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Manuscript/rsdft-cipsi-qmc.bib
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@ -1,13 +1,52 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-19 09:29:01 +0200
%% Created for Pierre-Francois Loos at 2020-08-19 14:05:12 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Karolewski_2013,
Author = {Andreas Karolewski and Leeor Kronik and Stephan Kummel},
Date-Added = {2020-08-19 14:04:10 +0200},
Date-Modified = {2020-08-19 14:05:08 +0200},
Doi = {10.1063/1.4807325},
Journal = {J. Chem. Phys.},
Pages = {204115},
Title = {Using optimally tuned range separated hybrid functionals in ground-state calculations: Consequences and caveats},
Volume = {138},
Year = {2013}}
@article{Kronik_2012,
Author = {Leeor Kronik and Tamar Stein and Sivan {Refaely-Abramson} and Roi Baer},
Date-Added = {2020-08-19 14:01:54 +0200},
Date-Modified = {2020-08-19 14:01:54 +0200},
Doi = {10.1021/ct2009363},
Journal = {J. Chem. Theory Comput.},
Pages = {1515--1531},
Title = {Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals},
Volume = {8},
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1021/ct2009363}}
@article{Stein_2009,
Author = {Stein, Tamar and Kronik, Leeor and Baer, Roi},
Date-Added = {2020-08-19 14:01:19 +0200},
Date-Modified = {2020-08-19 14:01:19 +0200},
Doi = {10.1021/ja8087482},
Eprint = {https://doi.org/10.1021/ja8087482},
Journal = {J. Am. Chem. Soc.},
Note = {PMID: 19239266},
Number = {8},
Pages = {2818-2820},
Title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory},
Url = {https://doi.org/10.1021/ja8087482},
Volume = {131},
Year = {2009},
Bdsk-Url-1 = {https://doi.org/10.1021/ja8087482}}
@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},
@ -1679,15 +1718,14 @@
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5116024}}
@misc{nist,
doi = {10.18434/T47C7Z},
url = {http://cccbdb.nist.gov/},
Note = {\url{http://cccbdb.nist.gov/}},
author = {Johnson, RD},
title = {Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database 101},
publisher = {National Institute of Standards and Technology},
year = {2002},
copyright = {License Information for NIST data}
}
Author = {Johnson, RD},
Copyright = {License Information for NIST data},
Doi = {10.18434/T47C7Z},
Note = {\url{http://cccbdb.nist.gov/}},
Publisher = {National Institute of Standards and Technology},
Title = {Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database 101},
Url = {http://cccbdb.nist.gov/},
Year = {2002},
Bdsk-Url-1 = {http://cccbdb.nist.gov/},
Bdsk-Url-2 = {https://doi.org/10.18434/T47C7Z}}

121
Manuscript/rsdft-cipsi-qmc.tex
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@ -424,17 +424,19 @@ the final trial wave function $\Psi^{\mu}$ is independent of the starting wave f
\label{sec:comp-details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For all the systems considered here, experimental geometries have been used and they have been extracted from the NIST website \titou{[REF]}.
Geometries for each system are reported in the {\SI}.
All reference data (geometries, atomization
energies, zero-point energy corrections, etc) were taken from the NIST
computational chemistry comparison and benchmark database
(CCCBDB).\cite{nist}
All the calculations have been performed using Burkatzki-Filippi-Dolg (BFD)
All calculations have been performed using Burkatzki-Filippi-Dolg (BFD)
pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double-,
triple-, and quadruple-$\zeta$ basis sets (VXZ-BFD).
The small-core BFD pseudopotentials include scalar relativistic effects.
Coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] and KS-DFT energies have been computed with
\emph{Gaussian09},\cite{g16} using the unrestricted formalism for open-shell systems.
All the CIPSI calculations have been performed with \emph{Quantum
The CIPSI calculations have been performed with \emph{Quantum
Package}.\cite{Garniron_2019,qp2_2020} We consider the short-range version
of the local-density approximation (LDA)\cite{Sav-INC-96a,TouSavFla-IJQC-04} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96}
xc functionals defined in
@ -728,7 +730,7 @@ produce a reasonable one-body density.
