CuCl OK
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# R CCSD(T) Stochastic CCSD(T)
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#---------------------------------------------------------
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1.55 -2099.590506349950 -2099.58890791 1.3630E-03
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1.65 -2099.671184187604 -2099.67175286 1.5710E-03
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1.75 -2099.720045862965 -2099.71877319 1.3199E-03
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1.85 -2099.747811193906 -2099.74897746 1.4668E-03
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1.95 -2099.761752030920 -2099.76232971 1.6105E-03
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2.05 -2099.766727898670 -2099.76565267 1.5202E-03
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2.15 -2099.765956694308 -2099.76485609 1.7470E-03
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2.25 -2099.761562105614 -2099.76237391 1.7474E-03
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2.35 -2099.754944906474 -2099.75681975 1.9951E-03
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2.45 -2099.747028328725 -2099.74813718 2.4288E-03
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2.55 -2099.738443175793 -2099.74031232 2.4057E-03
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2.65 -2099.729597826175 -2099.72866832 1.6894E-03
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@ -0,0 +1,30 @@
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#!/usr/bin/env gnuplot
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reset
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set grid
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set xlabel "Cu-Cl distance (bohr)"
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set ylabel "CCSD(T) energy (au)"
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set format y "%.2f"
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a0 = 1.8897161646321
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E(r) = De * (1-exp(-a*(r-re)))**2 + E0
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a = 0.84615 # +/- 0.03216 (3.8%)
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re = 3.92539 # +/- 0.01058 (0.2696%)
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De = 0.101589 # +/- 0.00932 (9.174%)
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E0 = -2099.77 # +/- 0.0008014 (3.817e-05%)
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set xrange [2.7:5.2]
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fit E(x) 'cucl.dat' using ($1*a0):3:4 via a, re, De, E0
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set xrange [3:5.2]
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set term pdfcairo enhanced font "Times,14" linewidth 2 rounded size 5.0in, 3.0in
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set output 'cucl.pdf'
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set pointsize 0.5
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plot \
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'cucl.dat' using ($1*a0):2 pointtype 7 lt 4 title "Full (T)", \
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E(x) title "" lt 3, \
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'cucl.dat' using ($1*a0):3:4 w err pt 0 lt 1 title "1% (T)"
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Binary file not shown.
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@ -132,7 +132,7 @@ i@article{watson_2016,
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month = sep,
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issn = {1870-249X},
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publisher = {Sociedad Qu{\'{\i}}mica de M{\'{e}}xico A.C.},
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url = {https://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1870-249X2012000300014}
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url = {https://www.scielo.org.mx/scielo.php?script=sci_arttext\&pid=S1870-249X2012000300014}
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}
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@article{du_2020,
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@ -170,7 +170,7 @@ volume = {118},
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number = {24},
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pages = {e1797915},
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year = {2020},
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publisher = {Taylor & Francis},
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publisher = {Taylor and Francis},
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doi = {10.1080/00268976.2020.1797915},
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URL = {https://doi.org/10.1080/00268976.2020.1797915},
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eprint = {https://doi.org/10.1080/00268976.2020.1797915}
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@ -1,5 +1,6 @@
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
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%\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
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\usepackage{graphicx,xcolor,physics,siunitx,xspace}
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\usepackage[version=4]{mhchem}
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\usepackage[utf8]{inputenc}
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@ -13,6 +14,9 @@
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urlcolor={red!80!black}
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}
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\DeclareSIUnit{\bohr}{\text{\ensuremath{a_{0}}}}
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\DeclareSIUnit{\hartree}{\text{\ensuremath{E_{\textup{h}}}}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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@ -69,7 +73,7 @@
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\begin{document}
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\title{Stochastically accelerated perturbative triples in coupled cluster calculations}
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\title{Stochastically accelerated perturbative triples correction in coupled cluster calculations}
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% Alphabetic order
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\author{Yann \surname{Damour}}
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@ -104,10 +108,10 @@ This work opens up new avenues for efficient and accurate
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computations, enabling investigations of complex molecular systems
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that were previously computationally prohibitive.
