diff --git a/Response_Letter/Response_Letter.tex b/Response_Letter/Response_Letter.tex index dcec984..ccfeaf3 100644 --- a/Response_Letter/Response_Letter.tex +++ b/Response_Letter/Response_Letter.tex @@ -48,7 +48,7 @@ We look forward to hearing from you. \\ \alert{The rPT2 correction corresponds to a partial resummation of some of the higher-order diagrams from many-body perturbation theory. We have mentioned this in the revised version of the manuscript. - As correctly pointed out by the reviewer, this correction has been thoroughly tested in Ref.~50 for weakly and strongly correlated systems, and we have then added this reference to the corresponding sentence as requested.} + As correctly pointed out by the reviewer, this correction has been thoroughly tested in Ref.~51 for weakly and strongly correlated systems, and we have then added this reference to the corresponding sentence as requested.} \item {It is not very clear how the extrapolation is actually done. @@ -91,12 +91,11 @@ We look forward to hearing from you. It would be valuable to the work if the authors were to indicate the variance with respect to the number of points used in the extrapolations; in the linear extrapolations, one could use, say, between 3-5 points, whereas between 4-6 points could be used in the quadratic fits? In any case, the authors should comment (in more detail) on their choice of fitting function and number of data points.} \\ -%TITOU : -863.1(11) ca veut dire -836.1 +/- 0.11 ou -836.1 +/- 1.1 ? \alert{Using the last 3, 4, 5, and 6 points (i.e., the largest wave functions), linear extrapolations yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, $-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively, where the fitting error is reported in parenthesis. These numbers vary by $1.4$ mE$_h$. The four-point extrapolated estimate that we have chosen to report as our best estimate corresponds to the smallest fitting error. Quadratic fits yield much larger variations and we never use them in practice. - All these additional information are now provided in the revised version of the manuscript (see footnote 67).} + All these additional information are now provided in the revised version of the manuscript (see footnote 71).} \item {As an aside, it would seem like the fifth last point differs ever so slightly from the general trend (regardless of the choice of (r)MP2)? @@ -112,8 +111,8 @@ We look forward to hearing from you. \item {It would be interesting if the authors could comment (even speculatively) on why results in the localized FB basis are significantly lower (and hence, in the authors' own words, more trustworthy) than the corresponding results in the NO basis.} \\ - \alert{Localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of the electron correlation. - We have mentioned this in the revised manuscript and added references discussing the use of localized orbitals \textit{vs} natural orbitals in CI calculations (Refs.~65--69).} + \alert{Localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of electron correlation. + We have mentioned this in the revised manuscript and added references discussing the use of localized orbitals \textit{vs} natural orbitals in CI calculations (Refs.~66--70).} \item {Why are the rMP2-based corrections considered superior to the corresponding corrections based on MP2? @@ -128,13 +127,13 @@ We look forward to hearing from you. Also, some references appear to be missing, e.g., for iCI and DMRG.} \\ \alert{We have more explicitly defined the method acronyms, corrected the description of FCCR, and added the missing references for iCI and DMRG. - An additional reference for CAD-FCIQMC has been also added for the sake of completeness.} + Additional references for FCCR and CAD-FCIQMC have been also added for the sake of completeness.} \item {Why are FB orbitals preferred over, e.g., PM orbitals or IBOs?