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benzene.bib
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benzene.bib
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@ -1,13 +1,24 @@
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-08-20 10:17:19 +0200
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%% Created for Pierre-Francois Loos at 2020-08-21 22:33:45 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Pipek_1989,
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Author = {Janos Pipek and Paul G. Mezey},
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Date-Added = {2020-08-21 22:32:52 +0200},
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Date-Modified = {2020-08-21 22:33:43 +0200},
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Doi = {10.1063/1.456588},
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Journal = {J. Chem. Phys.},
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Pages = {4916},
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Title = {A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions},
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Volume = {90},
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Year = {1989}}
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@article{Caffarel_2014,
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Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro},
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Date-Added = {2020-08-18 22:14:08 +0200},
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@ -815,11 +826,9 @@
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Bdsk-Url-1 = {https://doi.org/10.13140/RG.2.1.3187.9766}}
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@misc{Applencourt_2018,
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title={Spin adaptation with determinant-based selected configuration interaction},
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author={Thomas Applencourt and Kevin Gasperich and Anthony Scemama},
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year={2018},
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eprint={1812.06902},
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archivePrefix={arXiv},
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primaryClass={physics.chem-ph}
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}
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Archiveprefix = {arXiv},
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Author = {Thomas Applencourt and Kevin Gasperich and Anthony Scemama},
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Eprint = {1812.06902},
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Primaryclass = {physics.chem-ph},
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Title = {Spin adaptation with determinant-based selected configuration interaction},
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Year = {2018}}
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@ -0,0 +1,23 @@
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9 -230.71995654 -231.75487668 0.00000000 -231.38707847 0.00000000
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21 -230.75612470 -231.70098859 0.00000000 -231.41602784 0.00000000
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43 -230.77730679 -231.66964449 0.00000000 -231.42142233 0.00000000
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87 -230.79677324 -231.65037383 0.00000000 -231.42776597 0.00000000
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176 -230.81322284 -231.63873550 0.00000000 -231.43169967 0.00000000
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356 -230.83201378 -231.62756631 0.00000000 -231.43661910 0.00000000
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715 -230.86070351 -231.61048214 0.00000000 -231.44312189 0.00000000
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1435 -230.89813915 -231.59123406 0.00000000 -231.45082609 0.00000000
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1435 -230.95069206 -231.57171574 0.00014501 -231.46093651 0.00011914
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2876 -231.01492460 -231.55162453 0.00009651 -231.47204089 0.00008220
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5753 -231.08674981 -231.53926175 0.00012588 -231.48551511 0.00011093
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11507 -231.16199633 -231.53378267 0.00025395 -231.49947048 0.00023051
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23015 -231.23642786 -231.53378610 0.00052595 -231.51327852 0.00048967
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46032 -231.30376101 -231.53805882 0.00032788 -231.52616690 0.00031124
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92070 -231.35498272 -231.54178832 0.00061162 -231.53468765 0.00058838
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184142 -231.38593159 -231.54290366 0.00057977 -231.53809574 0.00056201
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368286 -231.40066490 -231.54576531 0.00051143 -231.54175009 0.00049728
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736613 -231.40995296 -231.54940407 0.00068226 -231.54577642 0.00066451
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1473232 -231.41811398 -231.55264853 0.00063964 -231.54932949 0.00062386
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2946480 -231.42614355 -231.55271703 0.00062093 -231.54983470 0.00060679
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5892976 -231.43454648 -231.55441275 0.00059892 -231.55185712 0.00058615
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11786019 -231.44370625 -231.55550485 0.00055109 -231.55332245 0.00054034
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23572080 -231.45375768 -231.55855354 0.00051764 -231.55667052 0.00050834
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benzene.nb
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benzene.tex
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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
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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}
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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}
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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}
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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}
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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).
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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}
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We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm.
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@ -123,16 +123,76 @@ As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially wh
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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.
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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.
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We believe that it provides a very safe estimate of the extrapolation error.
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Note that all the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}.
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%Note that all the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}.
