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\newcommand { \EPT } { E_ { \text { PT2} } }
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\newcommand { \EDMC } { E_ { \text { FN-DMC} } }
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\newcommand { \Ndet } { N_ { \text { det} } }
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\newcommand { \LCT } { Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}
\newcommand { \ANL } { Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
\newcommand { \LCPQ } { Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}
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\begin { document}
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\title { Enabling high accuracy diffusion Monte Carlo calculations with
range-separated density functional theory and selected configuration interaction}
\author { Anthony Scemama}
\affiliation { \LCPQ }
\author { Emmanuel Giner}
\email { emmanuel.giner@lct.jussieu.fr}
\affiliation { \LCT }
\author { Anouar Benali}
\email { benali@anl.gov}
\affiliation { \ANL }
\author { Pierre-Fran\c { c} ois Loos}
\email { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\begin { abstract}
\end { abstract}
\maketitle
\section { Introduction}
\label { sec:intro}
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The full configuration interaction (FCI) method within a finite atomic
basis set leads to an approximate solution of the Schrödinger
equation.
This solution is the eigenpair of an approximate Hamiltonian, which is
the projection of the exact Hamiltonian onto the finite basis of all
possible Slater determinants.
The FCI wave function can be interpreted as the constrained solution of the
true Hamiltonian, where the solution is forced to span the space
provided by the basis.
At the complete basis set (CBS) limit, the constraint vanishes and the
exact solution is obtained.
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Hence the FCI method enables a systematic improvement of the
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calculations by improving the quality of the basis set.
Nevertheless, its exponential scaling with the number of electrons and
with the size of the basis is prohibitive for large systems.
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In recent years, the introduction of new algorithms\cite { Booth_ 2009}
and the
revival\cite { Abrams_ 2005,Bytautas_ 2009,Roth_ 2009,Giner_ 2013,Knowles_ 2015,Holmes_ 2016,Liu_ 2016}
of selected configuration interaction (sCI)
methods\cite { Bender_ 1969,Huron_ 1973,Buenker_ 1974} pushed the limits of
the sizes of the systems that could be computed at the FCI level, but
the scaling remains exponential unless some bias is introduced leading
to a loss of size extensivity.
The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
the exact solution of the Schrödinger equation with an additional
constraint, imposing the solution to have the same nodal hypersurface
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as a given trial wave function.
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Within this so-called \emph { fixed-node} (FN) approximation,
the FN-DMC energy associated with a given trial wave function is an upper
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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 polynomial scaling with the number of electrons and with the size
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of the trial wave function makes the FN-DMC method attractive.
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In addition, the total energies obtained are usually far below
those obtained with the FCI method in computationally tractable basis
sets because the constraints imposed by the fixed-node approximation
are less severe than the constraints imposed by the finite-basis
approximation.
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
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determinant. This feature is in part responsible for the success of
density functional theory (DFT) and coupled cluster.
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DMC with a single-determinant trial wave function can be used as a
single-reference post-Hatree-Fock method, with an accuracy comparable
to coupled cluster.\cite { Dubecky_ 2014,Grossman_ 2002}
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The favorable scaling of QMC, its very low memory requirements and
its adequation with massively parallel architectures make it a
serious alternative for high-accuracy simulations on large systems.
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As it is not possible to minimize directly the FN-DMC energy with respect
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to the variational parameters of the trial wave function, the
fixed-node approximation is much more difficult to control than the
finite-basis approximation.
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The conventional approach consists in multiplying the trial wave
function by a positive function, the \emph { Jastrow factor} , taking
account of the electron-electron cusp and the short-range correlation
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effects. The 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. Using this technique, it has been
shown that the chemical accuracy could be reached within
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FN-DMC.\cite { Petruzielo_ 2012}
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Another approach consists in considering the FN-DMC method as a
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\emph { post-FCI method} . The trial wave function is obtained by
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approaching the FCI with a selected configuration interaction (sCI)
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method such as CIPSI for instance.\cite { Giner_ 2013,Caffarel_ 2016_ 2}
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When the basis set is increased, the trial wave function gets closer
to the exact wave function, so the nodal surface can be systematically
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improved.\cite { Caffarel_ 2016}
This technique has the advantage that using FCI nodes in a given basis
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set is well defined, so the calculations are reproducible in a
black-box way without needing any expertise in QMC.