} \\ \alert{Boys-Foster is the only localization criterion available at the moment but we are planning on implementing other schemes in the near future. - That being said, because we group the MOs by symmetry classes (see footnote 63), our localization procedure ensures the $\sigma$-$\pi$ separability, like in PM. + That being said, because we group the MOs by symmetry classes (see footnote 64), our localization procedure ensures the $\sigma$-$\pi$ separability, like in PM. This would not be possible in general but, thanks to the high symmetry of benzene, it is feasible in the present case. } diff --git a/benzene.bib b/benzene.bib index 891d7a4..974ad20 100644 --- a/benzene.bib +++ b/benzene.bib @@ -1,13 +1,35 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-10-09 21:38:45 +0200 +%% Created for Pierre-Francois Loos at 2020-10-10 13:54:12 +0200 %% Saved with string encoding Unicode (UTF-8) +@misc{Xu_2020, + Archiveprefix = {arXiv}, + Author = {Enhua Xu and Motoyuki Uejima and Seiichiro L. Ten-no}, + Date-Added = {2020-10-10 13:54:02 +0200}, + Date-Modified = {2020-10-10 13:54:09 +0200}, + Eprint = {2010.01850}, + Primaryclass = {physics.chem-ph}, + Title = {Towards near-exact solutions of molecular electronic structure: Full coupled-cluster reduction with a second-order perturbative correction}, + Year = {2020}} + +@inbook{Caffarel_2016b, + Author = {Caffarel, Michel and Applencourt, Thomas and Giner, Emmanuel and Scemama, Anthony}, + Booktitle = {Recent Progress in Quantum Monte Carlo}, + Chapter = {2}, + Date-Added = {2020-10-10 13:52:37 +0200}, + Date-Modified = {2020-10-10 13:53:07 +0200}, + Doi = {10.1021/bk-2016-1234.ch002}, + Pages = {15-46}, + Title = {Using CIPSI Nodes in Diffusion Monte Carlo}, + Bdsk-Url-1 = {https://pubs.acs.org/doi/abs/10.1021/bk-2016-1234.ch002}, + Bdsk-Url-2 = {https://doi.org/10.1021/bk-2016-1234.ch002}} + @article{BenAmor_2011, Author = {Ben Amor,Nadia and Bessac,Fabienne and Hoyau,Sophie and Maynau,Daniel}, Date-Added = {2020-10-09 21:28:40 +0200}, @@ -21,50 +43,46 @@ Bdsk-Url-1 = {https://doi.org/10.1063/1.3600351}} @article{Suaud_2017, - author = {Suaud, Nicolas and Malrieu, Jean-Paul}, - title = {{Natural molecular orbitals: limits of a Lowdin's conjecture}}, - journal = {Mol. Phys.}, - volume = {115}, - number = {21-22}, - pages = {2684--2695}, - year = {2017}, - month = {Nov}, - issn = {0026-8976}, - publisher = {Taylor {\&} Francis}, - doi = {10.1080/00268976.2017.1303207} -} + Author = {Suaud, Nicolas and Malrieu, Jean-Paul}, + Doi = {10.1080/00268976.2017.1303207}, + Issn = {0026-8976}, + Journal = {Mol. Phys.}, + Month = {Nov}, + Number = {21-22}, + Pages = {2684--2695}, + Publisher = {Taylor {\&} Francis}, + Title = {{Natural molecular orbitals: limits of a Lowdin's conjecture}}, + Volume = {115}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2017.1303207}} @article{Angeli_2009, - author = {Angeli, Celestino}, - title = {{On the nature of the π {$\rightarrow$} π{$\ast$} ionic excited states: The - V state of ethene as a prototype}}, - journal = {J. Comput. Chem.}, - volume = {30}, - number = {8}, - pages = {1319--1333}, - year = {2009}, - month = {Jun}, - issn = {0192-8651}, - publisher = {John Wiley {\&} Sons, Ltd}, - doi = {10.1002/jcc.21155} -} + Author = {Angeli, Celestino}, + Doi = {10.1002/jcc.21155}, + Issn = {0192-8651}, + Journal = {J. Comput. Chem.}, + Month = {Jun}, + Number = {8}, + Pages = {1319--1333}, + Publisher = {John Wiley {\&} Sons, Ltd}, + Title = {{On the nature of the π {$\rightarrow$} π{$\ast$} ionic excited states: The V state of ethene as a prototype}}, + Volume = {30}, + Year = {2009}, + Bdsk-Url-1 = {https://doi.org/10.1002/jcc.21155}} @article{Angeli_2003, - author = {Angeli, Celestino and Calzado, Carmen J. and - Cimiraglia, Renzo and Evangelisti, Stefano and Guih\'ery, Nathalie and - Leininger, Thierry and Malrieu, Jean-Paul and Maynau, Daniel and - Ruiz, Jos\'e Vicente Pitarch and Sparta, Manuel}, - title = {{The use of local orbitals in multireference calculations}}, - journal = {Mol. Phys.}, - volume = {101}, - number = {9}, - pages = {1389--1398}, - year = {2003}, - month = {May}, - issn = {0026-8976}, - publisher = {Taylor {\&} Francis}, - doi = {10.1080/0026897031000082149} -} + Author = {Angeli, Celestino and Calzado, Carmen J. and Cimiraglia, Renzo and Evangelisti, Stefano and Guih\'ery, Nathalie and Leininger, Thierry and Malrieu, Jean-Paul and Maynau, Daniel and Ruiz, Jos\'e Vicente Pitarch and Sparta, Manuel}, + Doi = {10.1080/0026897031000082149}, + Issn = {0026-8976}, + Journal = {Mol. Phys.