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27
benzene.bib
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@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-20 10:17:19 +0200 %% Created for Pierre-Francois Loos at 2020-08-21 22:33:45 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Pipek_1989,
Author = {Janos Pipek and Paul G. Mezey},
Date-Added = {2020-08-21 22:32:52 +0200},
Date-Modified = {2020-08-21 22:33:43 +0200},
Doi = {10.1063/1.456588},
Journal = {J. Chem. Phys.},
Pages = {4916},
Title = {A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions},
Volume = {90},
Year = {1989}}
@article{Caffarel_2014, @article{Caffarel_2014,
Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro}, Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro},
Date-Added = {2020-08-18 22:14:08 +0200}, Date-Added = {2020-08-18 22:14:08 +0200},
@ -815,11 +826,9 @@
Bdsk-Url-1 = {https://doi.org/10.13140/RG.2.1.3187.9766}} Bdsk-Url-1 = {https://doi.org/10.13140/RG.2.1.3187.9766}}
@misc{Applencourt_2018, @misc{Applencourt_2018,
title={Spin adaptation with determinant-based selected configuration interaction}, Archiveprefix = {arXiv},
author={Thomas Applencourt and Kevin Gasperich and Anthony Scemama}, Author = {Thomas Applencourt and Kevin Gasperich and Anthony Scemama},
year={2018}, Eprint = {1812.06902},
eprint={1812.06902}, Primaryclass = {physics.chem-ph},
archivePrefix={arXiv}, Title = {Spin adaptation with determinant-based selected configuration interaction},
primaryClass={physics.chem-ph} Year = {2018}}
}

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benzene.dat Normal file
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9 -230.71995654 -231.75487668 0.00000000 -231.38707847 0.00000000
21 -230.75612470 -231.70098859 0.00000000 -231.41602784 0.00000000
43 -230.77730679 -231.66964449 0.00000000 -231.42142233 0.00000000
87 -230.79677324 -231.65037383 0.00000000 -231.42776597 0.00000000
176 -230.81322284 -231.63873550 0.00000000 -231.43169967 0.00000000
356 -230.83201378 -231.62756631 0.00000000 -231.43661910 0.00000000
715 -230.86070351 -231.61048214 0.00000000 -231.44312189 0.00000000
1435 -230.89813915 -231.59123406 0.00000000 -231.45082609 0.00000000
1435 -230.95069206 -231.57171574 0.00014501 -231.46093651 0.00011914
2876 -231.01492460 -231.55162453 0.00009651 -231.47204089 0.00008220
5753 -231.08674981 -231.53926175 0.00012588 -231.48551511 0.00011093
11507 -231.16199633 -231.53378267 0.00025395 -231.49947048 0.00023051
23015 -231.23642786 -231.53378610 0.00052595 -231.51327852 0.00048967
46032 -231.30376101 -231.53805882 0.00032788 -231.52616690 0.00031124
92070 -231.35498272 -231.54178832 0.00061162 -231.53468765 0.00058838
184142 -231.38593159 -231.54290366 0.00057977 -231.53809574 0.00056201
368286 -231.40066490 -231.54576531 0.00051143 -231.54175009 0.00049728
736613 -231.40995296 -231.54940407 0.00068226 -231.54577642 0.00066451
1473232 -231.41811398 -231.55264853 0.00063964 -231.54932949 0.00062386
2946480 -231.42614355 -231.55271703 0.00062093 -231.54983470 0.00060679
5892976 -231.43454648 -231.55441275 0.00059892 -231.55185712 0.00058615
11786019 -231.44370625 -231.55550485 0.00055109 -231.55332245 0.00054034
23572080 -231.45375768 -231.55855354 0.00051764 -231.55667052 0.00050834

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benzene.tex
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@ -108,7 +108,7 @@ It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see
Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by one of us (AS) enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019} Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by one of us (AS) enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019}
In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
CIPSI is also frequently use to provide accurate trial wave function for QMC calculations. \cite{Caffarel_2014,Caffarel_2016a,Caffarel_2016b,Giner_2013,Giner_2015,Scemama_2015,Scemama_2016,Scemama_2018,Scemama_2018b,Scemama_2019,Dash_2018,Dash_2019} CIPSI is also frequently use to provide accurate trial wave function for QMC calculations. \cite{Caffarel_2014,Caffarel_2016a,Caffarel_2016b,Giner_2013,Giner_2015,Scemama_2015,Scemama_2016,Scemama_2018,Scemama_2018b,Scemama_2019,Dash_2018,Dash_2019}
The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm. \cite{Garniron_2017,Garniron_2019} The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm, \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
Moreover, a renormalized version of the PT2 correction (dubbed rPT2 in the following) has been recently implemented for a more efficient extrapolation to the FCI limit (see below). \cite{Garniron_2019} Moreover, a renormalized version of the PT2 correction (dubbed rPT2 in the following) has been recently implemented for a more efficient extrapolation to the FCI limit (see below). \cite{Garniron_2019}
We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm. We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm.