But this technique can't be applied to large systems because of the
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exponential scaling of the size of the trial wave function.
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Extrapolation techniques have been used to estimate the FN-DMC energies
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obtained with FCI wave functions,\cite { Scemama_ 2018} and other authors
have used a combination of the two approaches where highly truncated
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CIPSI trial wave functions are re-optimized in VMC under the presence
of a Jastrow factor to keep the number of determinants
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small,\cite { Giner_ 2016} and where the consistency between the
different wave functions is kept by imposing a constant energy
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
still an active field of research. The present paper falls
within this context.
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\section { Combining CIPSI with range-separated DFT}
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\label { sec:rsdft-cipsi}
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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 (NO) orbitals 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.
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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.
If one wants to have to exact CBS limit, a multi-determinant parameterization
of the wave functions is required.
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\subsection { CIPSI}
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Beyond the single-determinant representation, the best
multi-determinant wave function one can obtain is the FCI. FCI is
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a \emph { post-Hartree-Fock} method, and there exists several systematic
improvements between the Hartree-Fock and FCI wave functions:
increasing the maximum degree of excitation of CI methods (CISD, CISDT,
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CISDTQ, \emph { etc} ), or increasing the complete active space
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(CAS) wave functions until all the orbitals are in the active space.
Selected CI methods take a shorter path between the Hartree-Fock
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determinant and the FCI wave function by increasing iteratively the
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number of determinants on which the wave function is expanded,
selecting the determinants which are expected to contribute the most
to the FCI eigenvector. At every iteration, the lowest eigenpair is
extracted from the CI matrix expressed in the determinant subspace,
and the FCI energy can be estimated by computing a second-order
perturbative correction (PT2) to the variational energy, $ \EPT $ .
The magnitude of $ \EPT $ is a
measure of the distance to the exact eigenvalue, and is an adjustable
parameter controlling the quality of the wave function.
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Within the \emph { Configuration interaction using a perturbative
selection made iteratively} (CIPSI)\cite { Huron_ 1973} method, the PT2
correction is computed along with the determinant selection. So the
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magnitude of $ \EPT $ can be made the only parameter of the algorithm,
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and we choose this parameter as the convergence criterion of the CIPSI
algorithm.
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Considering that the perturbatively corrected energy is a reliable
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estimate of the FCI energy, using a fixed value of the PT2 correction
as a stopping criterion enforces a constant distance of all the
calculations to the FCI energy. In this work, we target the chemical
accuracy so all the CIPSI selections were made such that $ | \EPT | <
1$ ~mE $ _ h$ .
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\subsection { Range-separated DFT}
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\label { sec:rsdft}
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Following the seminal work of Savin,\cite { Savin_ 1996,Toulouse_ 2004}
the Coulomb electron-electron interaction is split into a short-range
(sr) and a long range (lr) interaction as
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\begin { equation}
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\frac { 1} { r_ { ij} } = w_ { \text { ee} } ^ { \text { lr} , \mu } (r_ { ij} ) + \qty (
\frac { 1} { r_ { ij} } - w_ { \text { ee} } ^ { \text { lr} , \mu } (r_ { ij} ) )
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\end { equation}
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where
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\begin { equation}
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w_ { \text { ee} } ^ { \text { lr} ,\mu } (r_ { ij} ) = \frac { \erf \qty ( \mu \, r_ { ij} )} { r_ { ij} }
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\end { equation}
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The main idea is to treat the short-range electron-electron
interaction with DFT, and the long range with wave function theory.
The parameter $ \mu $ controls the range of the separation, and allows
to go continuously from the Kohn-Sham Hamiltonian ($ \mu = 0 $ ) to
the FCI Hamiltinoan ($ \mu = \infty $ ).
