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