\subsection{Intermediate conclusions}
%============================
As a conclusion of the first part of this study, we can highlight the following observations:
As conclusions of the first part of this study, we can highlight the following observations:
\begin{itemize}
\item With respect to the nodes of a KS determinant or a FCI wave function,
one can obtain a multi-determinant trial wave function $\Psi^\mu$ with a smaller
@ -763,14 +765,17 @@ 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
molecules, leading to more accurate atomization energies.
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}.
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}, respectively.
%%% FIG 6 %%%
\begin{squeezetable}
\begin{table*}
\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.}
root mean square errors (RMSEs) with respect to the NIST reference values obtained with various methods and
basis sets.
All quantities are given in kcal/mol.}
\label{tab:mad}
\begin{ruledtabular}
\begin{tabular}{ll ddd ddd ddd}
@ -810,47 +815,55 @@ DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -
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. \toto{All reference data (geometries, atomization
energies, zero-point energy corrections) were taken from the NIST
computational chemistry comparison and benchmark database
(CCCBDB).}\cite{nist}
energy differences.
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 as a starting point (see Fig.~\ref{fig:algo}).
\footnote{At $\mu=0$, the number of determinants is not equal to one because
we have used the natural orbitals of a preliminary CIPSI calculation, and
not the srPBE orbitals.
So the Kohn-Sham determinant is expressed as a linear combination of
determinants built with NOs. It is possible to add
an extra step to the algorithm to compute the NOs from the
RS-DFT-CIPSI wave function, and re-do the RS-DFT-CIPSI calculation with
these orbitals to get an even more compact expansion. In that case, we would
have converged to the KS orbitals with $\mu=0$, and the
solution would have been the PBE single determinant.}
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 (MAEs), mean signed errors (MSEs), and root mean square errors (RMSEs).
absolute errors (MAEs), mean signed errors (MSEs), and root mean square errors (RMSEs)
with respect to the NIST reference values as explained in Sec.~\ref{sec:comp-details}.
For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have
provided the extrapolated values (\ie, when $\EPT \to 0$), and the error bars
correspond to the difference between the extrapolated energies computed with a
provided the extrapolated values (\ie, when $\EPT \to 0$), and, although one cannot provide theoretically sound error bars, they
correspond here 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
calculated at the DFT level are relatively accurate, even when
calculated at the KS-DFT level are relatively accurate, even when
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. Note that, due to the approximate nature of the xc functionals,
the MAEs associated with KS-DFT atomization energies do not converge towards zero and remain altered even in the CBS limit.
the statistical quantities associated with KS-DFT atomization energies do not converge towards zero and remain altered even in the CBS limit.
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 RMSE of CCSDT(T)
and FCI energies.
shown by the very good agreement of the MAE, MSE and RMSE of CCSD(T)
and FCI energies for each basis set.
The imbalance in the description of molecules compared
to atoms is exhibited by a very negative value of the MSE for
CCSD(T)/VDZ-BFD and FCI/VDZ-BFD, which is reduced by a factor of two
CCSD(T)/VDZ-BFD ($-23.96$ kcal/mol) and FCI/VDZ-BFD ($-23.49\pm0.04$ kcal/mol), 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 significant imbalance at the VDZ-BFD level affects the nodal
surfaces, because although the FN-DMC energies obtained with near-FCI
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
larger than the \titou{single-determinant} MAE ($4.61\pm0.34$ kcal/mol).
trial wave functions are much lower than the FN-DMC
energies at $\mu = 0$, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is
larger than the MAE at $\mu = 0$ ($4.61\pm0.34$ kcal/mol).
Using the FCI trial wave function the MSE is equal to the
negative MAE which confirms that the atomization energies are systematically
underestimated. This confirms that some of the basis set
underestimated. This corroborates that some of the basis set
incompleteness error is transferred in the fixed-node error.
