diff --git a/benzene.bib b/benzene.bib index 7cfd4b6..3465d03 100644 --- a/benzene.bib +++ b/benzene.bib @@ -1,13 +1,24 @@ %% This BibTeX bibliography file was created using BibDesk. %% 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) +@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, 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}, @@ -815,11 +826,9 @@ Bdsk-Url-1 = {https://doi.org/10.13140/RG.2.1.3187.9766}} @misc{Applencourt_2018, - title={Spin adaptation with determinant-based selected configuration interaction}, - author={Thomas Applencourt and Kevin Gasperich and Anthony Scemama}, - year={2018}, - eprint={1812.06902}, - archivePrefix={arXiv}, - primaryClass={physics.chem-ph} -} - + Archiveprefix = {arXiv}, + Author = {Thomas Applencourt and Kevin Gasperich and Anthony Scemama}, + Eprint = {1812.06902}, + Primaryclass = {physics.chem-ph}, + Title = {Spin adaptation with determinant-based selected configuration interaction}, + Year 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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} 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} 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. 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. -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 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 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 %%% \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. @@ -154,49 +214,6 @@ The CIPSI number is ? \end{ruledtabular} \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 diff --git a/benzene_noS2.dat b/benzene_noS2.dat new file mode 100644 index 0000000..54e8099 --- /dev/null +++ b/benzene_noS2.dat @@ -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 diff --git a/benzene_noS2_local.dat b/benzene_noS2_local.dat new file mode 100644 index 0000000..18a1e81 --- /dev/null +++ b/benzene_noS2_local.dat @@ -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 diff --git a/fig1a.pdf b/fig1a.pdf index 3415ebb..f24645e 100644 Binary files a/fig1a.pdf and b/fig1a.pdf differ diff --git a/fig1b.pdf b/fig1b.pdf index ea46fc4..daa716b 100644 Binary files a/fig1b.pdf and b/fig1b.pdf differ