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@article{Bressanini_2012,


Author = {D. Bressanini},


DateAdded = {20200816 22:13:51 +0200},


DateModified = {20200816 22:14:05 +0200},


Doi = {10.1103/PhysRevB.86.115120},


Journal = {Phys. Rev. B},


Pages = {115120},


Title = {Implications of the two nodal domains conjecture for ground state fermionic wave functions},


Volume = {86},


Year = {2012}}




@article{Giner_2013,


Author = {E. Giner and A. Scemama and M. Caffarel},


DateAdded = {20200816 15:38:03 +0200},



@ 89,13 +89,13 @@ Solving the Schr\"odinger equation for atoms and molecules is a complex task tha


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 (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.


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.


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 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.


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 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


@ 112,25 +112,31 @@ transfers the complexity of the manybody problem to the exchangecorrelation (x


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}


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


as a given (approximate) antisymmetric trial 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 trial wave function in FNDMC is then a key ingredient which dictates 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.\cite{Nakano_2020,Scemama_2013,Needs_2020,Kim_2018,Kent_2020}


of the trial wave function makes the FNDMC method particularly attractive.


This favorable scaling, its very low memory requirements and


its adequacy with massively parallel architectures make it a


serious alternative for highaccuracy 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 FN approximation


sets because the constraints imposed by the fixednode approximation


are less severe than the constraints imposed by the finitebasis


approximation.






%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


@ 143,16 +149,13 @@ 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


densityfunctional theory (DFT) and coupled cluster theory.


DFT and coupled cluster theory.


DMC with a singledeterminant trial wave function can be used as a


singlereference postHatreeFock method, with an accuracy comparable


singlereference postHartreeFock method, with an accuracy comparable


to coupled cluster.\cite{Dubecky_2014,Grossman_2002}


The favorable scaling of QMC, its very low memory requirements and


its adequacy with massively parallel architectures make it a


serious alternative for highaccuracy simulations of large systems.




As it is not possible to minimize directly the FNDMC energy with respect


to the variational parameters of the trial wave function, the


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


fixednode approximation is much more difficult to control than the


finitebasis approximation.


The conventional approach consists in multiplying the trial wave


@ 160,14 +163,13 @@ function by a positive function, the \emph{Jastrow factor}, taking


account of the electronelectron cusp and the shortrange correlation


effects. The wave function is then reoptimized 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


surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,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}




Another approach consists in considering the FNDMC method as a


\emph{postFCI method}. The trial wave function is obtained by


approaching the FCI with a selected configuration interaction (SCI)


method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}


``postFCI'' method. The trial wave function is obtained by


approaching the FCI with a SCI method such as CIPSI for instance.\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}


@ 191,6 +193,9 @@ calculations which can be reproduced systematically, and which is


applicable for large systems with a multiconfigurational character is


still an active field of research. The present paper falls


within this context.


The central idea of the present work, and the launchpad for the remainder of this study, is that one can combine the various strengths of these three philosophies in order to create a hybrid method with more attractive properties.


In particular, we show here that one can combine FBFCI and KSDFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FNDMC energies with compact multideterminant trial wave functions.








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


@ 229,7 +234,7 @@ Nevertheless, even when using the exact exchange correlation potential at the


CBS limit, a fixednode error necessarily remains because the


singledeterminant ansätz does not have enough flexibility to describe the


nodal surface of the exact correlated wave function of a generic $N$electron


system.


system. \cite{Bressanini_2012}


If one wants to recover the exact CBS limit, a multideterminant parameterization


of the wave functions is required.




@ 548,17 +553,13 @@ Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the


\end{equation}


Such a wave function $\Psi^J$ satisfies the generalized Hermitian eigenvalue equation


\begin{equation}


\toto{


e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E e^{2J} \Psi^J,


}


\label{eq:cij}


\end{equation}


but also the nonHermitian \toto{transcorrelated eigenvalue problem \cite{many_things} MANU:CITATIONS}


\begin{equation}


\label{eq:transcor}


\toto{


e^{J} \hat{H} \qty( e^{J} \Psi^J) = E \Psi^J,


}


\end{equation}


which is much easier to handle despite its nonHermiticity.


Of course, the FNDMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.


@ 571,7 +572,7 @@ function out of a large CIPSI calculation. Within this set of determinants,


we solve the selfconsistent equations of RSDFT [see Eq.~\eqref{rsdfteigenequation}]


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 twobody Jastrow factor \toto{$e^J$ of the form $\exp(J_{eN} + J_{ee})$ with


a simple one and twobody 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:jasteN} \\


@ 586,10 +587,10 @@ 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 Jastrowcorrelated determinants $e^J D_i$:


\begin{eqnarray}


H_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J} \right \rangle \\


S_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} \right \rangle


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:cij}.\cite{Nightingale_2001}}


and solving Eq.~\eqref{eq:cij}.\cite{Nightingale_2001}




We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed


on the same Slater determinant basis.



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