Modifs Mimi
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@ -287,61 +287,121 @@ The definition of the active space considered for each system as well as the num
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%------------------------------------------------
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In this section, we present our scheme to estimate the extrapolation error in SCI calculations.
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This new protocol is then applied to five- and six-membered ring molecules for which SCI calculations are particularly challenging even for small basis sets.
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Note that the present method does only apply to ``state-averaged'' SCI calculations where ground- and excited-state energies are produced during the same calculation with the same set of molecular orbitals, not to ``state-specific'' calculations where one computes solely the energy of a single state (like conventional ground-state calculations).
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Note that the present method does only apply to \emph{state-averaged} SCI calculations where ground- and excited-state energies are produced during the same calculation with the same set of molecular orbitals, not to \emph{state-specific} calculations where one computes solely the energy of a single state (like conventional ground-state calculations).
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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
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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 $E_{\text{rPT2}}^{(m)}$ \cite{Holmes_2017, Garniron_2019} using
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\begin{equation}
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E_\text{FCI}^{(m)} = E_{\text{var}}^{(m)} + \alpha^{(m)} E_{\text{rPT2}}^{(m)}
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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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$E_\text{var}^{(m)}$ 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.
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This implies that, at any iteration of the CIPSI algorithm, the estimated error on the CIPSI energy is
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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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Using Eq.(\ref{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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For the large systems considered here, $\abs{E_{\text{rPT2}}} > 2$ eV.
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Therefore, the accuracy of the excitation energy estimates will strongly depend on our ability to compensate the errors in the calculations.
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Because our selection procedure ensures that the rPT2 values of both states match as well as possible (a trick known as PT2 matching \cite{Dash_2018,Dash_2019}), i.e., $E_{\text{rPT2}} = E_{\text{rPT2}}^{(0)} \approx E_{\text{rPT2}}^{(m)}$, the extrapolated excitation energy associated with the $m$th excited state can be estimated as
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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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\begin{equation}
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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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+ \order{E_{\text{rPT2}}^2 }
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+ 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 $\qty(\alpha^{(m)}-\alpha^{(0)}) E_{\text{rPT2}} + \order{E_{\text{rPT2}}^2}$.
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Moreover, using a common set of state-averaged natural orbitals for the ground and excited states 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 practically of the order of $E_{\text{rPT2}}^2$.
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which evidences that the error in $\Delta E_{\text{FCI}}^{(m)}$ can be expressed as $\qty(\alpha^{(m)}-\alpha^{(0)}) E_{\text{rPT2}} + O\qty[{E_{\text{rPT2}}^2}]$.
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At the $n$th CIPSI iteration, we have access to the variational energies of both states, $E_\text{var}^{(0)}(n)$ and $E_\text{var}^{(m)}(n)$, as well as their rPT2 corrections, $E_{\text{rPT2}}^{(0)}(n)$ and $E_{\text{rPT2}}^{(m)}(n)$.
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The $m$th excitation energy at iteration $n$ is then assumed to be a Gaussian random variable with mean
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Now, for the largest systems considered here, $\qty|{E_{\text{rPT2}}}|$ can be as large as 2~eV and, thus,
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the accuracy of the excitation energy estimates strongly depends on our ability to compensate the errors in the calculations.
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Here, we greatly enhance the compensation of errors by making use of
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our selection procedure ensuring that the PT2 values of both states
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match as well as possible (a trick known as PT2 matching
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\cite{Dash_2018,Dash_2019}), i.e. $E_{\text{rPT2}} =
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E_{\text{rPT2}}^{(0)} \approx E_{\text{rPT2}}^{(m)}$, 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 for 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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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, the fluctuations are found to be (very weakly) dependent on the iteration number $n$ (see, Fig.\ref{fig2}), so
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this dependence will 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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\Delta E_\text{CIPSI}^{(m)}(n) = \qty[ E_\text{var}^{(m)}(n) + E_{\text{rPT2}}^{(m)}(n) ] - \qty[ E_\text{var}^{(0)}(n) + E_{\text{rPT2}}^{(0)}(n) ]
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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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and variance
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where
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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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\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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\end{equation}
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and we treat all CIPSI iterations as a set of Gaussian-distributed variables ($\mathcal{G}$) with weights $w(n) = 1/\sqrt{\sigma^2(n)}$.
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This choice ensures that the statistical uncertainty vanishes at the FCI limit.
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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., $P( \Delta E_{\text{FCI}}^{(m)} \in [ \Delta E_\text{CIPSI}^{(m)} \pm \sigma ] \; | \; \mathcal{G}) = 0.6827$.
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The probability that $\Delta E_{\text{FCI}}^{(m)}$ is in an interval $\mathcal{I}$ is
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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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P\qty( \Delta E_{\text{FCI}}^{(m)} \in \mathcal{I} ) = P\qty( \Delta E_{\text{FCI}}^{(m)} \in I \Big| \mathcal{G}) \times P(\mathcal{G})
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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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where the probability $P(\mathcal{G})$ that the random variables are normally distributed can be deduced from the Jarque-Bera test $J$ as
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which vanishes in the large-$n$ limit as it should be.
