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Manuscript/rsdft-cipsi-qmc.bib
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@ -1,13 +1,23 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% 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) %% 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, @article{Kutzelnigg_1985,
Author = {W. Kutzelnigg}, Author = {W. Kutzelnigg},
Date-Added = {2020-08-18 15:05:16 +0200}, Date-Added = {2020-08-18 15:05:16 +0200},

182
Manuscript/rsdft-cipsi-qmc.tex
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@ -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.}, stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
\cite{Assaraf_2000} with a time step of $5 \times 10^{-4}$ a.u. \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} \section{Influence of the range-separation parameter on the fixed-node error}
\label{sec:mu-dmc} \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. 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: 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 Atomization energies are challenging for post-HF methods
because their calculation requires a subtle balance in the 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 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. underestimated by variational methods.
In the context of FN-DMC calculations, the nodal surface is imposed by 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 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 related to set. Thus, we expect the fixed-node error to be also intimately connected to
the basis set incompleteness error. the basis set incompleteness error.
Increasing the size of the basis set improves the description of Increasing the size of the basis set improves the description of
the density and of the electron correlation, but also reduces the the density and of the electron correlation, but also reduces the
imbalance in the description of atoms and imbalance in the description of atoms and
molecule, leading to more accurate atomization energies. molecules, 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}. 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 %%% %%% FIG 6 %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{Mean absolute errors (MAE), mean signed errors (MSE), and \caption{Mean absolute errors (MAEs), mean signed errors (MSEs), and
root mean square errors (RMSE) obtained with various methods and root mean square errors (RMSEs) \titou{with respect to ??? (in kcal/mol)} obtained with various methods and
basis sets.} basis sets.}
\label{tab:mad} \label{tab:mad}
\begin{ruledtabular} \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 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 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 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 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 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} 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 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) shown by the very good agreement of the MAE, MSE and RMSE of CCSDT(T)
and FCI energies. 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 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 when going to the triple-$\zeta$ basis, and again by a factor of two when
going to the quadruple-$\zeta$ basis. 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 surfaces, because although the FN-DMC energies obtained with near-FCI
trial wave functions are much lower than the single-determinant FN-DMC 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 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 Using the FCI trial wave function the MSE is equal to the
negative MAE which confirms that all the atomization energies are negative MAE which confirms that the atomization energies are systematically
underestimated. This confirms that some of the basis-set underestimated. This confirms that some of the basis set
incompleteness error is transferred in the fixed-node error. incompleteness error is transferred in the fixed-node error.
Within the double-$\zeta$ basis set, the calculations could be performed for the 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 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 minimum of the spline interpolation curve of the FN-DMC energy as a
function of $\mu$. 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. The optimal $\mu$ value for each system is reported in the \SI.
Using the optimal value of $\mu$ clearly improves the 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 result is in line with the common knowledge that re-optimizing
the determinantal component of the trial wave function in the presence the determinantal component of the trial wave function in the presence
of electron correlation reduces the errors due to the basis set incompleteness. of electron correlation reduces the errors due to the basis set incompleteness.
These calculations were done only for the smallest basis set These calculations were done only for the smallest basis set
because of the expensive computational cost of the QMC calculations 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. determinants.
At the RS-DFT-CIPSI level, one can see that with the VTZ-BFD At the RS-DFT-CIPSI/VTZ-BFD level, one can see that
basis the MAEs are larger for $\mu=1$~bohr$^{-1}$ than for the the MAEs are larger for $\mu=1$~bohr$^{-1}$ ($9.06$ kcal/mol) than for
FCI. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet} FCI [$8.43(39)$ kcal/mol]. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet},
there are many systems which did not reach the threshold there are many systems for which we could not reach the threshold
$\EPT<1$~m\hartree{}, and the number of determinants exceeded $\EPT<1$~m\hartree{} as the number of determinants exceeded
10~million so the calculation stopped. In this regime, there is a 10~million before this threshold was reached. For these cases, there is then a
small size-consistency error originating from the imbalanced small size-consistency error originating from the imbalanced
truncation of the wave functions, which is not present in the truncation of the wave functions, which is not present in the
extrapolated FCI energies. The same comment applies to extrapolated FCI energies (see Appendix \ref{app:size}). The same comment applies to
$\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis set. $\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis.
\titou{T2: Fig. 6 is mentioned before Fig. 5.}
%%% FIG 5 %%% %%% FIG 5 %%%
\begin{figure*} \begin{figure*}
\centering \centering
\includegraphics[width=\textwidth]{g2-dmc.pdf} \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 trial wave functions. Each dot corresponds to an atomization
energy. energy.
