 \begin{equation}
@@ -330,7 +330,7 @@ This choice ensures that the statistical uncertainty vanishes at the FCI limit.
 We then 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., $P( \Delta E_{\text{FCI}}^{(m)} \in [ \Delta E_\text{CIPSI}^{(m)} \pm \sigma ] \; | \; \mathcal{G}) = 0.6827$. 
 The probability that $\Delta E_{\text{FCI}}^{(m)}$ is in an interval $\mathcal{I}$ is
 \begin{equation}
-   P( \Delta E_{\text{FCI}}^{(m)} \in \mathcal{I} ) = P( \Delta E_{\text{FCI}}^{(m)} \in I | \mathcal{G}) \times P(\mathcal{G}) 
+   P\qty( \Delta E_{\text{FCI}}^{(m)} \in \mathcal{I} ) = P\qty( \Delta E_{\text{FCI}}^{(m)} \in I \Big| \mathcal{G}) \times P(\mathcal{G}) 
 \end{equation}
 where the probability $P(\mathcal{G})$ that the random variables are normally distributed can be deduced from the Jarque-Bera test $J$ as
 \begin{equation}
@@ -343,24 +343,24 @@ The inverse of the cumulative distribution function of the $t$-distribution, $t_
    \beta = t_{\text{CDF}}^{-1} \qty[ 
    \frac{1}{2} \qty( 1 + \frac{0.6827}{P(\mathcal{G})}), M ]
 \end{equation}
-such that $P( \Delta E_{\text{FCI}}^{(m)} \in [ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p = 0.6827$.
+such that $P\qty( \Delta E_{\text{FCI}}^{(m)} \in \qty[ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p = 0.6827$.
 Only the last $M>2$ computed energy differences are considered. $M$ is chosen such that $P(\mathcal{G})>0.8$ and such that the error bar is minimal.
 If all the values of $P(\mathcal{G})$ are below $0.8$, $M$ is chosen such that $P(\mathcal{G})$ is maximal.
 A Python code associated with this procedure is provided in the {\SupInf}.
 
 The singlet and triplet FCI/6-31+G(d) excitation energies and their corresponding error bars estimated with the method presented above based on Gaussian random variables are reported in Table \ref{tab:cycles}.
-For the sake of comparison, we also report the CC3 and CCSDT vertical energies from Ref.~\cite{Loos_2020b} computed in the same basis. We note that there is for the vas majority of considered
+For the sake of comparison, we also report the CC3 and CCSDT vertical energies from Ref.~\cite{Loos_2020b} computed in the same basis. We note that there is for the vast majority of considered
 states a very good agreement between the CC3 and CCSDT values, indicating that the CC values can be trusted.
 The estimated values of the excitation energies obtained via a three-point linear extrapolation considering the three largest CIPSI wave functions are also gathered in Table \ref{tab:cycles}.
 In this case, the error bar is estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.
 This strategy has been considered in some of our previous works \cite{Loos_2020b,Loos_2020c,Loos_2020e}.
 The deviation from the CCSDT excitation energies for the same set of excitations are depicted in Fig.~\ref{fig:errors}, where the red dots correspond to the excitation energies and error bars estimated via the present method, and the blue dots correspond to the excitation energies obtained via a three-point linear fit and error bars estimated via the extrapolation distance.
-These results contains a good balance between well-behaved and ill-behaved cases.
+These results contain a good balance between well-behaved and ill-behaved cases.
 For example, cyclopentadiene and furan correspond to well-behaved scenarios where the two flavors of extrapolations yield nearly identical estimates and the error bars associated with these two methods nicely overlap.
 In these cases, one can observe that our method based on Gaussian random variables provides almost systematically smaller error bars. 
 Even in less idealistic situations (like in imidazole, pyrrole, and thiophene), the results are very satisfactory and stable.
 The six-membered rings represent much more challenging cases for SCI methods, and even for these systems the newly-developed method provides realistic error bars, and allows to easily detect problematic events (like pyridine for instance).
-The present scheme has also been tested on smaller systems when one can tightly converged the CIPSI calculations.
+The present scheme has also been tested on smaller systems when one can tightly converge the CIPSI calculations.
 In such cases, the agreement is nearly perfect in every scenario that we have encountered. 
 A selection of these results can be found in the {\SupInf}.