toto part

Manuscript/QUEST_WIREs.tex

@@ -243,7 +243,7 @@ The definition of the active space considered for each system as well as the num
 
 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}
-  E_\text{var}^{(m)} = E_{\text{FCI}}^{(m)} - \alpha^{(m)} E_{\text{rPT2}}^{(m)}
+  E_\text{FCI}^{(m)} = E_{\text{var}}^{(m)} + \alpha^{(m)} E_{\text{rPT2}}^{(m)}
 \end{equation}
 $E_\text{var}^{(m)}$ varies almost linearly as a function of $E_{\text{rPT2}}^{(m)}$, but with a coefficient $\alpha^{(m)}$ which deviates slightly from unity in well-behaved cases. 
 This implies that, at any iteration of the CIPSI algorithm, the estimated error on the CIPSI energy is 
@@ -273,7 +273,7 @@ The $m$th excitation energy at iteration $n$ is then assumed to be a Gaussian ra
   \sigma^2(n) \propto \qty[E_{\text{rPT2}}^{(m)}(n)]^2 + \qty[E_{\text{rPT2}}^{(0)}(n)]^2
 \end{gather}
 and we treat all CIPSI iterations as a set of Gaussian-distributed variables ($\mathcal{G}$) with weights $w(n) = 1/\sqrt{\sigma^2(n)}$.
-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., $P( \Delta E_{\text{FCI}} \in [ \Delta E_\text{CIPSI}^{(m)} \pm \sigma ] \; | \; \mathcal{G}) = 0.6827$. 
+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}) 
@@ -294,6 +294,7 @@ The inverse of the cumulative distribution function of the $t$-distribution allo
 such that $P( \Delta E_{\text{FCI}}^{(m)} \in [ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p$.
 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}.
 
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 \section{The QUEST database}