}, + Month = {May}, + Number = {9}, + Pages = {1389--1398}, + Publisher = {Taylor {\&} Francis}, + Title = {{The use of local orbitals in multireference calculations}}, + Volume = {101}, + Year = {2003}, + Bdsk-Url-1 = {https://doi.org/10.1080/0026897031000082149}} @article{Deustua_2017, Author = {Deustua, J. Emiliano and Shen, Jun and Piecuch, Piotr}, @@ -1040,16 +1058,6 @@ Year = {2016}, Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01099}} -@misc{Caffarel_2016b, - Date-Modified = {2020-08-18 20:50:22 +0200}, - Journal = {ACS Symp. Ser.}, - Month = {Jan}, - Note = {[Online; accessed 6. Jul. 2020]}, - Title = {{Recent Progress in Quantum Monte Carlo}}, - Url = {https://pubs.acs.org/doi/abs/10.1021/bk-2016-1234.ch002}, - Year = {2016}, - Bdsk-Url-1 = {https://pubs.acs.org/doi/abs/10.1021/bk-2016-1234.ch002}} - @misc{Scemama_2015, Author = {Scemama, Anthony and Giner, Emmanuel and Applencourt, Thomas and Caffarel, Michel}, Doi = {10.13140/RG.2.1.3187.9766}, diff --git a/benzene.tex b/benzene.tex index 856269b..ecc59fa 100644 --- a/benzene.tex +++ b/benzene.tex @@ -4,7 +4,7 @@ \newcommand{\ie}{\textit{i.e.}} \newcommand{\eg}{\textit{e.g.}} -\newcommand{\alert}[1]{\textcolor{red}{#1}} +\newcommand{\alert}[1]{\textcolor{black}{#1}} \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} @@ -61,7 +61,7 @@ In a recent preprint, \cite{Eriksen_2020} Eriksen \textit{et al.}~have proposed In addition to coupled cluster theory with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} a large panel of highly-accurate, emerging electronic structure methods were considered: (i) the many-body expansion FCI (MBE-FCI), \cite{Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b} (ii) three SCI methods including a second-order perturbative correction \alert{[adaptive sampling CI (ASCI), \cite{Tubman_2016,Tubman_2018,Tubman_2020} iterative CI (iCI), \cite{Liu_2014,Liu_2016,Lei_2017,Zhang_2020} and semistochastic heat-bath CI (SHCI) \cite{Holmes_2016,Holmes_2017,Sharma_2017}]}, - (iii) \alert{the full coupled-cluster reduction (FCCR) \cite{Xu_2018} which also includes a second-order perturbative correction}, + (iii) \alert{the full coupled-cluster reduction (FCCR) \cite{Xu_2018,Xu_2020} which also includes a second-order perturbative correction}, (iv) the density-matrix renornalization group approach (DMRG), \cite{White_1992,White_1993,Chan_2011} and (v) two flavors of FCI quantum Monte Carlo (FCIQMC), \cite{Booth_2009,Cleland_2010} namely AS-FCIQMC \cite{Ghanem_2019} and CAD-FCIQMC. \cite{Deustua_2017,Deustua_2018} We refer the interested reader to Ref.~\onlinecite{Eriksen_2020} and its supporting information for additional details on each method and the complete list of references. @@ -103,66 +103,6 @@ The outcome of this work is nicely summarized in the abstract of Ref.~\onlinecit \end{ruledtabular} \end{table} -% CIPSI -For the sake of completeness and our very own curiosity, we report in this Note the frozen-core correlation energy obtained with a fourth flavor of SCI known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), \cite{Huron_1973} which also includes a second-order perturbative (PT2) correction. -In short, the CIPSI algorithm belongs to the family of SCI+PT2 methods. -The idea behind such methods is to avoid the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, thanks to the use of a second-order energetic criterion to select perturbatively determinants in the FCI space. -However, performing SCI calculations rapidly becomes extremely tedious when one increases the system size as one hits the exponential wall inherently linked to these methods. - -From a historical point of view, CIPSI is probably one of the oldest SCI algorithm. -It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see also Ref.~\onlinecite{Evangelisti_1983}). -Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by our group enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019} -In particular, we were able to compute highly-accurate 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 used 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} (which explains the statistical error associated with the PT2 correction in the following). -\alert{Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \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 rPT2 correction and the CIPSI algorithm.} - -% Computational details -Being late to the party, we obviously cannot report blindly our CIPSI results. -However, following the philosophy of Eriksen \textit{et al.