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To rigorously connect wave function theory and DFT, the universal
Levy-Lieb density functional\cite { Lev-PNAS-79,Lie-IJQC-83} is
decomposed as
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\begin { equation}
\mathcal { F} [n] = \mathcal { F} ^ { \mathrm { lr} ,\mu } [n] + \bar { E} _ { \mathrm { Hxc} } ^ { \mathrm { sr,} \mu } [n],
\label { Fdecomp}
\end { equation}
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where $ n $ is a one-particle density,
$ \mathcal { F } ^ { \mathrm { lr } , \mu } $ is a long-range universal density
functional and $ \bar { E } _ { \mathrm { Hxc } } ^ { \mathrm { sr, } \mu } $ is the
complementary short-range Hartree-exchange-correlation (Hxc) density
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functional\cite { Savin_ 1996,Toulouse_ 2004} .
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One obtains the following expression for the ground-state
electronic energy
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\begin { equation}
\label { min_ rsdft} E_ 0= \min _ { \Psi } \left \{
\left
\langle \Psi |\hat { T} +\hat { W} _ \mathrm { { ee} } ^ { \mathrm { lr} ,\mu } +\hat { V} _ { \mathrm { ne} } |\Psi \right
\rangle
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+ \bar { E} ^ { \mathrm { sr} ,\mu } _ { \mathrm { Hxc} } [n_ \Psi ]\right \}
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\end { equation}
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with $ \hat { T } $ the kinetic energy operator,
$ \hat { W } _ \mathrm { ee } ^ { \mathrm { lr } } $ the long-range
electron-electron interaction,
$ n _ \Psi $ the one-particle density associated with $ \Psi $ ,
and $ \hat { V } _ { \mathrm { ne } } $ the electron-nucleus potential.
The minimizing multi-determinant wave function $ \Psi ^ \mu $
can be determined by the self-consistent eigenvalue equation
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\begin { equation}
\label { rs-dft-eigen-equation}
\hat { H} ^ \mu [n_{\Psi^{\mu}}] \ket { \Psi ^ { \mu } } = \mathcal { E} ^ { \mu } \ket { \Psi ^ { \mu } } ,
\end { equation}
with the long-range interacting Hamiltonian
\begin { equation}
\label { H_ mu}
\hat { H} ^ \mu [n_{\Psi^{\mu}}] = \hat { T} +\hat { W} _ { \mathrm { ee} } ^ { \mathrm { lr} ,\mu } +\hat { V} _ { \mathrm { ne} } + \hat { \bar { V} } _ { \mathrm { Hxc} } ^ { \mathrm { sr} ,\mu } [n_ { \Psi ^ { \mu } } ],
\end { equation}
where
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$ \hat { \bar { V } } _ { \mathrm { Hxc } } ^ { \mathrm { sr } , \mu } $
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is the complementary short-range Hartree-exchange-correlation
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potential operator.
Once $ \Psi ^ { \mu } $ has been calculated, the electronic ground-state
energy is obtained by
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\begin { equation}
\label { E-rsdft}
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E_ 0= \mel { \Psi ^ { \mu } } { \hat { T} +\hat { W} _ \mathrm { { ee} } ^ { \mathrm { lr} ,\mu } +\hat { V} _ { \mathrm { ne} } } { \Psi ^ { \mu } } +\bar { E} ^ { \mathrm { sr} ,\mu } _ { \mathrm { Hxc} } [n_ { \Psi ^ \mu } ].
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\end { equation}
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Note that, for $ \mu = 0 $ , the long-range interaction vanishes,
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$ w _ { \mathrm { ee } } ^ { \mathrm { lr } , \mu = 0 } ( r _ { 12 } ) = 0 $ , and thus
range-separated DFT (RS-DFT) reduces to standard KS-DFT and $ \Psi ^ \mu $
is the KS determinant. For $ \mu \to \infty $ , the long-range
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interaction becomes the standard Coulomb interaction,
$ w _ { \mathrm { ee } } ^ { \mathrm { lr } , \mu \to \infty } ( r _ { 12 } ) = 1 / r _ { 12 } $ , and
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thus RS-DFT reduces to standard wave-function theory and $ \Psi ^ \mu $ is
the FCI wave function.
