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@ 94,7 +94,7 @@ In order to achieve this formidable endeavour, various strategies have been care


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One of these strategies consists in relying on wave function theory \cite{Pople_1999} (WFT) and, in particular, on the full configuration interaction (FCI) method.


However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite basis (FB) of oneelectron basis functions, the FBFCI energy being an upper bound to the exact energy in accordance with the variational principle.


However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite basis (FB) of oneelectron functions, the FBFCI energy being an upper bound to the exact energy in accordance with the variational principle.


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.


@ 121,17 +121,17 @@ Presentday DFT calculations are almost exclusively done within the socalled Ko


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,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}}


\titou{However, there is no systematic way of refining the approximation of the unknown exact xc functional.}


Therefore, one faces, in practice, the unsettling choice of the \emph{approximate} xc functional. \cite{Becke_2014}


Moreover, because of the approximate nature of the xc functional, although the resolution of the KS equations is variational, the resulting KS energy does not have such property.




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\subsection{Stochastic methods}


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Diffusion Monte Carlo (DMC), which belongs to the family of stochastic methods, is yet another numerical scheme to obtain


Diffusion Monte Carlo (DMC) belongs to the family of stochastic methods and 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}


twist. \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 socalled fixednode (FN) approximation,


@ 139,7 +139,7 @@ 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 trial wave function, which can be single or multideterminantal in nature depending on the type of correlation at play and the target accuracy, is then the key ingredient dictating, via the quality of its nodal surface, the accuracy of the resulting energy and properties.


The trial wave function, which can be single or multideterminantal in nature depending on the type of correlation at play and the target accuracy, is 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 FNDMC method particularly attractive.


@ 155,12 +155,12 @@ However, because it is not possible to minimize directly the FNDMC energy with


to the linear and nonlinear parameters of the trial wave function, the


fixednode approximation is much more difficult to control than the


finitebasis 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.


The conventional approach consists in multiplying the determinantal part of the trial wave


function by a positive function, the Jastrow factor, which main assignment is to take into


account the bulk of the dynamical electron correlation and reduce the statistical fluctuation without altering the location of the nodes.


%electronelectron cusp and the shortrange correlation effects.


The trial wave function is then reoptimized within variational


Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal


The determinantal part of the trial wave function is then reoptimized within variational


Monte Carlo (VMC) in the presence of the Jastrow factor (which can also be simultaneously optimized) and the nodal


surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008}


Using this technique, it has been shown that the chemical accuracy could be reached within


FNDMC.\cite{Petruzielo_2012}


@ 192,7 +192,7 @@ Different molecular orbitals can be chosen:


HartreeFock (HF), KohnSham (KS), natural orbitals (NOs) of a


correlated wave function, or orbitals optimized in the


presence of a Jastrow factor.


The nodal surfaces obtained with a KS determinant are in general


Nodal surfaces obtained with a KS determinant are in general


better than those obtained with a HF determinant,\cite{Per_2012} and


of comparable quality to those obtained with a Slater determinant


built with NOs.\cite{Wang_2019} Orbitals obtained in the presence



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