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25
Manuscript/rsdft-cipsi-qmc.bib
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@ -1,13 +1,33 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-18 10:07:35 +0200
%% Created for Pierre-Francois Loos at 2020-08-18 15:08:15 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Kutzelnigg_1985,
Author = {W. Kutzelnigg},
Date-Added = {2020-08-18 15:05:16 +0200},
Date-Modified = {2020-08-18 15:05:51 +0200},
Journal = {Theor. Chim. Acta},
Pages = {445},
Title = {r12-Dependent terms in the wave function as closed sums of partial wave amplitudes for large l},
Volume = {68},
Year = {1985}}
@article{Kutzelnigg_1991,
Author = {W. Kutzelnigg and W. Klopper},
Date-Added = {2020-08-18 15:05:16 +0200},
Date-Modified = {2020-08-18 15:05:24 +0200},
Journal = {J. Chem. Phys.},
Pages = {1985},
Title = {Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory},
Volume = {94},
Year = {1991}}
@article{Pack_1966,
Author = {{R. T. Pack and W. Byers-Brown}},
Date-Added = {2020-08-18 09:51:18 +0200},
@ -943,8 +963,9 @@
Year = {2015},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4907920}}
@article{GinPraFerAssSavTou-JCP-18,
@article{Giner_2018,
Author = {Emmanuel Giner and Barth\'elemy Pradines and Anthony Fert\'e and Roland Assaraf and Andreas Savin and Julien Toulouse},
Date-Modified = {2020-08-18 15:08:15 +0200},
Journal = {J. Chem. Phys.},
Pages = {194301},
Title = {Curing basis-set convergence of wave-function theory using density-functional theory: A systematically improvable approach},

103
Manuscript/rsdft-cipsi-qmc.tex
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@ -120,7 +120,7 @@ Another route to solve the Schr\"odinger equation is density-functional theory (
Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, \cite{Kohn_1965} which
transfers the complexity of the many-body problem to the universal and yet unknown exchange-correlation (xc) functional thanks to a judicious mapping between a non-interacting reference system and its interacting analog which both have exactly the same one-electron density.
KS-DFT \cite{Hohenberg_1964,Kohn_1965} is now the workhorse of electronic structure calculations for atoms, molecules and solids thanks to its very favorable accuracy/cost ratio. \cite{ParrBook}
As compared to WFT, DFT has the indisputable advantage of converging much faster with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15,Loos_2019d,Giner_2020}
As compared to WFT, DFT has the indisputable advantage of converging much faster with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15,Giner_2018,Loos_2019d,Giner_2020}
However, unlike WFT where many-body perturbation theory provides a precious tool to go toward the exact wave function, there is no systematic way to improve approximate xc functionals toward the exact functional.
Therefore, one faces, in practice, the unsettling choice of the \emph{approximate} xc functional. \cite{Becke_2014}
Moreover, because of the approximate nature of the xc functional, although the resolution of the KS equations is variational, the resulting KS energy does not have such property.
@ -397,7 +397,7 @@ single- or multi-determinant wave function $\Psi^{(0)}$ which can
be obtained in many different ways depending on the system that one considers.
One of the particularity of the present work is that we
use the CIPSI algorithm to perform approximate FCI calculations
with the RS-DFT Hamiltonian $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
with the RS-DFT Hamiltonian $\hat{H}^\mu$. \cite{Giner_2018}
This provides a multi-determinant trial wave function $\Psi^{\mu}$ that one can ``feed'' to DMC.
In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selection is performed
to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$
@ -656,14 +656,14 @@ This is yet another key result of the present study.
\begin{figure*}
\includegraphics[width=\columnwidth]{density-mu.pdf}
\includegraphics[width=\columnwidth]{on-top-mu.pdf}
\caption{\toto{One-electron density $n(\br)$ (left) and on-top pair
\caption{One-electron density $n(\br)$ (left) and on-top pair
density $n_2(\br,\br)$ (right) along the \ce{O-H} axis of \ce{H2O}
as a function of $\mu$ for $\Psi^\mu$, and $\Psi^J$ (dashed
curve).
The integrated on-top pair density $\expval{P}$ is
given in the legend.
For all trial wave functions, the CI expansion consists of the 200 most important
determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}}
determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
\label{fig:densities}
\end{figure*}
%%% %%% %%% %%%
@ -671,9 +671,9 @@ This is yet another key result of the present study.