\begin { figure*}
\centering
\includegraphics [width=0.7\linewidth] { algorithm.pdf}
\caption { Algorithm showing the generation of the RS-DFT wave
function.}
\label { fig:algo}
\end { figure*}
Hence we have a continuous path connecting the KS determinant to the
FCI wave function, and as the KS nodes are of higher quality than the
HF nodes, we expect that using wave functions built along this path
will always provide reduced fixed-node errors compared to the path
connecting HF to FCI using an increasing number of selected
determinants.
We can follow this path by performing FCI calculations using the
RS-DFT Hamiltonian with different values of $ \mu $ . In this work, we
have used the CIPSI algorithm to peform approximate FCI calculations
with the RS-DFT Hamiltonians,\cite { GinPraFerAssSavTou-JCP-18} as shown
in figure~\ref { fig:algo} . In the outer loop (red), a CIPSI selection
is performed with a RS-Hamiltonian parameterized using the current
density. An inner loop (blue) is introduced to accelerate the
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convergence of the self-consistent calculation, in which the set of
determinants is kept fixed, and only the diagonalization of the
RS-Hamiltonian is performed iteratively.
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The convergence of the algorithm was further improved
by introducing a direct inversion in the iterative subspace (DIIS)
step to extrapolate the density both in the outer and inner loops.
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As mentioned above, the convergence criterion for CIPSI was set to
$ \EPT < 1 $ ~m$ E _ h $ .
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\section { Computational details}
\label { sec:comp-details}
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All the calculations were made using BFD
pseudopotentials\cite { Burkatzki_ 2008} with the associated double,
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triple and quadruple zeta basis sets (BFD-V$ n $ Z).
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CCSD(T) and DFT calculations were made with
\emph { Gaussian09} ,\cite { g16} using an unrestricted Hartree-Fock
determinant as a reference for open-shell systems.
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All the CIPSI calculations were made with \emph { Quantum
Package} .\cite { Garniron_ 2019,qp2_ 2020} We used the short-range version
of the Perdew-Burke-Ernzerhof (PBE)~\cite { PerBurErn-PRL-96} exchange
and correlation functionals of
Ref.~\onlinecite { GolWerStoLeiGorSav-CP-06} (see also
Refs.~\onlinecite { TouColSav-JCP-05,GolWerSto-PCCP-05} ).
The convergence criterion for stopping the CIPSI calculations
was $ \EPT < 1 $ ~m$ E _ h \vee \Ndet > 10 ^ 7 $ .
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Quantum Monte Carlo calculations were made with QMC=Chem,\cite { scemama_ 2013}
in the determinant localization approximation (DLA),\cite { Zen_ 2019}
where only the determinantal component of the trial wave
function is present in the expression of the wave function on which
the pseudopotential is localized. Hence, in the DLA the fixed-node
energy is independent of the Jatrow factor, as in all-electron
calculations.