@ -123,16 +123,76 @@ As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially wh
We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions. We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions.
Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value. Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value.
We believe that it provides a very safe estimate of the extrapolation error. We believe that it provides a very safe estimate of the extrapolation error.
Note that all the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}. %Note that all the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}.
A second run has been performed with localized orbitals.
A Pipek-Mezey localization procedure \cite{Pipek_1989} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, and vi) the rest. \titou{More information needed here.}
As one can see from the results of Table \ref{tab:NOvsLO}, the variational energy as well as the PT2 corrected energy is much lower with localized orbitals for a same number of determinants.
% Results and discussion % Results and discussion
The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer. The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
Each Irene's AMD node is a dual-socket \titou{Intel(R) Xeon(R) Platinum 8168 CPU@2.70 GHz with 192GiB of RAM}, with a total of 128 physical CPU cores. Each Irene's AMD node is a dual-socket AMD Rome (Epyc) CPU@2.60 GHz with 256GiB of RAM, with a total of 64 physical CPU cores.
These nodes are connected via Infiniband HDR100.
The three flavours of SCI fall into an interval ranging from $-863.7$ to $-862.8$ m$E_h$. The three flavours of SCI fall into an interval ranging from $-863.7$ to $-862.8$ m$E_h$.
The CIPSI number is ? The CIPSI number is ?
%%$ FIG. 1 %%%
\begin{figure*}
\includegraphics[width=0.4\linewidth]{fig1a}
\hspace{0.08\linewidth}
\includegraphics[width=0.4\linewidth]{fig1b}
\caption{
Convergence of the CIPSI correlation energy for 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.
Right: $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$.
The two-point linear extrapolation curves (dashed lines) are also reported.
The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.
\label{fig:CIPSI}
}
\end{figure*}
%%% TABLE II %%%
\begin{squeezetable}
\begin{table*}
\caption{Variational energy $E_\text{var.}$, second-order perturbative correction $E_\text{PT2}$ and its renormalized version $E_\text{rPT2}$ (in $E_h$) as a function of the number of determinants $N_\text{det}$ for the ground-state of the benzene molecule computed in the cc-pVDZ basis set.