In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
report several quantities related to the one- and two-body densities of
$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
report in the legend of Fig~\ref{fig:densities} the integrated on-top pair density
report in the legend of the right panel of Fig~\ref{fig:densities} the integrated on-top pair density
\begin{equation}
\expval{ P } = \int d\br \,\,n_2(\br,\br)
\expval{ P } = \int d\br \,\,n_2(\br,\br),
\end{equation}
where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$]
obtained for both $\Psi^\mu$ and $\Psi^J$.
@ -691,13 +691,13 @@ are much more important than that of the one-body density, the latter
being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the
former can vary by about 10$\%$ in some regions.
%TODO TOTO
\toto{In the high-density region of the \ce{O-H} bond, the value of the on-top
In the high-density region of the \ce{O-H} bond, the value of the on-top
pair density obtained from $\Psi^J$ is superimposed with
$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density of $\Psi^J$ is
the closest to $\mu=\infty$. The integrated on-top pair density
obtained with $\Psi^J$ is $\expval{P}=1.404$, between the values obtained with
obtained with $\Psi^J$ is $\expval{P}=1.404$, which nestles between the values obtained at
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
and the overlap curve depicted in Fig.~\ref{fig:overlap}.}
and the overlap curve depicted in Fig.~\ref{fig:overlap}.
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
@ -748,18 +748,18 @@ As a conclusion of the first part of this study, we can highlight the following
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Atomization energies are challenging for post-HF methods
because their calculation requires a perfect 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
calculations are atom-centered, so they are, by construction, better adapted to
atoms than molecules and atomization energies usually tend to be
underestimated by variational methods.
In the context of FN-DMC calculations, the nodal surface is imposed by
the trial wavefunction which is expanded in an atom-centered basis
set, so we expect the fixed-node error to be also intimately related to
the trial wavefunction which is expanded in the very same atom-centered basis
set. Thus, we expect the fixed-node error to be also intimately related to
the 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 quality of the description of the atoms and the
imbalance in the description of atoms and
molecule, leading to more accurate atomization energies.
%============================
@ -768,19 +768,21 @@ molecule, leading to more accurate atomization energies.
An extremely important feature required to get accurate
atomization energies is size-consistency (or strict separability),
since the numbers of correlated electron pairs in the isolated atoms
are different from those of the molecules.
The energy computed within DFT is size-consistent, and
as it is a mean-field method the convergence to the CBS limit
is relatively fast. Hence, DFT methods are very well adapted to
since the numbers of correlated electron pairs in the molecule and its isolated atoms
are different.
The energies computed within DFT are size-consistent, and
\titou{because it is a mean-field method the convergence to the CBS limit
is relatively fast}. \cite{FraMusLupTou-JCP-15}
Hence, DFT methods are very well adapted to
the calculation of atomization energies, especially with small basis
sets. But going to the CBS limit will converge to biased atomization
energies because of the use of approximate density functionals.
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.}
On the other hand, FCI is also size-consistent, but the convergence of
the FCI energies to the CBS limit is much slower because of the
Likewise, FCI is also size-consistent, but the convergence of
the FCI energies towards the CBS limit is much slower because of the
description of short-range electron correlation using atom-centered
functions. But ultimately the exact energy will be reached.
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
@ -805,7 +807,7 @@ one-electron, two-electron and one-nucleus-two-electron terms.
The problematic part is the two-electron term, whose simplest form can
be expressed as in Eq.~\eqref{eq:jast-ee}.
The parameter
$a$ is determined by cusp conditions, and $b$ is obtained by energy
$a$ is determined by the electron-electron 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 two-body
interaction is not size-consistent: the dissociation of a
@ -816,11 +818,11 @@ size-consistency error on a PES using this ans\"atz for $J_\text{ee}$,
one needs to impose that the parameters of $J_\text{ee}$ are fixed:
$b_A = b_B = b_{\ce{AB}}$.
When pseudopotentials are used in a QMC calculation, it is 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
complete wave function (determinantal component and Jastrow).
complete trial wave function $\Phi$.
If the wave function is not size-consistent,
so will be the locality approximation. Within, the DLA,\cite{Zen_2019} the Jastrow factor is
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 FN-DMC energy
only depends on the parameters of the determinantal component. Using a
@ -834,9 +836,9 @@ Ref.~\onlinecite{Scemama_2015}).
%%% TABLE III %%%
\begin{table}
\caption{FN-DMC energies (in hartree) using the VDZ-BFD basis set
and pseudopotential of the fluorine atom and the dissociated fluorine
dimer, and size-consistency error. }
\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}
@ -863,7 +865,7 @@ 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
case, so we can conclude that the proposed scheme provides
size-consistent FN-DMC energies for all values of $\mu$ (within
$2\times$ statistical error bars).
twice the statistical error bars).
