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Manuscript/rsdft-cipsi-qmc.bib
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@ -1,13 +1,122 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-16 22:14:09 +0200
%% Created for Pierre-Francois Loos at 2020-08-17 09:17:11 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Scemama_2019,
Author = {A. Scemama and M. Caffarel and A. Benali and D. Jacquemin and P. F. Loos.},
Date-Added = {2020-08-17 09:16:18 +0200},
Date-Modified = {2020-08-17 09:17:11 +0200},
Doi = {10.1016/j.rechem.2019.100002},
Journal = {Res. Chem.},
Pages = {100002},
Title = {Influence of pseudopotentials on excitation energies from selected configuration interaction and diffusion Monte Carlo},
Volume = {1},
Year = {2019}}
@article{Giner_2020,
Author = {E. Giner and A. Scemama and P. F. Loos and J. Toulouse},
Date-Added = {2020-08-17 09:05:35 +0200},
Date-Modified = {2020-08-17 09:06:48 +0200},
Doi = {10.1063/5.0002892},
Journal = {J. Chem. Phys.},
Pages = {174104},
Title = {A basis-set error correction based on density-functional theory for strongly correlated molecular systems},
Volume = {152},
Year = {2020}}
@article{Loos_2019d,
Author = {P. F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
Date-Added = {2020-08-17 09:03:41 +0200},
Date-Modified = {2020-08-17 09:03:41 +0200},
Doi = {10.1021/acs.jpclett.9b01176},
Journal = {J. Phys. Chem. Lett.},
Pages = {2931--2937},
Title = {A Density-Based Basis-Set Correction for Wave Function Theory},
Volume = {10},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01103}}
@article{Ohtsuka_2017,
Author = {Yuhki Ohtsuka and Jun-ya Hasegawa},
Date-Added = {2020-08-17 08:45:57 +0200},
Date-Modified = {2020-08-17 08:45:57 +0200},
Doi = {10.1063/1.4993214},
Journal = {J. Chem. Phys.},
Month = {jul},
Number = {3},
Pages = {034102},
Publisher = {{AIP} Publishing},
Title = {Selected configuration interaction method using sampled first-order corrections to wave functions},
Url = {https://doi.org/10.1063%2F1.4993214},
Volume = {147},
Year = 2017,
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4993214},
Bdsk-Url-2 = {https://doi.org/10.1063/1.4993214}}
@article{Zimmerman_2017,
Author = {Zimmerman, Paul M.},
Date-Added = {2020-08-17 08:45:48 +0200},
Date-Modified = {2020-08-17 08:45:48 +0200},
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Title = {Incremental full configuration interaction},
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Volume = {146},
Year = {2017},
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4977727}}
@article{Per_2017,
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Date-Added = {2020-08-17 08:45:37 +0200},
Date-Modified = {2020-08-17 08:45:37 +0200},
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Volume = {146},
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Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4981527}}
@article{Kohn_1999,
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Date-Added = {2020-08-17 08:38:17 +0200},
Date-Modified = {2020-08-17 08:47:47 +0200},
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Title = {{Nobel Lecture: Electronic structure of matter - wave functions and density functionals}},
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@article{Pople_1999,
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Date-Added = {2020-08-17 08:38:05 +0200},
Date-Modified = {2020-08-17 08:46:51 +0200},
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Pages = {1267},
Title = {{Nobel Lecture: Quantum chemical models}},
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@article{Bressanini_2012,
Author = {D. Bressanini},
Date-Added = {2020-08-16 22:13:51 +0200},
@ -17,7 +126,8 @@
Pages = {115120},
Title = {Implications of the two nodal domains conjecture for ground state fermionic wave functions},
Volume = {86},
Year = {2012}}
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.86.115120}}
@article{Giner_2013,
Author = {E. Giner and A. Scemama and M. Caffarel},
@ -32,12 +142,14 @@
@article{Becke_2014,
Author = {A. D. Becke},
Date-Added = {2020-08-16 14:00:56 +0200},
Date-Modified = {2020-08-16 14:01:02 +0200},
Date-Modified = {2020-08-17 08:51:31 +0200},
Doi = {10.1063/1.4869598},
Journal = {J. Chem. Phys.},
Pages = {18A301},
Title = {Perspective: Fifty years of density-functional theory in chemical physics},
Volume = {140},
Year = {2014}}
Year = {2014},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4869598}}
@book{ParrBook,
Address = {New York},
@ -51,22 +163,26 @@
@article{Kohn_1965,
Author = {W. Kohn and L. J. Sham},
Date-Added = {2020-08-16 13:59:30 +0200},
Date-Modified = {2020-08-16 13:59:40 +0200},
