From 03b35a14c8ee77a7d55878d1029e246e35044f1e Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 19 Aug 2020 09:42:13 +0200 Subject: [PATCH] saving work in Sec V and appendix A --- Manuscript/rsdft-cipsi-qmc.bib | 12 ++- Manuscript/rsdft-cipsi-qmc.tex | 182 +++++++++++++++++---------------- 2 files changed, 105 insertions(+), 89 deletions(-) diff --git a/Manuscript/rsdft-cipsi-qmc.bib b/Manuscript/rsdft-cipsi-qmc.bib index 590a14d..75faac8 100644 --- a/Manuscript/rsdft-cipsi-qmc.bib +++ b/Manuscript/rsdft-cipsi-qmc.bib @@ -1,13 +1,23 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-08-18 15:08:15 +0200 +%% Created for Pierre-Francois Loos at 2020-08-19 09:29:01 +0200 %% Saved with string encoding Unicode (UTF-8) +@article{Hattig_2012, + Author = {C. Hattig and W. Klopper and A. Kohn and D. P. Tew}, + Date-Added = {2020-08-19 09:28:48 +0200}, + Date-Modified = {2020-08-19 09:28:59 +0200}, + Journal = {Chem. Rev.}, + Pages = {4}, + Title = {Explicitly Correlated Electrons in Molecules}, + Volume = {112}, + Year = {2012}} + @article{Kutzelnigg_1985, Author = {W. Kutzelnigg}, Date-Added = {2020-08-18 15:05:16 +0200}, diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 96c562f..438465d 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -458,6 +458,8 @@ The FN-DMC simulations are performed with all-electron moves using the stochastic reconfiguration algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000} with a time step of $5 \times 10^{-4}$ a.u. +All the data related to the present study (geometries, basis sets, total energies, etc) can be found in the {\SI}. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Influence of the range-separation parameter on the fixed-node error} \label{sec:mu-dmc} @@ -723,7 +725,7 @@ they both deal with an effective non-divergent interaction but still produce a reasonable one-body density. %============================ -\subsection{Intermediate conclusion} +\subsection{Intermediate conclusions} %============================ As a conclusion of the first part of this study, we can highlight the following observations: @@ -749,25 +751,25 @@ As a conclusion of the first part of this study, we can highlight the following Atomization energies are challenging for post-HF methods because their calculation requires a subtle balance in the -description of atoms and molecules. The mainstream one-electron basis sets employed in molecular +description of atoms and molecules. The mainstream one-electron basis sets employed in molecular electronic structure calculations are atom-centered, so they are, by construction, better adapted to -atoms than molecules and atomization energies usually tend to be +atoms than molecules. Thus, atomization energies usually tend to be underestimated by variational methods. In the context of FN-DMC calculations, the nodal surface is imposed by -the trial wavefunction which is expanded in the very same atom-centered basis -set. Thus, we expect the fixed-node error to be also intimately related to +the determinantal part of the trial wavefunction which is expanded in the very same atom-centered basis +set. Thus, we expect the fixed-node error to be also intimately connected to the basis set incompleteness error. Increasing the size of the basis set improves the description of the density and of the electron correlation, but also reduces the imbalance in the description of atoms and -molecule, leading to more accurate atomization energies. -The size-consistency and the spin-invariance of the present scheme are discussed in Appendices \ref{app:size} and \ref{app:spin}. +molecules, leading to more accurate atomization energies. +The size-consistency and the spin-invariance of the present scheme, two key properties to obtain accurate atomization energies, are discussed in Appendices \ref{app:size} and \ref{app:spin}. %%% FIG 6 %%% \begin{squeezetable} \begin{table*} - \caption{Mean absolute errors (MAE), mean signed errors (MSE), and - root mean square errors (RMSE) obtained with various methods and + \caption{Mean absolute errors (MAEs), mean signed errors (MSEs), and + root mean square errors (RMSEs) \titou{with respect to ??? (in kcal/mol)} obtained with various methods and basis sets.} \label{tab:mad} \begin{ruledtabular} @@ -815,9 +817,9 @@ NOs from a preliminary CIPSI calculation as a starting point (see Fig.~\ref{fig: For comparison, we have computed the energies of all the atoms and molecules at the KS-DFT level with various semi-local and hybrid density functionals [PBE, BLYP, PBE0, and B3LYP], and at the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean -absolute errors (MAE), mean signed errors (MSE), and root mean square errors (RMSE). +absolute errors (MAEs), mean signed errors (MSEs), and root mean square errors (RMSEs). For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have -provided the extrapolated values at $\EPT \to 0$, and the error bars +provided the extrapolated values (\ie, when $\EPT \to 0$), and the error bars correspond to the difference between the extrapolated energies computed with a two-point and a three-point linear extrapolation. