From b57c6eedc6a2d27b5e90f92475b925c84051b7e6 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 23 Jul 2021 16:24:13 +0200 Subject: [PATCH] saving work in 2nd part of results --- Ec.nb | 7122 +++++++++++------------- Manuscript/Benzene_EvsNdet.pdf | Bin 48901 -> 48901 bytes Manuscript/Benzene_EvsNdetLO.pdf | Bin 56850 -> 56850 bytes Manuscript/Benzene_EvsPT2.pdf | Bin 38956 -> 39145 bytes Manuscript/Benzene_MPCC.pdf | Bin 39924 -> 39996 bytes Manuscript/Cyclopentadiene_EvsNdet.pdf | Bin 50196 -> 50196 bytes Manuscript/Cyclopentadiene_EvsPT2.pdf | Bin 40977 -> 41200 bytes Manuscript/Cyclopentadiene_MPCC.pdf | Bin 41923 -> 41838 bytes Manuscript/Ec.tex | 111 +- Manuscript/Furan_EvsNdet.pdf | Bin 48234 -> 48234 bytes Manuscript/Furan_EvsPT2.pdf | Bin 38713 -> 38964 bytes Manuscript/Furan_MPCC.pdf | Bin 40281 -> 40259 bytes Manuscript/Imidazole_EvsNdet.pdf | Bin 49241 -> 49241 bytes Manuscript/Imidazole_EvsPT2.pdf | Bin 39969 -> 40160 bytes Manuscript/Imidazole_MPCC.pdf | Bin 40914 -> 40907 bytes Manuscript/Pyrazine_EvsNdet.pdf | Bin 49840 -> 49840 bytes Manuscript/Pyrazine_EvsPT2.pdf | Bin 39764 -> 39975 bytes Manuscript/Pyrazine_MPCC.pdf | Bin 40188 -> 40188 bytes Manuscript/Pyridazine_EvsNdet.pdf | Bin 50215 -> 50215 bytes Manuscript/Pyridazine_EvsPT2.pdf | Bin 40155 -> 40370 bytes Manuscript/Pyridazine_MPCC.pdf | Bin 40672 -> 40672 bytes Manuscript/Pyridine_EvsNdet.pdf | Bin 49628 -> 49628 bytes Manuscript/Pyridine_EvsPT2.pdf | Bin 39838 -> 39994 bytes Manuscript/Pyridine_MPCC.pdf | Bin 40258 -> 40252 bytes Manuscript/Pyrimidine_EvsNdet.pdf | Bin 50259 -> 50259 bytes Manuscript/Pyrimidine_EvsPT2.pdf | Bin 40206 -> 40429 bytes Manuscript/Pyrimidine_MPCC.pdf | Bin 41061 -> 41077 bytes Manuscript/Pyrrole_EvsNdet.pdf | Bin 48434 -> 48434 bytes Manuscript/Pyrrole_EvsPT2.pdf | Bin 38979 -> 39210 bytes Manuscript/Pyrrole_MPCC.pdf | Bin 40456 -> 40438 bytes Manuscript/Tetrazine_EvsNdet.pdf | Bin 49808 -> 49808 bytes Manuscript/Tetrazine_EvsPT2.pdf | Bin 39716 -> 39971 bytes Manuscript/Tetrazine_MPCC.pdf | Bin 40873 -> 40872 bytes Manuscript/Thiophene_EvsNdet.pdf | Bin 48785 -> 48785 bytes Manuscript/Thiophene_EvsPT2.pdf | Bin 39458 -> 39684 bytes Manuscript/Thiophene_MPCC.pdf | Bin 40675 -> 40626 bytes Manuscript/Triazine_EvsNdet.pdf | Bin 49550 -> 49550 bytes Manuscript/Triazine_EvsPT2.pdf | Bin 39541 -> 39751 bytes Manuscript/Triazine_MPCC.pdf | Bin 40050 -> 40038 bytes 39 files changed, 3221 insertions(+), 4012 deletions(-) diff --git a/Ec.nb b/Ec.nb index bbc327d..b47732f 100644 --- a/Ec.nb +++ b/Ec.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 10747259, 199011] -NotebookOptionsPosition[ 10718425, 198521] -NotebookOutlinePosition[ 10718818, 198537] -CellTagsIndexPosition[ 10718775, 198534] +NotebookDataLength[ 10683804, 198181] +NotebookOptionsPosition[ 10653492, 197674] +NotebookOutlinePosition[ 10653915, 197691] +CellTagsIndexPosition[ 10653872, 197688] 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Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue). - The four-point weighted linear fit using the four largest variational wave functions for each set is depicted as a dashed black line. + The five-point weighted linear fit using the four largest variational wave functions for each set is depicted as a dashed black line. The weights are taken as the inverse square of the perturbative corrections. The CCSDTQ correlation energy is also represented as a thick black line. \label{fig:vsEPT2}} @@ -395,7 +395,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}. \begin{squeezetable} \begin{table*} \caption{Total energy $E$ (in \SI{}{\hartree}) and correlation energy $\Delta E$ (in \SI{}{\milli\hartree}) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set. - For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted four-point linear fit is reported in parenthesis. + For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted five-point linear fit is reported in parenthesis. \label{tab:Tab5-VDZ}} \begin{ruledtabular} \begin{tabular}{lcccccccccc} @@ -421,7 +421,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}. CCSD(T) & $-193.5439$ & $-735.6$ & $-229.4073$ & $-764.0$ & $-225.6099$ & $-774.5$ & $-209.5836$ & $-754.9$ & $-552.0458$ & $-724.8$ \\ \hline - FCI & & $-739.3(1)$ & & $-768.1(2)$ & & $-778.3(1)$ & & $-758.4(2)$ & & $-729.1(3)$\\ + FCI & & $-739.2(1)$ & & $-768.2(1)$ & & $-778.2(1)$ & & $-758.5(1)$ & & $-728.9(3)$\\ \end{tabular} \end{ruledtabular} \end{table*} @@ -432,7 +432,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}. \begin{squeezetable} \begin{table*} \caption{Total energy $E$ (in \SI{}{\hartree}) and correlation energy $\Delta E$ (in \SI{}{\milli\hartree}) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set. - For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted four-point linear fit is reported in parenthesis. + For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted five-point linear fit is reported in parenthesis. \label{tab:Tab6-VDZ}} \begin{ruledtabular} \begin{tabular}{lcccccccccccccc} @@ -458,7 +458,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}. \hline CCSD(T) & $-231.5798$ & $-857.5$ & $-263.6024$ & $-899.4$ & $-263.5740$ & $-904.1$ & $-247.5929$ & $-877.7$ & $-263.6099$ & $-896.2$ & $-295.5680$ & $-952.2$ & $-279.6305$ & $-913.1$ \\ \hline - FCI & & $-863.0(4)$ & & $-904.6(6)$ & & $-908.8(2)$ & & $-883.4(0)$ & & $-900.4(4)$ & & $-957.3(2)$ & & $-918.5(5)$\\ + FCI & & $-862.9(3)$ & & $-904.6(4)$ & & $-908.8(1)$ & & $-882.7(4)$ & & $-900.5(2)$ & & $-957.9(4)$ & & $-918.4(3)$\\ \end{tabular} \end{ruledtabular} \end{table*} @@ -482,87 +482,87 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}. \hline Cyclopentadiene & 3 & $-739.295$ & $0.199$ \\ & 4 & $-739.309$ & $0.088$ \\ - & 5 & $-739.230$ & $0.074$ \\ + & \bf5 & $\bf-739.230$ & $\bf0.074$ \\ & 6 & $-739.304$ & $0.072$ \\ & 7 & $-739.292$ & $0.055$ \\ - & 8 & $-739.119$ & $0.131$ \\ +% & 8 & $-739.119$ & $0.131$ \\ \hline Furan & 3 & $-767.790$ & $0.064$ \\ & 4 & $-768.104$ & $0.196$ \\ - & 5 & $-768.194$ & $0.135$ \\ + & \bf5 & $\bf-768.194$ & $\bf0.135$ \\ & 6 & $-768.060$ & $0.131$ \\ & 7 & $-768.086$ & $0.101$ \\ - & 8 & $-767.904$ & $0.154$ \\ +% & 8 & $-767.904$ & $0.154$ \\ \hline Imidazole & 3 & $-778.295$ & $0.356$ \\ & 4 & $-778.270$ & $0.150$ \\ - & 5 & $-778.178$ & $0.105$ \\ + & \bf5 & $\bf-778.178$ & $\bf0.105$ \\ & 6 & $-778.174$ & $0.072$ \\ & 7 & $-778.051$ & $0.099$ \\ - & 8 & $-777.992$ & $0.089$ \\ +% & 8 & $-777.992$ & $0.089$ \\ \hline Pyrrole & 3 & $-758.650$ & $0.321$ \\ & 4 & $-758.389$ & $0.174$ \\ - & 5 & $-758.460$ & $0.110$ \\ + & \bf5 & $\bf-758.460$ & $\bf0.110$ \\ & 6 & $-758.352$ & $0.100$ \\ & 7 & $-758.347$ & $0.075$ \\ - & 8 & $-758.357$ & $0.059$ \\ +% & 8 & $-758.357$ & $0.059$ \\ \hline Thiophene & 3 & $-728.744$ & $0.691$ \\ & 4 & $-729.052$ & $0.331$ \\ - & 5 & $-728.948$ & $0.203$ \\ + & \bf5 & $\bf-728.948$ & $\bf0.203$ \\ & 6 & $-728.987$ & $0.140$ \\ & 7 & $-729.067$ & $0.117$ \\ - & 8 & $-728.876$ & $0.162$ \\ +% & 8 & $-728.876$ & $0.162$ \\ \hline Benzene & 3 & $-862.325$ & $0.279$ \\ & 4 & $-863.024$ & $0.424$ \\ - & 5 & $-862.890$ & $0.266$ \\ + & \bf5 & $\bf-862.890$ & $\bf0.266$ \\ & 6 & $-862.360$ & $0.383$ \\ & 