Within the double-$\zeta$ basis set, the calculations could be performed for the
@ -861,40 +874,29 @@ function of $\mu$.
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 as compared to the FCI wave function. This
MAEs, MSEs, and RMSEs as 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.
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 contains more than a few million
determinants.
determinants. \cite{Scemama_2016}
At the RS-DFT-CIPSI/VTZ-BFD 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].
%TOTO TODO
For the largest systems, as shown in Fig.~\ref{fig:g2-ndet},
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 size-consistency error originating from the imbalanced
truncation of the wave functions, which is not present in the
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.}
FCI ($8.43\pm0.39$ kcal/mol).
The same comment applies to $\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis.
%%% FIG 5 %%%
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{g2-dmc.pdf}
\caption{Errors \titou{(with respect to ???)} in the FN-DMC atomization energies (in kcal/mol) with various
\caption{Errors in the FN-DMC 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.
\titou{T2: change basis set labels.}}
with a cross.}
\label{fig:g2-dmc}
\end{figure*}
%%% %%% %%% %%%
@ -910,7 +912,8 @@ are even lower than those obtained with the optimal value of
$\mu$. Although the FN-DMC 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 size-consistentcy when one uses different $\mu$ values for different systems like in optimally-tune range-separated hybrids. \cite{}}
This is another key result of the present study, and it can be explained by the lack of size-consistency when one considers different $\mu$ values for the molecule and the isolated atoms.
This observation was also mentioned in the context of optimally-tune range-separated hybrids. \cite{Stein_2009,Karolewski_2013,Kronik_2012}
%%% FIG 6 %%%
\begin{figure*}
@ -920,8 +923,7 @@ 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.
\titou{T2: change basis set labels.}}
with a cross.}
\label{fig:g2-ndet}
\end{figure*}
%%% %%% %%% %%%
@ -934,7 +936,7 @@ determinants when $\mu=0.5$~bohr$^{-1}$ is below $100\,000$ determinants
with the VQZ-BFD 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 \titou{two orders of magnitude}.
reduced by more than two orders of magnitude.
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
at the quadruple-$\zeta$ level for only a slight increase of the
@ -942,17 +944,14 @@ MAE. Hence, RS-DFT-CIPSI trial wave functions with small values of
$\mu$ could be very useful for large systems to go beyond the
single-determinant approximation at a very low computational cost
while ensuring size-consistency.
Note that when $\mu=0$ the number of determinants is not equal to one because
we have used the natural orbitals of a preliminary CIPSI calculation, and
not the srPBE orbitals.
So the Kohn-Sham determinant is expressed as a linear combination of
determinants built with NOs. It is possible to add
an extra step to the algorithm to compute the NOs from the
RS-DFT-CIPSI wave function, and re-do the RS-DFT-CIPSI calculation with
these orbitals to get an even more compact expansion. In that case, we would
have converged to the KS orbitals with $\mu=0$, and the
solution would have been the PBE single determinant.
For the largest systems, as shown in Fig.~\ref{fig:g2-ndet},
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 size-consistency error originating from the imbalanced
truncation of the wave functions, which is not present in the
extrapolated FCI energies (see Appendix \ref{app:size}).
%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
@ -991,7 +990,9 @@ cancellations of errors.
\begin{acknowledgments}
This work was performed using HPC resources from GENCI-TGCC (Grand
Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
2019-0510.
2020-18005.
Funding from \textit{``Projet International de Coop\'eration Scientifique''} (PICS08310) and from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This study has been (partially) supported through the EUR grant NanoX No.~ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.
\end{acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%
@ -1015,9 +1016,9 @@ atomization energies is size-consistency (or strict separability),
since the numbers of correlated electron pairs in the molecule and its isolated atoms
are different.
KS-DFT energies 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}
KS-DFT energies are size-consistent, and because xc functionals are
directly constructed in complete basis, their convergence with respect
to the size of the basis set is relatively fast. \cite{FraMusLupTou-JCP-15,Giner_2018,Loos_2019d,Giner_2020}
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}