Date-Modified = {2020-08-17 08:50:29 +0200},
Doi = {10.1103/PhysRev.140.A1133},
Journal = {Phys. Rev.},
Pages = {A1133},
Title = {Self-Consistent Equations Including Exchange and Correlation Effects},
Volume = {140},
Year = {1965}}
Year = {1965},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.140.A1133}}
@article{Hohenberg_1964,
Author = {P. Hohenberg and W. Kohn},
Date-Added = {2020-08-16 13:58:30 +0200},
Date-Modified = {2020-08-16 13:59:10 +0200},
Date-Modified = {2020-08-17 08:49:58 +0200},
Doi = {10.1103/PhysRev.136.B864},
Journal = {Phys. Rev.},
Pages = {B 864},
Title = {Inhomogeneous Electron Gas},
Volume = {{136}},
Year = {1964}}
Year = {1964},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.136.B864}}
@article{Ceperley_1991,
Author = {D. M. Ceperley},
@ -126,9 +242,9 @@
@article{Needs_2020,
Author = {R. J. Needs and M. D. Towler and N. D. Drummond and P. L{\'{o}}pez R{\'{\i}}os and J. R. Trail},
Date-Added = {2020-08-16 13:48:04 +0200},
Date-Modified = {2020-08-16 13:48:26 +0200},
Date-Modified = {2020-08-17 08:51:46 +0200},
Doi = {10.1063/1.5144288},
Journal = {The Journal of Chemical Physics},
Journal = {J. Chem. Phys.},
Month = {apr},
Number = {15},
Pages = {154106},
@ -253,18 +369,16 @@
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevX.7.031059},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevX.7.031059}}
@article{Scemama_2006c,
@article{Scemama_2006,
Author = {Scemama, Anthony and Filippi, Claudia},
Date-Added = {2020-08-09 15:41:04 +0200},
Date-Modified = {2020-08-09 15:41:04 +0200},
Date-Modified = {2020-08-17 08:55:57 +0200},
Doi = {10.1103/physrevb.73.241101},
Issn = {1550-235X},
Journal = {Phys. Rev. B},
Month = {Jun},
Number = {24},
Publisher = {American Physical Society (APS)},
Pages = {241101},
Title = {Simple and efficient approach to the optimization of correlated wave functions},
Url = {http://dx.doi.org/10.1103/PhysRevB.73.241101},
Volume = {73},
Year = {2006},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevB.73.241101},
@ -273,15 +387,13 @@
@article{Umrigar_2005,
Author = {Umrigar, C. J. and Filippi, Claudia},
Date-Added = {2020-08-09 15:37:17 +0200},
Date-Modified = {2020-08-09 15:37:17 +0200},
Date-Modified = {2020-08-17 08:55:30 +0200},
Doi = {10.1103/physrevlett.94.150201},
Issn = {1079-7114},
Journal = {Phys. Rev. Lett.},
Month = {Apr},
Number = {15},
Publisher = {American Physical Society (APS)},
Pages = {150201},
Title = {Energy and Variance Optimization of Many-Body Wave Functions},
Url = {http://dx.doi.org/10.1103/PhysRevLett.94.150201},
Volume = {94},
Year = {2005},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.94.150201},
@ -324,15 +436,13 @@
@article{Umrigar_2007,
Author = {Umrigar, C. J. and Toulouse, Julien and Filippi, Claudia and Sorella, S. and Hennig, R. G.},
Date-Added = {2020-08-09 15:36:54 +0200},
Date-Modified = {2020-08-09 15:36:54 +0200},
Date-Modified = {2020-08-17 08:55:19 +0200},
Doi = {10.1103/physrevlett.98.110201},
Issn = {1079-7114},
Journal = {Phys. Rev. Lett.},
Month = {Mar},
Number = {11},
Publisher = {American Physical Society (APS)},
Pages = {110201},
Title = {Alleviation of the Fermion-Sign Problem by Optimization of Many-Body Wave Functions},
Url = {http://dx.doi.org/10.1103/PhysRevLett.98.110201},
Volume = {98},
Year = {2007},
Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.98.110201},
@ -502,9 +612,9 @@
@article{Evangelisti_1983,
Author = {Stefano Evangelisti and Jean-Pierre Daudey and Jean-Paul Malrieu},
Date-Added = {2020-08-02 18:18:19 +0200},
Date-Modified = {2020-08-02 18:18:19 +0200},
Date-Modified = {2020-08-17 08:49:23 +0200},
Doi = {10.1016/0301-0104(83)85011-3},
Journal = {Chemical Physics},
Journal = {Chem. Phys.},
Month = {feb},
Number = {1},
Pages = {91--102},
@ -586,12 +696,14 @@
@article{Garniron_2018,
Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos},
Date-Added = {2020-08-02 17:40:11 +0200},
Date-Modified = {2020-08-02 17:40:11 +0200},
Date-Modified = {2020-08-17 08:48:49 +0200},
Doi = {10.1063/1.5044503},
Journal = {J. Chem. Phys.},
Pages = {064103},
Title = {Selected Configuration Interaction Dressed by Perturbation},
Volume = {149},
Year = {2018}}
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}}
@article{Loos_2018a,
Author = {P. F. Loos and A. Scemama and A. Blondel and Y. Garniron and M. Caffarel and D. Jacquemin},
@ -748,11 +860,14 @@
@article{FraMusLupTou-JCP-15,
Author = {O. Franck and B. Mussard and E. Luppi and J. Toulouse},
Date-Modified = {2020-08-17 08:51:04 +0200},
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Journal = {J. Chem. Phys.},
Pages = {074107},
Title = {Basis convergence of range-separated density-functional theory},
