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16
benzene.bib
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@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-21 22:33:45 +0200 %% Created for Pierre-Francois Loos at 2020-08-22 17:57:20 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Boys_1960,
Author = {J. M. Foster and S. F. Boys},
Date-Added = {2020-08-22 17:56:32 +0200},
Date-Modified = {2020-08-22 17:57:16 +0200},
Doi = {10.1103/RevModPhys.32.300},
Journal = {Rev. Mod. Phys.},
Pages = {300},
Title = {Canonical Configurational Interaction Procedure},
Volume = {32},
Year = {1960}}
@article{Pipek_1989, @article{Pipek_1989,
Author = {Janos Pipek and Paul G. Mezey}, Author = {Janos Pipek and Paul G. Mezey},
Date-Added = {2020-08-21 22:32:52 +0200}, Date-Added = {2020-08-21 22:32:52 +0200},
@ -17,7 +28,8 @@
Pages = {4916}, Pages = {4916},
Title = {A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions}, Title = {A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions},
Volume = {90}, Volume = {90},
Year = {1989}} Year = {1989},
Bdsk-Url-1 = {https://doi.org/10.1063/1.456588}}
@article{Caffarel_2014, @article{Caffarel_2014,
Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro}, Author = {Caffarel, Michel and Giner, Emmanuel and Scemama, Anthony and Ram{\'\i}rez-Sol{\'\i}s, Alejandro},

12
benzene.tex
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@ -32,6 +32,8 @@
\author{Pierre-Fran\c{c}ois Loos} \author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Yann Damour}
\affiliation{\LCPQ}
\author{Anthony Scemama} \author{Anthony Scemama}
\email{scemama@irsamc.ups-tlse.fr} \email{scemama@irsamc.ups-tlse.fr}
\affiliation{\LCPQ} \affiliation{\LCPQ}
@ -116,7 +118,7 @@ We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can
Being late to the party, we obviously cannot report blindly our CIPSI results. 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} we will report our results with the most neutral tone, leaving the freedom to the reader to make up his/her mind. However, following the philosophy of Eriksen \textit{et al.}, \cite{Eriksen_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. 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 (NOs) are then computed based on this wave function, and a new, larger SCI calculation is performed with this new set of orbitals. 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. 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 $E_\text{PT2}$ 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.
As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems. As mentioned above, SCI+PT2 methods rely heavily on extrapolation, especially when one deals with medium-sized systems.
@ -127,8 +129,8 @@ We believe that it provides a very safe estimate of the extrapolation error.
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 Hartree-Fock 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 higher virtual orbitals. \titou{More information needed here.} 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.}
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 NOs. We, therefore, consider these energies more trustworthy, and we will based our best estimate of the correlation energy of benzene on these calculations. 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.
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.
@ -139,9 +141,9 @@ The three flavours of SCI fall into an interval ranging from $-860.0$ m$E_h$ (AS
% Timings % Timings
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. 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 NOs, takes roughly 24 hours. 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.
%%$ FIG. 1 %%% %%$ FIG. 1 %%%
\begin{figure*} \begin{figure*}