%============================
@ -893,8 +895,8 @@ impacted by this spurious effect, as opposed to FCI.
%%% TABLE IV %%%
\begin{table}
\caption{FN-DMC energies (in hartree) of the triplet carbon atom (VDZ-BFD) with
different values of $m_s$.}
\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}
@ -937,15 +939,15 @@ noticeable with $\mu=5$~bohr$^{-1}$.
%%% FIG 6 %%%
\begin{squeezetable}
\begin{table*}
\caption{Mean absolute errors (MAE), mean signed errors (MSE) and
standard deviations (RMSD) obtained with various methods and
\caption{Mean absolute errors (MAE), mean signed errors (MSE), and
root mean square errors (RMSE) obtained with various methods and
basis sets.}
\label{tab:mad}
\begin{ruledtabular}
\begin{tabular}{ll ddd ddd ddd}
& & \mc{3}{c}{VDZ-BFD} & \mc{3}{c}{VTZ-BFD} & \mc{3}{c}{VQZ-BFD} \\
\cline{3-5} \cline{6-8} \cline{9-11}
Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\
Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} \\
\hline
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
@ -976,20 +978,21 @@ DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -
\end{squeezetable}
%%% %%% %%% %%%
The 55 molecules of the benchmark for the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the
The atomization energies of the 55 molecules of the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen as a benchmark set to test the
performance of the RS-DFT-CIPSI trial wave functions in the context of
energy differences. Calculations were made in the double-, triple-
energy differences. \titou{REFERENCES??}
Calculations were made in the double-, triple-
and quadruple-$\zeta$ basis sets with different values of $\mu$, and using
NOs from a preliminary CIPSI calculation \titou{as a starting point}.
For comparison, we have computed the energies of all atoms and
NOs from a preliminary CIPSI calculation as a starting point (see Fig.~\ref{fig:algo}).
For comparison, we have computed the energies of all the atoms and
molecules at the KS-DFT level with various semi-local and hybrid density functionals [PBE, BLYP, PBE0, and B3LYP], and at
the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean
absolute errors (MAE), mean signed errors (MSE) and \titou{standard
deviations (RMSE)}. For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have
absolute errors (MAE), mean signed errors (MSE), and root mean square errors (RMSE).
For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have
provided the extrapolated values at $\EPT \to 0$, and the error bars
correspond to the difference between the energies \titou{computed with a
two-point and with a three-point linear extrapolation}. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
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}
In this benchmark, the great majority of the systems are weakly correlated and are then well
described by a single determinant. Therefore, the atomization energies
@ -998,7 +1001,7 @@ the basis set is small. The introduction of exact exchange (B3LYP and
PBE0) make the results more sensitive to the basis set, and reduce the
accuracy. Thanks to the single-reference character of these systems,
the CCSD(T) energy is an excellent estimate of the FCI energy, as
shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T)
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
to atoms is exhibited by a very negative value of the MSE for
@ -1022,9 +1025,9 @@ trial wave function was estimated for each system by searching for the
minimum of the spline interpolation curve of the FN-DMC energy as a
function of $\mu$.
This corresponds the line of Table~\ref{tab:mad} labelled as ``Opt.''
\titou{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
MAE, the MSE an the RMSD compared to the FCI wave function. This
MAE, the MSE an the RMSE compared to the FCI wave function. This
result is in line with the common knowledge that re-optimizing
the determinantal component of the trial wave function in the presence
of electron correlation reduces the errors due to the basis set incompleteness.
@ -1059,11 +1062,11 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis set.
%%% %%% %%% %%%
Searching for the optimal value of $\mu$ may be too costly and time consuming, so we have
computed the MAD, MSE and RMSD for fixed values of $\mu$.
computed the MAD, MSE and RMSE for fixed values of $\mu$.
As illustrated in Fig.~\ref{fig:g2-dmc} and Table \ref{tab:mad},
the best choice for a fixed value of $\mu$ is
0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
the MAE [$3.74(35)$, $2.46(18)$, and $2.06(35)$ kcal/mol] and RMSD [$4.03(23)$,
the MAE [$3.74(35)$, $2.46(18)$, and $2.06(35)$ kcal/mol] and RMSE [$4.03(23)$,
$3.02(06)$, and $2.74(13)$ kcal/mol] are minimal. Note that these values
are even lower than those obtained with the optimal value of
$\mu$. Although the FN-DMC energies are higher, the numbers show that