7 & $-862.083$ & $0.339$ \\ - & 8 & $-861.711$ & $0.370$ \\ +% & 8 & $-861.711$ & $0.370$ \\ \hline Pyrazine & 3 & $-904.867$ & $1.420$ \\ & 4 & $-904.588$ & $0.650$ \\ - & 5 & $-904.550$ & $0.385$ \\ + & \bf5 & $\bf-904.550$ & $\bf0.385$ \\ & 6 & $-903.982$ & $0.439$ \\ & 7 & $-903.746$ & $0.359$ \\ - & 8 & $-903.549$ & $0.311$ \\ +% & 8 & $-903.549$ & $0.311$ \\ \hline Pyridazine & 3 & $-909.292$ & $0.024$ \\ & 4 & $-908.808$ & $0.230$ \\ - & 5 & $-908.820$ & $0.133$ \\ + & \bf5 & $\bf-908.820$ & $\bf0.133$ \\ & 6 & $-908.342$ & $0.303$ \\ & 7 & $-908.368$ & $0.224$ \\ - & 8 & $-908.229$ & $0.198$ \\ +% & 8 & $-908.229$ & $0.198$ \\ \hline Pyridine & 3 & $-883.363$ & $0.047$ \\ & 4 & $-883.413$ & $0.029$ \\ - & 5 & $-882.700$ & $0.405$ \\ + & \bf5 & $\bf-882.700$ & $\bf0.405$ \\ & 6 & $-882.361$ & $0.341$ \\ & 7 & $-882.023$ & $0.330$ \\ - & 8 & $-881.732$ & $0.322$ \\ +% & 8 & $-881.732$ & $0.322$ \\ \hline Pyrimidine & 3 & $-900.817$ & $0.726$ \\ & 4 & $-900.383$ & $0.356$ \\ - & 5 & $-900.496$ & $0.214$ \\ + & \bf5 & $\bf-900.496$ & $\bf0.214$ \\ & 6 & $-900.698$ & $0.190$ \\ & 7 & $-900.464$ & $0.206$ \\ - & 8 & $-900.226$ & $0.227$ \\ +% & 8 & $-900.226$ & $0.227$ \\ \hline Tetrazine & 3 & $-957.559$ & $0.246$ \\ & 4 & $-957.299$ & $0.160$ \\ - & 5 & $-957.869$ & $0.349$ \\ + & \bf5 & $\bf-957.869$ & $\bf0.349$ \\ & 6 & $-957.744$ & $0.247$ \\ & 7 & $-957.709$ & $0.183$ \\ - & 8 & $-957.558$ & $0.176$ \\ +% & 8 & $-957.558$ & $0.176$ \\ \hline Triazine & 3 & $-919.596$ & $0.105$ \\ & 4 & $-918.457$ & $0.538$ \\ - & 5 & $-918.355$ & $0.312$ \\ + & \bf5 & $\bf-918.355$ & $\bf0.312$ \\ & 6 & $-918.206$ & $0.226$ \\ & 7 & $-917.876$ & $0.267$ \\ - & 8 & $-917.533$ & $0.308$ \\ +% & 8 & $-917.533$ & $0.308$ \\ \end{tabular} \end{ruledtabular} \end{table} @@ -581,7 +581,7 @@ As compared to natural orbitals (solid red lines), one can see that, for a given Adding the perturbative correction $\EPT$ yields similar curves for both sets of orbitals (dashed lines). This indicates that, for a given number of determinants, $\EPT$ (which, we recall, provides a qualitative idea to the distance to the FCI limit) is much smaller for optimized orbitals than for natural orbitals. This is further evidenced in Fig.~\ref{fig:vsEPT2} where we show the behavior of $\Delta \Evar$ as a function of $\EPT$ for both sets of orbitals. -The four-point weighted linear fit using the four largest variational wave functions are also represented (dashed black lines). +The five-point weighted linear fit using the four largest variational wave functions are also represented (dashed black lines). The CCSDTQ correlation energy (solid black line) is also reported for comparison purposes in Figs.~\ref{fig:vsNdet} and \ref{fig:vsEPT2} From Fig.~\ref{fig:vsEPT2}, it is clear that the behavior of $\Delta \Evar$ is much more linear and produces smaller $\EPT$ values, hence facilitating the extrapolation procedure to the FCI limit (see below). @@ -602,7 +602,7 @@ A similar improvement is observed going from natural to localized orbitals. \titou{Comment on PT2 for localized orbitals.} Accordingly to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals. -To do so, we have then extrapolated the orbital-optimized variational CIPSI correlation energies to $\EPT = 0$ via a weighted four-point linear fit using the four largest variational wave functions (see Fig.~\ref{fig:vsEPT2}). +To do so, we have then extrapolated the orbital-optimized variational CIPSI correlation energies to $\EPT = 0$ via a weighted five-point linear fit using the four largest variational wave functions (see Fig.