} \cite{Eriksen_2020} and Lee \textit{et al.}, \cite{Lee_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind. -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 natural 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 $E_\text{(r)PT2}$ 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 $E_\text{(r)PT2} = 0$ (which effectively corresponds to the FCI limit). -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 singlet 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. -Starting from the same natural orbitals, a Boys-Foster localization procedure \cite{Boys_1960} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals. -\footnote{Indices of molecular orbitals for Boys-Foster localization procedure: -core [1--6]; -$\sigma$ [7--18]; -$\pi$ [19--21]; -$\pi^*$ [22--24]; -$\sigma^*$ [25--36]; -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. -\alert{Indeed, localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of electron correlation.\cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020}} -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 four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy. -\alert{(In other words, the four largest variational wave functions are considered to perform the linear extrapolation.)} -From this figure, one clearly sees that the rPT2-based 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 non-SCI 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 four-point 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). -\footnote{\alert{Using the last 3, 4, 5, and 6 largest wave functions to perform the linear extrapolation yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, $-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively. -These numbers vary by $1.4$ mE$_h$. -The four-point extrapolated value of $-863.4(5)$ mE$_h$ that we have chosen to report as our best estimate corresponds to the smallest fitting error. -Quadratic fits yield much larger variations and are discarded in practice. -Due to the stochastic nature of $E_\text{rPT2}$, the fifth point is slightly off as compared to the others. -Taking into account this fifth point yield a slightly smaller estimate of the correlation energy [$-862.1(8)$ mE$_h$], while adding a sixth point settles down the correlation energy estimate at $-863.5(11)$ mE$_h$ -}} -For comparison, the best post blind test SHCI estimate is $-863.3$ m$E_h$, which agrees almost perfectly with our best CIPSI estimate, while the best post blind test ASCI and iCI correlation energies are $-861.3$ and $-864.15$ m$E_h$, respectively (see Table \ref{tab:extrap_dist_table}). - %%$ FIG. 1 %%% \begin{figure*} \includegraphics[width=0.4\linewidth]{fig1a} @@ -224,7 +164,7 @@ The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviat %\end{squeezetable} %%% %%% %%% %%% -%%% TABLE II %%% +%%% TABLE III %%% \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 ASCI, iCI, SHCI, DMRG, and CIPSI for the best blind-test and post-blind-test estimates of the correlation energy of benzene in the cc-pVDZ basis. The final variational energies $\Delta E_{\text{var.}}$ are also reported. @@ -252,6 +192,66 @@ The statistical error on $E_\text{(r)PT2}$, corresponding to one standard deviat \end{ruledtabular} \end{table} +% CIPSI +For the sake of completeness and our very own curiosity, we report in this Note the frozen-core correlation energy obtained with a fourth flavor of SCI known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), \cite{Huron_1973} which also includes a second-order perturbative (PT2) correction. +In short, the CIPSI algorithm belongs to the family of SCI+PT2 methods. +The idea behind such methods is to avoid the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, thanks to the use of a second-order energetic criterion to select perturbatively determinants in the FCI space. +However, performing SCI calculations rapidly becomes extremely tedious when one increases the system size as one hits the exponential wall inherently linked to these methods. + +From a historical point of view, CIPSI is probably one of the oldest SCI algorithm. +It was developed in 1973 by Huron, Rancurel, and Malrieu \cite{Huron_1973} (see also Ref.