From 03b35a14c8ee77a7d55878d1029e246e35044f1e Mon Sep 17 00:00:00 2001
From: PierreFrancois Loos
Date: Wed, 19 Aug 2020 09:42:13 +0200
Subject: [PATCH] saving work in Sec V and appendix A

Manuscript/rsdftcipsiqmc.bib  12 ++
Manuscript/rsdftcipsiqmc.tex  182 +++++++++++++++++
2 files changed, 105 insertions(+), 89 deletions()
diff git a/Manuscript/rsdftcipsiqmc.bib b/Manuscript/rsdftcipsiqmc.bib
index 590a14d..75faac8 100644
 a/Manuscript/rsdftcipsiqmc.bib
+++ b/Manuscript/rsdftcipsiqmc.bib
@@ 1,13 +1,23 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for PierreFrancois Loos at 20200818 15:08:15 +0200
+%% Created for PierreFrancois Loos at 20200819 09:29:01 +0200
%% Saved with string encoding Unicode (UTF8)
+@article{Hattig_2012,
+ Author = {C. Hattig and W. Klopper and A. Kohn and D. P. Tew},
+ DateAdded = {20200819 09:28:48 +0200},
+ DateModified = {20200819 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},
DateAdded = {20200818 15:05:16 +0200},
diff git a/Manuscript/rsdftcipsiqmc.tex b/Manuscript/rsdftcipsiqmc.tex
index 96c562f..438465d 100644
 a/Manuscript/rsdftcipsiqmc.tex
+++ b/Manuscript/rsdftcipsiqmc.tex
@@ 458,6 +458,8 @@ The FNDMC simulations are performed with allelectron 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 rangeseparation parameter on the fixednode error}
\label{sec:mudmc}
@@ 723,7 +725,7 @@ they both deal with an effective nondivergent interaction but still
produce a reasonable onebody 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 postHF methods
because their calculation requires a subtle balance in the
description of atoms and molecules. The mainstream oneelectron basis sets employed in molecular
+description of atoms and molecules. The mainstream oneelectron basis sets employed in molecular electronic structure
calculations are atomcentered, 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 FNDMC calculations, the nodal surface is imposed by
the trial wavefunction which is expanded in the very same atomcentered basis
set. Thus, we expect the fixednode error to be also intimately related to
+the determinantal part of the trial wavefunction which is expanded in the very same atomcentered basis
+set. Thus, we expect the fixednode 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 sizeconsistency and the spininvariance of the present scheme are discussed in Appendices \ref{app:size} and \ref{app:spin}.
+molecules, leading to more accurate atomization energies.
+The sizeconsistency and the spininvariance 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 KSDFT level with various semilocal 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 (RSDFTCIPSI, $\mu=\infty$) 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
twopoint and a threepoint linear extrapolation. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
@@ 830,20 +832,20 @@ accuracy. Thanks to the singlereference 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/VDZBFD, which is reduced by a factor of two
+CCSD(T)/VDZBFD and FCI/VDZBFD, 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 VDZBFD level affects the nodal
+This significant imbalance at the VDZBFD level affects the nodal
surfaces, because although the FNDMC energies obtained with nearFCI
trial wave functions are much lower than the singledeterminant FNDMC
energies, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is
larger than the singledeterminant MAE ($4.61\pm 0.34$ kcal/mol).
+larger than the \titou{singledeterminant} 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 basisset
+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 fixednode 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 FNDMC 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 reoptimizing
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 RSDFTCIPSI level, one can see that with the VTZBFD
basis the MAEs are larger for $\mu=1$~bohr$^{1}$ than for the
FCI. For the largest systems, as shown in Fig.~\ref{fig:g2ndet}
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 RSDFTCIPSI/VTZBFD 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:g2ndet},
+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 sizeconsistency 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]{g2dmc.pdf}
 \caption{Errors in the FNDMC atomization energies with various
+ \caption{Errors \titou{(with respect to ???)} in the FNDMC 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:g2dmc}
\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:g2dmc} 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 FNDMC 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 sizeconsistentcy when one uses different $\mu$ values for different systems like in optimallytune rangeseparated 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:g2ndet}
\end{figure*}
%%% %%% %%% %%%
@@ 916,12 +921,12 @@ cancellations of errors.
The number of determinants in the trial wave functions are shown in
Fig.~\ref{fig:g2ndet}. 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 VQZBFD 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 VDZBFD 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{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
@@ 1030,13 +1013,13 @@ 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.}
+However, in the CBS, KSDFT atomization energies do not match the exact values due to the approximate nature of the xc 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.
+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 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:
$b_A = b_B = b_{\ce{AB}}$.
+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 nonlocal part of the pseudopotential on the
@@ 1085,46 +1068,47 @@ 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
+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 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
+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:sizecons} shows that it is indeed the
+The data in Table~\ref{tab:sizecons} shows that this 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).
+%%% 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}
+%%% %%% %%% %%%
+
%============================
\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
@@ 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 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
+$2$ 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$^{1}$.
+%%% 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.
+ The spininvariance error is also reported.}
+ \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}
+%%% %%% %%% %%%
+
\titou{T2: what do you conclude from this section? What value of $m_s$ do you use to compute the atoms?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%