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c} @@ -830,20 +832,20 @@ accuracy. Thanks to the single-reference character of these systems, the CCSD(T) energy is an excellent estimate of the FCI energy, as shown by the very good agreement of the MAE, MSE and RMSE of CCSDT(T) and FCI energies. -The imbalance of the quality of description of molecules compared +The imbalance in the description of molecules compared to atoms is exhibited by a very negative value of the MSE for -CCSD(T) and FCI/VDZ-BFD, which is reduced by a factor of two +CCSD(T)/VDZ-BFD and FCI/VDZ-BFD, which is reduced by a factor of two when going to the triple-$\zeta$ basis, and again by a factor of two when going to the quadruple-$\zeta$ basis. -This large imbalance at the VDZ-BFD level affects the nodal +This significant imbalance at the VDZ-BFD level affects the nodal surfaces, because although the FN-DMC energies obtained with near-FCI trial wave functions are much lower than the single-determinant FN-DMC energies, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is -larger than the single-determinant MAE ($4.61\pm 0.34$ kcal/mol). +larger than the \titou{single-determinant} MAE ($4.61\pm 0.34$ kcal/mol). Using the FCI trial wave function the MSE is equal to the -negative MAE which confirms that all the atomization energies are -underestimated. This confirms that some of the basis-set +negative MAE which confirms that the atomization energies are systematically +underestimated. This confirms that some of the basis set incompleteness error is transferred in the fixed-node error. Within the double-$\zeta$ basis set, the calculations could be performed for the @@ -851,45 +853,46 @@ whole range of values of $\mu$, and the optimal value of $\mu$ for the trial wave function was estimated for each system by searching for the minimum of the spline interpolation curve of the FN-DMC energy as a function of $\mu$. -This corresponds the line of Table~\ref{tab:mad} labelled as ``Opt.'' +This corresponds to the line labelled as ``Opt.'' in Table~\ref{tab:mad}. The optimal $\mu$ value for each system is reported in the \SI. Using the optimal value of $\mu$ clearly improves the -MAE, the MSE an the RMSE compared to the FCI wave function. This +MAE, the MSE an the RMSE as compared to the FCI wave function. This result is in line with the common knowledge that re-optimizing the determinantal component of the trial wave function in the presence of electron correlation reduces the errors due to the basis set incompleteness. These calculations were done only for the smallest basis set because of the expensive computational cost of the QMC calculations -when the trial wave function is expanded on more than a few million +when the trial wave function contains more than a few million determinants. -At the RS-DFT-CIPSI level, one can see that with the VTZ-BFD -basis the MAEs are larger for $\mu=1$~bohr$^{-1}$ than for the -FCI. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet} -there are many systems which did not reach the threshold -$\EPT<1$~m\hartree{}, and the number of determinants exceeded -10~million so the calculation stopped. In this regime, there is a +At the RS-DFT-CIPSI/VTZ-BFD level, one can see that +the MAEs are larger for $\mu=1$~bohr$^{-1}$ ($9.06$ kcal/mol) than for +FCI [$8.43(39)$ kcal/mol]. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet}, +there are many systems for which we could not reach the threshold +$\EPT<1$~m\hartree{} as the number of determinants exceeded +10~million before this threshold was reached. For these cases, there is then a small size-consistency error originating from the imbalanced truncation of the wave functions, which is not present in the -extrapolated FCI energies. The same comment applies to -$\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis set. - +extrapolated FCI energies (see Appendix \ref{app:size}). The same comment applies to +$\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis. +\titou{T2: Fig. 6 is mentioned before Fig. 5.} %%% FIG 5 %%% \begin{figure*} \centering \includegraphics[width=\textwidth]{g2-dmc.pdf} - \caption{Errors in the FN-DMC atomization energies with various + \caption{Errors \titou{(with respect to ???)} in the FN-DMC atomization energies (in kcal/mol) with various trial wave functions. Each dot corresponds to an atomization energy. The boxes contain the data between first and third quartiles, and the line in the box represents the median. The outliers are shown - with a cross.} + with a cross. + \titou{T2: change basis set labels.