take 2 Mimi
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@ -294,16 +294,15 @@ For the $m$th excited state (where $m = 0$ corresponds to the ground state), we
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E_{\text{var}}^{(m)} \approx E_\text{FCI}^{(m)} - \alpha^{(m)} E_{\text{rPT2}}^{(m)},
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\label{eqx}
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\end{equation}
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where $E_{\text{var}}^{(m)}$ and $E_{\text{rPT2}}^{(m)}$ are calculated with CIPSI and $E_\text{FCI}^{(m)}$ is the FCI energy
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to be extrapolated. This relation is valid in the regime of a sufficiently large number of determinants where the second-order perturbational
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correction largely dominates.
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However, in practice, due to the residual higher-order terms, the coefficient $\alpha^{(m)}$ deviates slightly from unity.
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where $E_{\text{var}}^{(m)}$ and $E_{\text{rPT2}}^{(m)}$ are calculated with CIPSI and $E_\text{FCI}^{(m)}$ is the FCI energy to be extrapolated.
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This relation is valid in the regime of a sufficiently large number of determinants where the second-order perturbational correction largely dominates.
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In theory, the coefficient $\alpha^{(m)}$ should be equal to one but, in practice, due to the residual higher-order terms, it deviates slightly from unity.
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Using Eq.~\eqref{eqx} the estimated error on the CIPSI energy is calculated as
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\begin{equation}
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E_{\text{CIPSI}}^{(m)} - E_{\text{FCI}}^{(m)}
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= \qty(E_\text{var}^{(m)}+E_{\text{rPT2}}^{(m)}) - E_{\text{FCI}}^{(m)}
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= \qty(1-\alpha^{(m)}) E_{\text{rPT2}}^{(m)},
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= \qty(1-\alpha^{(m)}) E_{\text{rPT2}}^{(m)}
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\end{equation}
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and thus the extrapolated excitation energy associated with the $m$th
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state is given by
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@ -311,7 +310,7 @@ state is given by
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\Delta E_{\text{FCI}}^{(m)}
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= \qty[ E_\text{var}^{(m)} + E_{\text{rPT2}} + \qty(\alpha^{(m)}-1) E_{\text{rPT2}} ]
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- \qty[ E_\text{var}^{(0)} + E_{\text{rPT2}} + \qty(\alpha^{(0)}-1) E_{\text{rPT2}} ]
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+ \mathcal{O}\qty[{E_{\text{rPT2}}^2 }],
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+ \mathcal{O}\qty[{E_{\text{rPT2}}^2 }]
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\end{equation}
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which evidences that the error in $\Delta E_{\text{FCI}}^{(m)}$ can be expressed as $\qty(\alpha^{(m)}-\alpha^{(0)}) E_{\text{rPT2}} + \mathcal{O}\qty[{E_{\text{rPT2}}^2}]$.
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@ -325,65 +324,63 @@ E_{\text{rPT2}}^{(0)} \approx E_{\text{rPT2}}^{(m)}$, and
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by using a common set of state-averaged natural orbitals with equal weights for the ground and excited states.
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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
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is decreased.
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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
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$\Delta E_\text{CIPSI}^{(m)}(n)$ as a function of $n$ would completely vanish and the exact excitation energy would be obtained from the first CIPSI iterations.
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In the ideal case where one is able to fully correlate the CIPSI calculations associated with the ground and excited states, the fluctuations of
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$\Delta E_\text{CIPSI}^{(m)}(n)$ as a function of the iteration number $n$ would completely vanish and the exact excitation energy would be obtained from the first CIPSI iterations.
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Quite remarkably, in practice, numerical experience shows that the fluctuations with respect to the extrapolated value $\Delta E_\text{FCI}^{(m)}$ are small,
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zero-centered, almost independent of $n$ when not too close iteration
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numbers are considered, and display a Gaussian-like distribution.
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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
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this dependence will not significantly alter our results and will not be considered here.
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zero-centered, and display a Gaussian-like distribution.
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In addition, as evidenced in Fig.~\ref{fig:histo}, these fluctuations are found to be (very weakly) dependent on the iteration number $n$ (as far as not too close $n$ values are considered).
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Hence, this weak dependency does not significantly alter our results and will not be considered here.
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We thus introduce the following random variable
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\begin{equation}
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X^{(m)}= \frac{\Delta E_\text{CIPSI}^{(m)}(n)- \Delta E_\text{FCI}^{(m)}}{\sigma(n)}
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\label{eq:X}
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X^{(m)}= \frac{\Delta E_\text{CIPSI}^{(m)}(n)- \Delta E_\text{FCI}^{(m)}}{\sigma(n)}
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\end{equation}
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where
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\begin{equation}
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\Delta E_\text{CIPSI}^{(m)}(n) = \qty[ E_\text{var}^{(m)}(n) +
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E_{\text{rPT2}}^{(m)}(n) ]
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- \qty[ E_\text{var}^{(0)}(n) + E_{\text{rPT2}}^{(0)}(n) ],
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- \qty[ E_\text{var}^{(0)}(n) + E_{\text{rPT2}}^{(0)}(n) ]
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\end{equation}
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and
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${\sigma(n)}$ is a quantity proportional to the average fluctuations of $\Delta E_\text{CIPSI}^{(m)}$.
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A natural choice for $\sigma^2(n)$, playing here the role of a variance, is
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\begin{equation}
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\sigma^2(n) \propto \qty[E_{\text{rPT2}}^{(m)}(n)]^2 + \qty[E_{\text{rPT2}}^{(0)}(n)]^2,
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\sigma^2(n) \propto \qty[E_{\text{rPT2}}^{(m)}(n)]^2 + \qty[E_{\text{rPT2}}^{(0)}(n)]^2
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\end{equation}
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which vanishes in the large-$n$ limit as it should.
