saving work. One last number missing

benzene.bib

@@ -1,13 +1,69 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-08-22 17:57:20 +0200 
+%% Created for Pierre-Francois Loos at 2020-08-24 16:21:19 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Sauer_2009,
+	Author = {Sauer, Stephan P. A. and Schreiber, Marko and Silva-Junior, Mario R. and Thiel, Walter},
+	Date-Added = {2020-08-24 16:15:18 +0200},
+	Date-Modified = {2020-08-24 16:20:35 +0200},
+	Doi = {10.1021/ct800256j},
+	Journal = {J. Chem. Theory Comput.},
+	Number = {3},
+	Pages = {555--564},
+	Title = {Benchmarks for Electronically Excited States: A Comparison of Noniterative and Iterative Triples Corrections in Linear Response Coupled Cluster Methods: CCSDR(3) versus CC3},
+	Volume = {5},
+	Year = {2009}}
+
+@article{Schreiber_2008,
+	Author = {Schreiber, M. and Silva-Junior, M. R. and Sauer, S. P. A. and Thiel, W.},
+	Date-Added = {2020-08-24 16:15:18 +0200},
+	Date-Modified = {2020-08-24 16:21:02 +0200},
+	Doi = {10.1063/1.2889385},
+	Journal = {J. Chem. Phys.},
+	Pages = {134110},
+	Title = {Benchmarks for Electronically Excited States: CASPT2, CC2, CCSD and CC3},
+	Volume = 128,
+	Year = 2008}
+
+@article{Silva-Junior_2010a,
+	Author = {Silva-Junior, M. R. and Schreiber, M. and Sauer, S. P. A. and Thiel, W.},
+	Date-Added = {2020-08-24 16:15:18 +0200},
+	Date-Modified = {2020-08-24 16:19:10 +0200},
+	Doi = {10.1063/1.2973541},
+	Journal = {J. Chem. Phys.},
+	Pages = {104103},
+	Title = {Benchmarks for Electronically Excited States: Time-Dependent Density Functional Theory and Density Functional Theory Based Multireference Configuration Interaction},
+	Volume = 129,
+	Year = 2008}
+
+@article{Silva-Junior_2010b,
+	Author = {Silva-Junior, M. R. and Sauer, S. P. A. and Schreiber, M. and Thiel, W.},
+	Date-Added = {2020-08-24 16:15:18 +0200},
+	Date-Modified = {2020-08-24 16:19:37 +0200},
+	Doi = {10.1080/00268970903549047},
+	Journal = {Mol. Phys.},
+	Pages = {453--465},
+	Title = {Basis Set Effects on Coupled Cluster Benchmarks of Electronically Excited States: CC3, CCSDR(3) and CC2},
+	Volume = 108,
+	Year = 2010}
+
+@article{Silva-Junior_2010c,
+	Author = {Silva-Junior, M. R. and Schreiber, M. and Sauer, S. P. A. and Thiel, W.},
+	Date-Added = {2020-08-24 16:15:18 +0200},
+	Date-Modified = {2020-08-24 16:20:03 +0200},
+	Doi = {10.1063/1.3499598},
+	Journal = {J. Chem. Phys.},
+	Pages = {174318},
+	Title = {Benchmarks of Electronically Excited States: Basis Set Effecs on {{CASPT2}} Results},
+	Volume = 133,
+	Year = 2010}
+
 @article{Boys_1960,
 	Author = {J. M. Foster and S. F. Boys},
 	Date-Added = {2020-08-22 17:56:32 +0200},
@@ -17,7 +73,8 @@
 	Pages = {300},
 	Title = {Canonical Configurational Interaction Procedure},
 	Volume = {32},
-	Year = {1960}}
+	Year = {1960},
+	Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.32.300}}
 
 @article{Pipek_1989,
 	Author = {Janos Pipek and Paul G. Mezey},

benzene.tex

@@ -1,5 +1,5 @@
 \documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,wrapfig}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,wrapfig,txfonts}
 \usepackage[version=4]{mhchem}
 
 \newcommand{\ie}{\textit{i.e.}}
@@ -43,7 +43,7 @@
 
 % Abstract
 \begin{abstract}
-Following the recent work of Eriksen \textit{et al.} [\href{https://arxiv.org/abs/2008.02678}{arXiv:2008.02678 [physics.chem-ph]}], we report the performance of the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method on the non-relativistic frozen-core correlation energy of the ground state of the benzene molecule in the cc-pVDZ basis. Following our usual protocol, we obtain a correlation energy of \titou{$-8xx.xx$} m$E_h$ which agrees with the best theoretical estimate of $-863$ m$E_h$ proposed by Eriksen \textit{et al.} using an extensive array of highly-accurate new electronic structure methods.
+Following the recent work of Eriksen \textit{et al.} [\href{https://arxiv.org/abs/2008.02678}{arXiv:2008.02678 [physics.chem-ph]}], we report the performance of the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method on the non-relativistic frozen-core correlation energy of the ground state of the benzene molecule in the cc-pVDZ basis. Following our usual protocol, we obtain a correlation energy of \titou{$-86x.x(x)$} m$E_h$ which agrees with the best theoretical estimate of $-863$ m$E_h$ proposed by Eriksen \textit{et al.} using an extensive array of highly-accurate new electronic structure methods.
 \end{abstract}
 
