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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, 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), Universit\'e de Toulouse, CNRS, UPS, France}
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\begin{document}
\title{Taming the fixed-node error in diffusion Monte Carlo via range separation}
%\title{Enabling high accuracy diffusion Monte Carlo calculations with
% range-separated density functional theory and selected configuration interaction}
\author{Anthony Scemama}
\email{scemama@irsamc.ups-tlse.fr}
\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}
By combining density-functional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multi-determinant trial wave functions.
These compact trial wave functions are generated via the diagonalization of the RS-DFT Hamiltonian.
In particular, we combine here short-range correlation functionals with selected configuration interaction (SCI).
As the WFT method 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 SCI calculation.
Having low energies does not mean that they are good for chemical properties.
\titou{T2: work in progress.}
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 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
the projection of the exact Hamiltonian onto the finite many-electron basis of
all possible Slater determinants generated within this finite one-electron basis.
The FB-FCI wave function can then be interpreted as a constrained solution of the
true Hamiltonian forced to span the restricted space provided by the finite one-electron basis.
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,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,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,Loos_2019d,Giner_2020}
\titou{However, there is no systematic way of refining the approximation of the unknown exact xc functional, and therefore in practice
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.
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)
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 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.
This favorable scaling, its very low memory requirements and
its adequacy with massively parallel architectures make it a
serious alternative for high-accuracy simulations of large systems. \cite{Nakano_2020,Scemama_2013,Needs_2020,Kim_2018,Kent_2020}
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.
However, because it is not possible to minimize directly the FN-DMC energy with respect
to the linear and non-linear parameters of the trial wave function, the
fixed-node approximation is much more difficult to control than the
finite-basis approximation, especially to compute energy differences.
The conventional approach consists in multiplying the trial wave
function by a positive function, the \emph{Jastrow factor}, taking
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_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}
%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
%function can be a tricky job as \trashAS{there is no variational
%principle for the nodes}\toto{the derivatives of the FN-DMC energy
%with respect to the variational parameters of the wave function can't
%be computed}.
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.
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.
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}
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}
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
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
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,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
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 SCI wave function.\cite{Dash_2018,Dash_2019}
Nevertheless, finding a robust protocol to obtain high accuracy
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 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}
\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}
%====================
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:
increasing the maximum degree of excitation of CI methods (CISD, CISDT,
CISDTQ, \emph{etc}), 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
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
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.
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
magnitude of $\EPT$ can be made the only parameter of the algorithm,
and we choose this parameter as the convergence criterion of the CIPSI
algorithm.
Considering that the perturbatively corrected energy is a reliable
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 $\abs{\EPT} <
1$ m\hartree{}.
%=================================
\subsection{Range-separated DFT}
\label{sec:rsdft}
%=================================
Range-separated DFT (RS-DFT) was introduced in the seminal work of Savin,\cite{Sav-INC-96a,Toulouse_2004}
where the Coulomb operator entering the electron-electron repulsion is split into two pieces:
\begin{equation}
\frac{1}{r}
= w_{\text{ee}}^{\text{sr}, \mu}(r)
+ w_{\text{ee}}^{\text{lr}, \mu}(r)
\end{equation}
where
\begin{align}
w_{\text{ee}}^{\text{sr},\mu}(r) & = \frac{\erfc \qty( \mu\, r)}{r},
&
w_{\text{ee}}^{\text{lr},\mu}(r) & = \frac{\erf \qty( \mu\, r)}{r}
\end{align}
are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version.
The main idea behind RS-DFT is to treat the short-range part of the
interaction using a density functional, and the long-range part within a WFT method like FCI in the present case.
The parameter $\mu$ controls the range of the separation, and allows
to go continuously from the KS Hamiltonian ($\mu=0$) to
the FCI Hamiltonian ($\mu = \infty$).