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which vanishes in the large-$n$ limit (as it should).
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%%% FIGURE 2 %%%
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\begin{figure}
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\centering
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\includegraphics[width=0.9\linewidth]{fig2/fig2}
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\caption{Histogram of the random variable $X^{(m)}$ (see, text). About 200 values of the transition energies
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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.
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The number $M$ of iterations kept is chosen according to the statistical test presented in the text.}
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\caption{Histogram of the random variable $X^{(m)}$ [see Eq.~\eqref{eq:X} in the main text for its definition].
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About 200 values of singlet and triplet excitation energies taken at various iteration number $n$ for the 13 five- and six-membered ring molecules have been considered to build the present histogram.
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The number $M$ of iterations kept at each calculation is chosen according to the statistical test presented in the text.}
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\label{fig:histo}
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\end{figure}
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The histogram of $X^{(m)}$ resulting from the excitation energies
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obtained at different values of the CIPSI iterations $n$
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and for the 13 five- and six-membered ring molecules, both for the singlet and triplet transitions,
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is shown in Fig.~\ref{fig:histo}. To avoid transient effects, only excitation energies at sufficiently large $n$ are retained in the data set.
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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
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of values employed to make the histogram is about 200. The dashed line of Fig.~\ref{fig:histo} represents the best Gaussian fit
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(in the sense of least-squares) reproducing the data.
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As seen, the distribution can be described by the Gaussian probability
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The histogram of $X^{(m)}$ resulting from the singlet and triplet excitation energies obtained at various iteration number $n$ for the 13 five- and six-membered ring molecules is shown in Fig.~\ref{fig:histo}.
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To avoid transient effects, only excitation energies at sufficiently large $n$ are retained in the data set.
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The statistical criterion used to decide from which precise value of $n$ the data should be kept is presented below.
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In the present example, the total number of values employed to construct the histogram of Fig.~\ref{fig:histo} is about 200.
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The dashed line represents the best (in a least-squares sense) Gaussian fit reproducing the data.
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As clearly seen from Fig.~\ref{fig:histo}, the distribution can be fairly well described by a Gaussian probability distribution
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\begin{equation}
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P\qty[X^{(m)}] \propto \exp[-\frac{{X^{(m)}}^2} {2{\sigma^{*}}^2} ]
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P\qty[X^{(m)}] \propto \exp[-\frac{{X^{(m)}}^2} {2{\sigma^{*}}^2} ]
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\end{equation}
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where $\sigma^{*2}$ is some "universal" variance depending only
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on the way the correlated selection of both states is done, not on the molecule considered in our set.
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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.
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An estimate of $\Delta E_{\text{FCI}}^{(m)}$ as the average excitation energy of $\Delta E_\text{CIPSI}^{(m)}$ is thus
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$$\Delta E_\text{FCI}^{(m)} = \frac{ \sum_{n=1}^M \frac{\Delta E_\text{CIPSI}^{(m)}(n)} {\sigma(n)} }
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{ \sum_{n=1}^M \frac{1}{\sigma(n)} },
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$$
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where $M$ is the number of data kept.
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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
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(see Fig.~\ref{fig:histo}) there exists
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some significant departure from it and we need to take this feature into account.
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For each CIPSI calculation, an estimate of $\Delta E_{\text{FCI}}^{(m)}$ is thus
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\begin{equation}
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\Delta E_\text{FCI}^{(m)} = \frac{ \sum_{n=1}^M \frac{\Delta E_\text{CIPSI}^{(m)}(n)} {\sigma(n)} }
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{ \sum_{n=1}^M \frac{1}{\sigma(n)} }
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\end{equation}
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where $M$ is the number of iterations that has been retained to compute the statistical quantities.
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Regarding the estimate of the error on $\Delta E_\text{FCI}^{(m)}$ some caution is required since, although the distribution is globally Gaussian-like
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(see Fig.~\ref{fig:histo}), there exists some significant deviation from it and we must to take this feature into account.
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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., $P\qty( \Delta E_{\text{FCI}}^{(m)} \in \qty[ \Delta E_\text{CIPSI}^{(m)} \pm \sigma ] \; \Big| \; \mathcal{G}) = 0.6827$.
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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., $P\qty( \Delta E_{\text{FCI}}^{(m)} \in \qty[ \Delta E_\text{CIPSI}^{(m)} \pm \sigma ] \; \Big| \; \mathcal{G}) = p = 0.6827$.
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In a Bayesian framework, the probability that $\Delta E_{\text{FCI}}^{(m)}$ is in an interval $\mathcal{I}$ is
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\begin{equation}
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P\qty( \Delta E_{\text{FCI}}^{(m)} \in \mathcal{I} ) = P\qty( \Delta E_{\text{FCI}}^{(m)} \in I \Big| \mathcal{G}) \times P\qty(\mathcal{G})
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@ -400,7 +397,7 @@ The inverse of the cumulative distribution function of the $t$-distribution, $t_
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\beta = t_{\text{CDF}}^{-1} \qty[
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\frac{1}{2} \qty( 1 + \frac{0.6827}{P(\mathcal{G})}), M ]
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\end{equation}
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such that $P\qty( \Delta E_{\text{FCI}}^{(m)} \in \qty[ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p = 0.6827$.
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such that $P\qty( \Delta E_{\text{FCI}}^{(m)} \in \qty[ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p $.
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Only the last $M>2$ computed transition energies are considered. $M$ is chosen such that $P(\mathcal{G})>0.8$ and such that the error bar is minimal.
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If all the values of $P(\mathcal{G})$ are below $0.8$, $M$ is chosen such that $P(\mathcal{G})$ is maximal.
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A Python code associated with this procedure is provided in the {\SupInf}.
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