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168
Manuscript/rsdft-cipsi-qmc.bib
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@ -1,13 +1,29 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-17 10:19:43 +0200
%% Created for Pierre-Francois Loos at 2020-08-17 12:55:12 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2015b,
Author = {Loos, Pierre-Fran{\c c}ois and Bressanini, Dario},
Date-Added = {2020-08-17 12:54:30 +0200},
Date-Modified = {2020-08-17 12:54:30 +0200},
Doi = {10.1063/1.4922159},
File = {/Users/loos/Zotero/storage/UBTEMR7G/45.pdf},
Issn = {0021-9606, 1089-7690},
Journal = {J. Chem. Phys.},
Month = jun,
Number = {21},
Pages = {214112},
Title = {Nodal Surfaces and Interdimensional Degeneracies},
Volume = {142},
Year = {2015},
Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4922159}}
@article{Scemama_2016,
Author = {Scemama, Anthony and Applencourt, Thomas and Giner, Emmanuel and Caffarel, Michel},
Date-Added = {2020-08-17 10:18:21 +0200},
@ -51,7 +67,8 @@
Pages = {100002},
Title = {Influence of pseudopotentials on excitation energies from selected configuration interaction and diffusion Monte Carlo},
Volume = {1},
Year = {2019}}
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1016/j.rechem.2019.100002}}
@article{Giner_2020,
Author = {E. Giner and A. Scemama and P. F. Loos and J. Toulouse},
@ -62,7 +79,8 @@
Pages = {174104},
Title = {A basis-set error correction based on density-functional theory for strongly correlated molecular systems},
Volume = {152},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1063/5.0002892}}
@article{Loos_2019d,
Author = {P. F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
@ -1527,88 +1545,82 @@
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.87.043401}}
@article{BoyHan-PRSLA-69,
author = {Francis, Boys Samuel and Charles, Handy Nicholas},
title = {{A condition to remove the indeterminacy in interelectronic correlation functions}},
journal = {Proc. R. Soc. Lond. A.},
volume = {309},
number = {1497},
pages = {209--220},
year = {1969},
month = {Mar},
publisher = {The Royal Society},
doi = {10.1098/rspa.1969.0038}
}
Author = {Francis, Boys Samuel and Charles, Handy Nicholas},
Doi = {10.1098/rspa.1969.0038},
Journal = {Proc. R. Soc. Lond. A.},
Month = {Mar},
Number = {1497},
Pages = {209--220},
Publisher = {The Royal Society},
Title = {{A condition to remove the indeterminacy in interelectronic correlation functions}},
Volume = {309},
Year = {1969},
Bdsk-Url-1 = {https://doi.org/10.1098/rspa.1969.0038}}
@article{BoyHanLin-1-PRSLA-69,
author = {Francis, Boys Samuel and Charles, Handy Nicholas and
Wilfrid, Linnett John},
title = {{The determination of energies and wavefunctions with full electronic correlation}},
journal = {Proc. R. Soc. Lond. A.},
volume = {310},
number = {1500},
pages = {43--61},
year = {1969},
month = {Apr},
publisher = {The Royal Society},
doi = {10.1098/rspa.1969.0061}
}
Author = {Francis, Boys Samuel and Charles, Handy Nicholas and Wilfrid, Linnett John},
Doi = {10.1098/rspa.1969.0061},
Journal = {Proc. R. Soc. Lond. A.},
Month = {Apr},
Number = {1500},
Pages = {43--61},
Publisher = {The Royal Society},
Title = {{The determination of energies and wavefunctions with full electronic correlation}},
Volume = {310},
Year = {1969},
Bdsk-Url-1 = {https://doi.org/10.1098/rspa.1969.0061}}
@article{BoyHanLin-2-PRSLA-69,
author = {Francis, Boys Samuel and Charles, Handy Nicholas and
Wilfrid, Linnett John},
title = {{A calculation for the energies and wavefunctions for states of neon with
full electronic correlation accuracy}},
journal = {Proc. R. Soc. Lond. A.},
volume = {310},
number = {1500},
pages = {63--78},
year = {1969},
month = {Apr},
publisher = {The Royal Society},
doi = {10.1098/rspa.1969.0062}
}
Author = {Francis, Boys Samuel and Charles, Handy Nicholas and Wilfrid, Linnett John},
Doi = {10.1098/rspa.1969.0062},
Journal = {Proc. R. Soc. Lond. A.},
Month = {Apr},
Number = {1500},
Pages = {63--78},
Publisher = {The Royal Society},
Title = {{A calculation for the energies and wavefunctions for states of neon with full electronic correlation accuracy}},
Volume = {310},
Year = {1969},
Bdsk-Url-1 = {https://doi.org/10.1098/rspa.1969.0062}}
@article{Luo-JCP-10,
author = {Luo, Hongjun},
title = {{Variational transcorrelated method}},
journal = {J. Chem. Phys.},
volume = {133},
number = {15},
pages = {154109},
year = {2010},
month = {Oct},
issn = {0021-9606},
publisher = {American Institute of Physics},
doi = {10.1063/1.3505037}
}
Author = {Luo, Hongjun},
Doi = {10.1063/1.3505037},
Issn = {0021-9606},
Journal = {J. Chem. Phys.},
Month = {Oct},
Number = {15},
Pages = {154109},