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A second run has been performed with localized orbitals.
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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.}
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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.
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% Results and discussion
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The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
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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.
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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.
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These nodes are connected via Infiniband HDR100.
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The three flavours of SCI fall into an interval ranging from $-863.7$ to $-862.8$ m$E_h$.
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The CIPSI number is ?
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%%$ FIG. 1 %%%
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\begin{figure*}
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\includegraphics[width=0.4\linewidth]{fig1a}
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\hspace{0.08\linewidth}
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\includegraphics[width=0.4\linewidth]{fig1b}
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\caption{
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Convergence of the CIPSI correlation energy for benzene using localized orbitals.
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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.
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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 $E_\text{rPT2}$.
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The two-point linear extrapolation curves (dashed lines) are also reported.
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The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.
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\label{fig:CIPSI}
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}
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\end{figure*}
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%%% TABLE II %%%
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\begin{squeezetable}
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\begin{table*}
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\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.
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The statistical error on $E_\text{PT2}$, corresponding to one standard deviation, are reported in parenthesis.}
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\label{tab:NOvsLO}
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\begin{ruledtabular}
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\begin{tabular}{rcccccc}
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& \mc{3}{c}{Natural orbitals} & \mc{3}{c}{Localized orbitals} \\
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\cline{2-4} \cline{5-7}
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\tabc{$N_\text{det}$} & \tabc{$E_\text{var.}$} & \tabc{$E_\text{var.}+E_\text{PT2}$} & \tabc{$E_\text{var.}+E_\text{rPT2}$}
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& \tabc{$E_\text{var.}$} & \tabc{$E_\text{var.}+E_\text{PT2}$} & \tabc{$E_\text{var.}+E_\text{rPT2}$} \\
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\hline
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% 5 & $-230.719\,957$ & $-231.754\,877(0)$ & $-231.387\,078(0)$ & $-230.719\,928$ & $-231.718\,694(0)$ & $-231.385\,276(0)$ \\
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% 10 & $-230.750\,520$ & $-231.707\,154(0)$ & $-231.411\,127(0)$ & $-230.760\,937$ & $-231.670\,043(0)$ & $-231.402\,535(0)$ \\
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% 20 & $-230.767\,479$ & $-231.681\,740(0)$ & $-231.416\,388(0)$ & $-230.807\,967$ & $-231.636\,613(0)$ & $-231.415\,300(0)$ \\
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% 40 & $-230.782\,929$ & $-231.664\,092(0)$ & $-231.423\,696(0)$ & $-230.863\,737$ & $-231.588\,750(0)$ & $-231.421\,131(0)$ \\
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% 80 & $-230.800\,057$ & $-231.649\,643(0)$ & $-231.429\,207(0)$ & $-230.892\,146$ & $-231.567\,290(0)$ & $-231.427\,639(0)$ \\
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% 160 & $-230.818\,727$ & $-231.635\,786(0)$ & $-231.433\,153(0)$ & $-230.922\,173$ & $-231.556\,426(0)$ & $-231.435\,952(0)$ \\
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% 320 & $-230.842\,915$ & $-231.619\,618(0)$ & $-231.438\,459(0)$ & $-230.957\,439$ & $-231.541\,301(0)$ & $-231.440\,416(0)$ \\
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% 640 & $-230.875\,821$ & $-231.601\,124(0)$ & $-231.445\,624(0)$ & $-231.000\,058$ & $-231.529\,709(0)$ & $-231.449\,099(0)$ \\