~\onlinecite{Evangelisti_1983}). +Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by our group enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019} +In particular, we were able to compute highly-accurate 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 used 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} (which explains the statistical error associated with the PT2 correction in the following). +\alert{Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \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 rPT2 correction and the CIPSI algorithm.} + +% Computational details +Being late to the party, we obviously cannot report blindly our CIPSI results. +However, following the philosophy of Eriksen \textit{et al.} \cite{Eriksen_2020} and Lee \textit{et al.}, \cite{Lee_2020} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind. +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 natural 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 $E_\text{(r)PT2}$ 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 $E_\text{(r)PT2} = 0$ (which effectively corresponds to the FCI limit). +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 singlet 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. +Starting from the same natural orbitals, a Boys-Foster localization procedure \cite{Boys_1960} was performed in several orbital windows: i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals. +\footnote{Indices of molecular orbitals for Boys-Foster localization procedure: +core [1--6]; +$\sigma$ [7--18]; +$\pi$ [19--21]; +$\pi^*$ [22--24]; +$\sigma^*$ [25--36]; +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. +\alert{Indeed, localized orbitals significantly speed up the convergence of SCI calculations by taking benefit of the local character of electron correlation.\cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020}} +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 four-point linear extrapolation curves that we have used to get our final estimate of the correlation energy. +\alert{(In other words, the four largest variational wave functions are considered to perform the linear extrapolation.)} +From this figure, one clearly sees that the rPT2-based 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 non-SCI 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 four-point 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). +\footnote{\alert{Using the last 3, 4, 5, and 6 largest wave functions to perform the linear extrapolation yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, $-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively. +These numbers vary by $1.4$ mE$_h$. +The four-point extrapolated value of $-863.4(5)$ mE$_h$ that we have chosen to report as our best estimate corresponds to the smallest fitting error. +Quadratic fits yield much larger variations and are discarded in practice. +Due to the stochastic nature of $E_\text{rPT2}$, the fifth point is slightly off as compared to the others. +Taking into account this fifth point yield a slightly smaller estimate of the correlation energy [$-862.1(8)$ mE$_h$], while adding a sixth point settles down the correlation energy estimate at $-863.5(11)$ mE$_h$ +}} +For comparison, the best post blind test SHCI estimate is $-863.3$ m$E_h$, which agrees almost perfectly with our best CIPSI estimate, while the best post blind test ASCI and iCI correlation energies are $-861.3$ and $-864.15$ m$E_h$, respectively (see Table \ref{tab:extrap_dist_table}). + % 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. @@ -263,10 +263,869 @@ In total, the present calculation has required 150k core hours, most of it being % Acknowledgements We thank Janus Eriksen and Cyrus Umrigar for useful comments. This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and from CALMIP (Toulouse) under allocation 2020-18005. +PFL and AS have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481). % Data availability statement \alert{The data that support the findings of this study are openly available in Zenodo at http://doi.org/10.5281/zenodo.4075286.