~\ref{fig:vsEPT2}). The fitting weights have been taken as the inverse square of the perturbative correction at each point. Our final FCI correlation energy estimates are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively, alongside their corresponding fitting error. The stability of these estimates are illustrated by the results gathered in Table \ref{tab:fit} where we report, for each system, the extrapolated correlation energies $\Delta \Eextrap$ and their associated fitting errors obtained via weighted linear fits varying the number of fitting points from 3 to 8. @@ -639,21 +639,60 @@ Note that it is pleasing to see that, although different geometries are consider \end{figure*} %%% %%% %%% +%%% TABLE III %%% +\begin{squeezetable} +\begin{table} + \caption{ + Mean absolute error (MAE), mean signed error (MSE), and minimum (Min) and maximum (Max) absolute errors (in \SI{}{\milli\hartree}) with respect to the FCI correlation energy for various methods. + The formal computational scaling of each method is also reported. + \label{tab:stats}} + \begin{ruledtabular} + \begin{tabular}{lcdddd} + Method & Scaling & \tabc{MAE} & \tabc{MSE} & \tabc{Min} & \tabc{Max} \\ + \hline + MP2 & $\order{N^5}$ & 68.4 & 68.4 & 80.6 & 57.8 \\ + MP3 & $\order{N^6}$ & 46.5 & 46.5 & 58.4 & 37.9 \\ + MP4 & $\order{N^7}$ & 2.1 & 2.0 & 4.7 & 0.7 \\ + MP5 & $\order{N^8}$ & 9.4 & 9.4 & 13.6 & 5.8 \\ + \hline + CC2 & $\order{N^5}$ & 58.9 & 58.9 & 73.5 & 48.9 \\ + CC3 & $\order{N^7}$ & 2.7 & 2.7 & 3.8 & 2.1 \\ + CC4 & $\order{N^9}$ & 1.5 & 1.5 & 2.3 & 0.8 \\ + \hline + CCSD & $\order{N^6}$ & 39.4 & 39.4 & 48.8 & 32.0 \\ + CCSDT & $\order{N^8}$ & 4.5 & 4.5 & 6.3 & 3.0 \\ + CCSDTQ & $\order{N^{10}}$& 1.8 & 1.8 & 2.6 & 1.0 \\ + \hline + CCSD(T) & $\order{N^7}$ & 4.5 & 4.5 & 5.7 & 3.6 \\ + \end{tabular} + \end{ruledtabular} +\end{table} +\end{squeezetable} +%%% %%% %%% + Using the CIPSI estimates of the FCI correlation energy produced in Sec.~\ref{sec:cipsi_res}, we now study the performance and convergence properties of three series of methods: i) MP2, MP3, MP4, and MP5, ii) CC2, CC3, and CC4, and iii) CCSD, CCSDT, and CCSDTQ. Additionally, we also report CCSD(T) correlation energies. All these data are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively. In Fig.~\ref{fig:MPCC}, we show, for each molecule, the convergence of the correlation energy for each series of methods as a function of the computational cost of the corresponding method. The FCI correlation energy estimate is represented as a black line for reference. +Key statistical quantities [mean absolute error (MAE), mean signed error (MSE), and minimum (Min) and maximum (Max) absolute errors with respect to the FCI reference values] are also reported in Table \ref{tab:stats} for each method as well as their formal computational scaling. -First, let us investigate the series of MP approximations which is known, as mentioned in Sec.~\ref{sec:intro}, to potentially exhibit ``surprising'' behavior depending on the type of correlation at play.\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003} +First, we investigate the ``complete'' and well-established series of methods CCSD, CCSDT, and CCSDTQ. +Unfortunately, CC with singles, doubles, triples, quadruples and pentuples (CCSDTQP) calculations are out of reach here. \cite{Hirata_2000,Kallay_2001} +As expected, going from CCSD to CCSDTQ, one improves systematically and quickly the correlation energies with MAEs of $39.4$, $4.5$, \SI{1.8}{\milli\hartree} for CCSD, CCSDT, and CCSDTQ, respectively. +As usually observed, CCSD(T) (MAE of \SI{4.5}{\milli\hartree}) provides similar correlation energies than the more expensive CCSDT method by computing perturbatively the triple excitations. + +Second, let us look into the series of MP approximations which is known, as mentioned in Sec.