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1\,280 & $-230.978\,056$ & $-231.559\,025(212)$ & $-231.463\,633(177)$ & $-231.101\,676$ & $-231.519\,522(149)$ & $-231.472\,224(132)$ \\
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2\,560 & $-231.043\,712$ & $-231.542\,344(139)$ & $-231.474\,885(120)$ & $-231.161\,264$ & $-231.515\,577(155)$ & $-231.482\,477(140)$ \\
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5\,120 & $-231.115\,142$ & $-231.534\,122(213)$ & $-231.488\,815(190)$ & $-231.224\,632$ & $-231.516\,375(191)$ & $-231.495\,022(177)$ \\
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10\,240 & $-231.188\,813$ & $-231.531\,660(516)$ & $-231.502\,992(473)$ & $-231.283\,295$ & $-231.520\,907(271)$ & $-231.507\,708(255)$ \\
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20\,480 & $-231.260\,065$ & $-231.534\,172(611)$ & $-231.517\,063(573)$ & $-231.330\,209$ & $-231.526\,433(586)$ & $-231.518\,045(561)$ \\
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40\,960 & $-231.321\,906$ & $-231.538\,269(501)$ & $-231.528\,301(478)$ & $-231.366\,008$ & $-231.532\,288(303)$ & $-231.526\,639(293)$ \\
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81\,920 & $-231.366\,895$ & $-231.541\,945(813)$ & $-231.535\,785(785)$ & $-231.392\,888$ & $-231.536\,578(614)$ & $-231.532\,575(597)$ \\
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163\,840 & $-231.392\,866$ & $-231.545\,499(761)$ & $-231.541\,010(739)$ & $-231.414\,132$ & $-231.541\,400(624)$ & $-231.538\,378(609)$ \\
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327\,680 & $-231.407\,802$ & $-231.548\,699(662)$ & $-231.544\,980(645)$ & $-231.431\,952$ & $-231.545\,873(557)$ & $-231.543\,532(545)$ \\
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655\,360 & $-231.418\,752$ & $-231.551\,208(661)$ & $-231.548\,004(645)$ & $-231.447\,007$ & $-231.548\,856(498)$ & $-231.547\,043(489)$ \\
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1\,310\,720 & $-231.428\,852$ & $-231.552\,760(616)$ & $-231.550\,006(603)$ & $-231.460\,970$ & $-231.552\,137(453)$ & $-231.550\,723(446)$ \\
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2\,621\,440 & $-231.439\,324$ & $-231.553\,845(572)$ & $-231.551\,544(560)$ & $-231.473\,751$ & $-231.555\,261(403)$ & $-231.554\,159(397)$ \\
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5\,242\,880 & $-231.450\,156$ & $-231.557\,541(534)$ & $-231.555\,558(524)$ & $-231.485\,829$ & $-231.558\,303(362)$ & $-231.557\,451(358)$ \\
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10\,485\,760 & $-231.461\,927$ & $-231.559\,390(481)$ & $-231.557\,796(474)$ & $-231.497\,515$ & $-231.562\,568(322)$ & $-231.561\,901(319)$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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\end{squeezetable}
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%%% %%% %%% %%%
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%%% TABLE II %%%
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\begin{table}
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\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.
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@ -154,49 +214,6 @@ The CIPSI number is ?
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\end{ruledtabular}
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\end{table}
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%%$ FIG. 1 %%%
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\begin{figure*}
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\includegraphics[width=0.4\linewidth]{fig1a}
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\hspace{0.08\linewidth}
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\includegraphics[width=0.4\linewidth]{fig1b}
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\caption{
|
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Convergence of the CIPSI correlation energy for benzene.
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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.
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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 $E_\text{rPT2}$.
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The two-point linear extrapolation curves (dashed lines) are also reported.
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The theoretical best estimate of $-863$ m$E_h$ from Ref.~\onlinecite{Eriksen_2020} is marked by a black line for comparison purposes.
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\label{fig:CIPSI}
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}
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\end{figure*}
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%%% TABLE 2 %%%
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%\begin{squeezetable}
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\begin{table*}
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\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.