The statistical error on $E_\text{PT2}$, corresponding to one standard deviation, are reported in parenthesis.}
\label{tab:NOvsLO}
\begin{ruledtabular}
\begin{tabular}{rcccccc}
& \mc{3}{c}{Natural orbitals} & \mc{3}{c}{Localized orbitals} \\
\cline{2-4} \cline{5-7}
\tabc{$N_\text{det}$} & \tabc{$E_\text{var.}$} & \tabc{$E_\text{var.}+E_\text{PT2}$} & \tabc{$E_\text{var.}+E_\text{rPT2}$}
& \tabc{$E_\text{var.}$} & \tabc{$E_\text{var.}+E_\text{PT2}$} & \tabc{$E_\text{var.}+E_\text{rPT2}$} \\
\hline
% 5 & $-230.719\,957$ & $-231.754\,877(0)$ & $-231.387\,078(0)$ & $-230.719\,928$ & $-231.718\,694(0)$ & $-231.385\,276(0)$ \\
% 10 & $-230.750\,520$ & $-231.707\,154(0)$ & $-231.411\,127(0)$ & $-230.760\,937$ & $-231.670\,043(0)$ & $-231.402\,535(0)$ \\
% 20 & $-230.767\,479$ & $-231.681\,740(0)$ & $-231.416\,388(0)$ & $-230.807\,967$ & $-231.636\,613(0)$ & $-231.415\,300(0)$ \\
% 40 & $-230.782\,929$ & $-231.664\,092(0)$ & $-231.423\,696(0)$ & $-230.863\,737$ & $-231.588\,750(0)$ & $-231.421\,131(0)$ \\
% 80 & $-230.800\,057$ & $-231.649\,643(0)$ & $-231.429\,207(0)$ & $-230.892\,146$ & $-231.567\,290(0)$ & $-231.427\,639(0)$ \\
% 160 & $-230.818\,727$ & $-231.635\,786(0)$ & $-231.433\,153(0)$ & $-230.922\,173$ & $-231.556\,426(0)$ & $-231.435\,952(0)$ \\
% 320 & $-230.842\,915$ & $-231.619\,618(0)$ & $-231.438\,459(0)$ & $-230.957\,439$ & $-231.541\,301(0)$ & $-231.440\,416(0)$ \\
% 640 & $-230.875\,821$ & $-231.601\,124(0)$ & $-231.445\,624(0)$ & $-231.000\,058$ & $-231.529\,709(0)$ & $-231.449\,099(0)$ \\
1\,280 & $-230.978\,056$ & $-231.559\,025(212)$ & $-231.463\,633(177)$ & $-231.101\,676$ & $-231.519\,522(149)$ & $-231.472\,224(132)$ \\
2\,560 & $-231.043\,712$ & $-231.542\,344(139)$ & $-231.474\,885(120)$ & $-231.161\,264$ & $-231.515\,577(155)$ & $-231.482\,477(140)$ \\
5\,120 & $-231.115\,142$ & $-231.534\,122(213)$ & $-231.488\,815(190)$ & $-231.224\,632$ & $-231.516\,375(191)$ & $-231.495\,022(177)$ \\
10\,240 & $-231.188\,813$ & $-231.531\,660(516)$ & $-231.502\,992(473)$ & $-231.283\,295$ & $-231.520\,907(271)$ & $-231.507\,708(255)$ \\
20\,480 & $-231.260\,065$ & $-231.534\,172(611)$ & $-231.517\,063(573)$ & $-231.330\,209$ & $-231.526\,433(586)$ & $-231.518\,045(561)$ \\
40\,960 & $-231.321\,906$ & $-231.538\,269(501)$ & $-231.528\,301(478)$ & $-231.366\,008$ & $-231.532\,288(303)$ & $-231.526\,639(293)$ \\
81\,920 & $-231.366\,895$ & $-231.541\,945(813)$ & $-231.535\,785(785)$ & $-231.392\,888$ & $-231.536\,578(614)$ & $-231.532\,575(597)$ \\
163\,840 & $-231.392\,866$ & $-231.545\,499(761)$ & $-231.541\,010(739)$ & $-231.414\,132$ & $-231.541\,400(624)$ & $-231.538\,378(609)$ \\
327\,680 & $-231.407\,802$ & $-231.548\,699(662)$ & $-231.544\,980(645)$ & $-231.431\,952$ & $-231.545\,873(557)$ & $-231.543\,532(545)$ \\
655\,360 & $-231.418\,752$ & $-231.551\,208(661)$ & $-231.548\,004(645)$ & $-231.447\,007$ & $-231.548\,856(498)$ & $-231.547\,043(489)$ \\
1\,310\,720 & $-231.428\,852$ & $-231.552\,760(616)$ & $-231.550\,006(603)$ & $-231.460\,970$ & $-231.552\,137(453)$ & $-231.550\,723(446)$ \\
2\,621\,440 & $-231.439\,324$ & $-231.553\,845(572)$ & $-231.551\,544(560)$ & $-231.473\,751$ & $-231.555\,261(403)$ & $-231.554\,159(397)$ \\
5\,242\,880 & $-231.450\,156$ & $-231.557\,541(534)$ & $-231.555\,558(524)$ & $-231.485\,829$ & $-231.558\,303(362)$ & $-231.557\,451(358)$ \\
10\,485\,760 & $-231.461\,927$ & $-231.559\,390(481)$ & $-231.557\,796(474)$ & $-231.497\,515$ & $-231.562\,568(322)$ & $-231.561\,901(319)$ \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
%%% TABLE II %%% %%% TABLE II %%%
\begin{table} \begin{table}
\caption{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 the ASCI, iCI, SHCI, CIPSI, and DMRG results. \caption{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 the ASCI, iCI, SHCI, CIPSI, and DMRG results.