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\bigskip
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\begin{center}
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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\end{center}
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\bigskip
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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@ -203,41 +207,44 @@ accelerators.\cite{ma_2011,haidar_2015,dinapoli_2014,springer_2018}
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% - Benzene TZ
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% - Streptocyanine QZ: Small molecule in a large basis set
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% - Caffeine def2-svp: Large molecule in a small basis set
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% - Vibrational frequency of F2/cc-pvqz
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% - Vibrational frequency of CuCl/cc-pvqz
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%b. Discussion of the obtained results, comparing against other methods
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% - Measure flops and compare to the peak
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%c. Analysis of the algorithm's accuracy, efficiency, and scalability
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%d. Discussion of any observed limitations or challenges
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\subsection{Vibrational frequency of \ce{F2}}
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\subsection{Vibrational frequency of copper chloride}
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In this example, we compute the vibrational frequency of \ce{F2} by
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computing the potential energy curve, and fitting it with a Morse
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potential
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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.
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In this section, we discuss the application of our novel algorithm within the context of computing vibrational frequencies, specifically through the example of copper chloride (\ce{CuCl}).
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A demonstrative application presented here involves the determination of the equilibrium bond length and the computation of the vibrational frequency of \ce{CuCl} using the CCSD(T)/cc-pVQZ level of theory.
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The procedure involves determining the CCSD(T) potential energy curve for \ce{CuCl}, followed by its analytical representation through a Morse potential fitting:
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\begin{equation}
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E(r) = D_e \left( 1 - e^{-a (r - r_e)} \right)^2 + E(r_e)
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E(r) = D_e \left( 1 - e^{-a (r - r_e)} \right)^2 + E_0
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\end{equation}
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where $E(r)$ is the energy at distance $r$, $D_e$ is the well depth,
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$r_e$ is the equilibrium bond distance, and $a$ is a parameter
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controlling the width of the potential well.
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The vibrational frequency $\nu$ is calculated as
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where $E(r)$ represents the energy at a bond length $r$, $D_e$ the depth of the potential well, $r_e$ the equilibrium bond length, $a$ the parameter defining the potential well's width, and $E_0$ the energy at the equilibrium bond length. The vibrational frequency, $\nu$, is derived as follows:
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\begin{equation}
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\nu = \frac{1}{2 \pi c} \sqrt{\frac{2D_e a^2}{\mu}
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\nu = \frac{1}{2 \pi c} \sqrt{\frac{2D_e a^2}{\mu}}
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\end{equation}
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where $\mu$ is the mass of the Fluorine atom, and $c$ is the speed of
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light in cm/s.
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with $\mu$ denoting the reduced mass of the \ce{CuCl} molecule, and $c$ the speed of light.
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% CCSD
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%a = 2.2936 +/- 0.006318 (0.2755%)
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%De = 0.125888 +/- 0.0005213 (0.4141%)
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%re = 1.3893 +/- 0.0003428 (0.02468%)
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%E0 = -199.338 +/- 6.422e-05 (3.222e-05%)
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\begin{figure}
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\includegraphics[width=\columnwidth]{cucl.pdf}
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\caption{\label{fig:cucl} CCSD(T) energies of CuCl obtained with the exact CCSD(T) algorithm (dots), the stochastic algorithm using only 1\% of the contributions (error bars), and the Morse potential fitting the points obtained with the stochastic algorithm.}
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\end{figure}
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The initial step involved the precise calculation of the CCSD(T) energy across various points along the potential curve.
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We froze the six lowest molecular orbitals, specifically the $1s$ orbital of \ce{Cl} and the $1s$, $2s$, and $2p$ orbitals of \ce{Cu}, and correlated 34 electrons within 157 molecular orbitals.
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The fitted Morse potential revealed a vibrational frequency of $\nu = \SI{414.7}{\per\centi\meter}$ and an equilibrium bond length of $r_e = \SI{3.92}{\bohr}$, aligning remarkably well with experimental values $\nu = \SI{414}{\per\centi\meter}$ and $r_e = \SI{3.88}{\bohr}$.
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Subsequently, we applied our semi-stochastic algorithm to estimate the perturbative triples correction, utilizing merely 1\% of the total contributions.
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This approach yielded a hundredfold acceleration in computational efficiency, achieving statistical uncertainty within the range of \SI{1.3} to \SI{2.5}{\milli\hartree}.
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The vibrational frequency and equilibrium distance estimated using this data, $\nu = \SI{415.0}{\per\centi\meter}$ and $r_e = \SI{3.92}{\bohr}$, demonstrated comparable precision to the full computational results.