The boxes contain the data between first and third quartiles, and The boxes contain the data between first and third quartiles, and
the line in the box represents the median. The outliers are shown 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} \label{fig:g2-dmc}
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%% %%% %%%
Searching for the optimal value of $\mu$ may be too costly and time consuming, so we have 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}, As illustrated in Fig.~\ref{fig:g2-dmc} and Table \ref{tab:mad},
the best choice for a fixed value of $\mu$ is 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 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 $\mu$. Although the FN-DMC energies are higher, the numbers show that
they are more consistent from one system to another, giving improved they are more consistent from one system to another, giving improved
cancellations of errors. 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 %%% %%% FIG 6 %%%
\begin{figure*} \begin{figure*}
@ -908,7 +912,8 @@ cancellations of errors.
functions. Each dot corresponds to an atomization energy. functions. Each dot corresponds to an atomization energy.
The boxes contain the data between first and third quartiles, and The boxes contain the data between first and third quartiles, and
the line in the box represents the median. The outliers are shown 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} \label{fig:g2-ndet}
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -916,12 +921,12 @@ cancellations of errors.
The number of determinants in the trial wave functions are shown in The number of determinants in the trial wave functions are shown in
Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants
is smaller when $\mu$ is small and larger when $\mu$ is large. 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 determinants when $\mu=0.5$~bohr$^{-1}$ is below $100\,000$ determinants
with the VQZ-BFD basis, making these calculations feasible with the VQZ-BFD basis, making these calculations feasible
with such a large basis set. At the double-$\zeta$ level, compared to the 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 FCI trial wave functions, the median of the number of determinants is
reduced by more than two orders of magnitude. reduced by more than \titou{two orders of magnitude}.
Moreover, going to $\mu=0.25$~bohr$^{-1}$ gives a median close to 100 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 determinants at the VDZ-BFD level, and close to $1\,000$ determinants
at the quadruple-$\zeta$ level for only a slight increase of the 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} \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 An extremely important feature required to get accurate
atomization energies is size-consistency (or strict separability), atomization energies is size-consistency (or strict separability),
since the numbers of correlated electron pairs in the molecule and its isolated atoms 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 Hence, DFT methods are very well adapted to
the calculation of atomization energies, especially with small basis the calculation of atomization energies, especially with small basis
sets. \cite{Giner_2018,Loos_2019d,Giner_2020} sets. \cite{Giner_2018,Loos_2019d,Giner_2020}
\titou{But going to the CBS limit will converge to biased atomization However, in the CBS, KS-DFT atomization energies do not match the exact values due to the approximate nature of the xc functionals.
energies because of the use of approximate density functionals.}
Likewise, FCI is also size-consistent, but the convergence of Likewise, FCI is also size-consistent, but the convergence of
the FCI energies towards the CBS limit is much slower because of the the FCI energies towards the CBS limit is much slower because of the
description of short-range electron correlation using atom-centered 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 In the context of SCI calculations, when the variational energy is
extrapolated to the FCI energy \cite{Holmes_2017} there is no 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 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 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 $J_\text{ee}$, 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: one needs to impose that the parameters of $J_\text{ee}$ are fixed, \ie,
$b_A = b_B = b_{\ce{AB}}$. $b_{\ce{A}} = b_{\ce{B}} = b_{\ce{AB}}$.
When pseudopotentials are used in a QMC calculation, it is of common When pseudopotentials are used in a QMC calculation, it is of common
practice to localize the non-local part of the pseudopotential on the 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 will reduce the statistical errors by reducing the variance of the
local energy. Moreover, the integrals involved in the pseudopotential local energy. Moreover, the integrals involved in the pseudopotential
are computed analytically and the computational cost of the 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}). Ref.~\onlinecite{Scemama_2015}).
In this section, we make a numerical verification that the produced In this section, we make a numerical verification that the produced
wave functions are size-consistent for a given range-separation wave functions are size-consistent for a given range-separation
parameter. parameter.
We have computed the FN-DMC energy of the dissociated fluorine dimer, where 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. 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 case, so we can conclude that the proposed scheme provides
size-consistent FN-DMC energies for all values of $\mu$ (within size-consistent FN-DMC energies for all values of $\mu$ (within
twice the statistical error bars). 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} \section{Spin invariance}
\label{app:spin} \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 Closed-shell molecules often dissociate into open-shell
fragments. To get reliable atomization energies, it is important to fragments. To get reliable atomization energies, it is important to
have a theory which is of comparable quality for open- and 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 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 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 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 $\mu=\infty$ 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 we observe a perfect spin-invariance of the energy (within the error bars), and the bias is not
noticeable for $\mu=5$~bohr$^{-1}$. 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?} \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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