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\begin{abstract}
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\begin{abstract}
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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 multideterminant trial wave functions.
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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.
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These compact trial wave functions are generated via the diagonalization of the RS-DFT Hamiltonian.
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These compact trial wave functions are generated via the diagonalization of the RS-DFT Hamiltonian.
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In particular, we combine here short-range correlation functionals with selected configuration interaction (SCI).
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In particular, we combine here short-range correlation functionals with selected configuration interaction (SCI).
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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.
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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.
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@ -124,7 +124,8 @@ the FN-DMC energy associated with a given trial wave function is an upper
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bound to the exact energy, and the latter is recovered only when the
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bound to the exact energy, and the latter is recovered only when the
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nodes of the trial wave function coincide with the nodes of the exact
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nodes of the trial wave function coincide with the nodes of the exact
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wave function.
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wave function.
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The trial wave function in FN-DMC is then a key ingredient which dictates via the quality of its nodal surface the accuracy of the resulting energy and properties.
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The trial wave function, which can be single- or multi-determinantal in nature depending on the type of correlation at play, is then a key ingredient dictating via the quality of its nodal surface the accuracy of the resulting energy and properties.
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The polynomial scaling of its computational cost with respect to the number of electrons and with the size
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The polynomial scaling of its computational cost with respect to the number of electrons and with the size
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of the trial wave function makes the FN-DMC method particularly attractive.
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of the trial wave function makes the FN-DMC method particularly attractive.
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This favorable scaling, its very low memory requirements and
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This favorable scaling, its very low memory requirements and
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@ -135,6 +136,19 @@ those obtained with the FCI method in computationally tractable basis
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sets because the constraints imposed by the fixed-node approximation
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sets because the constraints imposed by the fixed-node approximation
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are less severe than the constraints imposed by the finite-basis
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are less severe than the constraints imposed by the finite-basis
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approximation.
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approximation.
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However, because it is not possible to minimize directly the FN-DMC energy with respect
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to the linear and non-linear parameters of the trial wave function, the
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fixed-node approximation is much more difficult to control than the
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finite-basis approximation.
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The conventional approach consists in multiplying the trial wave
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function by a positive function, the \emph{Jastrow factor}, taking
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account of the bulk of the dynamical correlation.
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%electron-electron cusp and the short-range correlation effects.
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The trial wave function is then re-optimized within variational
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Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
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surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
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Using this technique, it has been shown that the chemical accuracy could be reached within
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FN-DMC.\cite{Petruzielo_2012}
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%However, it is usually harder to control the FN error in DMC, and this
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%However, it is usually harder to control the FN error in DMC, and this
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@ -150,26 +164,14 @@ correlated systems, such as organic molecules near their equilibrium
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geometry, is usually well represented with a single Slater
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geometry, is usually well represented with a single Slater
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determinant. This feature is in part responsible for the success of
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determinant. This feature is in part responsible for the success of
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DFT and coupled cluster theory.
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DFT and coupled cluster theory.
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DMC with a single-determinant trial wave function can be used as a
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Likewise, DMC with a single-determinant trial wave function can be used as a
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single-reference post-Hartree-Fock method, with an accuracy comparable
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single-reference post-Hartree-Fock method for weakly correlated systems, with an accuracy comparable
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to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
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to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
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This approach obviously fails in the presence of strong correlation, like in
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As it is not possible to minimize directly the FN-DMC energy with respect
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transition metal complexes, low-spin open-shell systems, and covalent bond breaking situations which cannot be even qualitatively described by a single electronic configuration.
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to the linear and non-linear parameters of the trial wave function, the
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In such cases, a viable alternative is to consider the FN-DMC method as a
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fixed-node approximation is much more difficult to control than the
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``post-FCI'' method. The multi-determinant trial wave function is then produced by
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finite-basis approximation.
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approaching FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
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The conventional approach consists in multiplying the trial wave
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function by a positive function, the \emph{Jastrow factor}, taking
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account of the electron-electron cusp and the short-range correlation
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effects. The wave function is then re-optimized within variational
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Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
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surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
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Using this technique, it has been shown that the chemical accuracy could be reached within
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FN-DMC.\cite{Petruzielo_2012}
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Another approach consists in considering the FN-DMC method as a
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``post-FCI'' method. The trial wave function is obtained by
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approaching the FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
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When the basis set is enlarged, the trial wave function gets closer to
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When the basis set is enlarged, the trial wave function gets closer to
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the exact wave function, so we expect the nodal surface to be
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the exact wave function, so we expect the nodal surface to be
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improved.\cite{Caffarel_2016}
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improved.\cite{Caffarel_2016}
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@ -194,12 +196,12 @@ applicable for large systems with a multi-configurational character is
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still an active field of research. The present paper falls
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still an active field of research. The present paper falls
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within this context.
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within this context.
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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 these three philosophies in order to create a hybrid method with more attractive properties.
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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 these three philosophies in order to create a hybrid method with more attractive properties.
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In particular, we show here that one can combine FB-FCI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FN-DMC energies with compact multideterminant trial wave functions.
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In particular, we show here that one can combine FB-FCI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Combining CIPSI with RS-DFT}
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\section{Combining WFT and DFT}
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\label{sec:rsdft-cipsi}
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\label{sec:rsdft-cipsi}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -326,7 +328,7 @@ $\hat{W}_\text{ee}^{\text{lr},\mu}$ the long-range
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electron-electron interaction,
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electron-electron interaction,
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$n_\Psi$ the one-electron density associated with $\Psi$,
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$n_\Psi$ the one-electron density associated with $\Psi$,
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and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
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and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
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The minimizing multideterminant wave function $\Psi^\mu$
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The minimizing multi-determinant wave function $\Psi^\mu$
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can be determined by the self-consistent eigenvalue equation
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can be determined by the self-consistent eigenvalue equation
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\begin{equation}
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\begin{equation}
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\label{rs-dft-eigen-equation}
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\label{rs-dft-eigen-equation}
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@ -528,7 +530,7 @@ tens of m\hartree{}.
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An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:
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An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:
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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 $\mu = \infty$.
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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 $\mu = \infty$.
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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 multideterminant expansion as compared to FCI.
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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.
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%======================================================
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%======================================================
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\subsection{Link between RS-DFT and Jastrow factors }
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\subsection{Link between RS-DFT and Jastrow factors }
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@ -537,7 +539,7 @@ The take-home message of this numerical study is that RS-DFT trial wave function
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The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
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The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
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trial wave functions with better nodes than FCI wave function.
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trial wave functions with better nodes than FCI wave function.
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Such behaviour can be directly compared to the common practice of
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Such behaviour can be directly compared to the common practice of
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re-optimizing the multideterminant 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_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
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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_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
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Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
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Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
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and wave function optimization in the presence of a Jastrow factor.
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and wave function optimization in the presence of a Jastrow factor.
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For simplicity in the comparison, the molecular orbitals and the Jastrow
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For simplicity in the comparison, the molecular orbitals and the Jastrow
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