From e72c1b9751858de2c081ed7b8332d1917e7cb4d0 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 8 Sep 2020 15:30:37 +0200 Subject: [PATCH] toto part --- Manuscript/QUEST_WIREs.tex | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/Manuscript/QUEST_WIREs.tex b/Manuscript/QUEST_WIREs.tex index 7befc1d..22fac74 100644 --- a/Manuscript/QUEST_WIREs.tex +++ b/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}) @@ -294,7 +294,8 @@ 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}. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The QUEST database} \label{sec:QUEST}