}} \label{fig:g2-dmc} \end{figure*} %%% %%% %%% %%% Searching for the optimal value of $\mu$ may be too costly and time consuming, so we have -computed the MAD, MSE and RMSE for fixed values of $\mu$. +computed the MAEs, MSEs and RMSEs for fixed values of $\mu$. As illustrated in Fig.~\ref{fig:g2-dmc} and Table \ref{tab:mad}, the best choice for a fixed value of $\mu$ is 0.5~bohr$^{-1}$ for all three basis sets. It is the value for which @@ -899,6 +902,7 @@ are even lower than those obtained with the optimal value of $\mu$. Although the FN-DMC energies are higher, the numbers show that they are more consistent from one system to another, giving improved cancellations of errors. +\titou{This is another key result of the present study and it can be explained by the lack of size-consistentcy when one uses different $\mu$ values for different systems like in optimally-tune range-separated hybrids. \cite{}} %%% FIG 6 %%% \begin{figure*} @@ -908,7 +912,8 @@ cancellations of errors. functions. Each dot corresponds to an atomization energy. The boxes contain the data between first and third quartiles, and the line in the box represents the median. The outliers are shown - with a cross.} + with a cross. + \titou{T2: change basis set labels.}} \label{fig:g2-ndet} \end{figure*} %%% %%% %%% %%% @@ -916,12 +921,12 @@ cancellations of errors. The number of determinants in the trial wave functions are shown in Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants is smaller when $\mu$ is small and larger when $\mu$ is large. -It is important to remark that the median of the number of +It is important to note that the median of the number of determinants when $\mu=0.5$~bohr$^{-1}$ is below $100\,000$ determinants with the VQZ-BFD basis, making these calculations feasible with such a large basis set. At the double-$\zeta$ level, compared to the -FCI trial wave functions the median of the number of determinants is -reduced by more than two orders of magnitude. +FCI trial wave functions, the median of the number of determinants is +reduced by more than \titou{two orders of magnitude}. Moreover, going to $\mu=0.25$~bohr$^{-1}$ gives a median close to 100 determinants at the VDZ-BFD level, and close to $1\,000$ determinants at the quadruple-$\zeta$ level for only a slight increase of the @@ -997,28 +1002,6 @@ The data that support the findings of this study are available within the articl \label{app:size} %============================ -%%% TABLE III %%% -\begin{table} - \caption{FN-DMC energy (in hartree) using the VDZ-BFD basis set and the srPBE functional - of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values. - The size-consistency error is also reported.} - \label{tab:size-cons} - \begin{ruledtabular} - \begin{tabular}{cccc} - $\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\ - \hline - 0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\ - 0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\ - 0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\ - 1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\ - 2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\ - 5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\ - $\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\ - \end{tabular} - \end{ruledtabular} -\end{table} -%%% %%% %%% %%% - An extremely important feature required to get accurate atomization energies is size-consistency (or strict separability), since the numbers of correlated electron pairs in the molecule and its isolated atoms @@ -1030,13 +1013,13 @@ is relatively fast}. \cite{FraMusLupTou-JCP-15} Hence, DFT methods are very well adapted to the calculation of atomization energies, especially with small basis sets. \cite{Giner_2018,Loos_2019d,Giner_2020} -\titou{But going to the CBS limit will converge to biased atomization -energies because of the use of approximate density functionals.} +However, in the CBS, KS-DFT atomization energies do not match the exact values due to the approximate nature of the xc functionals. Likewise, FCI is also size-consistent, but the convergence of the FCI energies towards the CBS limit is much slower because of the description of short-range electron correlation using atom-centered -functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991} But ultimately the exact energy will be reached. +functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991,Hattig_2012} +Eventually though, the exact atomization energies will be reached. In the context of SCI calculations, when the variational energy is extrapolated to the FCI energy \cite{Holmes_2017} there is no @@ -1069,8 +1052,8 @@ diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$ will lead to two different two-body Jastrow factors, each with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the size-consistency error on a PES using this ans\"atz for $J_\text{ee}$, -one needs to impose that the parameters of $J_\text{ee}$ are fixed: -$b_A = b_B = b_{\ce{AB}}$. +one needs to impose that the parameters of $J_\text{ee}$ are fixed, \ie, +$b_{\ce{A}} = b_{\ce{B}} = b_{\ce{AB}}$. When pseudopotentials are used in a QMC calculation, it is of common practice to localize the non-local part of the pseudopotential on the @@ -1085,46 +1068,47 @@ not introduce an additional error in FN-DMC calculations, although it will reduce the statistical errors by reducing the variance of the local energy. Moreover, the integrals involved in the pseudopotential are computed analytically and the computational cost of the -pseudopotential is dramatically reduced (for more detail, see +pseudopotential is dramatically reduced (for more details, see Ref.