To rigorously connect WFT and DFT, the universal
Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
decomposed as
\begin{equation}
\mathcal{F}[n] = \mathcal{F}^{\text{lr},\mu}[n] + \bar{E}_{\text{Hxc}}^{\text{sr,}\mu}[n],
\label{Fdecomp}
\end{equation}
where $n$ is a one-electron density,
$\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density
functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the
complementary short-range Hartree-exchange-correlation (Hxc) density
functional. \cite{Savin_1996,Toulouse_2004}
The exact ground state energy can be therefore obtained as a minimization
over a multi-determinant wave function as follows:
\begin{equation}
\label{min_rsdft} E_0= \min_{\Psi} \qty{
\mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi}
+ \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi]
},
\end{equation}
with $\hat{T}$ the kinetic energy operator,
$\hat{W}_\text{ee}^{\text{lr},\mu}$ the long-range
electron-electron interaction,
$n_\Psi$ the one-electron density associated with $\Psi$,
and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
The minimizing multi-determinant wave function $\Psi^\mu$
can be determined by the self-consistent eigenvalue equation
\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}_{\text{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}+ \hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}],
\end{equation}
where
$\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}$
is the complementary short-range Hartree-exchange-correlation
potential operator.
Once $\Psi^{\mu}$ has been calculated, the electronic ground-state
energy is obtained as
\begin{equation}
\label{E-rsdft}
E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi^{\mu}}+\bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_{\Psi^\mu}].
\end{equation}
Note that for $\mu=0$ the long-range interaction vanishes, \ie,
$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus RS-DFT reduces to standard
KS-DFT and $\Psi^\mu$ is the KS determinant. For $\mu = \infty$, the long-range
interaction becomes the standard Coulomb interaction, \ie,
$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and thus RS-DFT reduces
to standard WFT and $\Psi^\mu$ is the FCI wave function.
%%% FIG 1 %%%
\begin{figure*}
\centering
\includegraphics[width=0.7\linewidth]{algorithm.pdf}
\caption{Algorithm showing the generation of the RS-DFT wave
function $\Psi^{\mu}$ starting from $\Psi^{(0)}$.
The outer (macro-iteration) and inner (micro-iteration) loops are represented in red and blue, respectively.
The steps common to both loops are represented in purple.
DIIS extrapolations of the one-electron density are introduced in both the outer and inner loops in order to speed up convergence of the self-consistent process.}
\label{fig:algo}
\end{figure*}
%%% %%% %%% %%%
Hence, range separation creates a continuous path connecting smoothly the KS determinant to the
FCI wave function. Because 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 which consists in increasing the number of determinants.
We follow the KS-to-FCI path by performing FCI calculations using the
RS-DFT Hamiltonian with different values of $\mu$.
Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a
single- or multi-determinant wave function $\Psi^{(0)}$ which can
be obtained in many different ways depending on the system that one considers.
One of the particularity of the present work is that we
use the CIPSI algorithm to perform approximate FCI calculations
with the RS-DFT Hamiltonian $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
This provides a multi-determinant trial wave function $\Psi^{\mu}$ that one can ``feed'' to DMC.
In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selection is performed
to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$
parameterized using the current one-electron density $n^{(k)}$.
At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to $10^{-3}$ \hartree{} in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}).
An inner (micro-iteration) loop (blue) is introduced to accelerate the
convergence of the self-consistent calculation, in which the set of
determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to $10^{-2} \times \tau_1$ \hartree{}.
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. \cite{Pulay_1980,Pulay_1982}
Note that any range-separated post-HF method can be
implemented using this scheme by just replacing the CIPSI step by the
post-HF method of interest.
Note that, thanks to the self-consistent nature of the algorithm,
the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:comp-details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For all the systems considered here, experimental geometries have been used and they have been extracted from the NIST website \titou{[REF]}.
Geometries for each system are reported in the {\SI}.
All 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
\emph{Gaussian09},\cite{g16} using the unrestricted formalism for open-shell systems.
All the CIPSI calculations have been performed with \emph{Quantum
Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
of the local-density approximation (LDA)\cite{Sav-INC-96a,TouSavFla-IJQC-04} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
and correlation functionals defined in
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
The convergence criterion for stopping the CIPSI calculations
has been set to $\EPT < 10^{-3}$ \hartree{} or $ \Ndet > 10^7$.
All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
described in Ref.~\onlinecite{Applencourt_2018}.
