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