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%% Created for Pierre-Francois Loos at 2020-08-20 10:30:22 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@ -75,7 +75,7 @@ In particular, we combine here short-range exchange-correlation functionals with
|
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One of the take-home messages of the present study is that RS-DFT-CIPSI trial wave functions yield lower fixed-node energies with more compact multi-determinant expansions than CIPSI, especially for small basis sets.
|
||||
Indeed, as the CIPSI component of RS-DFT-CIPSI is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional CIPSI calculation.
|
||||
Importantly, by performing various numerical experiments, we evidence that the RS-DFT scheme essentially plays the role of a simple Jastrow factor by mimicking short-range correlation effects, hence avoiding the burden of performing a stochastic optimization.
|
||||
Considering the 55 atomization energies of the Gaussian-1 benchmark set of molecules, we show that using a fixed value of $\mu=0.5$~bohr$^{-1}$ provides an effective cancellation of errors as well as compact trial wave functions, making the present method a good candidate for the accurate description of large chemical systems.
|
||||
Considering the 55 atomization energies of the Gaussian-1 benchmark set of molecules, we show that using a fixed value of $\mu=0.5$~bohr$^{-1}$ provides effective error cancellations as well as compact trial wave functions, making the present method a good candidate for the accurate description of large chemical systems.
|
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\end{abstract}
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\maketitle
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@ -86,17 +86,15 @@ Considering the 55 atomization energies of the Gaussian-1 benchmark set of molec
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
||||
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 a 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.
|
||||
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 a hundred years now. \cite{Schrodinger_1926}
|
||||
In order to achieve this formidable endeavor, various strategies have been carefully designed and efficiently implemented in various quantum chemistry software packages.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Wave function-based methods}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
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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 functions, the FB-FCI energy being an upper bound to the exact energy in accordance with the variational principle.
|
||||
However, FCI delivers only the exact solution of the Schr\"odinger equation within a finite basis (FB) of one-electron 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
|
||||
all possible Slater determinants generated within this finite one-electron basis.
|
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@ -104,15 +102,15 @@ The FB-FCI wave function can then be interpreted as a constrained solution of th
|
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true Hamiltonian forced to span the restricted space provided by the finite one-electron basis.
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||||
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.
|
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Nevertheless, the exponential growth of its computational cost with the number of electrons and with the basis set size is prohibitive for most chemical systems.
|
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||||
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,Evangelista_2014,Liu_2016,Per_2017,Zimmerman_2017,Ohtsuka_2017,Garniron_2018}
|
||||
In recent years, the introduction of new algorithms \cite{White_1992,Booth_2009,Thom_2010,Xu_2018,Motta_2018,Deustua_2018,Eriksen_2018,Eriksen_2019,Ghanem_2019} and the
|
||||
revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Holmes_2017,Sharma_2017,Evangelista_2014,Liu_2016,Tubman_2016,Tubman_2020,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}
|
||||
to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017,Ghanem_2019}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Density-based methods}
|
||||
|
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@ -123,7 +121,7 @@ Present-day DFT calculations are almost exclusively done within the so-called Ko
|
|||
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.
|
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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,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.
|
||||
However, unlike WFT where, for example, 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.
|
||||
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||||
|
|
@ -161,7 +159,7 @@ The conventional approach consists in multiplying the determinantal part of the
|
|||
function by a positive function, the Jastrow factor, which main assignment is to take into
|
||||
account the bulk of the dynamical electron correlation and reduce the statistical fluctuation without altering the location of the nodes.
|
||||
%electron-electron cusp and the short-range correlation effects.
|
||||
The determinantal part of the trial wave function is then re-optimized within variational
|
||||
The determinantal part of the trial wave function is then stochastically re-optimized within variational
|
||||
Monte Carlo (VMC) in the presence of the Jastrow factor (which can also be simultaneously optimized) and the nodal
|
||||
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
|
||||
|
|
@ -179,7 +177,7 @@ determinant. This feature is in part responsible for the success of
|
|||
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 CC.\cite{Dubecky_2014,Grossman_2002}
|
||||
to CCSD(T), \cite{Dubecky_2014,Grossman_2002} the gold standard of WFT for ground state energies.
|
||||
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.
|
||||
|
|
@ -208,8 +206,8 @@ 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.
|
||||
Yet, 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
|
||||
single-determinant ans\"atz does not have enough flexibility for describing the
|
||||
nodal surface of the exact correlated wave function for 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.