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 Jastrow factor, as in all-electron
calculations. Simple Jastrow factors were used to reduce the
fluctuations of the local energy.
The FN-DMC simulations are performed with all-electron moves using the
stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
\cite{Assaraf_2000} with a time step of $5 \times 10^{-4}$ a.u.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Influence of the range-separation parameter on the fixed-node error}
\label{sec:mu-dmc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% TABLE I %%%
\begin{table}
\caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$ obtained with the sr-PBE density functional.}
\label{tab:h2o-dmc}
\centering
\begin{ruledtabular}
\begin{tabular}{crlrl}
& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
\cline{2-3} \cline{4-5}
$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
\hline
$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
%%% FIG 2 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
\caption{Fixed-node energy of \ce{H2O} as a function
of $\mu$ for various levels of theory to generate
the trial wave function.}
\label{fig:h2o-dmc}
\end{figure}
%%% %%% %%% %%%
The first question we would like to address is the quality of the
nodes of the wave function $\Psi^{\mu}$ obtained with an intermediate
range separation parameter (\ie, $0 < \mu < +\infty$).
For this purpose, we consider a weakly correlated molecular system, namely the water
molecule near its equilibrium geometry. \cite{Caffarel_2016}
We then generate trial wave functions $\Psi^\mu$ for multiple values of
$\mu$, and compute the associated fixed-node energy keeping fixed all the
parameters such as the CI coefficients and molecular orbitals impacting the
nodal surface.
%======================================================
\subsection{Fixed-node energy of $\Psi^\mu$}
\label{sec:fndmc_mu}
%======================================================
From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc}, where we report the fixed-node energy of \ce{H2O} as a function of $\mu$ for various short-range density functionals and basis sets,
one can clearly observe that relying on FCI trial
wave functions ($\mu = \infty$) give FN-DMC energies lower
than the energies obtained with a single KS determinant ($\mu=0$):
a lowering of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm
0.3$~m\hartree{} at the triple-$\zeta$ level are obtained.
Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
a smooth behaviour is obtained:
starting from $\mu=0$ (\ie, the KS determinant),
the FN-DMC error is reduced continuously until it reaches a minimum
for an optimal value of $\mu$ (which is obviously basis set and functional dependent),
and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$,
one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
with an optimal value of $\mu=1.75$~bohr$^{-1}$.
This lowering in FN-DMC energy is to be compared with the $3.2 \pm
0.7$~m\hartree{} gain in FN-DMC energy between the KS wave function ($\mu=0$)
and the FCI wave function ($\mu=\infty$). When the basis set is increased, the
gain in FN-DMC energy with respect to the FCI trial wave function is reduced,
and the optimal value of $\mu$ is slightly shifted towards large $\mu$.
Last but not least, the nodes of the wave functions $\Psi^\mu$ obtained with the srLDA
exchange-correlation functional give very similar FN-DMC energies with respect
to those obtained with the srPBE functional, even if the
RS-DFT energies obtained with these two functionals differ by several
tens of m\hartree{}.
An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:
at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
The take-home message of this numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multi-determinant expansion as compared to FCI.
%======================================================
\subsection{Link between RS-DFT and Jastrow factors }
\label{sec:rsdft-j}
%======================================================
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_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
factor are kept fixed: only the CI coefficients are modified.
Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_N)$,
and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$,
where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determinants $D_I$.
The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$.
Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the variational energy
\begin{equation}
\Psi^J = \argmin_{\Psi}\frac{ \mel{ \Psi }{ e^{J} \hat{H} e^{J} }{ \Psi } }{\mel{ \Psi }{ e^{2J} }{ \Psi } }.
\end{equation}
Such a wave function $\Psi^J$ satisfies the generalized Hermitian eigenvalue equation
\begin{equation}
e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E e^{2J} \Psi^J,
\label{eq:ci-j}
\end{equation}
but also the non-Hermitian transcorrelated eigenvalue problem\cite{BoyHan-PRSLA-69,BoyHanLin-1-PRSLA-69,BoyHanLin-2-PRSLA-69,Ten-no2000Nov,Luo-JCP-10,YanShi-JCP-12,CohLuoGutDowTewAla-JCP-19}
\begin{equation}
\label{eq:transcor}
e^{-J} \hat{H} \qty( e^{J} \Psi^J) = E \Psi^J,
\end{equation}
which is much easier to handle despite its non-Hermiticity.
Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.
In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes
of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.
To do so, we have made the following numerical experiment.
First, we extract the 200 determinants with the largest weights in the FCI wave
function out of a large CIPSI calculation. Within this set of determinants,
we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
with different values of $\mu$. This gives the CI expansions $\Psi^\mu$.
Then, within the same set of determinants we optimize the CI coefficients in the presence of
a simple one- and two-body Jastrow factor $e^J$ of the form $\exp(J_{eN} + J_{ee})$ with
\begin{eqnarray}
J_\text{eN} & = & - \sum_{A=1}^{\Nat} \sum_{i=1}^{\Nelec} \left( \frac{\alpha_A\, r_{iA}}{1 + \alpha_A\, r_{iA}} \right)^2
\label{eq:jast-eN} \\
J_\text{ee} & = & \sum_{i=1}^{\Nelec} \sum_{j=1}^{i-1} \frac{a\, r_{ij}}{1 + b\, r_{ij}}. \label{eq:jast-ee}
\end{eqnarray}
$J_\text{eN}$ contains the electron-nucleus terms with a single parameter
$\alpha_A$ per atom, and $J_\text{ee}$ contains the electron-electron terms
where the indices $i$ and $j$ loop over all electron pairs. The parameters $a=1/2$
and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$
were obtained by energy minimization with a single determinant.
The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements
of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the
basis of Jastrow-correlated determinants $e^J D_i$:
\begin{eqnarray}
H_{ij} & = & \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J} } \\
S_{ij} & = & \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} }
\end{eqnarray}
and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}
We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
on the same Slater determinant basis.
In Fig.~\ref{fig:overlap}, we plot the overlaps
$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water,
and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions
$\Psi^\mu$ together with that of $\Psi^J$.
%%% FIG 3 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{overlap.pdf}
\caption{\ce{H2O}, double-zeta basis set, 200 most important
determinants of the FCI expansion (see Sec.~\ref{sec:rsdft-j}).
Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI
expansion optimized in the presence of a Jastrow factor $\Psi^J$.}
\label{fig:overlap}
\end{figure}
%%% %%% %%% %%%
%%% FIG 4 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
\caption{\ce{H2O}, double-zeta basis set, 200 most important
determinants of the FCI expansion (see Sec.~\ref{sec:rsdft-j}).
FN-DMC energies of $\Psi^\mu$ (red curve), together with
the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines
represent the statistical error bars.}
\label{dmc_small}
\end{figure}
%%% %%% %%% %%%
There is a clear maximum of overlap at $\mu=1$~bohr$^{-1}$, which
coincides with the minimum of the FN-DMC energy of $\Psi^\mu$.
Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is compatible
with that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
introducing short-range correlation with DFT has
an impact on the CI coefficients similar to the Jastrow factor.
%%% TABLE II %%%
\begin{table}
\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2(\br,\br) }$
for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
\titou{Please remove table and merge data in the Fig. 5.}}
\label{table_on_top}
\begin{ruledtabular}
\begin{tabular}{cc}
$\mu$ & $\expval{ n_2(\br,\br) }$ \\
\hline
0.00 & 1.443 \\
0.25 & 1.438 \\
0.50 & 1.423 \\
1.00 & 1.378 \\
2.00 & 1.325 \\
5.00 & 1.288 \\
$\infty$ & 1.277 \\
\hline
$\Psi^J$ & 1.404 \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
%%% FIG 5 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{density-mu.pdf}
\caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. One-electron density $n(\br)$ along
the \ce{O-H} axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
\label{fig:n1}
\end{figure}
%%% %%% %%% %%%
%%% FIG 6 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{on-top-mu.pdf}
\caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. On-top pair
density $n_2(\br,\br)$ along the \ce{O-H} axis,
for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
\label{fig:n2}
\end{figure}
%%% %%% %%% %%%
In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
report several quantities related to the one- and two-body density of
$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
report in Table~\ref{table_on_top} the integrated on-top pair density
\begin{equation}
\expval{ n_2(\br,\br) } = \int d\br \,\,n_2(\br,\br)
\end{equation}
where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons]
obtained for both $\Psi^\mu$ and $\Psi^J$.
