diff --git a/Manuscript/QUEST_WIREs.tex b/Manuscript/QUEST_WIREs.tex index 22fac74..33e0e23 100644 --- a/Manuscript/QUEST_WIREs.tex +++ b/Manuscript/QUEST_WIREs.tex @@ -282,14 +282,12 @@ where the probability $P(\mathcal{G})$ that the random variables are normally di \begin{equation} P(\mathcal{G}) = 1 - \chi^2_{\text{CDF}}(J,2) \end{equation} -where $\chi^2_{\text{CDF}}(x,k)$ is the cumulative distribution function of the $\chi^2$-distribution with $k$ degrees of freedom. +where $\chi^2_{\text{CDF}}(x,k)$ is the cumulative distribution function (CDF) of the $\chi^2$-distribution with $k$ degrees of freedom. As the number of samples is usually small, we use Student's $t$-distribution to estimate the statistical error. -The inverse of the cumulative distribution function of the $t$-distribution allows us to find how to scale the interval by a parameter +The inverse of the cumulative distribution function of the $t$-distribution, $t_{\text{CDF}}^{-1}$, allows us to find how to scale the interval by a parameter \begin{equation} - %\beta = t_{\text{CDF}}^{-1} \left[ - %\frac{1}{2} \left( 1 + \frac{P( \Delta E_{\text{FCI}} \in [ \Delta E \pm \sigma ] \; | \; \mathcal{G}) }{P(\mathcal{G})}\right), n \right] - \beta = t_{\text{CDF}}^{-1} \left[ - \frac{1}{2} \left( 1 + \frac{0.6827}{P(\mathcal{G})}\right), n \right] + \beta = t_{\text{CDF}}^{-1} \qty[ + \frac{1}{2} \qty( 1 + \frac{0.6827}{P(\mathcal{G})}), M ] \end{equation} such that $P( \Delta E_{\text{FCI}}^{(m)} \in [ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p$. 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.