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\section { Influence of the range-separation parameter on the fixed-node
error}
\label { sec:mu-dmc}
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\begin { table}
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\caption { Fixed-node energies and number of determinants in the water
molecule and the fluorine dimer with different trial wave functions.}
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\label { tab:h2o-dmc}
\centering
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\begin { ruledtabular}
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\begin { tabular} { crlrl}
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& \multicolumn { 2} { c} { BFD-VDZ} & \multicolumn { 2} { c} { BFD-VTZ} \\
$ \mu $ & $ \Ndet $ & $ \EDMC $ & $ \Ndet $ & $ \EDMC $ \\
\hline
& \multicolumn { 4} { c} { H$ _ 2 $ O} \\
$ 0 . 00 $ & $ 11 $ & $ - 17 . 253 \, 59 ( 6 ) $ & $ 23 $ & $ - 17 . 256 \, 74 ( 7 ) $ \\
$ 0 . 20 $ & $ 23 $ & $ - 17 . 253 \, 73 ( 7 ) $ & $ 23 $ & $ - 17 . 256 \, 73 ( 8 ) $ \\
$ 0 . 30 $ & $ 53 $ & $ - 17 . 253 \, 4 ( 2 ) $ & $ 219 $ & $ - 17 . 253 \, 7 ( 5 ) $ \\
$ 0 . 50 $ & $ 1 \, 442 $ & $ - 17 . 253 \, 9 ( 2 ) $ & $ 16 \, 99 $ & $ - 17 . 257 \, 7 ( 2 ) $ \\
$ 0 . 75 $ & $ 3 \, 213 $ & $ - 17 . 255 \, 1 ( 2 ) $ & $ 13 \, 362 $ & $ - 17 . 258 \, 4 ( 3 ) $ \\
$ 1 . 00 $ & $ 6 \, 743 $ & $ - 17 . 256 \, 6 ( 2 ) $ & $ 256 \, 73 $ & $ - 17 . 261 \, 0 ( 2 ) $ \\
$ 1 . 75 $ & $ 54 \, 540 $ & $ - 17 . 259 \, 5 ( 3 ) $ & $ 207 \, 475 $ & $ - 17 . 263 \, 5 ( 2 ) $ \\
$ 2 . 50 $ & $ 51 \, 691 $ & $ - 17 . 259 \, 4 ( 3 ) $ & $ 858 \, 123 $ & $ - 17 . 264 \, 3 ( 3 ) $ \\
$ 3 . 80 $ & $ 103 \, 059 $ & $ - 17 . 258 \, 7 ( 3 ) $ & $ 1 \, 621 \, 513 $ & $ - 17 . 263 \, 7 ( 3 ) $ \\
$ 5 . 70 $ & $ 102 \, 599 $ & $ - 17 . 257 \, 7 ( 3 ) $ & $ 1 \, 629 \, 655 $ & $ - 17 . 263 \, 2 ( 3 ) $ \\
$ 8 . 50 $ & $ 101 \, 803 $ & $ - 17 . 257 \, 3 ( 3 ) $ & $ 1 \, 643 \, 301 $ & $ - 17 . 263 \, 3 ( 4 ) $ \\
$ \infty $ & $ 200 \, 521 $ & $ - 17 . 256 \, 8 ( 6 ) $ & $ 1 \, 631 \, 982 $ & $ - 17 . 263 \, 9 ( 3 ) $ \\
& \multicolumn { 3} { c} { F$ _ 2 $ } \\
$ 0 . 00 $ & $ 23 $ & $ - 48 . 419 \, 5 ( 4 ) $ \\
$ 0 . 25 $ & $ 8 $ & $ - 48 . 421 \, 9 ( 4 ) $ \\
$ 0 . 50 $ & $ 1743 $ & $ - 48 . 424 \, 8 ( 8 ) $ \\
$ 1 . 00 $ & $ 11952 $ & $ - 48 . 432 \, 4 ( 3 ) $ \\
$ 2 . 00 $ & $ 829438 $ & $ - 48 . 441 \, 0 ( 7 ) $ \\
$ 5 . 00 $ & $ 5326459 $ & $ - 48 . 445 ( 2 ) $ \\
$ \infty $ & $ 8302442 $ & $ - 48 . 437 ( 3 ) $ \\
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\end { tabular}
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\end { ruledtabular}
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\end { table}
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\begin { figure}
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\centering
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\includegraphics [width=\columnwidth] { h2o-dmc.pdf}
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\caption { Fixed-node energies of the water molecule for different
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values of $ \mu $ .}
\label { fig:h2o-dmc}
\end { figure}
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\begin { figure}
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\centering
\includegraphics [width=\columnwidth] { f2-dmc.pdf}
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\caption { Fixed-node energies of difluorine for different
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values of $ \mu $ .}
\label { fig:f2-dmc}
\end { figure}
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The first question we would like to address is the quality of the
nodes of the wave functions $ \Psi ^ { \mu } $ obtained with an intermediate
range separation parameter $ \mu $ (\textit { i.e.} $ 0 < \mu < + \infty $ ).
We generated trial wave functions $ \Psi ^ \mu $ with multiple
values of $ \mu $ , and computed the associated fixed node energy
keeping fixed all the parameters having an impact on the nodal surface.