Then, in order to have a pictorial representation of both the on-top
pair density and the density, we report in Figs.~\ref{fig:n1} and \ref{fig:n2}
the plots of the total density $n(\br)$ and on-top pair density
$n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule.
From these data, one can clearly notice several trends.
First, from Table~\ref{table_on_top}, we can observe that the overall
on-top pair density decreases when $\mu$ increases, which is expected
as the two-electron interaction increases in $H^\mu[n]$.
Second, the relative variations of the on-top pair density with $\mu$
are much more important than that of the one-body density, the latter
being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the
former can vary by about 10$\%$ in some regions.
%TODO TOTO
In the high-density region of the \ce{O-H} bond, the value of the on-top
pair density obtained from $\Psi^J$ is superimposed with
$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
the closest to $\mu=\infty$. The integrated on-top pair density
obtained with $\Psi^J$ lies between the values obtained with
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
and the overlap curve.
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}],
one can notice that the differences with respect to the usual Hamiltonian come
from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
The roles of these two terms are therefore very different: with respect
to the exact ground state wave function $\Psi$, the non divergent two body interaction
increases the probability to find electrons at short distances in $\Psi^\mu$,
while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
provided that it is exact, maintains the exact one-body density.
This is clearly what has been observed from the plots in
Figs.~\ref{fig:n1} and \ref{fig:n2}.
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No,\cite{Ten-no2000Nov}
the effective two-body interaction induced by the presence of a Jastrow factor
can be non-divergent when a proper Jastrow factor is chosen.
Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Jastrow-Slater optimization:
they both deal with an effective non-divergent interaction but still
produce a reasonable one-body density.
As a conclusion of the first part of this study, we can notice that:
\begin{itemize}
\item with respect to the nodes of a KS determinant or a FCI wave function,
one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller
fixed node error by properly choosing an optimal value of $\mu$
in RS-DFT calculations,
\item the optimal value of $\mu$ depends on the system and the
basis set, and the larger the basis set, the larger the optimal value
of $\mu$,
\item numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$,
one-body density, on-top pair density, and FN-DMC energy) indicate
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
\ie, mimicking short-range correlation effects. The latter
statement can be qualitatively understood by noticing that both RS-DFT
and transcorrelated approaches deal with an effective non-divergent
electron-electron interaction, while keeping the density constant.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Energy differences in FN-DMC: atomization energies}
\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 with variational methods.
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 tightly 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.
%============================
\subsection{Size consistency}
%============================
An extremely important feature required to get accurate
atomization energies is size-consistency (or strict separability),
since the numbers of correlated electron pairs in the isolated atoms
are different from those of the molecules.
The energy computed within density functional theory is size-consistent, 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, FCI is also size-consistent, but the convergence of
the FCI energies to the CBS limit is much slower because of the
description of short-range electron correlation using atom-centered
functions. But ultimately the exact energy will be reached.
In the context of selected CI calculations, when the variational energy is
extrapolated to the FCI energy\cite{Holmes_2017} there is no
size-consistency error. But when the truncated SCI wave function is used
as a reference for post-Hartree-Fock methods such as SCI+PT2
or for QMC calculations, there is a residual size-consistency error
originating from the truncation of the wave function.
QMC energies 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-consistency error can be reduced by choosing the number of
selected determinants such that the sum of the PT2 corrections on the
fragments is equal to the PT2 correction of the molecule, enforcing that
the variational potential energy surface (PES) is
parallel to the perturbatively corrected PES, which is a relatively
accurate estimate of the FCI PES.\cite{Giner_2015}
Another source of size-consistency error in QMC calculations originates
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 in Eq.~\eqref{eq:jast-ee}.