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The statistical errors ($\delta E_\text{PT2}$ and $\delta E_\text{rPT2}$), corresponding to one standard deviation, are also reported.}
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\label{tab:Cr2}
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\begin{ruledtabular}
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\begin{tabular}{rddddd}
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\tabc{$N_\text{det}$} & \tabc{$E_\text{var.}$} & \tabc{$E_\text{PT2}$} & \tabc{$\delta E_\text{PT2}$} & \tabc{$E_\text{rPT2}$} & \tabc{$\delta E_\text{rPT2}$} \\
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\hline
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12418 & -231.17094496 & -0.36188641 & 0.00040193 & -0.32968372 & 0.00036616 \\
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24841 & -231.24480922 & -0.28851015 & 0.00046895 & -0.26939524 & 0.00043788 \\
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49683 & -231.31129020 & -0.22584693 & 0.00043741 & -0.21493067 & 0.00041627 \\
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99367 & -231.35989232 & -0.18072875 & 0.00030762 & -0.17412788 & 0.00029638 \\
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198754 & -231.38819148 & -0.15638840 & 0.00023911 & -0.15163900 & 0.00023185 \\
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397561 & -231.40076198 & -0.14599792 & 0.00024280 & -0.14195331 & 0.00023608 \\
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795126 & -231.40804005 & -0.14060686 & 0.00023393 & -0.13690453 & 0.00022777 \\
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1590261 & -231.41441889 & -0.13567145 & 0.00024941 & -0.13227230 & 0.00024316 \\
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3180532 & -231.41959399 & -0.13129538 & 0.00023846 & -0.12814025 & 0.00023273 \\
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6361098 & -231.42421895 & -0.12727083 & 0.00024438 & -0.12432787 & 0.00023873 \\
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12722217 & -231.43050041 & -0.12234780 & 0.00023326 & -0.11965826 & 0.00022813 \\
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25444447 & -231.43883900 & -0.11574659 & 0.00022627 & -0.11336968 & 0.00022163 \\
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Extrap. & & & & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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%\end{squeezetable}
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%%% %%% %%% %%%
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% Acknowledgements
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@ -0,0 +1,22 @@
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5 -230.71995654 -231.75487668 0.00000000 -231.38707847 0.00000000
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10 -230.75051967 -231.70715390 0.00000000 -231.41112682 0.00000000
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20 -230.76747889 -231.68173967 0.00000000 -231.41638814 0.00000000
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40 -230.78292911 -231.66409160 0.00000000 -231.42369561 0.00000000
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80 -230.80005735 -231.64964348 0.00000000 -231.42920688 0.00000000
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160 -230.81872680 -231.63578564 0.00000000 -231.43315312 0.00000000
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||||
320 -230.84291515 -231.61961822 0.00000000 -231.43845911 0.00000000
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640 -230.87582073 -231.60112402 0.00000000 -231.44562420 0.00000000
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1280 -230.97805555 -231.55902542 0.00021154 -231.46363343 0.00017681
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2560 -231.04371160 -231.54234414 0.00013867 -231.47488484 0.00011991
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5120 -231.11514169 -231.53412204 0.00021309 -231.48881480 0.00019005
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10240 -231.18881346 -231.53165989 0.00051578 -231.50299249 0.00047265
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20480 -231.26006467 -231.53417161 0.00061117 -231.51706341 0.00057302
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40960 -231.32190639 -231.53826889 0.00050086 -231.52830130 0.00047778
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81920 -231.36689492 -231.54194540 0.00081330 -231.53578515 0.00078468
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163840 -231.39286579 -231.54549865 0.00076102 -231.54101042 0.00073864
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327680 -231.40780236 -231.54869903 0.00066218 -231.54498032 0.00064471
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655360 -231.41875248 -231.55120849 0.00066129 -231.54800405 0.00064529
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1310720 -231.42885192 -231.55276044 0.00061627 -231.55000591 0.00060257
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2621440 -231.43932438 -231.55384542 0.00057169 -231.55154435 0.00056021
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5242880 -231.45015608 -231.55754133 0.00053356 -231.55555800 0.00052371
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10485760 -231.46192749 -231.55939032 0.00048147 -231.55779568 0.00047359
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@ -0,0 +1,22 @@
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5 -230.71992820 -231.71869363 0.00000000 -231.38527572 0.00000000
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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
|
||||
Loading…
Reference in New Issue