@ -154,49 +214,6 @@ The CIPSI number is ?
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
%%$ FIG. 1 %%%
\begin{figure*}
\includegraphics[width=0.4\linewidth]{fig1a}
\hspace{0.08\linewidth}
\includegraphics[width=0.4\linewidth]{fig1b}
\caption{
Convergence of the CIPSI correlation energy for benzene.
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.
Right: $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$.
The two-point linear extrapolation curves (dashed lines) are also reported.
The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.
\label{fig:CIPSI}
}
\end{figure*}
%%% TABLE 2 %%%
%\begin{squeezetable}
\begin{table*}
\caption{Variational energy $E_\text{var.}$, second-order perturbative correction $E_\text{PT2}$ and its renormalized version $E_\text{rPT2}$ (in $E_h$) as a function of the number of determinants $N_\text{det}$ for the ground-state of the benzene molecule computed in the cc-pVDZ basis set.
The statistical errors ($\delta E_\text{PT2}$ and $\delta E_\text{rPT2}$), corresponding to one standard deviation, are also reported.}
\label{tab:Cr2}
\begin{ruledtabular}
\begin{tabular}{rddddd}
\tabc{$N_\text{det}$} & \tabc{$E_\text{var.}$} & \tabc{$E_\text{PT2}$} & \tabc{$\delta E_\text{PT2}$} & \tabc{$E_\text{rPT2}$} & \tabc{$\delta E_\text{rPT2}$} \\
\hline
12418 & -231.17094496 & -0.36188641 & 0.00040193 & -0.32968372 & 0.00036616 \\
24841 & -231.24480922 & -0.28851015 & 0.00046895 & -0.26939524 & 0.00043788 \\
49683 & -231.31129020 & -0.22584693 & 0.00043741 & -0.21493067 & 0.00041627 \\
99367 & -231.35989232 & -0.18072875 & 0.00030762 & -0.17412788 & 0.00029638 \\
198754 & -231.38819148 & -0.15638840 & 0.00023911 & -0.15163900 & 0.00023185 \\
397561 & -231.40076198 & -0.14599792 & 0.00024280 & -0.14195331 & 0.00023608 \\
795126 & -231.40804005 & -0.14060686 & 0.00023393 & -0.13690453 & 0.00022777 \\
1590261 & -231.41441889 & -0.13567145 & 0.00024941 & -0.13227230 & 0.00024316 \\
3180532 & -231.41959399 & -0.13129538 & 0.00023846 & -0.12814025 & 0.00023273 \\
6361098 & -231.42421895 & -0.12727083 & 0.00024438 & -0.12432787 & 0.00023873 \\
12722217 & -231.43050041 & -0.12234780 & 0.00023326 & -0.11965826 & 0.00022813 \\
25444447 & -231.43883900 & -0.11574659 & 0.00022627 & -0.11336968 & 0.00022163 \\
Extrap. & & & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%\end{squeezetable}
%%% %%% %%% %%%
% Acknowledgements % Acknowledgements

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@ -0,0 +1,22 @@
5 -230.71995654 -231.75487668 0.00000000 -231.38707847 0.00000000
10 -230.75051967 -231.70715390 0.00000000 -231.41112682 0.00000000
20 -230.76747889 -231.68173967 0.00000000 -231.41638814 0.00000000
40 -230.78292911 -231.66409160 0.00000000 -231.42369561 0.00000000
80 -230.80005735 -231.64964348 0.00000000 -231.42920688 0.00000000