~\ref{sec:intro}, to potentially exhibit ``surprising'' behavior depending on the type of correlation at play.\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003} (See Ref.~\onlinecite{Marie_2021} for a detailed discussion). For each system, the MP series decreases monotonically up to MP4 but raises quite significantly when one takes into account the fifth-order correction. -We note that the MP4 correlation energy is always quite accurate and is only a few millihartree higher than the FCI value (except in the case of tetrazine where the MP4 number is very slightly below the reference value): MP5 is thus systematically worse than MP4 for these systems. -Importantly here, one notices that MP4 (which scales as $\order*{N^7}$) is systematically on par with the more expensive $\order*{N^10}$ CCSDTQ method. +We note that the MP4 correlation energy is always quite accurate (MAE of \SI{2.1}{\milli\hartree}) and is only a few millihartree higher than the FCI value (except in the sole case of tetrazine where the MP4 number is very slightly below the reference value): MP5 (MAE of \SI{9.4}{\milli\hartree}) is thus systematically worse than MP4 for these systems. +Importantly here, one notices that MP4 (which scales as $\order*{N^7}$) is systematically on par with the more expensive $\order*{N^10}$ CCSDTQ method which exhibits a slightly smaller MAE of \SI{1.8}{\milli\hartree}. Second, we investigate the approximate CC series of methods CC2, CC3, and CC4. -As observed in our recent study on excitation energies, \cite{Loos_2021} CC4 is an outstanding approximation to its CCSDTQ parent. -Moreover, we observe here that CC3 and CC4 provide correlation energies only one or two millihartree different, which shows that CC3 is particularly effective for ground-state energetics. +As observed in our recent study on excitation energies, \cite{Loos_2021} CC4, which returns a MAE of \SI{1.5}{\milli\hartree}, is an outstanding approximation to its CCSDTQ parent and is, in the present case, even slightly more accurate. +Moreover, we observe that CC3 (MAE of \SI{2.7}{\milli\hartree}) and CC4 provide correlation energies that only deviate by one or two millihartree, which shows that CC3 is particularly effective for ground-state energetics and outperforms both CCSD(T) and CCSDT. + +As a final remark, we would like to mention that even if the present perturbative and coupled-cluster methods are known to non-variational, they very rarely produce a lower energy than the FCI estimate as illustrated by the systematic equality between MAEs and MSEs. %%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} diff --git a/Manuscript/Furan_EvsNdet.pdf b/Manuscript/Furan_EvsNdet.pdf index a1e26a15afebd27efe9ea579370b921217ad2f31..bcf23e00a435c146d683c512473e6a8b6ea05d4b 100644 GIT binary patch delta 34 kcmaF$gXz@|rVUj)*^LcN4UCPACO7UBhp{)$-+7t|0R9RMH~;_u delta 34 kcmaF$gXz@|rVUj)*$oXWP0b8UCO7UBhp{)$-+7t|0RHR^NB{r; diff --git a/Manuscript/Furan_EvsPT2.pdf b/Manuscript/Furan_EvsPT2.pdf index cc20e3b23c1ad8e7f372dc968732bbafd9a11417..68ec9c1697255b477f651ad2f32f2c2255e127d0 100644 GIT binary patch delta 24355 zcmV)lK%c+4t^%~M0~vGxde zv&~W?uFm!hWso-ef6({?J;(EQ8RiK1JHsrkDi)c5Yhz`t92Gv^k2f>>ibaXJTA8aT zzs(yC^F>ZZ{b!M@XuWm}I8KNcT5G?>kfuYW){{2x*0j3^!p#+Sf5Z|mYsK|pE|&HD zn!DWDtb$Ko{bue*eBkMTSeX}r*sJvm?9+Q0u^C&R8%yg2cSL~(urRN=`I3DWW!7J# 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