~\onlinecite{Scemama_2015}). In this section, we make a numerical verification that the produced wave functions are size-consistent for a given range-separation parameter. We have computed the FN-DMC energy of the dissociated fluorine dimer, where -the two atoms are at a distance of 50~\AA. We expect that the energy +the two atoms are separated by 50~\AA. We expect that the energy of this system is equal to twice the energy of the fluorine atom. -The data in Table~\ref{tab:size-cons} shows that it is indeed the +The data in Table~\ref{tab:size-cons} shows that this is indeed the case, so we can conclude that the proposed scheme provides size-consistent FN-DMC energies for all values of $\mu$ (within twice the statistical error bars). +%%% TABLE III %%% +\begin{table} + \caption{FN-DMC energy (in \hartree{}) using the VDZ-BFD basis set and the srPBE functional + of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values. + The size-consistency error is also reported.} + \label{tab:size-cons} + \begin{ruledtabular} + \begin{tabular}{cccc} + $\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\ + \hline + 0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\ + 0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\ + 0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\ + 1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\ + 2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\ + 5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\ + $\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} +%%% %%% %%% %%% + %============================ \section{Spin invariance} \label{app:spin} %============================ -%%% TABLE IV %%% -\begin{table} - \caption{FN-DMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with - different values of $m_s$ computed with the VDZ-BFD basis set and the srPBE functional.} - \label{tab:spin} - \begin{ruledtabular} - \begin{tabular}{cccc} - $\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\ - \hline - 0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\ - 0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\ - 0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\ - 1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\ - 2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\ - 5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\ - $\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\ - \end{tabular} - \end{ruledtabular} -\end{table} -%%% %%% %%% %%% - Closed-shell molecules often dissociate into open-shell fragments. To get reliable atomization energies, it is important to have a theory which is of comparable quality for open- and @@ -1159,11 +1143,33 @@ Although the energy obtained with $m_s=0$ is higher than the one obtained with $ bias is relatively small, \ie, more than one order of magnitude smaller than the energy gained by reducing the fixed-node error going from the single determinant to the FCI trial wave function. The largest bias, close to -$2$ m\hartree, is obtained for $\mu=0$, but this bias decreases quickly +$2$ m\hartree{}, is obtained for $\mu=0$, but this bias decreases quickly below $1$ m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$ we observe a perfect spin-invariance of the energy (within the error bars), and the bias is not noticeable for $\mu=5$~bohr$^{-1}$. +%%% TABLE IV %%% +\begin{table} + \caption{FN-DMC energy (in \hartree{}) for various $\mu$ values of the triplet carbon atom with + different values of $m_s$ computed with the VDZ-BFD basis set and the srPBE functional. + The spin-invariance error is also reported.} + \label{tab:spin} + \begin{ruledtabular} + \begin{tabular}{cccc} + $\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\ + \hline + 0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\ + 0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\ + 0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\ + 1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\ + 2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\ + 5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\ + $\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} +%%% %%% %%% %%% + \titou{T2: what do you conclude from this section? What value of $m_s$ do you use to compute the atoms?} %%%%%%%%%%%%%%%%%%%%%%%%%%%%