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% Abstract


\begin{abstract}


Following the recent work of Eriksen \textit{et al.}~[\href{https://arxiv.org/abs/2008.02678}{arXiv:2008.02678 [physics.chemph]}], we report the performance of the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method on the nonrelativistic frozencore correlation energy of the benzene molecule in the ccpVDZ basis. Following our usual protocol, we obtain a correlation energy of \titou{$86x.x(x)$} m$E_h$ which agrees with the theoretical estimate of $863$ m$E_h$ proposed by Eriksen \textit{et al.}~using an extensive array of highlyaccurate new electronic structure methods.


Following the recent work of Eriksen \textit{et al.}~[\href{https://arxiv.org/abs/2008.02678}{arXiv:2008.02678 [physics.chemph]}], we report the performance of the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method on the nonrelativistic frozencore correlation energy of the benzene molecule in the ccpVDZ basis. Following our usual protocol, we obtain a correlation energy of $863.4(5)$ m$E_h$ which agrees with the theoretical estimate of $863$ m$E_h$ proposed by Eriksen \textit{et al.}~using an extensive array of highlyaccurate new electronic structure methods.


\end{abstract}




% Title


@ 97,7 +97,7 @@ The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecit


\hline


phAFQMC & $864.3(4)$ & \onlinecite{Lee_2020} \\


\hline


CIPSI & \titou{$86x.x(x)$} & This work \\


CIPSI & $863.4(5)$ & This work \\


\end{tabular}


\end{ruledtabular}


\end{table}


@ 142,25 +142,14 @@ higherlying $\sigma$ [37,38,40,44,45,47,48,51,52,5870,7581,86,8891,9497


Like PipekMezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$$\pi$ separation in planar systems like benzene.


As one can see from the energies of Table \ref{tab:NOvsLO}, for a given value of $N_\text{det}$, the variational energy as well as the PT2corrected energies are much lower with localized orbitals than with natural orbitals. We, therefore, consider these energies more trustworthy, and we will base our best estimate of the correlation energy of benzene on these calculations.


The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of the correlation energy, $\Delta E_\text{var.}$ and $\Delta E_\text{var.} + E_\text{(r)PT2}$, as a function of $N_\text{det}$ (left panel).


The right panel of Fig.~\ref{fig:CIPSI} is more instructive as it shows $\Delta E_\text{var.}$ as a function of $E_\text{(r)PT2}$, and their corresponding \titou{five}point linear extrapolation curves that we have used to get our final estimate of the correlation energy.


The right panel of Fig.~\ref{fig:CIPSI} is more instructive as it shows $\Delta E_\text{var.}$ as a function of $E_\text{(r)PT2}$, and their corresponding fourpoint linear extrapolation curves that we have used to get our final estimate of the correlation energy.


From this figure, one clearly sees that the rPT2based correction behaves more linearly than its corresponding PT2 version, and is thus systematically employed in the following.




% Results


Our final number are gathered in Table \ref{tab:extrap_dist_table}, where, following the notations of Ref.~\onlinecite{Eriksen_2020}, we report, in addition to the final variational energies $\Delta E_{\text{var.}}$, the


extrapolation distances, $\Delta E_{\text{dist}}$, defined as the difference between the final computed energy, $\Delta E_{\text{final}}$, and the extrapolated energy, $\Delta E_{\text{extrap.}}$ associated with ASCI, iCI, SHCI, DMRS, and CIPSI.


The three flavours of SCI fall into an interval ranging from $860.0$ m$E_h$ (ASCI) to $864.2$ m$E_h$ (SHCI), while the other nonSCI methods yield correlation energies ranging from $863.7$ to $862.8$ m$E_h$ (see Table \ref{tab:energy}). Our final CIPSI number (obtained with localized orbitals and rPT2 correction via a \titou{five}point linear extrapolation) is \titou{$86x.x(x)$} m$E_h$.


The three flavours of SCI fall into an interval ranging from $860.0$ m$E_h$ (ASCI) to $864.2$ m$E_h$ (SHCI), while the other nonSCI methods yield correlation energies ranging from $863.7$ to $862.8$ m$E_h$ (see Table \ref{tab:energy}). Our final CIPSI number (obtained with localized orbitals and rPT2 correction via a fourpoint linear extrapolation) is $863.4(5)$ m$E_h$, where the error reported in parenthesis represents the fitting error (not the extrapolation error for which it is much harder to provide a theoretically sound estimate).


