From e9f2120a30d28e4ff7de440fbd52bd62aaa3c51e Mon Sep 17 00:00:00 2001
From: PierreFrancois Loos
Date: Tue, 18 Aug 2020 15:55:15 +0200
Subject: [PATCH] saving work in spin part

Manuscript/rsdftcipsiqmc.tex  353 +++++++++++++++++
1 file changed, 178 insertions(+), 175 deletions()
diff git a/Manuscript/rsdftcipsiqmc.tex b/Manuscript/rsdftcipsiqmc.tex
index 80f5179..9434444 100644
 a/Manuscript/rsdftcipsiqmc.tex
+++ b/Manuscript/rsdftcipsiqmc.tex
@@ 761,180 +761,7 @@ 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.

%============================
\subsection{Size consistency}
%============================

An extremely important feature required to get accurate
atomization energies is sizeconsistency (or strict separability),
since the numbers of correlated electron pairs in the molecule and its isolated atoms
are different.
The energies computed within DFT are sizeconsistent, and
\titou{because it is a meanfield method the convergence to the CBS limit
is relatively fast}. \cite{FraMusLupTouJCP15}
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.}

Likewise, FCI is also sizeconsistent, but the convergence of
the FCI energies towards the CBS limit is much slower because of the
description of shortrange electron correlation using atomcentered
functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991} But ultimately the exact energy 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
sizeconsistency error. But when the truncated SCI wave function is used
as a reference for postHF methods such as SCI+PT2
or for QMC calculations, there is a residual sizeconsistency error
originating from the truncation of the wave function.

QMC energies can be made sizeconsistent by extrapolating the
FNDMC energy to estimate the energy obtained with the FCI as a trial
wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the
sizeconsistency error can be reduced by choosing the number of
selected determinants such that the sum of the PT2 corrections on the
fragments is equal to the PT2 correction of the molecule, enforcing that
the variational potential energy surface (PES) is
parallel to the perturbatively corrected PES, which is a relatively
accurate estimate of the FCI PES.\cite{Giner_2015}

Another source of sizeconsistency error in QMC calculations originates
from the Jastrow factor. Usually, the Jastrow factor contains
oneelectron, twoelectron and onenucleustwoelectron terms.
The problematic part is the twoelectron term, whose simplest form can
be expressed as in Eq.~\eqref{eq:jastee}.
The parameter
$a$ is determined by the electronelectron cusp condition, \cite{Kato_1957,Pack_1966} and $b$ is obtained by energy
or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
One can easily see that this parameterization of the twobody
interaction is not sizeconsistent: the dissociation of a
diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$
will lead to two different twobody Jastrow factors, each
with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the
sizeconsistency 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:
$b_A = b_B = b_{\ce{AB}}$.

When pseudopotentials are used in a QMC calculation, it is of common
practice to localize the nonlocal part of the pseudopotential on the
complete trial wave function $\Phi$.
If the wave function is not sizeconsistent,
so will be the locality approximation. Within the DLA,\cite{Zen_2019} the Jastrow factor is
removed from the wave function on which the pseudopotential is localized.
The great advantage of this approximation is that the FNDMC energy
only depends on the parameters of the determinantal component. Using a
nonsizeconsistent Jastrow factor, or a nonoptimal Jastrow factor will
not introduce an additional error in FNDMC 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
Ref.~\onlinecite{Scemama_2015}).

%%% TABLE III %%%
\begin{table}
 \caption{FNDMC energy (in hartree) using the VDZBFD basis set and the srPBE functional
 of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values.
 The sizeconsistency error is also reported.}
 \label{tab:sizecons}
 \begin{ruledtabular}
 \begin{tabular}{cccc}
 $\mu$ & \ce{F} & Dissociated \ce{F2} & Sizeconsistency 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}
%%% %%% %%% %%%

In this section, we make a numerical verification that the produced
wave functions are sizeconsistent for a given rangeseparation
parameter.
We have computed the FNDMC energy of the dissociated fluorine dimer, where
the two atoms are at a distance of 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:sizecons} shows that it is indeed the
case, so we can conclude that the proposed scheme provides
sizeconsistent FNDMC energies for all values of $\mu$ (within
twice the statistical error bars).