We considered two weakly correlated molecular systems: the water
molecule and fluorine dimer, near their equilibrium
geometry\cite { Caffarel_ 2016} .
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We report the FN-DMC energies of the water molecule in
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table~\ref { tab:h2o-dmc} and figure~\ref { fig:h2o-dmc} .
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\eg { From table~\ref { tab:h2o-dmc} and figure~\ref { fig:h2o-dmc} one can clearly observe that } using FCI trial wave functions gives FN-DMC energies which are lower
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than the energies obtained with a single Kohn-Sham determinant:
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\eg { a gain of} 3~m$ E _ h $ at the double-zeta level and 7~m$ E _ h $ at the triple-zeta
level \eg { are obtained} . Interestingly, with the double-zeta basis one can obtain a
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FN-DMC energy 2.5~m$ E _ h $ lower than the energy obtained with the FCI
trial wave function, using the RSDFT-CIPSI with a range-separation
parameter $ \mu = 1 . 75 $ . This can be explained by the inability of the
basis set to properly describe short-range correlation, shifting
the nodes from their optimal position. Using DFT to take account of
short-range correlation frees the determinant expansion from describing
short-range effects, and enables a better placement of the nodes.
At the triple-zeta level, the short-range correlations can be better
described, and the improvement due to DFT is insignificant. However,
it is important to note that the same FN-DMC energy can be
obtained with a CI expansion which is eight times smaller when sr-DFT
is introduced. One can also remark that the minimum has been
shifted towards the FCI, which is consistent with the fact that
in the CBS limit we expect the minimum of the FN-DMC energy to be
obtained for the FCI wave function, at $ \mu = \infty $ .
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\eg { To further study the behaviour of the FN-DMC energy as a function of $ \mu $ ,
we report in table ???? and figure~\ref { fig:f2-dmc} the FN-DMC energies obtained
with a similar procedure for the fluorine dimer.
The global behaviour and shape of the curves show a very similar behaviour
with respect to that obtained on the water molecule: there exist an "optimal" value of $ \mu $ which provides a lower FN-DMC energy than both the KS determinant (\textit { i.e.} $ \mu = 0 $ ) and the FCI wave function (\textit { i.e.} $ \mu = \infty $ ).
Nevertheless, one can notice that the value of such optimal $ \mu $ is sensibly larger in F$ _ 2 $ than H$ _ 2 $ O: this is probably the signature of the fact that the average inter electronic distance in the valence is smaller in F$ _ 2 $ than in H$ _ 2 $ O due to the larger nuclear charge and corresponding shrinking of the electronic density.
}
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\begin { figure}
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\centering
\includegraphics [width=\columnwidth] { overlap.pdf}
\caption { Overlap of the RSDFT-CIPSI wave functions with the
wave function reoptimized in the presence of a Jastrow factor.}
\label { fig:overlap}
\end { figure}
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\section { Atomization energy benchmarks}
\label { sec:atomization}
Atomization energies are challenging for post-Hartree-Fock methods
because their calculation requires a perfect balance in the
description of atoms and molecules. Basis sets used in molecular
calculations are atom-centered, so they are always better adapted to
atoms than molecules and atomization energies usually tend to be
underestimated.
In the context of FN-DMC calculations, the nodal surface is imposed by
the trial wavefunction which is expanded on an atom-centered basis
set. So we expect the fixed-node error to be also related to the basis
set incompleteness error.
Increasing the size of the basis set improves the description of
the density and of electron correlation, but also reduces the
imbalance in the quality of the description of the atoms and the
molecule, leading to more accurate atomization energies.
Another important feature required to get accurate atomization
energies is size-extensivity, since the numbers of correlated electrons
in the isolated atoms are different from the number of correlated
electrons in the molecule.
In the context of selected CI calculations, when the variational energy is
extrapolated to the FCI energy\cite { Holmes_ 2017} there is obviously no
size-consistence error. But when the selected wave function is used
for as a reference for post-Hartree-Fock methods or QMC calculations,
there is a residual size-consistence error originating from the
truncation of the determinant space.