160 -230.81872680 -231.63578564 0.00000000 -231.43315312 0.00000000
320 -230.84291515 -231.61961822 0.00000000 -231.43845911 0.00000000
640 -230.87582073 -231.60112402 0.00000000 -231.44562420 0.00000000
1280 -230.97805555 -231.55902542 0.00021154 -231.46363343 0.00017681
2560 -231.04371160 -231.54234414 0.00013867 -231.47488484 0.00011991
5120 -231.11514169 -231.53412204 0.00021309 -231.48881480 0.00019005
10240 -231.18881346 -231.53165989 0.00051578 -231.50299249 0.00047265
20480 -231.26006467 -231.53417161 0.00061117 -231.51706341 0.00057302
40960 -231.32190639 -231.53826889 0.00050086 -231.52830130 0.00047778
81920 -231.36689492 -231.54194540 0.00081330 -231.53578515 0.00078468
163840 -231.39286579 -231.54549865 0.00076102 -231.54101042 0.00073864
327680 -231.40780236 -231.54869903 0.00066218 -231.54498032 0.00064471
655360 -231.41875248 -231.55120849 0.00066129 -231.54800405 0.00064529
1310720 -231.42885192 -231.55276044 0.00061627 -231.55000591 0.00060257
2621440 -231.43932438 -231.55384542 0.00057169 -231.55154435 0.00056021
5242880 -231.45015608 -231.55754133 0.00053356 -231.55555800 0.00052371
10485760 -231.46192749 -231.55939032 0.00048147 -231.55779568 0.00047359

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@ -0,0 +1,22 @@
5 -230.71992820 -231.71869363 0.00000000 -231.38527572 0.00000000
10 -230.76093652 -231.67004325 0.00000000 -231.40253519 0.00000000
20 -230.80796668 -231.63661254 0.00000000 -231.41530045 0.00000000
40 -230.86373652 -231.58874997 0.00000000 -231.42113105 0.00000000
80 -230.89214599 -231.56729008 0.00000000 -231.42763934 0.00000000
160 -230.92217332 -231.55642581 0.00000000 -231.43595199 0.00000000
320 -230.95743944 -231.54130089 0.00000000 -231.44041552 0.00000000
640 -231.00005789 -231.52970891 0.00000000 -231.44909877 0.00000000
1280 -231.10167630 -231.51952172 0.00014913 -231.47222377 0.00013225
2560 -231.16126352 -231.51557692 0.00015473 -231.48247698 0.00014028
5120 -231.22463235 -231.51637458 0.00019132 -231.49502240 0.00017731
10240 -231.28329507 -231.52090717 0.00027047 -231.50770840 0.00025544
20480 -231.33020881 -231.52643331 0.00058628 -231.51804546 0.00056122
40960 -231.36600803 -231.53228847 0.00030301 -231.52663905 0.00029271
81920 -231.39288791 -231.53657828 0.00061435 -231.53257506 0.00059723
163840 -231.41413240 -231.54140029 0.00062380 -231.53837841 0.00060899
327680 -231.43195196 -231.54587265 0.00055658 -231.54353242 0.00054515
655360 -231.44700671 -231.54885563 0.00049762 -231.54704342 0.00048876
1310720 -231.46097026 -231.55213717 0.00045322 -231.55072252 0.00044619
2621440 -231.47375051 -231.55526062 0.00040253 -231.55415883 0.00039709
5242880 -231.48582938 -231.55830266 0.00036203 -231.55745147 0.00035778
10485760 -231.49751473 -231.56256801 0.00032228 -231.56190078 0.00031897

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