|
||||
|
|
@ -230,7 +228,7 @@ improved.\cite{Caffarel_2016}
|
|||
Note that, as discussed in Ref.~\onlinecite{Caffarel_2016_2}, there is no mathematical guarantee that increasing the size of the one-electron basis lowers the FN-DMC energy, because the variational principle does not explicitly optimize the nodal surface, nor the FN-DMC energy.
|
||||
However, in all applications performed so far, \cite{Giner_2013,Scemama_2014,Scemama_2016,Giner_2015,Caffarel_2016,Scemama_2018,Scemama_2018b,Scemama_2019} a systematic decrease of the FN-DMC energy has been observed whenever the SCI trial wave function is improved variationally upon enlargement of the basis set.
|
||||
|
||||
The technique relying on SCI multi-determinant trial wave functions described above has the advantage of using near-FCI quality nodes in a given basis
|
||||
The technique relying on CIPSI multi-determinant trial wave functions described above has the advantage of using near-FCI quality nodes in a given basis
|
||||
set, which is perfectly well defined and therefore makes the calculations systematically improvable and reproducible in a
|
||||
black-box way without needing any QMC expertise.
|
||||
Nevertheless, this procedure cannot be applied to large systems because of the
|
||||
|
|
@ -238,7 +236,7 @@ exponential growth of the number of Slater determinants in the trial wave functi
|
|||
Extrapolation techniques have been employed to estimate the FN-DMC energies
|
||||
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
|
||||
CIPSI trial wave functions are stochastically re-optimized in VMC under the presence
|
||||
of a Jastrow factor to keep the number of determinants
|
||||
small,\cite{Giner_2016} and where the consistency between the
|
||||
different wave functions is kept by imposing a constant energy
|
||||
|
|
@ -252,7 +250,8 @@ 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 QMC in order to create a new hybrid method with more attractive features and higher accuracy.
|
||||
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} --- a scheme that we label RS-DFT-CIPSI in the following --- to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
|
||||
|
||||
An important take-home message from the present study is that the RS-DFT scheme essentially plays the role of a simple Jastrow factor by mimicking short-range correlation effects.
|
||||
Thanks to this, RS-DFT-CIPSI multi-determinant trial wave functions yield lower fixed-node energies with more compact multi-determinant expansion than CIPSI, especially for small basis sets, and can be produced in a completely deterministic and systematic way, without the burden of the stochastic optimization.
|
||||
|
||||
The present manuscript is organized as follows.
|
||||
In Sec.~\ref{sec:rsdft-cipsi}, we provide theoretical details about the CIPSI algorithm (Sec.~\ref{sec:CIPSI}) and range-separated DFT (Sec.~\ref{sec:rsdft}).
|
||||
|
|
@ -761,7 +760,7 @@ calculations are atom-centered, so they are, by construction, better adapted to
|
|||
atoms than molecules. Thus, atomization energies usually tend to be
|
||||
underestimated by variational methods.
|
||||
In the context of FN-DMC calculations, the nodal surface is imposed by
|
||||
the determinantal part of the trial wavefunction which is expanded in the very same atom-centered basis
|
||||
the determinantal part of the trial wave function which is expanded in the very same atom-centered basis
|
||||
set. Thus, we expect the fixed-node error to be also intimately connected to
|
||||
the basis set incompleteness error.
|
||||
Increasing the size of the basis set improves the description of
|
||||
|
|
@ -833,7 +832,7 @@ 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
|
||||
molecules at the KS-DFT level with various semi-local and hybrid density functionals [PBE, \cite{PerBurErn-PRL-96} BLYP, \cite{Becke_1988,Lee_1988} PBE0, \cite{Perdew_1996} and B3LYP \cite{Becke_1993}], 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)
|
||||
with respect to the NIST reference values as explained in Sec.~\ref{sec:comp-details}.
|
||||
|
|
@ -846,7 +845,7 @@ In this benchmark, the great majority of the systems are weakly correlated and a
|
|||
described by a single determinant. Therefore, the atomization energies
|
||||
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
|
||||
PBE) 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 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,
|
||||
|
|
@ -908,7 +907,7 @@ Searching for the optimal value of $\mu$ may be too costly and time consuming, s
|
|||
computed the MAEs, MSEs and RMSEs 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
|
||||
$0.5$ 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 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
|
||||
|
|
|
|||
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Reference in New Issue