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% Invariance with m_s
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QMC calculations can be made size-consistent by extrapolating the
FN-DMC energy to estimate the energy obtained with the FCI as a trial
wave function.\cite { Scemama_ 2018,Scemama_ 2018b} Alternatively, the
size-consistence error can be reduced by choosing the number of
selected determinants such that the sum of the PT2 corrections on the
atoms is equal to the PT2 correction of the molecule, enforcing that
the variational dissociation potential energy surface (PES) is
parallel to the perturbatively corrected PES, which is an accurate
estimate of the FCI PES.\cite { Giner_ 2015}
Another source of size-consistence error in QMC calculation may originate
from the Jastrow factor. Usually, the Jastrow factor contains
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
\begin { equation}
J_ \text { ee} = \sum _ i \sum _ { j<i} \frac { a r_ { ij} } { 1 + b r_ { ij} } .
\end { equation}
$ a $ is determined by cusp conditions, 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
interation is not size-consistent. The dissociation of a
heteroatomic diatomic molecule $ AB $ with a parameter $ b _ { AB } $
will lead to two different two-body Jastrow factors on each atom, each
with its own optimal value $ b _ A $ and $ b _ B $ . To remove the
size-consistence error on a PES using this ansätz for $ J _ \text { ee } $ ,
one needs to impose that the parameters of $ J _ \text { ee } $ are fixed:
$ b _ A = b _ B = b _ { AB } $ .
When pseudopotentials are used in a QMC calculation, it is common
practice to localize the pseudopotential on the complete wave
function. If the wave function is not size-consistent, so will be the
locality approximation. Recently, the determinant localization
approximation was introduced.\cite { Zen_ 2019} This approximation
consists in removing the Jastrow factor from the wave function on
which the pseudopotential is localized.
The great advantage of this approximation is that the FN-DMC energy
within this approximation only depends on the parameters of the
determinantal component. Using a size-inconsistent Jastrow factor, or
a non-optimal Jastrow factor will not introduce an additional
size-consistence error in FN-DMC calculations, although it will
reduce the statistical errors by reducing the variance of the local energy.
The energy computed within density functional theory is extensive, and
as it is a mean-field method the convergence to the complete basis set
(CBS) limit is relatively fast. 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.
On the other hand, the convergence of the FCI energies to the CBS
limit will be slower because of the description of short-range electron
correlation with atom-centered functions, but ultimately the exact
energy will be reached.
The 55 molecules of the benchmark for the Gaussian-1
theory\cite { Pople_ 1989,Curtiss_ 1990} were chosen to test the quality
of the RSDFT-CIPSI trial wave functions for energy differences.
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%\begin{squeezetable}
\begin { table*}
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\caption { Mean absolute error (MAE), mean signed errors (MSE) and
standard deviations (RMSD) obtained with the different methods and
basis sets.}
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\label { tab:mad}
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\begin { ruledtabular}
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\begin { tabular} { ll rrr rrr rrr}
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Method & \( \mu \) & \phantom { } & VDZ-BFD & \phantom { } & \phantom { } & VTZ-BFD & \phantom { } & \phantom { } & VQZ-BFD & \phantom { } \\
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\phantom { } & \phantom { } & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\
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\hline
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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 \\
PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
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\hline
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CCSD(T) & \( \infty \) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
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\hline
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RSDFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
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\phantom { } & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
\phantom { } & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
\phantom { } & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
\phantom { } & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\
\phantom { } & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
\phantom { } & \( \infty \) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\
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\hline
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DMC@RSDFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom { 0.} 34) & 5.30 & 3.52(19) & -1.03(19) & 4.39 & 3.16(26) & -0.12(26) & 4.12 \\
