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benzene.tex
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@ -9,6 +9,9 @@
\newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\toto}[1]{\textcolor{green}{#1}}
\newcommand{\trashAS}[1]{\textcolor{green}{\sout{#1}}}
\newcommand{\AS}[1]{\toto{(\underline{\bf AS}: #1)}}
\newcommand{\mc}{\multicolumn} \newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark} \newcommand{\fnm}{\footnotemark}
@ -105,11 +108,11 @@ 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. 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. 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 an historical point of view, CIPSI is probably one of the oldest SCI algorithm. 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}). 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 one of us (AS) enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019} Recently, the determinant-driven CIPSI algorithm has been efficiently implemented \cite{Giner_2013,Giner_2015} in the open-source programming environment {\QP} by our group enabling to perform massively parallel computations. \cite{Garniron_2017,Garniron_2018,Garniron_2019}
In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c} In particular, we were able to compute highly-accurate calculations of ground- and excited-state energies for small- and medium-sized molecules (including benzene). \cite{Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c}
CIPSI is also frequently use to provide accurate trial wave function for QMC calculations. \cite{Caffarel_2014,Caffarel_2016a,Caffarel_2016b,Giner_2013,Giner_2015,Scemama_2015,Scemama_2016,Scemama_2018,Scemama_2018b,Scemama_2019,Dash_2018,Dash_2019} CIPSI is also frequently 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). The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm, \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
Moreover, a renormalized version of the PT2 correction (dubbed rPT2 in the following) has been recently implemented for a more efficient extrapolation to the FCI limit (see below). \cite{Garniron_2019} Moreover, a renormalized version of the PT2 correction (dubbed rPT2 in the following) has been recently implemented for a more efficient extrapolation to the FCI limit (see below). \cite{Garniron_2019}
We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm. We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the CIPSI algorithm.
@ -125,12 +128,20 @@ As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially wh
We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions. We then linearly extrapolate the total SCI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wave functions.
Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by \titou{the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value.} Although it is not possible to provide a theoretically sound error bar, we estimate the extrapolation error by \titou{the difference in excitation energy between the largest SCI wave function and its corresponding extrapolated value.}
We believe that it provides a very safe estimate of the extrapolation error. We believe that it provides a very safe estimate of the extrapolation error.
Note that, 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 the $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero. 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 \toto{($\sim 5 \times 10^{-3}$ a.u.)}.
The corresponding energies are reported in Table \ref{tab:NOvsLO} as functions of the number of determinants in the variational space $N_\text{det}$. 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. 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. Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene. \titou{T2: add MO indices.} 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. Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like benzene. \toto{T2: add MO indices (see comments in tex file).}
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 based our best estimate of the correlation energy of benzene on these calculations. % MO Indices:
%[1-6] # Core
%[7,8,9,10,11,12,13,14,15,16,17,18] # Sigma occ
%[19,20,21] # Pi occ
%[22,23,24] # Pi virt 1
%[25,26,27,28,29,30,31,32,33,34,35,36] # Sigma virt 1
%[39,41,42,43,46,49,50,53,54,55,56,57,71,72,73,74,82,83,84,85,87,92,93,98] # Pi virt 2
%[37,38,40,44,45,47,48,51,52,58,59,60,61,62,63,64,65,66,67,68,69,70,75,76,77,78,79,80,81,86,88,89,90,91,94,95,96,97,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114] # Sigma virt 2
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 $\Delta E_\text{var.}$, $\Delta E_\text{var.} + E_\text{PT2}$, and $\Delta E_\text{var.} + E_\text{rPT2}$ as a function of $N_\text{det}$ (left panel).
The 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} 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.
@ -143,7 +154,7 @@ The three flavours of SCI fall into an interval ranging from $-860.0$ m$E_h$ (AS
The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer. The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
Each Irene's AMD node is a dual-socket AMD Rome (Epyc) CPU@2.60 GHz with 256GiB of RAM, with a total of 64 physical CPU cores per socket. 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. 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, and reaching the same number of determinants with natural orbitals or localized orbitals takes roughly the same amount of time. 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 \toto{on a single node}, and reaching the same number of determinants with natural orbitals or localized orbitals takes roughly the same amount of time. \toto{A second 24-hour run on 10 distributed nodes was performed to push the selection to 80M determinants, and a third distributed run using 40 nodes was used to reach 160M determinants.}
%%$ FIG. 1 %%% %%$ FIG. 1 %%%
\begin{figure*} \begin{figure*}
@ -228,7 +239,7 @@ The statistical error on $E_\text{PT2}$, corresponding to one standard deviation
% Acknowledgements % Acknowledgements
This work was performed using HPC resources from GENCI-TGCC (Grand Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation 2020-18005. This work was performed using HPC resources from GENCI-TGCC (2020-gen1738) and from CALMIP (Toulouse) under allocation 2020-18005.\toto{L'allocation Grand-Challenge est finie!}
\bibliography{benzene} \bibliography{benzene}