Publisher = {American Institute of Physics},
Title = {{Variational transcorrelated method}},
Volume = {133},
Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3505037}}
@article{YanShi-JCP-12,
author = {Yanai, Takeshi and Shiozaki, Toru},
title = {{Canonical transcorrelated theory with projected Slater-type geminals}},
journal = {J. Chem. Phys.},
volume = {136},
number = {8},
pages = {084107},
year = {2012},
month = {Feb},
issn = {0021-9606},
publisher = {American Institute of Physics},
doi = {10.1063/1.3688225}
}
Author = {Yanai, Takeshi and Shiozaki, Toru},
Doi = {10.1063/1.3688225},
Issn = {0021-9606},
Journal = {J. Chem. Phys.},
Month = {Feb},
Number = {8},
Pages = {084107},
Publisher = {American Institute of Physics},
Title = {{Canonical transcorrelated theory with projected Slater-type geminals}},
Volume = {136},
Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3688225}}
@article{CohLuoGutDowTewAla-JCP-19,
author = {Cohen, Aron J. and Luo, Hongjun and
Guther, Kai and Dobrautz, Werner and Tew, David P. and Alavi, Ali},
title = {{Similarity transformation of the electronic Schrödinger equation via Jastrow
factorization}},
journal = {J. Chem. Phys.},
volume = {151},
number = {6},
pages = {061101},
year = {2019},
month = {Aug},
issn = {0021-9606},
publisher = {American Institute of Physics},
doi = {10.1063/1.5116024}
}
Author = {Cohen, Aron J. and Luo, Hongjun and Guther, Kai and Dobrautz, Werner and Tew, David P. and Alavi, Ali},
Doi = {10.1063/1.5116024},
Issn = {0021-9606},
Journal = {J. Chem. Phys.},
Month = {Aug},
Number = {6},
Pages = {061101},
Publisher = {American Institute of Physics},
Title = {{Similarity transformation of the electronic Schr{\"o}dinger equation via Jastrow factorization}},
Volume = {151},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5116024}}

117
Manuscript/rsdft-cipsi-qmc.tex
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@ -89,6 +89,10 @@ Having low energies does not mean that they are good for chemical properties.
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Wave function-based methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One of these 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
@ -108,6 +112,10 @@ Importantly, one can now routinely compute the ground- and excited-state energie
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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Density-based methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.
@ -117,6 +125,10 @@ As compared to WFT, DFT has the indisputable advantage of converging much faster
one faces 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stochastic methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}
@ -127,7 +139,7 @@ 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 the 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 and the target accuracy, 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.
@ -153,7 +165,6 @@ surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007
Using this technique, it has been shown that the chemical accuracy could be reached within
FN-DMC.\cite{Petruzielo_2012}
%However, it is usually harder to control the FN error in DMC, and this
%might affect energy differences such as atomization energies.
%Moreover, improving systematically the nodal surface of the trial wave
@ -162,16 +173,57 @@ FN-DMC.\cite{Petruzielo_2012}
%with respect to the variational parameters of the wave function can't
%be computed}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Single-determinant trial wave functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The qualitative picture of the electronic structure of weakly
correlated systems, such as organic molecules near their equilibrium
geometry, is usually well represented with a single Slater
determinant. This feature is in part responsible for the success of
DFT and coupled cluster theory.
DFT and coupled cluster (CC) theory.
Likewise, DMC with a single-determinant trial wave function can be used as a
single-reference post-Hartree-Fock method for weakly correlated systems, with an accuracy comparable
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 \trashtoto{even} qualitatively described by a single electronic configuration.
to CC.\cite{Dubecky_2014,Grossman_2002}
In single-determinant DMC calculations, the only degree of freedom available to
reduce the fixed-node error are the molecular orbitals with which the
Slater determinant is built.
Different molecular orbitals can be chosen:
Hartree-Fock (HF), Kohn-Sham (KS), natural orbitals (NOs) of a
correlated wave function, or orbitals optimized in the
presence of a Jastrow factor.