The parameter
$a$ 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
interaction is not size-consistent: the dissociation of a
diatomic molecule $AB$ with a parameter $b_{AB}$
will lead to two different two-body Jastrow factors, each
with its own optimal value $b_A$ and $b_B$. To remove the
size-consistency 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 non-local part of the pseudopotential on the
complete wave function (determinantal component and Jastrow).
If the wave function is not size-consistent,
so will be the locality approximation. Within, the DLA,\cite{Zen_2019} the Jastrow factor is
removed from the wave function on which the pseudopotential is localized.
The great advantage of this approximation is that the FN-DMC energy
only depends on the parameters of the determinantal component. Using a
non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will
not introduce an additional error in FN-DMC calculations, although it
will reduce the statistical errors by reducing the variance of the
local energy. Moreover, the integrals involved in the pseudopotential
are computed analytically and the computational cost of the
pseudopotential is dramatically reduced (for more detail, see
Ref.~\onlinecite{Scemama_2015}).
%%% TABLE III %%%
\begin{table}
\caption{FN-DMC energies (in hartree) using the VDZ-BFD basis set
and pseudopotential of the fluorine atom and the dissociated fluorine
dimer, and size-consistency error. }
\label{tab:size-cons}
\begin{ruledtabular}
\begin{tabular}{cccc}
$\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
\hline
0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\
0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\
0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\
1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\
2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\
5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\
$\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
In this section, we make a numerical verification that the produced
wave functions are size-consistent for a given range-separation
parameter.
We have computed the FN-DMC energy of the dissociated fluorine dimer, where
the two atoms are at a distance of 50~\AA. We expect that the energy
of this system is equal to twice the energy of the fluorine atom.
The data in Table~\ref{tab:size-cons} shows that it is indeed the
case, so we can conclude that the proposed scheme provides
size-consistent FN-DMC energies for all values of $\mu$ (within
$2\times$ statistical error bars).
%============================
\subsection{Spin invariance}
%============================
Closed-shell molecules often dissociate into open-shell
fragments. To get reliable atomization energies, it is important to
have a theory which is of comparable quality for open-shell and
closed-shell systems. A good test is to check that all the components
of a spin multiplet are degenerate, as expected from exact solutions.
FCI wave functions have this property and give degenerate energies with
respect to the spin quantum number $m_s$, but the multiplication by a
Jastrow factor introduces spin contamination if the parameters
for the same-spin electron pairs are different from those
for the opposite-spin pairs.\cite{Tenno_2004}
Again, when pseudopotentials are used this tiny error is transferred
in the FN-DMC energy unless the DLA is used.
Within DFT, the common density functionals make a difference for
same-spin and opposite-spin interactions. As DFT is a
single-determinant theory, the density functionals are designed to be
used with the highest value of $m_s$, and therefore different values
of $m_s$ lead to different energies.
So in the context of RS-DFT, the determinantal expansions will be
impacted by this spurious effect, as opposed to FCI.
%%% TABLE IV %%%
\begin{table}
\caption{FN-DMC energies (in hartree) of the triplet carbon atom (BFD-VDZ) with
different values of $m_s$.}
\label{tab:spin}
\begin{ruledtabular}
\begin{tabular}{cccc}
$\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\
\hline
0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\
0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\
0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\
1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\
2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\
5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\
$\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
In this section, we investigate the impact of the spin contamination
due to the short-range density functional on the FN-DMC energy. We have
computed the energies of the carbon atom in its triplet state
with BFD pseudopotentials and the corresponding double-zeta basis
set. The calculation was done with $m_s=1$ (3 spin-up electrons
and 1 spin-down electrons) and with $m_s=0$ (2 spin-up and 2
spin-down electrons).
The results are presented in Table~\ref{tab:spin}.
Although using $m_s=0$ the energy is higher than with $m_s=1$, the
bias is relatively small, more than one order of magnitude smaller
than the energy gained by reducing the fixed-node error going from the single
determinant to the FCI trial wave function. The highest bias, close to
2~m\hartree, is obtained for $\mu=0$, but the bias decreases rapidly
below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
there is no bias (within the error bars), and the bias is not
noticeable with $\mu=5$~bohr$^{-1}$.
