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Benzene

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Anthony Scemama 2024-03-28 12:25:44 +01:00
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Data/benzene.dat Normal file
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#!/usr/bin/env gnuplot
reset
set grid
set xlabel "% Contributions"
set ylabel "Statistical Error (au)"
set format y "%g"
set log y
set term pdfcairo enhanced font "Times,14" linewidth 2 rounded size 5.0in, 3.0in
set output 'benzene_err.pdf'
data='benzene.dat'
tmax=2812.08030200005 * 0.01
set xrange [0:100.]
set yrange [1.e-7:1.e-2]
plot data i 0 u ($3):($2) w l title "cc-pVTZ", \
data i 1 u ($3):($2) w l title "cc-pVQZ"

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Data/benzene_qz.plt Normal file
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#!/usr/bin/env gnuplot
reset
set grid
set xlabel "Wall clock time (seconds)"
set ylabel "Energy (au)"
set format y "%10.4f"
set term pdfcairo enhanced font "Times,14" linewidth 2 rounded size 5.0in, 3.0in
set output 'benzene_qz.pdf'
set style fill transparent solid 0.50 border
#set yrange [-231.8075:-231.8040]
data='benzene.dat'
tmax=2812.08030200005 * 0.01
set xrange [0:tmax*100.]
set yrange [-231.8724:-231.8712]
plot data i 1 u ($3*tmax):($1+$2):($1-$2) w filledcurves ls 1 notitle, \
data i 1 u ($3*tmax):($1) w l ls 1 notitle , \
-231.871740549698 notitle ls 3

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Data/benzene_tz.plt Normal file
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#!/usr/bin/env gnuplot
reset
set grid
set xlabel "Wall clock time (seconds)"
set ylabel "Energy (au)"
set format y "%10.4f"
set xrange [0:240]
set term pdfcairo enhanced font "Times,14" linewidth 2 rounded size 5.0in, 3.0in
set output 'benzene_tz.pdf'
set style fill transparent solid 0.50 border
set yrange [-231.8072:-231.8046]
data='benzene.dat'
tmax=239.890999078751 * 0.01
plot data i 0 u ($3*tmax):($1+$2):($1-$2) w filledcurves ls 1 notitle, \
data i 0 u ($3*tmax):($1) w l ls 1 notitle , \
-231.805729365546 notitle ls 3

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Manuscript/benzene_err.pdf Normal file
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Manuscript/stochastic_triples.tex
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@ -213,6 +213,33 @@ accelerators.\cite{ma_2011,haidar_2015,dinapoli_2014,springer_2018}
%c. Analysis of the algorithm's accuracy, efficiency, and scalability
%d. Discussion of any observed limitations or challenges
\subsection{Convergence of the statistical error in benzene}
In this section we illustrate the convergence of the statistical error of the perturbative triples correction as a function of the computational cost.
The benzene molecule serves as our reference system for conducting frozen-core CCSD(T) calculations with the cc-pVTZ and cc-pVQZ basis sets.
Essentially, this involves the correlation of 30 electrons across either 258 or 503 molecular orbitals.
\begin{figure}
\includegraphics[width=\columnwidth]{benzene_tz.pdf}
\includegraphics[width=\columnwidth]{benzene_qz.pdf}
\caption{\label{fig:benzene} Convergence of the energy of benzene as a function of the execution time of the program. The top curve corresponds to the cc-pVTZ basis set and the bottom curve to cc-pVQZ. The blue line represents the exact CCSD(T) energy.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{benzene_err.pdf}
\caption{\label{fig:benzene_err} Convergence of the statistical error of the perturbative triples contribution in benzene as a function of the percentage of computed contributions, in the cc-pVTZ and cc-pVQZ basis sets.}
\end{figure}
Figure~\ref{fig:benzene} shows the convergence of the CCSD(T) energy as a function of the execution time of the program with both basis sets. We observe that the exact CCSD(T) energy is alwasy within $2\sigma$, showing that the statistical error is reliable.
Figure~\ref{fig:benzene_err} displays the statistical error as a function of the percentage of computed contributions.
In both figures, the discontinuities in the curves are due to changes in the splitting between deterministic and stochastic components, leading to a change in the estimated value, and a reduction of the statistical error.
For the both basis sets, the chemical accuracy (\SI{1.6}{\milli \hartree}) is reached using less than 1\% of the contributions.
A precision of (\SI{0.1}{\milli \hartree}) is obtained using respectively 32\% and 15\% for the cc-pVTZ and cc-pVQZ basis sets.
The faster convergence for the largest basis set was expected: using a larger basis set increases the number of tiny contributions, and keeps the number of large contributions rather constant. Hence, this curve shows that the proposed algorithm is better adapted to fewer electrons in large basis sets than many electrons in small basis sets.
% Discuter les cassures dans les courbes
% Ajouter une courbe de convergence de l'erreur
\subsection{Vibrational frequency of copper chloride}
Our methodology proves especially advantageous for scenarios requiring the aggregation of numerous CCSD(T) energies, such as neural network training or the exploration of potential energy surfaces.
@ -252,6 +279,9 @@ Figure \ref{fig:cucl} illustrates the potential energy surface of \ce{CuCl}, dis
%a. Summary of the algorithm and its advantages
%b. Recapitulation of the key findings and contributions
%c. Final remarks and encouragement for further research
%
% Works better for few electrons in large basis sets
% Interesting for ML or PES exploration
%=================================================================%