%============================
\subsection{Spin invariance}
%============================

Closedshell molecules often dissociate into openshell
fragments. To get reliable atomization energies, it is important to
have a theory which is of comparable quality for openshell and
closedshell systems. A good test is to check that all the components
of a spin multiplet are degenerate, as expected from exact solutions.
FCI wave functions have this property and give degenerate energies with
respect to the spin quantum number $m_s$, but the multiplication by a
Jastrow factor introduces spin contamination if the parameters
for the samespin electron pairs are different from those
for the oppositespin pairs.\cite{Tenno_2004}
Again, when pseudopotentials are used this tiny error is transferred
in the FNDMC energy unless the DLA is used.

Within DFT, the common density functionals make a difference for
samespin and oppositespin interactions. As DFT is a
singledeterminant theory, the density functionals are designed to be
used with the highest value of $m_s$, and therefore different values
of $m_s$ lead to different energies.
So in the context of RSDFT, the determinantal expansions will be
impacted by this spurious effect, as opposed to FCI.

%%% TABLE IV %%%
\begin{table}
 \caption{FNDMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with
 different values of $m_s$ computed with the VDZBFD basis set and the srPBE functional.}
 \label{tab:spin}
 \begin{ruledtabular}
 \begin{tabular}{cccc}
 $\mu$ & $m_s=1$ & $m_s=0$ & Spininvariance 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}
%%% %%% %%% %%%

In this section, we investigate the impact of the spin contamination
due to the shortrange density functional on the FNDMC energy. We have
computed the energies of the carbon atom in its triplet state
with BFD pseudopotentials and the corresponding double$\zeta$ basis
set. The calculation was done with $m_s=1$ (3 spinup electrons
and 1 spindown electrons) and with $m_s=0$ (2 spinup and 2
spindown electrons).

The results are presented in Table~\ref{tab:spin}.
Although using $m_s=0$ the energy is higher than with $m_s=1$, the
bias is relatively small, more than one order of magnitude smaller
than the energy gained by reducing the fixednode error going from the single
determinant to the FCI trial wave function. The highest bias, close to
2~m\hartree, is obtained for $\mu=0$, but the bias decreases rapidly
below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
there is no bias (within the error bars), and the bias is not
noticeable with $\mu=5$~bohr$^{1}$.