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Figure \ref{fig:cucl} illustrates the potential energy surface of \ce{CuCl}, displaying both the exact CCSD(T) energies and those estimated via the semi-stochastic method.
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%%%
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% CCSD(T) exact
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%a = 2.65592 +/- 0.0403 (1.518%)
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%De = 0.0718253 +/- 0.001879 (2.617%)
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%re = 1.4105 +/- 0.00215 (0.1524%)
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%E0 = -199.358 +/- 0.0003179 (0.0001595%)
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\section{Conclusion}
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\label{sec:conclusion}
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@ -248,20 +255,20 @@ light in cm/s.
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%=================================================================%
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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This work was supported by the European Centre of Excellence in Exascale Computing TREX --- Targeting Real Chemical Accuracy at the Exascale.
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This project has received funding from the European Union's Horizon 2020 — Research and Innovation program --- under grant agreement No.~952165.
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This work was performed using HPC resourced from CALMIP (Toulouse) under allocations p18005 and p22001.}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Data availability statement}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The data that supports the findings of this study are available within the article and its supplementary material.
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{stochastic_triples}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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36
triples.org
36
triples.org
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@ -2,7 +2,7 @@
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* Stochastic formulation
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The perturbative correction reads:
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The perturbative correction reads:2te
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\[
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E_{(T)} = \sum_{ijkabc} E_{ijk}^{abc} = \sum_{ijkabc} \frac{(4 W_{ijk}^{abc} +
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@ -7697,7 +7697,7 @@ re = 3.9
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De = 0.1
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E0 = -2099.767
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set xrange [2.7:5.2]
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fit E(x) data using ($1*a0):2 via a, re, De, E0
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fit E(x) data using ($1*a0):2:3 via a, re, De, E0
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plot E(x), data using ($1*a0):2:3 w err
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#+end_src
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@ -7717,14 +7717,8 @@ E0 = -2099.77 +/- 0.0008014 (3.817e-05%)
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#+RESULTS:
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: 413.5302408975902
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#+CALL:freq(0.895573,0.0854261)
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#+RESULTS:
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: 401.3588602143032
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** CCSD(T) exact
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#+name:cucl_ccsdt_ex
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| 1.50 | -2099.533616067071 |
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| 1.55 | -2099.590506349950 |
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| 1.60 | -2099.635662051331 |
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@ -7750,6 +7744,20 @@ E0 = -2099.77 +/- 0.0008014 (3.817e-05%)
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| 2.60 | -2099.734033236772 |
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| 2.65 | -2099.729597826175 |
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#+name:cucl_ccsdt_ex
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| 1.55 | -2099.590506349950 |
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| 1.65 | -2099.671184187604 |
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| 1.75 | -2099.720045862965 |
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| 1.85 | -2099.747811193906 |
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| 1.95 | -2099.761752030920 |
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| 2.05 | -2099.766727898670 |
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| 2.15 | -2099.765956694308 |
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| 2.25 | -2099.761562105614 |
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| 2.35 | -2099.754944906474 |
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| 2.45 | -2099.747028328725 |
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| 2.55 | -2099.738443175793 |
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| 2.65 | -2099.729597826175 |
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#+begin_src gnuplot :var data=cucl_ccsdt_ex :results file :file cucl_ccsdt_ex.png
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reset
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a0 = 1.8897161646321
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@ -7767,16 +7775,16 @@ plot E(x), data using ($1*a0):2
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[[file:cucl_ccsdt_ex.png]]
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#+begin_example
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a = 0.853035 +/- 0.006204 (0.7273%)
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re = 3.91994 +/- 0.002032 (0.05183%)
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De = 0.100264 +/- 0.001906 (1.901%)
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E0 = -2099.77 +/- 0.0001757 (8.367e-06%)
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a = 0.840928 +/- 0.009924 (1.18%)
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re = 3.92266 +/- 0.003278 (0.08358%)
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De = 0.103446 +/- 0.002926 (2.829%)
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E0 = -2099.77 +/- 0.0002473 (1.178e-05%)
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#+end_example
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#+CALL:freq(0.853035,0.100264)
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#+CALL:freq(0.840928,0.103446)
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#+RESULTS:
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: 414.16742408686565
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: 414.7173810736408
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#+begin_src gnuplot :var data=cucl_ccsdt :var data2=cucl_ccsdt_ex :results file :file cucl_ccsdt2.png
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