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%% This BibTeX bibliography file was created using BibDesk.


%% http://bibdesk.sourceforge.net/




%% Created for PierreFrancois Loos at 20200809 15:41:06 +0200


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@article{Kim_2018,


Annote = {QMCPACK is an open source quantum Monte Carlo package for ab initio electronic structure calculations. It supports calculations of metallic and insulating solids, molecules, atoms, and some model Hamiltonians. Implemented real space quantum Monte Carlo algorithms include variational, diffusion, and reptation Monte Carlo. QMCPACK uses SlaterJastrow type trial wavefunctions in conjunction with a sophisticated optimizer capable of optimizing tens of thousands of parameters. The orbital space auxiliaryfield quantum Monte Carlo method is also implemented, enabling cross validation between different highly accurate methods. The code is specifically optimized for calculations with large numbers of electrons on the latest high performance computing architectures, including multicore central processing unit and graphical processing unit systems. We detail the program's capabilities, outline its structure, and give examples of its use in current research calculations. The package is available at http://qmcpack.org.},


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Title = {{QMCPACK}: an open source ab initio quantum Monte Carlo package for the electronic structure of atoms, molecules and solids},


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@article{Eriksen_2019,


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@misc{Eriksen_2020,


Archiveprefix = {arXiv},


Author = {Janus J. Eriksen and Tyler A. Anderson and J. Emiliano Deustua and Khaldoon Ghanem and Diptarka Hait and Mark R. Hoffmann and Seunghoon Lee and Daniel S. Levine and Ilias Magoulas and Jun Shen and Norman M. Tubman and K. Birgitta Whaley and Enhua Xu and Yuan Yao and Ning Zhang and Ali Alavi and Garnet KinLic Chan and Martin HeadGordon and Wenjian Liu and Piotr Piecuch and Sandeep Sharma and Seiichiro L. Tenno and C. J. Umrigar and J{\"u}rgen Gauss},


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Primaryclass = {physics.chemph},


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Year = {2020}}




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Publisher = {American Physical Society},


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@article{Scemama_2006c,


Author = {Scemama, Anthony and Filippi, Claudia},


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@ 569,13 +793,10 @@




@article{Scemama_2013,


Author = {Scemama, Anthony and Caffarel, Michel and Oseret, Emmanuel and Jalby, William},


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@ 1058,16 +1279,13 @@


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@incollection{SavINC96a,


author = {A. Savin},


title = {Beyond the KohnSham Determinant},


booktitle = {Recent Advances in Density Functional Theory},


publisher = {World Scientific},


address = {},


editor = {D. P. Chong},


pages = {129148},


year = {1996}


}




Author = {A. Savin},


Booktitle = {Recent Advances in Density Functional Theory},


Editor = {D. P. Chong},


Pages = {129148},


Publisher = {World Scientific},


Title = {Beyond the KohnSham Determinant},


Year = {1996}}




@article{Toulouse_2004,


Author = {Toulouse, Julien and Colonna, Fran{\c c}ois and Savin, Andreas},


@ 1142,14 +1360,14 @@


BdskUrl1 = {https://doi.org/10.5281/zenodo.3677565}}




@article{TouSavFlaIJQC04,


author = {J. Toulouse and A. Savin and H.J. Flad},


title = {Shortrange exchangecorrelation energy of a uniform electron gas with modified electronelectron interaction},


journal = {Int. J. Quantum Chem.},


volume = {100},


pages = {1047},


year = {2004},


note = {}


}


Author = {J. Toulouse and A. Savin and H.J. Flad},


Journal = {Int. J. Quantum Chem.},


Pages = {1047},


Title = {Shortrange exchangecorrelation energy of a uniform electron gas with modified electronelectron interaction},


Volume = {100},


Year = {2004}}








@article{Nightingale_2001,


author = {Nightingale, M. P. and MelikAlaverdian, Vilen},



@ 87,36 +87,43 @@ As the WFT method is relieved from describing the shortrange part of the correl


Solving the Schr\"odinger equation for atoms and molecules is a complex task that has kept theoretical and computational chemists busy for almost hundred years now. \cite{Schrodinger_1926}


In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.




One of this strategies consists in relying on wave function theory 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 oneelectron basis.


This solution is the eigenpair of an approximate Hamiltonian defined as


One of this strategies consists in relying on wave function theory (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 oneelectron basis functions.


The FBFCI wave function and its corresponding energy form the eigenpair of an approximate Hamiltonian defined as


the projection of the exact Hamiltonian onto the finite manyelectron basis of


all possible Slater determinants generated within this finite oneelectron basis.


The FCI wave function can be interpreted as a constrained solution of the


true Hamiltonian forced to span the restricted space provided by the oneelectron basis.


In the complete basis set (CBS) limit, the constraint is lifted and the


exact solution is recovered.


Hence, the accuracy of a FCI calculation can be systematically improved by increasing the size of the oneelectron basis set.


Nevertheless, its exponential scaling with the number of electrons and with the size of the basis is prohibitive for most chemical systems.


In recent years, the introduction of new algorithms \cite{Booth_2009} and the


The FBFCI wave function can then be interpreted as a constrained solution of the


true Hamiltonian forced to span the restricted space provided by the finite oneelectron basis.


In the complete basis set (CBS) limit, the constraint is lifted and the exact solution is recovered.