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\phantom { } & 1/4 & 4.04(37) & -3.13(\phantom { 0.} 37) & 4.88 & 3.39(77) & -0.59(77) & 4.44 & 2.90(25) & 0.25(25) & 3.74 \\
\phantom { } & 1/2 & 3.74(35) & -3.53(\phantom { 0.} 35) & 4.03 & 2.46(18) & -1.72(18) & 3.02 & 2.06(35) & -0.44(35) & 2.74 \\
\phantom { } & 1 & 5.42(29) & -5.14(\phantom { 0.} 29) & 4.55 & 4.38(94) & -4.24(94) & 5.11 & --- & --- & --- \\
\phantom { } & 2 & 5.98(83) & -5.91(\phantom { 0.} 83) & 4.79 & --- & --- & --- & --- & --- & --- \\
\phantom { } & 5 & 6.18(84) & -6.13(\phantom { 0.} 84) & 4.87 & --- & --- & --- & --- & --- & --- \\
\phantom { } & \( \infty \) & 7.38(1.08) & -7.38(1.08) & 5.67 & --- & --- & --- & --- & --- & --- \\
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\phantom { } & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79 & --- & --- & --- & --- & --- & --- \\
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\end { tabular}
\end { ruledtabular}
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\end { table*}
%\end{squeezetable}
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\begin { figure}
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\centering
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\includegraphics [width=\columnwidth] { g2-dmc.pdf}
\caption { Errors in the DMC atomization energies with the different
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.}
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\label { fig:g2-dmc}
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\end { figure}
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\begin { figure}
\centering
\includegraphics [width=\columnwidth] { g2-ndet.pdf}
\caption { Number of determinants in the different 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.}
\label { fig:g2-ndet}
\end { figure}
The number of determinants in the wave functions are shown in
figure~\ref { fig:n2-ndet} . For all the calculations, the stopping
criterion of the CIPSI algorithm was $ \EPT < 1 $ ~m$ E _ h $ or $ \Ndet >
10^ 7$ .
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For FCI, we have given extrapolated values at $ \EPT \rightarrow 0 $ .
At $ \mu = 0 $ the number of determinants is not equal to one because
we have used the natural orbitals of a first CIPSI calculation.
So the Kohn-Sham determinant is expressed as a linear combination of
determinants built with natural orbitals. Note that it is possible to add
an extra step to the algorithm to compute the natural orbitals 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 obtained the Kohn-Sham orbitals with $ \mu = 0 $ , and the solution would have
been the KS single determinant.
We could have obtained
single-determinant wave functions by using the natural orbitals of a first
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%%---------------------------------------
\begin { acknowledgments}
An award of computer time was provided by the Innovative and Novel
Computational Impact on Theory and Experiment (INCITE) program. This
research has used resources of the Argonne Leadership Computing
Facility, which is a DOE Office of Science User Facility supported
under Contract DE-AC02-06CH11357. AB, was supported by the
U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division, as part of the
Computational Materials Sciences Program and Center for Predictive
Simulation of Functional Materials.
\end { acknowledgments}
\bibliography { rsdft-cipsi-qmc}
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\begin { enumerate}
\item Total energies and nodal quality:
\begin { itemize}
\item Facts: KS occupied orbitals closer to NOs than HF
\item Even if exact functional, complete basis set, still approximated nodes for KS
\item KS -> exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)
\item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS
\item With FCI, good limit at CBS ==> exact energy
\item But slow convergence with basis set because of divergence of the e-e interaction not well represented in atom centered basis set
\item Exponential increase of number of Slater determinants
\item Cite papiers RS-DFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)
\item Question: does such a scheme provide better nodal quality ?
\item In RSDFT we cannot optimize energy with $ \mu $ , but in FNDMC
\item Factual stuffs: with optimal $ \mu $ , lower FNDMC energy than HF/KS/FCI
\begin { itemize}
\item less determinants $ \Rightarrow $ large systems
\item only one parameter to optimize $ \Rightarrow $ deterministic
\item $ \Rightarrow $ reproducible
\end { itemize}
\item with the optimal $ \mu $ :
\begin { itemize}
\item Direct optimization of FNDMC with one parameter
\item Do we improve energy differences ?
\item system dependent
\item basis set dependent: $ \mu \rightarrow \infty $ when $ \mathcal { B } \rightarrow \text { CBS } $
\item large wave functions
\end { itemize}
\begin { itemize}
\item plot $ N _ { det } $ en fonction de $ \mu $
\end { itemize}
\end { itemize}
\end { enumerate}
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\end { document}