Volume = {142},
Year = {2015}}
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@article{GinPraFerAssSavTou-JCP-18,
Author = {Emmanuel Giner and Barth\'elemy Pradines and Anthony Fert\'e and Roland Assaraf and Andreas Savin and Julien Toulouse},
@ -1248,29 +1363,14 @@
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpca.7b05798}}
@article{Scemama_2006,
Author = {Scemama, Anthony and Filippi, Claudia},
Doi = {10.1103/PhysRevB.73.241101},
Issn = {2469-9969},
Journal = {Phys. Rev. B},
Month = {Jun},
Number = {24},
Pages = {241101},
Publisher = {American Physical Society},
Title = {{Simple and efficient approach to the optimization of correlated wave functions}},
Volume = {73},
Year = {2006},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.73.241101}}
@article{Filippi_2000,
Author = {Filippi, Claudia and Fahy, Stephen},
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Pages = {3523--3531},
Publisher = {American Institute of Physics},
Title = {{Optimal orbitals from energy fluctuations in correlated wave functions}},
Volume = {112},
Year = {2000},

48
Manuscript/rsdft-cipsi-qmc.tex
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@ -85,10 +85,10 @@ Having low energies does not mean that they are good for chemical properties.
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Solving the Schr\"odinger equation for atoms and molecules is a complex task that has kept theoretical and computational chemists busy for almost hundred years now. \cite{Schrodinger_1926}
In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.
Solving the Schr\"odinger equation for the ground state of atoms and molecules is a complex task that has kept theoretical and computational chemists busy for almost hundred years now. \cite{Schrodinger_1926}
In order to achieve this formidable endeavour, various strategies have been carefully designed and efficiently implemented in various quantum chemistry software packages.
One of this strategies consists in relying on wave function theory (WFT) and, in particular, on the full configuration interaction (FCI) method.
One of this strategies consists in relying on wave function theory \cite{Pople_1999} (WFT) and, in particular, on the full configuration interaction (FCI) method.
However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite basis (FB) of one-electron basis functions, the FB-FCI energy being an upper bound to the exact energy in accordance with the variational principle.
The FB-FCI wave function and its corresponding energy form the eigenpair of an approximate Hamiltonian defined as
the projection of the exact Hamiltonian onto the finite many-electron basis of
@ -98,33 +98,34 @@ true Hamiltonian forced to span the restricted space provided by the finite one-
In the complete basis set (CBS) limit, the constraint is lifted and the exact energy and wave function are recovered.
Hence, the accuracy of a FB-FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
Nevertheless, the exponential growth of its computational scaling with the number of electrons and with the basis set size is prohibitive for most chemical systems.
In recent years, the introduction of new algorithms \cite{Booth_2009,Xu_2018,Eriksen_2018,Eriksen_2019} and the
revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016,Garniron_2018}
revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Evangelista_2014,Liu_2016,Per_2017,Zimmerman_2017,Ohtsuka_2017,Garniron_2018}
of selected configuration interaction (SCI)
methods \cite{Bender_1969,Huron_1973,Buenker_1974} significantly expanded the range of applicability of this family of methods.
Importantly, one can now routinely compute the ground- and excited-state energies of small- and medium-sized molecular systems with near-FCI accuracy. \cite{Booth_2010,Cleland_2010,Daday_2012,Motta_2017,Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c,Williams_2020,Eriksen_2020}
However, although the prefactor is reduced, the overall computational scaling remains exponential unless some bias is introduced leading
to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}
Another route to solve the Schr\"odinger equation is density-functional theory (DFT). \cite{Hohenberg_1964}
Another route to solve the Schr\"odinger equation is density-functional theory (DFT). \cite{Hohenberg_1964,Kohn_1999}
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 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}
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}
However, one faces the unsettling choice of the \emph{approximate} xc functional which makes inexorably KS-DFT hard to systematically improve. \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.
