diff --git a/Data/benzene.dat b/Data/benzene.dat index 54bcb92..fbb376b 100644 --- a/Data/benzene.dat +++ b/Data/benzene.dat @@ -1,2940 +1,2117 @@ # TZ # E(CCSD(T)) Error % # ---------------+------------+------- - -231.80582073 0.1702E-02 0.43 - -231.80568602 0.1182E-02 0.86 - -231.80638784 0.9730E-03 1.28 - -231.80583046 0.8243E-03 1.70 - -231.80564026 0.7246E-03 2.13 - -231.80549184 0.6546E-03 2.55 - -231.80528021 0.5917E-03 2.97 - -231.80533305 0.5513E-03 3.39 - -231.80552546 0.5262E-03 3.81 - -231.80531583 0.4945E-03 4.23 - -231.80535472 0.4705E-03 4.66 - -231.80557527 0.4565E-03 5.08 - -231.80557303 0.4355E-03 5.50 - -231.80548029 0.4202E-03 5.92 - -231.80556655 0.4050E-03 6.34 - -231.80563017 0.3936E-03 6.77 - -231.80566114 0.3827E-03 7.19 - -231.80558699 0.3701E-03 7.61 - -231.80567134 0.3635E-03 8.03 - -231.80560792 0.3524E-03 8.45 - -231.80599656 0.2819E-03 8.87 - -231.80600921 0.2748E-03 9.29 - -231.80602737 0.2682E-03 9.71 - -231.80602047 0.2619E-03 10.13 - -231.80600268 0.2562E-03 10.55 - -231.80605143 0.2532E-03 10.97 - -231.80603473 0.2490E-03 11.39 - -231.80606029 0.2452E-03 11.81 - -231.80608868 0.2406E-03 12.23 - -231.80608311 0.2367E-03 12.65 - -231.80603067 0.2316E-03 13.07 - -231.80602845 0.2276E-03 13.49 - -231.80598091 0.2235E-03 13.91 - -231.80592227 0.2203E-03 14.33 - -231.80586276 0.2164E-03 14.75 - -231.80585569 0.2125E-03 15.17 - -231.80588786 0.2088E-03 15.59 - -231.80580990 0.1836E-03 16.01 - -231.80575077 0.1801E-03 16.43 - -231.80575959 0.1777E-03 16.85 - -231.80577139 0.1757E-03 17.27 - -231.80573497 0.1730E-03 17.69 - -231.80571827 0.1703E-03 18.11 - -231.80570941 0.1679E-03 18.53 - -231.80568515 0.1654E-03 18.95 - -231.80565681 0.1628E-03 19.37 - -231.80563140 0.1606E-03 19.79 - -231.80564907 0.1587E-03 20.21 - -231.80561730 0.1562E-03 20.63 - -231.80558851 0.1539E-03 21.05 - -231.80559825 0.1525E-03 21.48 - -231.80563699 0.1512E-03 21.90 - -231.80562532 0.1326E-03 22.32 - -231.80558771 0.1310E-03 22.74 - -231.80558484 0.1299E-03 23.16 - -231.80559053 0.1284E-03 23.58 - -231.80560391 0.1271E-03 24.00 - -231.80561828 0.1261E-03 24.42 - -231.80562601 0.1249E-03 24.84 - -231.80562503 0.1235E-03 25.26 - -231.80561025 0.1221E-03 25.68 - -231.80561714 0.1209E-03 26.10 - -231.80561468 0.1197E-03 26.52 - -231.80562579 0.1185E-03 26.94 - -231.80561713 0.1103E-03 27.36 - -231.80560273 0.1091E-03 27.78 - -231.80563253 0.1084E-03 28.20 - -231.80564377 0.1075E-03 28.62 - -231.80564167 0.1064E-03 29.04 - -231.80563093 0.1054E-03 29.46 - -231.80563660 0.1049E-03 29.88 - -231.80563212 0.1040E-03 30.30 - -231.80563092 0.1030E-03 30.72 - -231.80563308 0.1021E-03 31.14 - -231.80563106 0.1011E-03 31.56 - -231.80567721 0.9500E-04 31.98 - -231.80566323 0.9423E-04 32.40 - -231.80565084 0.9338E-04 32.82 - -231.80564784 0.9250E-04 33.24 - -231.80565170 0.9174E-04 33.66 - -231.80566332 0.9103E-04 34.08 - -231.80566231 0.9026E-04 34.50 - -231.80567450 0.8967E-04 34.92 - -231.80567643 0.8886E-04 35.34 - -231.80567236 0.8800E-04 35.76 - 