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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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\label{fig2}
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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{fig2}. 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{fig2} 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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\begin{equation}
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P(\mathcal{G}) = 1 - \chi^2_{\text{CDF}}(J,2)
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P\qty[X^{(m)}] \propto e^{-\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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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{fig2}) there exists
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some significant departure from it and we need 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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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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\end{equation}
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where $P\qty(\mathcal{G})$ is the probability that the random variables considered in the latest CIPSI iterations are normally distributed.
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A common test in statistics of the normality of a distribution is the Jarque-Bera test $J$ and we have
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\begin{equation}
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P\qty(\mathcal{G}) = 1 - \chi^2_{\text{CDF}}(J,2)
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\end{equation}
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where $\chi^2_{\text{CDF}}(x,k)$ is the cumulative distribution function (CDF) of the $\chi^2$-distribution with $k$ degrees of freedom.
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As the number of samples is usually small, we use Student's $t$-distribution to estimate the statistical error.
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As the number of samples $M$ is usually small, we use Student's $t$-distribution to estimate the statistical error.
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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
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\begin{equation}
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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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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.
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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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@ -409,10 +469,10 @@ The error bars reported in parenthesis correspond to one standard deviation.
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\end{threeparttable}
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\end{table}
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%%% FIGURE 2 %%%
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%%% FIGURE 3 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{fig2}
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\includegraphics[width=\linewidth]{fig3}
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\caption{Deviation from the CCSDT excitation energies for the lowest singlet and triplet excitation energies (in eV) of five- and six-membered rings obtained at the CIPSI/6-31+G(d) level of theory. Red dots: excitation energies and error bars estimated via the present method (see Sec.~\ref{sec:error}). Blue dots: excitation energies obtained via a three-point linear fit using the three largest CIPSI wave functions, and error bars estimated via the extrapolation distance, \ie, the difference in excitation energies obtained with the three-point linear extrapolation and the largest CIPSI wave function.}
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\label{fig:errors}
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\end{figure}
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@ -430,10 +490,10 @@ from diatomics to molecules as large as naphthalene (see Fig.~\ref{fig:molecules
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pure hydrocarbons and various heteroatomic structures, etc. Each of the five subsets making up the QUEST dataset is detailed below. Throughout the present review, we report several statistical indicators: the mean signed
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error (MSE), mean absolute error (MAE), root-mean square error (RMSE), and standard deviation of the errors (SDE), as well as the maximum positive [Max(+)] and maximum negative [Max($-$)] errors.
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%%% FIGURE 3 %%%
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%%% FIGURE 4 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{fig3}
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\includegraphics[width=\linewidth]{fig4}
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\caption{Molecules from each of the five subsets making up the present QUEST dataset of highly-accurate vertical excitation energies:
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QUEST\#1 (red), QUEST\#2 (magenta and/or underlined), QUEST\#3 (black), QUEST\#4 (green), and QUEST\#5 (blue).}
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\label{fig:molecules}
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Binary file not shown.
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Manuscript/fig2/data_gaussian_histogram_paper
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59
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59
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** Initialize R packages
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#+begin_src R :results output :session *R* :exports code
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library(ggplot2)
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library(latex2exp)
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library(extrafont)
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library(RColorBrewer)
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loadfonts()
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#+end_src
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#+RESULTS:
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:
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: Registering fonts with R
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** Read data
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#+begin_src R :results output :session *R* :exports both
|
||||
df <- read.table("data_histogram_paper");
|
||||
df$x <- df$V1
|
||||
df$y <- df$V2
|
||||
df2 <- read.table("data_gaussian_histogram_paper");
|
||||
spline.d <- as.data.frame(spline(df2$V1, df2$V2))
|
||||
summary(spline.d)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
:
|
||||
: x y
|
||||
: Min. :-0.05818 Min. :0.000e+00
|
||||
: 1st Qu.:-0.02909 1st Qu.:2.000e-08
|
||||
: Median : 0.00000 Median :1.213e-04
|
||||
: Mean : 0.00000 Mean :3.093e-02
|
||||
: 3rd Qu.: 0.02909 3rd Qu.:3.011e-02
|
||||
: Max. : 0.05818 Max. :1.873e-01
|
||||
|
||||
#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
|
||||
p <- ggplot(data=df, aes(x, y)) +
|
||||
geom_bar(stat="identity", fill="steelblue")
|
||||
p <- p+ geom_line(data=spline.d, lwd=1, linetype="dashed")
|
||||
p <- p + scale_x_continuous(name=TeX("$X^{(m)}$"))
|
||||
p <- p + scale_y_continuous(name=TeX("Frequency"))
|
||||
p <- p + theme(text = element_text(size = 20, family="Times"),
|
||||
legend.position = c(.20, .20),
|
||||
legend.title = element_blank())
|
||||
p
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:/tmp/babel-nBBwmV/figureJJu58N.png]]
|
||||
|
||||
* Export to pdf
|
||||
#+begin_src R :results output :session *R* :exports code
|
||||
pdf("fig2.pdf", family="Times", width=8, height=5)
|
||||
p
|
||||
dev.off()
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
:
|
||||
: png
|
||||
: 2
|
||||
BIN
Manuscript/fig2/fig2.pdf
Normal file
BIN
Manuscript/fig2/fig2.pdf
Normal file
Binary file not shown.
Binary file not shown.
BIN
Manuscript/fig4.pdf
Normal file
BIN
Manuscript/fig4.pdf
Normal file
Binary file not shown.
Loading…
Reference in New Issue
Block a user