%============================
\subsection{Benchmark}
%============================
+The sizeconsistency and the spininvariance of the present scheme are discussed in Appendices \ref{app:size} and \ref{app:spin}.
%%% FIG 6 %%%
\begin{squeezetable}
@@ 1161,10 +988,186 @@ Challenge 2019gch0418) and from CALMIP (Toulouse) under allocation
The data that support the findings of this study are available within the article and its {\SI}, and are openly available in [repository name] at \url{http://doi.org/[doi]}, reference number [reference number].
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\appendix
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%============================
+\section{Size consistency}
+\label{app:size}
+%============================
+
+%%% TABLE III %%%
+\begin{table}
+ \caption{FNDMC energy (in hartree) using the VDZBFD basis set and the srPBE functional
+ of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values.
+ The sizeconsistency error is also reported.}
+ \label{tab:sizecons}
+ \begin{ruledtabular}
+ \begin{tabular}{cccc}
+ $\mu$ & \ce{F} & Dissociated \ce{F2} & Sizeconsistency 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 sizeconsistency (or strict separability),
+since the numbers of correlated electron pairs in the molecule and its isolated atoms
+are different.
+The energies computed within DFT are sizeconsistent, and
+\titou{because it is a meanfield method the convergence to the CBS limit
+is relatively fast}. \cite{FraMusLupTouJCP15}
+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.}
+
+Likewise, FCI is also sizeconsistent, but the convergence of
+the FCI energies towards the CBS limit is much slower because of the
+description of shortrange electron correlation using atomcentered
+functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991} But ultimately the exact energy 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
+sizeconsistency error. But when the truncated SCI wave function is used
+as a reference for postHF methods such as SCI+PT2
+or for QMC calculations, there is a residual sizeconsistency error
+originating from the truncation of the wave function.
+
+QMC energies can be made sizeconsistent by extrapolating the
+FNDMC energy to estimate the energy obtained with the FCI as a trial
+wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the
+sizeconsistency error can be reduced by choosing the number of
+selected determinants such that the sum of the PT2 corrections on the
+fragments is equal to the PT2 correction of the molecule, enforcing that
+the variational potential energy surface (PES) is
+parallel to the perturbatively corrected PES, which is a relatively
+accurate estimate of the FCI PES.\cite{Giner_2015}
+
+Another source of sizeconsistency error in QMC calculations originates
+from the Jastrow factor. Usually, the Jastrow factor contains
+oneelectron, twoelectron and onenucleustwoelectron terms.
+The problematic part is the twoelectron term, whose simplest form can
+be expressed as in Eq.~\eqref{eq:jastee}.
+The parameter
+$a$ is determined by the electronelectron cusp condition, \cite{Kato_1957,Pack_1966} and $b$ is obtained by energy
+or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
+One can easily see that this parameterization of the twobody
+interaction is not sizeconsistent: the dissociation of a
+diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$
+will lead to two different twobody Jastrow factors, each
+with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the
+sizeconsistency 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:
+$b_A = b_B = b_{\ce{AB}}$.
+
+When pseudopotentials are used in a QMC calculation, it is of common
+practice to localize the nonlocal part of the pseudopotential on the
+complete trial wave function $\Phi$.
+If the wave function is not sizeconsistent,
+so will be the locality approximation. Within the DLA,\cite{Zen_2019} the Jastrow factor is
+removed from the wave function on which the pseudopotential is localized.
+The great advantage of this approximation is that the FNDMC energy
+only depends on the parameters of the determinantal component. Using a
+nonsizeconsistent Jastrow factor, or a nonoptimal Jastrow factor will
+not introduce an additional error in FNDMC 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
+Ref.~\onlinecite{Scemama_2015}).
+
+In this section, we make a numerical verification that the produced
+wave functions are sizeconsistent for a given rangeseparation
+parameter.
+We have computed the FNDMC energy of the dissociated fluorine dimer, where
+the two atoms are at a distance of 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:sizecons} shows that it is indeed the
+case, so we can conclude that the proposed scheme provides
+sizeconsistent FNDMC energies for all values of $\mu$ (within
+twice the statistical error bars).
+
+%============================
+\section{Spin invariance}
+\label{app:spin}
+%============================
+
+%%% TABLE IV %%%
+\begin{table}
+ \caption{FNDMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with
+ different values of $m_s$ computed with the VDZBFD basis set and the srPBE functional.}
+ \label{tab:spin}
+ \begin{ruledtabular}
+ \begin{tabular}{cccc}
+ $\mu$ & $m_s=1$ & $m_s=0$ & Spininvariance 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}
+%%% %%% %%% %%%
+
+Closedshell molecules often dissociate into openshell
+fragments. To get reliable atomization energies, it is important to
+have a theory which is of comparable quality for open and
+closedshell systems. A good test is to check that all the components
+of a spin multiplet are degenerate, as expected from exact solutions.
+
+FCI wave functions have this property and yield degenerate energies with
+respect to the spin quantum number $m_s$.
+However, multiplying the determinantal part of the trial wave function by a
+Jastrow factor introduces spin contamination if the Jastrow parameters
+for the samespin electron pairs are different from those
+for the oppositespin pairs.\cite{Tenno_2004}
+Again, when pseudopotentials are employed, this tiny error is transferred
+to the FNDMC energy unless the DLA is enforced.
+
+The context is rather different within DFT.
+Indeed, mainstream density functionals have distinct functional forms to take
+into account correlation effects of samespin and oppositespin electron pairs.
+Therefore, KS determinants corresponding to different values of $m_s$ lead to different total energies.
+Consequently, in the context of RSDFT, the determinant expansions is impacted by this spurious effect, as opposed to FCI.
+
+In this Appendix, we investigate the impact of the spin contamination on the FNDMC energy
+originating from the shortrange density functional. We have
+computed the energies of the carbon atom in its triplet state
+with the VDZBFD basis set and the srPBE functional.
+The calculations are performed for $m_s=1$ (3 spinup
+and 1 spindown electrons) and for $m_s=0$ (2 spinup and 2
+spindown electrons).
+
+The results are reported in Table~\ref{tab:spin}.
+Although the energy obtained with $m_s=0$ is higher than the one obtained with $m_s=1$, the
+bias is relatively small, \ie, more than one order of magnitude smaller
+than the energy gained by reducing the fixednode 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
+below $1$ m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
+we observe a perfect spininvariance of the energy (within the error bars), and the bias is not
+noticeable for $\mu=5$~bohr$^{1}$.
+
+\titou{T2: what do you conclude from this section? What value of $m_s$ do you use to compute the atoms?}
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{rsdftcipsiqmc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