%============================
\subsection{Benchmark}
%============================
%%% FIG 6 %%%
\begin{squeezetable}
\begin{table*}
\caption{Mean absolute errors (MAE), mean signed errors (MSE) and
standard deviations (RMSD) obtained with various methods and
basis sets.}
\label{tab:mad}
\begin{ruledtabular}
\begin{tabular}{ll ddd ddd ddd}
& & \mc{3}{c}{VDZ-BFD} & \mc{3}{c}{VTZ-BFD} & \mc{3}{c}{VQZ-BFD} \\
\cline{3-5} \cline{6-8} \cline{9-11}
Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\
\hline
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 \\
\\
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
\\
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
& 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
& 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
& 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & & & \\
& 2 & 19.20 & -19.05 & 10.91 & & & & & & \\
& 5 & 22.93 & -22.79 & 13.24 & & & & & & \\
& \(\infty\) & 23.63(4) & -23.49(4) & 12.81(4) & 8.43(39) & -8.43(39) & 4.87(7) & 4.51(78) & -4.18(78) & 4.19(20) \\
\\
DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
RS-DFT-CIPSI & 1/4 & 4.04(37) & -3.13(37) & 4.88(10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
& 1/2 & 3.74(35) & -3.53(35) & 4.03(23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
& 1 & 5.42(29) & -5.14(29) & 4.55(03) & 4.38(94) & -4.24(94) & 5.11(31) & & & \\
& 2 & 5.98(83) & -5.91(83) & 4.79(71) & & & & & & \\
& 5 & 6.18(84) & -6.13(84) & 4.87(55) & & & & & & \\
& \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(68) & & & & & & \\
& Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & & & & & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
The 55 molecules of the benchmark for the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the
performance of the RS-DFT-CIPSI trial wave functions in the context of
energy differences. Calculations were made in the double-, triple-
and quadruple-zeta basis sets with different values of $\mu$, and using
natural orbitals of a preliminary CIPSI calculation.
For comparison, we have computed the energies of all the atoms and
molecules at the DFT level with different density functionals, and at
the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean
absolute errors (MAE), mean signed errors (MSE) and standard
deviations (RMSD). For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have
given extrapolated values at $\EPT\rightarrow 0$, and the error bars
correspond to the difference between the energies computed with a
two-point and with a three-point linear extrapolation.
In this benchmark, the great majority of the systems are well
described by a single determinant. Therefore, the atomization energies
calculated at the DFT level are relatively accurate, even when
the basis set is small. The introduction of exact exchange (B3LYP and
PBE0) make the results more sensitive to the basis set, and reduce the
accuracy. Thanks to the single-reference character of these systems,
the CCSD(T) energy is an excellent estimate of the FCI energy, as
shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T)
and FCI energies.
The imbalance of the quality of description of molecules compared
to atoms is exhibited by a very negative value of the MSE for
CCSD(T) and FCI/VDZ-BFD, which is reduced by a factor of two
when going to the triple-zeta basis, and again by a factor of two when
going to the quadruple-zeta basis.
This large imbalance at the double-zeta level affects the nodal
surfaces, because although the FN-DMC energies obtained with near-FCI
trial wave functions are much lower than the single-determinant FN-DMC
energies, the MAE obtained with FCI (7.38~$\pm$ 1.08~kcal/mol) is
larger than the single-determinant MAE (4.61~$\pm$ 0.34 kcal/mol).
Using the FCI trial wave function the MSE is equal to the
negative MAE which confirms that all the atomization energies are
underestimated. This confirms that some of the basis-set
incompleteness error is transferred in the fixed-node error.
Within the double-zeta basis set, the calculations could be done for the
whole range of values of $\mu$, and the optimal value of $\mu$ for the
trial wave function was estimated for each system by searching for the
minimum of the spline interpolation curve of the FN-DMC energy as a
function of $\mu$.
This corresponds the the line of the table labelled by the \emph{Opt}
value of $\mu$. Using the optimal value of $\mu$ clearly improves the
MAE, the MSE an the RMSD compared the the FCI wave function. This
result is in line with the common knowledge that re-optimizing
the determinantal component of the trial wave function in the presence
of electron correlation reduces the errors due to the basis set incompleteness.