} -\bibliography{benzene} +%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked +%Control: key (0) +%Control: author (8) initials jnrlst +%Control: editor formatted (1) identically to author +%Control: production of article title (-1) disabled +%Control: page (0) single +%Control: year (1) truncated +%Control: production of eprint (0) enabled +\begin{thebibliography}{71}% +\makeatletter +\providecommand \@ifxundefined [1]{% + \@ifx{#1\undefined} +}% +\providecommand \@ifnum [1]{% + \ifnum #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \@ifx [1]{% + \ifx #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \natexlab [1]{#1}% +\providecommand \enquote [1]{``#1''}% +\providecommand \bibnamefont [1]{#1}% +\providecommand \bibfnamefont [1]{#1}% +\providecommand \citenamefont [1]{#1}% +\providecommand \href@noop [0]{\@secondoftwo}% +\providecommand \href [0]{\begingroup \@sanitize@url \@href}% +\providecommand \@href[1]{\@@startlink{#1}\@@href}% +\providecommand \@@href[1]{\endgroup#1\@@endlink}% +\providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode + `\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax}% +\providecommand \@@startlink[1]{}% +\providecommand \@@endlink[0]{}% +\providecommand \url [0]{\begingroup\@sanitize@url \@url }% +\providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }}% +\providecommand \urlprefix [0]{URL }% +\providecommand \Eprint [0]{\href }% +\providecommand \doibase [0]{http://dx.doi.org/}% +\providecommand \selectlanguage [0]{\@gobble}% +\providecommand \bibinfo [0]{\@secondoftwo}% +\providecommand \bibfield [0]{\@secondoftwo}% +\providecommand \translation [1]{[#1]}% +\providecommand \BibitemOpen [0]{}% +\providecommand \bibitemStop [0]{}% +\providecommand \bibitemNoStop [0]{.\EOS\space}% +\providecommand \EOS [0]{\spacefactor3000\relax}% +\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}% +\let\auto@bib@innerbib\@empty +% +\bibitem [{\citenamefont {LeBlanc}\ \emph {et~al.}(2015)\citenamefont + {LeBlanc}, \citenamefont {Antipov}, \citenamefont {Becca}, \citenamefont + {Bulik}, \citenamefont {Chan}, \citenamefont {Chung}, \citenamefont {Deng}, + \citenamefont {Ferrero}, \citenamefont {Henderson}, \citenamefont + {Jim{\'e}nez-Hoyos} \emph {et~al.}}]{Leblanc_2015}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~P.~F.}\ + \bibnamefont {LeBlanc}}, \bibinfo {author} {\bibfnamefont {A.~E.}\ + \bibnamefont {Antipov}}, \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont + {Becca}}, \bibinfo {author} {\bibfnamefont {I.~W.}\ \bibnamefont {Bulik}}, + \bibinfo {author} {\bibfnamefont {G.~K.-L.}\ \bibnamefont {Chan}}, \bibinfo + {author} {\bibfnamefont {C.-M.}\ \bibnamefont {Chung}}, \bibinfo {author} + {\bibfnamefont {Y.}~\bibnamefont {Deng}}, \bibinfo {author} {\bibfnamefont + {M.}~\bibnamefont {Ferrero}}, \bibinfo {author} {\bibfnamefont {T.~M.}\ + \bibnamefont {Henderson}}, \bibinfo {author} {\bibfnamefont {C.~A.}\ + \bibnamefont {Jim{\'e}nez-Hoyos}}, \emph {et~al.},\ }\href {\doibase + 10.1103/PhysRevX.5.041041} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. 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Theory Comput.}\ }\textbf {\bibinfo {volume} {15}},\ \bibinfo {pages} + {4896} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Applencourt}\ \emph {et~al.}(2018)\citenamefont + {Applencourt}, \citenamefont {Gasperich},\ and\ \citenamefont + {Scemama}}]{Applencourt_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont + {Applencourt}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont + {Gasperich}}, \ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Scemama}},\ }\href@noop {} {\enquote {\bibinfo {title} {Spin adaptation with + determinant-based selected configuration interaction},}\ } (\bibinfo {year} + {2018}),\ \Eprint {http://arxiv.org/abs/1812.06902} {arXiv:1812.06902 + [physics.chem-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Foster}\ and\ \citenamefont + {Boys}(1960)}]{Boys_1960}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont + {Foster}}\ and\ \bibinfo {author} {\bibfnamefont {S.~F.}\ \bibnamefont + {Boys}},\ }\href {\doibase 10.1103/RevModPhys.32.300} {\bibfield {journal} + {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {32}},\ + \bibinfo {pages} {300} (\bibinfo {year} {1960})}\BibitemShut {NoStop}% +\bibitem [{Note1()}]{Note1}% + \BibitemOpen + \bibinfo {note} {Indices of molecular orbitals for Boys-Foster localization + procedure: core [1--6]; $\sigma $ [7--18]; $\pi $ [19--21]; $\pi ^*$ + [22--24]; $\sigma ^*$ [25--36]; 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].