Hence, the accuracy of a FBFCI calculation can be systematically improved by increasing the size of the oneelectron basis set.


Nevertheless, the exponential growth of its computational scaling with the number of electrons and with the basis set size is prohibitive for most chemical systems.


In recent years, the introduction of new algorithms \cite{Booth_2009,Xu_2018,Eriksen_2018,Eriksen_2019} and the


revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016,Garniron_2018}


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. \cite{Booth_2010,Cleland_2010,Daday_2012,Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}


However, the scaling remains exponential unless some bias is introduced leading


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 excitedstate energies of small and mediumsized molecular systems with nearFCI 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}




Diffusion Monte Carlo (DMC) is another numerical scheme to obtain


Another route to solve the Schr\"odinger equation is densityfunctional theory (DFT). \cite{Hohenberg_1964}


Presentday DFT calculations are almost exclusively done within the socalled KohnSham (KS) formalism, \cite{Kohn_1965} which


transfers the complexity of the manybody problem to the exchangecorrelation (xc) functional thanks to a judicious mapping between a noninteracting reference system and its interacting analog which both have exactly the same oneelectron density.


KSDFT \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{FraMusLupTouJCP15}


However, one faces the unsettling choice of the \emph{approximate} xc functional which makes inexorably KSDFT hard to systematically improve. \cite{Becke_2014}




Diffusion Monte Carlo (DMC) is yet another numerical scheme to obtain


the exact solution of the Schr\"odinger equation with a different


constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)


as a given trial (approximate) wave function.


Within this socalled \emph{fixednode} (FN) approximation,


constraint. \cite{Foulkes_2001,Austin_2012,Needs_2020}


In DMC, the solution is imposed to have the same nodes (or zeroes)


as a given trial (approximate) wave function. \cite{Reynolds_1982,Ceperley_1991}


Within this socalled fixednode (FN) approximation,


the FNDMC 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 polynomial scaling with the number of electrons and with the size


of {\manu{in what sense is it polynomial?}the trial wave function makes the FNDMC method particularly attractive.


of {\manu{in what sense is it polynomial?}the trial wave function makes the FNDMC method particularly attractive.\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 FN approximation


@ 194,7 +201,7 @@ In singledeterminant DMC calculations, the degrees of freedom used to


reduce the fixednode error are the molecular orbitals on which the


Slater determinant is built.


Different molecular orbitals can be chosen:


HartreeFock (HF), KohnSham (KS), natural (NO) orbitals of a


HartreeFock (HF), KohnSham (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


@ 267,7 +274,7 @@ accuracy so all the CIPSI selections were made such that $\abs{\EPT} <


\label{sec:rsdft}


%=================================




The rangeseparated DFT (RSDFT) was introduced in the seminal work of Savin,\cite{SavINC96a,Toulouse_2004}


Rangeseparated DFT (RSDFT) was introduced in the seminal work of Savin,\cite{SavINC96a,Toulouse_2004}


where the Coulomb operator entering the electronelectron repulsion is split into two pieces:


\begin{equation}


\frac{1}{r}


@ 300,7 +307,7 @@ $\mathcal{F}^{\text{lr},\mu}$ is a longrange universal density


functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the


complementary shortrange Hartreeexchangecorrelation (Hxc) density


functional. \cite{Savin_1996,Toulouse_2004}


The exact ground state energy can therefore be obtained as a minimization


The exact ground state energy can be therefore obtained as a minimization


over a multideterminant wave function as follows:


\begin{equation}


\label{min_rsdft} E_0= \min_{\Psi} \qty{


@ 416,7 +423,7 @@ 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}


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


@ 478,9 +485,9 @@ 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 fixednode energy keeping fixed all the


\toto{$\mu$, and compute the associated fixednode energy keeping fixed all the


parameters such as the CI coefficients and molecular orbitals impacting the


nodal surface.


nodal surface.}




%======================================================


\subsection{Fixednode energy of $\Psi^\mu$}


@ 513,7 +520,7 @@ to those obtained with the srPBE functional, even if the


RSDFT 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 $\Psi^\mu$:


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/VDZBFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2odmc}). Even at the srPBE/VTZBFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.


The takehome message of this numerical study is that RSDFT trial wave functions can yield a lower fixednode energy with more compact multideterminant expansion as compared to FCI.




@ 848,7 +855,7 @@ Ref.~\onlinecite{Scemama_2015}).


In this section, we make a numerical verification that the produced


wave functions are sizeconsistent for a given rangeseparation


parameter.


We have computed the \manu{FNDMC} energy of the dissociated fluorine dimer, where


We have computed the FNDMC 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:sizecons} shows that it is indeed the


@ 865,7 +872,7 @@ Closedshell molecules often dissociate into openshell


fragments. To get reliable atomization energies, it is important to


have a theory which is of comparable quality for openshell and


closedshell systems. A good test is to check that all the components


of a spin multiplet are degenerate\manu{, as expected from exact solutions}.


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


@ 1105,7 +1112,7 @@ solution would have been the PBE single determinant.


\section{Conclusion}


%%%%%%%%%%%%%%%%%%%%




\manu{In the present work} we have shown that introducing shortrange correation via


In the present work, we have shown that introducing shortrange correlation via


a rangeseparated Hamiltonian in a full CI expansion yields improved


nodal surfaces, especially with small basis sets. The effect of srDFT


on the determinant expansion is similar to the effect of reoptimizing



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