These calculations were done only for the smallest basis set
because of the expensive computational cost of the QMC calculations
when the trial wave function is expanded on more than a few million
determinants.
At the RS-DFT-CIPSI level, we can remark that with the triple-zeta
basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the
FCI. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet}
there are many systems which did not reach the threshold
$\EPT<1$~m\hartree{}, and the number of determinants exceeded
10~million so the calculation stopped. In this regime, there is a
small size-consistency error originating from the imbalanced
truncation of the wave functions, which is not present in the
extrapolated FCI energies. The same comment applies to
$\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
%%% FIG 7 %%%
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{g2-dmc.pdf}
\caption{Errors in the FN-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.}
\label{fig:g2-dmc}
\end{figure*}
%%% %%% %%% %%%
Searching for the optimal value of $\mu$ may be too costly, so we have
computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
are illustrated in Fig.~\ref{fig:g2-dmc}. As seen on the figure and
in 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
the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (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
$\mu$. Although the FN-DMC energies are higher, the numbers show that
they are more consistent from one system to another, giving improved
cancellations of errors.
%%% FIG 8 %%%
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{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 trial wave functions are shown in
Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants
is smaller when $\mu$ is small and larger when $\mu$ is large.
It is important to remark that the median of the number of
determinants when $\mu=0.5$~bohr$^{-1}$ is below 100~000 determinants
with the quadruple-zeta basis set, making these calculations feasilble
with such a large basis set. At the double-zeta level, compared to the
FCI trial wave functions the median of the number of determinants is
reduced by more than two orders of magnitude.
Moreover, going to $\mu=0.25$~bohr$^{-1}$ gives a median close to 100
determinants at the double-zeta level, and close to 1~000 determinants
at the quadruple-zeta level for only a slight increase of the
MAE. Hence, RS-DFT-CIPSI trial wave functions with small values of
$\mu$ could be very useful for large systems to go beyond the
single-determinant approximation at a very low computational cost
while keeping the size-consistency.
Note that when $\mu=0$ the number of determinants is not equal to one because
we have used the natural orbitals of a first CIPSI calculation, and
not the sr-PBE orbitals.
So the Kohn-Sham determinant is expressed as a linear combination of
determinants built with natural orbitals. 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 converged to the Kohn-Sham orbitals with $\mu=0$, and the
solution would have been the PBE single determinant.
%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%
In the present work, we have shown that introducing short-range correlation via
a range-separated Hamiltonian in a full CI expansion yields improved
nodal surfaces, especially with small basis sets. The effect of sr-DFT
on the determinant expansion is similar to the effect of re-optimizing
the CI coefficients in the presence of a Jastrow factor, but without
the burden of performing a stochastic optimization.
Varying the range-separation parameter $\mu$ and approaching the
RS-DFT-FCI with CIPSI provides a way to adapt the number of
determinants in the trial wave function, leading always to
size-consistent FN-DMC energies.
We propose two methods. The first one is for the computation of
accurate total energies by a one-parameter optimization of the FN-DMC
energy via the variation of the parameter $\mu$.
The second method is for the computation of energy differences, where
the target is not the lowest possible FN-DMC energies but the best
possible cancellation of errors. Using a fixed value of $\mu$
increases the consistency of the trial wave functions, and we have found
that $\mu=0.5$~bohr$^{-1}$ is the value where the cancellation of
errors is the most effective.
Moreover, such a small value of $\mu$ gives extremely
compact wave functions, making this recipe a good candidate for
the accurate description of the whole potential energy surfaces of
large systems. If the number of determinants is still too large, the
value of $\mu$ can be further reduced to $0.25$~bohr$^{-1}$ to get
extremely compact wave functions at the price of less efficient
cancellations of errors.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgments}
This work was performed using HPC resources from GENCI-TGCC (Grand
Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
2019-0510.
\end{acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that support the findings of this study are available within the article and its {\SI}, and are openly available in [repository name] at \url{http://doi.org/[doi]}, reference number [reference number].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{rsdft-cipsi-qmc}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
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