}\BibitemShut + {Stop}% +\bibitem [{\citenamefont {Pipek}\ and\ \citenamefont + {Mezey}(1989)}]{Pipek_1989}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Pipek}}\ and\ \bibinfo {author} {\bibfnamefont {P.~G.}\ \bibnamefont + {Mezey}},\ }\href {\doibase 10.1063/1.456588} {\bibfield {journal} {\bibinfo + {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {90}},\ \bibinfo + {pages} {4916} (\bibinfo {year} {1989})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Angeli}\ \emph {et~al.}(2003)\citenamefont {Angeli}, + \citenamefont {Calzado}, \citenamefont {Cimiraglia}, \citenamefont + {Evangelisti}, \citenamefont {Guih\'ery}, \citenamefont {Leininger}, + \citenamefont {Malrieu}, \citenamefont {Maynau}, \citenamefont {Ruiz},\ and\ + \citenamefont {Sparta}}]{Angeli_2003}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Angeli}}, \bibinfo {author} {\bibfnamefont {C.~J.}\ \bibnamefont {Calzado}}, + \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Cimiraglia}}, \bibinfo + {author} {\bibfnamefont {S.}~\bibnamefont {Evangelisti}}, \bibinfo {author} + {\bibfnamefont {N.}~\bibnamefont {Guih\'ery}}, \bibinfo {author} + {\bibfnamefont {T.}~\bibnamefont {Leininger}}, \bibinfo {author} + {\bibfnamefont {J.-P.}\ \bibnamefont {Malrieu}}, \bibinfo {author} + {\bibfnamefont {D.}~\bibnamefont {Maynau}}, \bibinfo {author} {\bibfnamefont + {J.~V.~P.}\ \bibnamefont {Ruiz}}, \ and\ \bibinfo {author} {\bibfnamefont + {M.}~\bibnamefont {Sparta}},\ }\href {\doibase 10.1080/0026897031000082149} + {\bibfield {journal} {\bibinfo {journal} {Mol. Phys.}\ }\textbf {\bibinfo + {volume} {101}},\ \bibinfo {pages} {1389} (\bibinfo {year} + {2003})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Angeli}(2009)}]{Angeli_2009}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Angeli}},\ }\href {\doibase 10.1002/jcc.21155} {\bibfield {journal} + {\bibinfo {journal} {J. Comput. 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Chem. + Phys.}\ }\textbf {\bibinfo {volume} {135}},\ \bibinfo {pages} {014101} + (\bibinfo {year} {2011})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Suaud}\ and\ \citenamefont + {Malrieu}(2017)}]{Suaud_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont + {Suaud}}\ and\ \bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont + {Malrieu}},\ }\href {\doibase 10.1080/00268976.2017.1303207} {\bibfield + {journal} {\bibinfo {journal} {Mol. Phys.}\ }\textbf {\bibinfo {volume} + {115}},\ \bibinfo {pages} {2684} (\bibinfo {year} {2017})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Chien}\ \emph {et~al.}(2018)\citenamefont {Chien}, + \citenamefont {Holmes}, \citenamefont {Otten}, \citenamefont {Umrigar}, + \citenamefont {Sharma},\ and\ \citenamefont {Zimmerman}}]{Chien_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~D.}\ \bibnamefont + {Chien}}, \bibinfo {author} {\bibfnamefont {A.~A.}\ \bibnamefont {Holmes}}, + \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Otten}}, \bibinfo + {author} {\bibfnamefont {C.~J.}\ \bibnamefont {Umrigar}}, \bibinfo {author} + {\bibfnamefont {S.}~\bibnamefont {Sharma}}, \ and\ \bibinfo {author} + {\bibfnamefont {P.~M.}\ \bibnamefont {Zimmerman}},\ }\href {\doibase + 10.1021/acs.jpca.8b01554} {\bibfield {journal} {\bibinfo {journal} {J. + Phys. Chem. A}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages} {2714} + (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{Note2()}]{Note2}% + \BibitemOpen + \bibinfo {note} {\protect \leavevmode {\protect \color {black}Using the last + 3, 4, 5, and 6 largest wave functions to perform the linear extrapolation + yield the following correlation energy estimates: $-863.1(11)$, $-863.4(5)$, + $-862.1(8)$, and $-863.5(11)$ mE$_h$, respectively. These numbers vary by + $1.4$ mE$_h$. The four-point extrapolated value of $-863.4(5)$ mE$_h$ that we + have chosen to report as our best estimate corresponds to the smallest + fitting error. Quadratic fits yield much larger variations and are discarded + in practice. Due to the stochastic nature of $E_\protect \text {rPT2}$, the + fifth point is slightly off as compared to the others. Taking into account + this fifth point yield a slightly smaller estimate of the correlation energy + [$-862.1(8)$ mE$_h$], while adding a sixth point settles down the correlation + energy estimate at $-863.5(11)$ mE$_h$ }}\BibitemShut {NoStop}% +\end{thebibliography}% \end{document}