 % Title
@@ -53,6 +53,7 @@ Following the recent work of Eriksen \textit{et al.} [\href{https://arxiv.org/ab
 Although sometimes decried, one cannot deny the usefulness of benchmark sets and their corresponding reference data for the electronic structure community.
 These are indeed essential for the validation of existing theoretical models and to bring to light and subsequently understand their strengths and, more importantly, their weaknesses.
 In that regard, the previous benchmark datasets provided by the \textit{Simons Collaboration on the Many-Electron Problem} have been extremely valuable. \cite{Leblanc_2015,Motta_2017,Williams_2020}
+The same comment applies to the excited-state benchmark set of Thiel and coworkers. \cite{Sauer_2009,Schreiber_2008,Silva-Junior_2010a,Silva-Junior_2010b,Silva-Junior_2010c}
 Following a similar goal, we have recently proposed a large set of highly-accurate vertical transition energies for various types of excited states thanks to the renaissance of selected configuration interaction (SCI) methods \cite{Bender_1969,Huron_1973,Buenker_1974} which can now routinely produce near full configuration interaction (FCI) quality excitation energies for small- and medium-sized organic molecules. \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
 
 % The context
@@ -123,12 +124,10 @@ However, following the philosophy of Eriksen \textit{et al.}, \cite{Eriksen_2020
 We then follow our usual ``protocol'' \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} by performing a preliminary SCI calculation using Hartree-Fock orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
 Natural orbitals are then computed based on this wave function, and a new, larger SCI calculation is performed with this new set of orbitals. 
 This has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
-The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element. The magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
+The total SCI energy is defined as the sum of the variational energy $E_\text{var.}$ (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction $E_\text{(r)PT2}$ which takes into account the external determinants, \ie, the determinants which do not belong to the variational space but are linked to the reference space via a nonzero matrix element. The magnitude of $E_\text{(r)PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
 As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems.
-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 $E_\text{(r)PT2} = 0$ (which effectively corresponds to the FCI limit). 
-Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by \titou{the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value.}
+Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Applencourt_2018} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a single state. We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero ($\sim 5 \times 10^{-3}$ a.u.).
-We believe that it provides a very safe estimate of the extrapolation error. 
-Note that, unlike excited-state calculations where it is important to enforce that the wave functions are `eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Applencourt_2018} the present wave functions do not fulfil this property as we aim for the lowest energy of a single state. We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero ($\sim 5 \times 10^{-3}$ a.u.).
 The corresponding energies are reported in Table \ref{tab:NOvsLO} as functions of the number of determinants in the variational space $N_\text{det}$.
 
 A second run has been performed with localized orbitals. 
@@ -143,19 +142,26 @@ higher-lying $\pi$ [39,41--43,46,49,50,53--57,71--74,82--85,87,92,93,98];
 higher-lying $\sigma$ [37,38,40,44,45,47,48,51,52,58--70,75--81,86,88--91,94--97,99--114].}
 Like Pipek-Mezey, \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 PT2-corrected 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 $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ as a function of $N_\text{det}$ (left panel).
+The convergence of the CIPSI correlation energy using localized orbitals is illustrated in Fig.~\ref{fig:CIPSI}, where one can see the behavior of $\Delta E_\text{var.}$ 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} shows $\Delta E_\text{var.} + E_\text{PT2}$ and $\Delta E_\text{var.} + E_\text{rPT2}$ (in m$E_h$) as functions of $E_\text{PT2}$ or $E_\text{rPT2}$, and their corresponding \titou{two}-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 $\Delta E_\text{var.}$ (in m$E_h$) 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.
+From this figure, one clearly sees that the rPT2-based correction behaves more linearly than its corresponding PT2 version, and thus systematically employed in the following.
 
 % Results
 Our final numbers 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 the ASCI, iCI, SHCI, CIPSI, and DMRG results. 
-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 methods yield correlation energies ranging from $-863.7$ to $-862.8$ m$E_h$. Our final CIPSI number is \titou{$-86x.xx$} 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 methods yield correlation energies ranging from $-863.7$ to $-862.8$ m$E_h$. Our final CIPSI number (obtained with localized orbitals and rPT2 correction via a \titou{five}-point linear extrapolation) is \titou{$-86x.x(x)$} m$E_h$.
+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 dual-socket 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 Hartree-Fock 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 24-hour 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 160M determinants.
+The first step of the calculation, \ie, performing a CIPSI calculation up to $N_\text{det} \sim 10^7$ with Hartree-Fock 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 24-hour 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 24-hour 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*}
@@ -166,7 +172,7 @@ The first step of the calculation, \ie, performing a CIPSI calculation up to $N_
 	Convergence of the CIPSI correlation energy 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 \titou{two}-point linear extrapolation curves (dashed lines) are also reported.
+	The \titou{five}-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}
 	}
@@ -211,6 +217,7 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
     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*}  
@@ -232,8 +239,9 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
 			ASCI	&	$-737.1$	&	$-835.4$	&	$-860.0$	&	$-24.6$	\\
 			iCI		&	$-730.0$	&	$-833.7$	&	$-861.1$	&	$-27.4$	\\
 			SHCI	&	$-827.2$	&	$-852.8$	&	$-864.2$	&	$-11.4$	\\
-			CIPSI	&	\titou{$-8xx.x$}	&	\titou{$-8xx.x$}	&	\titou{$-86x.x$}	&	\titou{$-xx.x$}	\\
 			DMRG	&	$-859.2$	&	$-859.2$	&	$-862.8$	&	$-3.6$	\\
+			\hline
+			CIPSI	&	\titou{$-8xx.x$}	&	\titou{$-8xx.x$}	&	\titou{$-86x.x$}	&	\titou{$-xx.x$}	\\
 		\end{tabular}
 	\end{ruledtabular}
 \end{table}