The nodal surfaces obtained with a KS determinant are in general
better than those obtained with a HF determinant,\cite{Per_2012} and
of comparable quality to those obtained with a Slater determinant
built with NOs.\cite{Wang_2019} Orbitals obtained in the presence
of a Jastrow factor are generally superior to KS
orbitals.\cite{Filippi_2000,Scemama_2006,HaghighiMood_2017,Ludovicy_2019}
The description of electron correlation within DFT is very different
from correlated methods such as FCI or CC.
\titou{As mentioned above, within KS-DFT, one solves a mean-field problem with a modified potential
incorporating the effects of electron correlation, whereas in
correlated methods the real Hamiltonian is used and the
electron-electron interactions are considered.}
Nevertheless, as the orbitals are one-electron functions,
the procedure of orbital optimization in the presence of a
Jastrow factor can be interpreted as a self-consistent field procedure
with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.
So KS-DFT can be viewed as a very cheap way of introducing the effect of
correlation in the orbital coefficients dictating the location of the nodes of a single Slater determinant.
Nevertheless, even when employing the exact xc potential in a complete basis set, a fixed-node error necessarily remains because the
single-determinant ans\"atz does not have enough flexibility to describe the
nodal surface of the exact correlated wave function of a generic many-electron
system. \cite{Ceperley_1991,Bressanini_2012,Loos_2015b}
If one wants to recover the exact energy, a multi-determinant parameterization
of the wave functions must be considered.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Multi-determinant trial wave functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The single-determinant trial wave function 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 qualitatively described by a single electronic configuration.
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
@ -205,59 +257,22 @@ 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 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Combining WFT and DFT}
\section{Theory}
\label{sec:rsdft-cipsi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In single-determinant DMC calculations, the degrees of freedom used to
reduce the fixed-node error are the molecular orbitals on which the
Slater determinant is built.
Different molecular orbitals can be chosen:
Hartree-Fock (HF), Kohn-Sham (KS), natural orbitals (NOs) of a
correlated wave function, or orbitals optimized under the
presence of a Jastrow factor.
The nodal surfaces obtained with the KS determinant are in general
better than those obtained with the HF determinant,\cite{Per_2012} and
of comparable quality to those obtained with a Slater determinant
built with NOs.\cite{Wang_2019} Orbitals obtained in the presence
of a Jastrow factor are generally superior to KS
orbitals.\cite{Filippi_2000,Scemama_2006,HaghighiMood_2017,Ludovicy_2019}
The description of electron correlation within DFT is very different
from correlated methods.
In DFT, one solves a mean field problem with a modified potential
incorporating the effects of electron correlation, whereas in
correlated methods the real Hamiltonian is used and the
electron-electron interactions are considered.
Nevertheless, as the orbitals are one-electron functions,
the procedure of orbital optimization in the presence of the
Jastrow factor can be interpreted as a self-consistent field procedure
with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.
So KS-DFT can be viewed as a very cheap way of introducing the effect of
correlation in the orbital parameters determining the nodal surface
of a single Slater determinant.
Nevertheless, even when using the exact exchange correlation potential at the
CBS limit, a fixed-node error necessarily remains because the
single-determinant ansätz does not have enough flexibility to describe the
nodal surface of the exact correlated wave function of a generic $N$-electron
system. \cite{Bressanini_2012}
If one wants to recover the exact CBS limit, a multi-determinant parameterization
of the wave functions is required.
%====================
\subsection{CIPSI}
\subsection{The CIPSI algorithm}
%====================
Beyond the single-determinant representation, the best
multi-determinant wave function one can obtain in a given basis set is the FCI.
FCI is the ultimate goal of \emph{post-Hartree-Fock} methods, and there exists several systematic
improvements between the Hartree-Fock and FCI wave functions:
multi-determinant wave function one can wish for --- in a given basis set --- is the FCI wave function.
FCI is the ultimate goal of post-HF methods, and there exists several systematic
improvements on the path from HF to FCI:
increasing the maximum degree of excitation of CI methods (CISD, CISDT,
CISDTQ, \emph{etc}), or increasing the complete active space
CISDTQ, \ldots), or increasing the complete active space
(CAS) wave functions until all the orbitals are in the active space.
Selected CI methods take a shorter path between the Hartree-Fock
Selected CI methods take a shortcut between the HF
determinant and the FCI wave function by increasing iteratively the
number of determinants on which the wave function is expanded,
selecting the determinants which are expected to contribute the most
@ -424,7 +439,7 @@ All the 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 (BFD-VXZ).
The small-core BFD pseudopotentials include scalar relativistic effects.
CCSD(T) and KS-DFT energies have been computed with
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