For comparison, the best post blind test SHCI estimate is $863.3$ m$E_h$, which agrees almost perfectly with our best CIPSI estimate.


\titou{Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by ...


We believe that it provides a very safe estimate of the extrapolation error.}




% Timings


The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.


Each Irene's AMD node is a dualsocket AMD Rome (Epyc) CPU@2.60 GHz with 256GiB of RAM, with a total of 64 physical CPU cores per socket.


These nodes are connected via Infiniband HDR100.


The first step of the calculation, \ie, performing a CIPSI calculation up to $N_\text{det} \sim 10^7$ with HartreeFock orbitals in order to produce natural orbitals, takes roughly 24 hours on a single node, and reaching the same number of determinants with natural orbitals or localized orbitals takes roughly the same amount of time.


A second 24hour run on 10 distributed nodes was performed to push the selection to $8 \times 10^7$ determinants, and a third distributed run using 40 nodes was used to reach $16 \times 10^7$ determinants.


A last 24hour run on 60 distributed nodes was performed to obtain the largest wave function of $25 \times 10^7$ determinants and its PT2 correction.


In total, the present calculation has required \titou{340k} core hours, almost half of it being spent in the last stage of the computation.




%%$ FIG. 1 %%%


\begin{figure*}


@ 171,7 +160,7 @@ In total, the present calculation has required \titou{340k} core hours, almost h


Convergence of the CIPSI correlation energy of benzene using localized orbitals.


Left: $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of the number of determinants in the variational space $N_\text{det}$.


Right: $\Delta E_\text{var.}$ (in m$E_h$) as a function of $E_\text{PT2}$ or $E_\text{rPT2}$.


The \titou{five}point linear extrapolation curves (dashed lines) are also reported.


The fourpoint linear extrapolation curves (dashed lines) are also reported.


The theoretical estimate of $863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.


\label{fig:CIPSI}


}


@ 216,7 +205,6 @@ The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviat


41\,943\,040 & $231.487\,978$ & $231.564\,529(382)$ & $231.563\,593(377)$ & $231.519\,122$ & $231.567\,419(240)$ & $231.567\,069(238)$ \\


83\,886\,080 & $231.501\,334$ & $231.566\,994(317)$ & $231.566\,325(314)$ & $231.528\,568$ & $231.570\,084(199)$ & $231.569\,832(198)$ \\


167\,772\,160 & $231.514\,009$ & $231.569\,939(273)$ & $231.569\,467(271)$ & $231.536\,655$ & $231.571\,981(175)$ & $231.571\,804(174)$ \\


251\,658\,240 & \\


\end{tabular}


\end{ruledtabular}


\end{table*}


@ 240,11 +228,18 @@ The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviat


SHCI & $827.2$ & $852.8$ & $864.2$ & $11.4$ \\


DMRG & $859.2$ & $859.2$ & $862.8$ & $3.6$ \\


\hline


CIPSI & \titou{$8xx.x$} & \titou{$8xx.x$} & \titou{$86x.x$} & \titou{$xx.x$} \\


CIPSI & $814.8$ & $850.2$ & $863.4$ & $13.2$ \\


\end{tabular}


\end{ruledtabular}


\end{table}




% Timings


The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.


Each Irene's AMD node is a dualsocket AMD Rome (Epyc) CPU@2.60 GHz with 256GiB of RAM, with a total of 64 physical CPU cores per socket.


These nodes are connected via Infiniband HDR100.


The first step of the calculation, \ie, performing a CIPSI calculation up to $N_\text{det} \sim 10^7$ with HartreeFock orbitals in order to produce natural orbitals, takes roughly 24 hours on a single node, and reaching the same number of determinants with natural orbitals or localized orbitals takes roughly the same amount of time.


A second 24hour run on 10 distributed nodes was performed to push the selection to $8 \times 10^7$ determinants, and a third distributed run using 40 nodes was used to reach $16 \times 10^7$ determinants.


In total, the present calculation has required 150k core hours, most of it being spent in the last stage of the computation.






% Acknowledgements



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