Diffusion Monte Carlo (DMC) is yet another numerical scheme to obtain
Diffusion Monte Carlo (DMC), which belongs to the family of stochastic methods, is yet another numerical scheme to obtain
the exact solution of the Schr\"odinger equation with a different
constraint. \cite{Foulkes_2001,Austin_2012,Needs_2020}
In DMC, the solution is imposed to have the same nodes (or zeroes)
In DMC, solution is imposed to have the same nodes (or zeroes)
as a given (approximate) antisymmetric trial wave function. \cite{Reynolds_1982,Ceperley_1991}
Within this so-called fixed-node (FN) approximation,
the FN-DMC energy associated with a given trial wave function is an upper
bound to the exact energy, and the latter is recovered only when the
nodes of the trial wave function coincide with the nodes of the exact
wave function.
The trial wave function, which can be single- or multi-determinantal in nature depending on the type of correlation at play, is then a key ingredient dictating via the quality of its nodal surface the accuracy of the resulting energy and properties.
The trial wave function, which can be single- or multi-determinantal in nature depending on the type of correlation at play, is then the key ingredient dictating, via the quality of its nodal surface, the accuracy of the resulting energy and properties.
The polynomial scaling of its computational cost with respect to the number of electrons and with the size
of the trial wave function makes the FN-DMC method particularly attractive.
@ -146,7 +147,7 @@ account of the bulk of the dynamical correlation.
%electron-electron cusp and the short-range correlation effects.
The trial wave function is then re-optimized within variational
Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008}
Using this technique, it has been shown that the chemical accuracy could be reached within
FN-DMC.\cite{Petruzielo_2012}
@ -169,19 +170,20 @@ single-reference post-Hartree-Fock method for weakly correlated systems, with an
to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
This approach obviously fails in the presence of strong correlation, like in
transition metal complexes, low-spin open-shell systems, and covalent bond breaking situations which cannot be even qualitatively described by a single electronic configuration.
In such cases, a viable alternative is to consider the FN-DMC method as a
``post-FCI'' method. The multi-determinant trial wave function is then produced by
approaching FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
When the basis set is enlarged, the trial wave function gets closer to
In such cases or when very high accuracy is required, a viable alternative is to consider the FN-DMC method as a
``post-FCI'' method. A multi-determinant trial wave function is then produced by
approaching FCI with a SCI method such as the \emph{configuration interaction using a perturbative
selection made iteratively} (CIPSI) method.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
\titou{When the basis set is enlarged, the trial wave function gets closer to
the exact wave function, so we expect the nodal surface to be
improved.\cite{Caffarel_2016}
improved.\cite{Caffarel_2016}}
This technique has the advantage of using the FCI nodes in a given basis
set, which is perfectly well defined and therefore makes the calculations reproducible in a
black-box way without needing any expertise in QMC.
black-box way without needing any QMC expertise.
Nevertheless, this technique cannot be applied to large systems because of the
exponential scaling of the size of the trial wave function.
exponential growth of the number of Slater determinants in the trial wave function.
Extrapolation techniques have been used to estimate the FN-DMC energies
obtained with FCI wave functions,\cite{Scemama_2018} and other authors
obtained with FCI wave functions,\cite{Scemama_2018,Scemama_2018b,Scemama_2019} and other authors
have used a combination of the two approaches where highly truncated
CIPSI trial wave functions are re-optimized in VMC under the presence
of a Jastrow factor to keep the number of determinants
@ -191,12 +193,12 @@ difference between the estimated FCI energy and the variational energy
of the CI wave function.\cite{Dash_2018,Dash_2019}
Nevertheless, finding a robust protocol to obtain high accuracy
calculations which can be reproduced systematically, and which is
applicable for large systems with a multi-configurational character is
calculations which can be reproduced systematically and
applicable to large systems with a multi-configurational character is
still an active field of research. The present paper falls
within this context.
The central idea of the present work, and the launch-pad for the remainder of this study, is that one can combine the various strengths of these three philosophies in order to create a hybrid method with more attractive properties.
In particular, we show here that one can combine FB-FCI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
The central idea of the present work, and the launch-pad for the remainder of this study, is that one can combine the various strengths of WFT, DFT, and DMC in order to create a new hybrid method with more attractive properties.
In particular, we show here that one can combine CIPSI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator \cite{Sav-INC-96a,Toulouse_2004} to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
@ -539,7 +541,7 @@ The take-home message of this numerical study is that RS-DFT trial wave function
The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
trial wave functions with better nodes than FCI wave function.
Such behaviour can be directly compared to the common practice of
re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008}
Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
and wave function optimization in the presence of a Jastrow factor.
For simplicity in the comparison, the molecular orbitals and the Jastrow