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-231.80573038 0.8139E-06 98.95 - -231.80573031 0.4206E-06 99.38 - -231.80572937 0.2185E-06 99.82 - +# ======================= ============== ========== + -231.80600113 1.1096E-03 0.96 + -231.80526641 7.5354E-04 1.96 + -231.80485336 5.8772E-04 2.95 + -231.80482981 5.0487E-04 3.94 + -231.80502318 4.5959E-04 4.94 + -231.80514239 4.1918E-04 5.93 + -231.80537695 3.9050E-04 6.92 + -231.80527265 3.6123E-04 7.91 + -231.80571625 2.7896E-04 8.89 + -231.80570511 2.6248E-04 9.89 + -231.80579627 2.5255E-04 10.88 + -231.80581490 2.4260E-04 11.88 + -231.80581200 2.3196E-04 12.87 + -231.80574192 2.2274E-04 13.86 + -231.80562502 2.1393E-04 14.86 + -231.80558975 2.0485E-04 15.84 + -231.80553042 1.7558E-04 16.83 + -231.80553942 1.7033E-04 17.82 + -231.80549697 1.6456E-04 18.81 + -231.80547086 1.5909E-04 19.81 + -231.80546770 1.5417E-04 20.80 + -231.80552013 1.5061E-04 21.80 + -231.80547549 1.2978E-04 22.79 + -231.80549774 1.2683E-04 23.78 + -231.80552164 1.2399E-04 24.78 + -231.80550146 1.2080E-04 25.77 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-231.87174561 2.3437E-07 99.14 + -231.87174564 2.3355E-07 99.19 + -231.87174567 2.3228E-07 99.24 + -231.87174564 2.3127E-07 99.29 + -231.87174564 2.3210E-07 99.34 + -231.87174564 2.3134E-07 99.39 + -231.87174565 2.2173E-07 99.44 + -231.87174564 2.2067E-07 99.50 + -231.87174560 2.1954E-07 99.55 + -231.87174561 2.1852E-07 99.60 + -231.87174575 1.8075E-07 99.66 + -231.87174575 1.7996E-07 99.71 + -231.87174575 1.7907E-07 99.77 + -231.87174574 1.7804E-07 99.83 + -231.87174572 1.7695E-07 99.88 + -231.87174571 1.7592E-07 99.94 + -231.87174571 1.7496E-07 100.00 + -231.87174565 0. 100.00 diff --git a/Data/benzene_qz.plt b/Data/benzene_qz.plt index 36a3bc2..594ae6f 100644 --- a/Data/benzene_qz.plt +++ b/Data/benzene_qz.plt @@ -14,13 +14,13 @@ set style fill transparent solid 0.50 border #set yrange [-231.8075:-231.8040] data='benzene.dat' -tmax=2812.08030200005 * 0.01 +tmax=2070.63480687141 * 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 + -231.87174565 notitle ls 3 diff --git a/Data/benzene_tz.plt b/Data/benzene_tz.plt index dd4b1fd..0b760d9 100644 --- a/Data/benzene_tz.plt +++ b/Data/benzene_tz.plt @@ -7,19 +7,20 @@ 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] +set yrange [-231.8068:-231.8042] data='benzene.dat' -tmax=239.890999078751 * 0.01 +tmax=102.592093944550 * 0.01 + +set xrange [0:tmax*100] 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 + -231.805731383683 notitle ls 3 diff --git a/Data/cucl.dat b/Data/cucl.dat index ceda691..32aa885 100644 --- a/Data/cucl.dat +++ b/Data/cucl.dat @@ -1,15 +1,15 @@ # R CCSD(T) Stochastic CCSD(T) #--------------------------------------------------------- - 1.55 -2099.590506349950 -2099.58890791 1.3630E-03 - 1.65 -2099.671184187604 -2099.67175286 1.5710E-03 - 1.75 -2099.720045862965 -2099.71877319 1.3199E-03 - 1.85 -2099.747811193906 -2099.74897746 1.4668E-03 - 1.95 -2099.761752030920 -2099.76232971 1.6105E-03 - 2.05 -2099.766727898670 -2099.76565267 1.5202E-03 - 2.15 -2099.765956694308 -2099.76485609 1.7470E-03 - 2.25 -2099.761562105614 -2099.76237391 1.7474E-03 - 2.35 -2099.754944906474 -2099.75681975 1.9951E-03 - 2.45 -2099.747028328725 -2099.74813718 2.4288E-03 - 2.55 -2099.738443175793 -2099.74031232 2.4057E-03 - 2.65 -2099.729597826175 -2099.72866832 1.6894E-03 + 1.55 -2099.590506349950 -2099.58900041 1.2867E-03 + 1.65 -2099.671184187604 -2099.67111214 1.4130E-03 + 1.75 -2099.720045862965 -2099.72247666 1.8605E-03 + 1.85 -2099.747811193906 -2099.74783889 1.7384E-03 + 1.95 -2099.761752030920 -2099.76294620 1.5644E-03 + 2.05 -2099.766727898670 -2099.76607775 1.6396E-03 + 2.15 -2099.765956694308 -2099.76889281 2.0352E-03 + 2.25 -2099.761562105614 -2099.76003424 1.4932E-03 + 2.35 -2099.754944906474 -2099.75495657 1.8955E-03 + 2.45 -2099.747028328725 -2099.74878665 1.8202E-03 + 2.55 -2099.738451107727 -2099.73743548 1.6473E-03 + 2.65 -2099.729597826175 -2099.73119406 1.7203E-03 diff --git a/Data/cucl.plt b/Data/cucl.plt index 26055d3..7ba0a28 100644 --- a/Data/cucl.plt +++ b/Data/cucl.plt @@ -22,9 +22,9 @@ set term pdfcairo enhanced font "Times,14" linewidth 2 rounded size 5.0in, 3.0in set output 'cucl.pdf' set pointsize 0.5 plot \ -'cucl.dat' using ($1*a0):2 pointtype 7 lt 4 title "Full (T)", \ -E(x) title "" lt 3, \ -'cucl.dat' using ($1*a0):3:4 w err pt 0 lt 1 title "1% (T)" +'cucl.dat' using ($1*a0):2 pointtype 3 lc rgb "red" title "Full (T)", \ +E(x) title "" lt 3 lc rgb "grey", \ +'cucl.dat' using ($1*a0):3:4 w err pt 0 lc rgb "blue" title "1% (T)" diff --git a/Manuscript/benzene_err.pdf b/Manuscript/benzene_err.pdf index 0570f6b..8d3d5f3 100644 Binary files a/Manuscript/benzene_err.pdf and b/Manuscript/benzene_err.pdf differ diff --git a/Manuscript/benzene_qz.pdf b/Manuscript/benzene_qz.pdf index a9b1032..29d1ddd 100644 Binary files a/Manuscript/benzene_qz.pdf and b/Manuscript/benzene_qz.pdf differ diff --git a/Manuscript/benzene_tz.pdf b/Manuscript/benzene_tz.pdf index 4763c08..243d85c 100644 Binary files a/Manuscript/benzene_tz.pdf and b/Manuscript/benzene_tz.pdf differ diff --git a/Manuscript/cucl.pdf b/Manuscript/cucl.pdf index 66edd9e..ae55720 100644 Binary files a/Manuscript/cucl.pdf and b/Manuscript/cucl.pdf differ diff --git a/Manuscript/stochastic_triples.tex b/Manuscript/stochastic_triples.tex index fea1509..be38915 100644 --- a/Manuscript/stochastic_triples.tex +++ b/Manuscript/stochastic_triples.tex @@ -221,9 +221,9 @@ Consequently, employing a sufficient number of Monte Carlo samples to ensure tha To reduce the variance, the samples are drawn using the probability \begin{equation} -P^{abc} = \frac{1}{\mathcal{N}} \frac{1}{|\epsilon_a + \epsilon_b + \epsilon_c|} +P^{abc} = \frac{1}{\mathcal{N}} \frac{1}{\max \left(\epsilon_{\min}, \epsilon_a + \epsilon_b + \epsilon_c \right)} \end{equation} -where $\mathcal{N}$ normalizes the sum such that $\sum_{abc} P^{abc} = 1$. +where $\mathcal{N}$ normalizes the sum such that $\sum_{abc} P^{abc} = 1$, and $\epsilon_{\min}$ is an arbitrary minimal denominator to ensure that $P^{abc}$ does not diverge. In our calculations, we have set $\epsilon_{\min}$ to 0.2~a.u. The perturbative contribution is then evaluated as an average over $M$ samples \begin{equation} E_{(T)} = \left\langle \frac{E^{abc}}{P^{abc}} \right \rangle_{P^{abc}} = @@ -370,7 +370,7 @@ $t_0 \leftarrow \text{WallClockTime}()$ \; 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 using either 258 or 503 molecular orbitals. -The calculations were performed on an Intel Xeon Gold 6130 dual socket server (32 cores in total). +The calculations were performed on an AMD \textsc{Epyc} 7513 dual socket server (64 cores in total). \begin{figure} \includegraphics[width=\columnwidth]{benzene_tz.pdf} @@ -413,7 +413,7 @@ with $\mu$ denoting the reduced mass of the \ce{CuCl} molecule, and $c$ the spee \begin{figure} \includegraphics[width=\columnwidth]{cucl.pdf} -\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.} +\caption{\label{fig:cucl} CCSD(T) energies of CuCl obtained with the exact CCSD(T) algorithm (stars), the stochastic algorithm using only 1\% of the contributions (error bars), and the Morse potential fitting the points obtained with the stochastic algorithm.} \end{figure} The initial step involved the precise calculation of the CCSD(T) energy across various points along the potential curve. @@ -421,8 +421,8 @@ We froze the six lowest molecular orbitals, specifically the $1s$ orbital of \ce 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 from the NIST database\cite{nist_2022} $\nu = \SI{417.6}{\per\centi\meter}$ and $r_e = \SI{3.88}{\bohr}$. Subsequently, we applied our semi-stochastic algorithm to estimate the perturbative triples correction, utilizing merely 1\% of the total contributions. -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}. -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. +This approach yielded a hundredfold acceleration in computational efficiency, achieving statistical uncertainty within the range of \SI{1.2} to \SI{2.0}{\milli\hartree}. +The vibrational frequency and equilibrium distance estimated using this data, $\nu = \SI{415.1}{\per\centi\meter}$ and $r_e = \SI{3.91}{\bohr}$, demonstrated comparable precision to the full computational results. 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. diff --git a/triples.org b/triples.org index e2130de..b8b1a77 100644 --- a/triples.org +++ b/triples.org @@ -7635,16 +7635,16 @@ plot E(x), data using ($1*a0):2 w p [[file:cucl_ccsd.png]] #+begin_example -a = 0.850836 +/- 0.005939 (0.698%) -re = 3.94582 +/- 0.001786 (0.04526%) -De = 0.0961907 +/- 0.00189 (1.965%) -E0 = -2099.73 +/- 0.0001574 (7.496e-06%) +a = 0.879992 +/- 0.02833 (3.219%) +re = 3.91095 +/- 0.009327 (0.2385%) +De = 0.0946485 +/- 0.007497 (7.92%) +E0 = -2099.77 +/- 0.0007291 (3.472e-05%) #+end_example - #+CALL:freq(0.8637,0.0912735) + #+CALL:freq(0.879992,0.0946485) #+RESULTS: - : 400.10303409950683 + : 415.11857299491743 ** CCSD(T) 1%