modifications in first part of results
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@ -251,7 +251,7 @@ estimate of the FCI energy, using a fixed value of the PT2 correction
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as a stopping criterion enforces a constant distance of all the
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as a stopping criterion enforces a constant distance of all the
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calculations to the FCI energy. In this work, we target the chemical
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calculations to the FCI energy. In this work, we target the chemical
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accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
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accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
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1$ millihartree.
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1$ m\hartree{}.
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@ -279,7 +279,7 @@ The main idea behind RS-DFT is to treat the short-range part of the
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interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case.
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interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case.
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The parameter $\mu$ controls the range of the separation, and allows
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The parameter $\mu$ controls the range of the separation, and allows
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to go continuously from the KS Hamiltonian ($\mu=0$) to
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to go continuously from the KS Hamiltonian ($\mu=0$) to
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the FCI Hamiltonian ($\mu \to \infty$).
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the FCI Hamiltonian ($\mu = \infty$).
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To rigorously connect WFT and DFT, the universal
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To rigorously connect WFT and DFT, the universal
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Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
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Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
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@ -331,7 +331,7 @@ energy is obtained as
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Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie,
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Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie,
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$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus
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$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus
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RS-DFT reduces to standard KS-DFT and $\Psi^\mu$
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RS-DFT reduces to standard KS-DFT and $\Psi^\mu$
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is the KS determinant. For $\mu\to\infty$, the long-range
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is the KS determinant. For $\mu = \infty$, the long-range
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interaction becomes the standard Coulomb interaction, \ie,
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interaction becomes the standard Coulomb interaction, \ie,
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$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and
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$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and
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thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is
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thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is
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@ -369,12 +369,12 @@ In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selec
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to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$
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to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$
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parameterized using the current one-electron density $n^{(k)}$.
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parameterized using the current one-electron density $n^{(k)}$.
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At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
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At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
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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}).
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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}).
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An inner (micro-iteration) loop (blue) is introduced to accelerate the
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An inner (micro-iteration) loop (blue) is introduced to accelerate the
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convergence of the self-consistent calculation, in which the set of
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convergence of the self-consistent calculation, in which the set of
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determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
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determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
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$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
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$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
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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.
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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{}.
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The convergence of the algorithm was further improved
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The convergence of the algorithm was further improved
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by introducing a direct inversion in the iterative subspace (DIIS)
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by introducing a direct inversion in the iterative subspace (DIIS)
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step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
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step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
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@ -407,7 +407,7 @@ and correlation functionals defined in
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Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
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Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
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Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
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Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
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The convergence criterion for stopping the CIPSI calculations
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The convergence criterion for stopping the CIPSI calculations
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has been set to $\EPT < 10^{-3}$ hartree or $ \Ndet > 10^7$.
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has been set to $\EPT < 10^{-3}$ \hartree{} or $ \Ndet > 10^7$.
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All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
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All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
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described in Ref.~\onlinecite{Applencourt_2018}.
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described in Ref.~\onlinecite{Applencourt_2018}.
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@ -431,7 +431,8 @@ with a time step of $5 \times 10^{-4}$ a.u.
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%%% TABLE I %%%
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%%% TABLE I %%%
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\begin{table}
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\begin{table}
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\caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} with various trial wave functions $\Psi^{\mu}$.}
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\caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$.
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\titou{srPBE?}.}
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\label{tab:h2o-dmc}
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\label{tab:h2o-dmc}
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\centering
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\centering
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\begin{ruledtabular}
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\begin{ruledtabular}
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@ -461,9 +462,9 @@ with a time step of $5 \times 10^{-4}$ a.u.
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\begin{figure}
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\begin{figure}
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\centering
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\centering
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\caption{Fixed-node energies of \ce{H2O} for different
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\caption{Fixed-node energy of \ce{H2O} as a function
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values of $\mu$, using the srLDA or srPBE density
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of $\mu$ for various levels of theory to generate
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functionals to build the trial wave function.}
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the trial wave function.}
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\label{fig:h2o-dmc}
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\label{fig:h2o-dmc}
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\end{figure}
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\end{figure}
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%%% %%% %%% %%%
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%%% %%% %%% %%%
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@ -481,31 +482,34 @@ parameters having an impact on the nodal surface fixed (\titou{such as ??}).
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\subsection{Fixed-node energy of $\Psi^\mu$}
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\subsection{Fixed-node energy of $\Psi^\mu$}
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\label{sec:fndmc_mu}
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\label{sec:fndmc_mu}
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%======================================================
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%======================================================
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From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc},
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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,
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one can clearly observe that relying on FCI trial
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one can clearly observe that relying on FCI trial
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wave functions ($\mu \to \infty$) give FN-DMC energies lower
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wave functions ($\mu = \infty$) give FN-DMC energies lower
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than the energies obtained with a single KS determinant ($\mu=0$):
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than the energies obtained with a single KS determinant ($\mu=0$):
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a gain of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm
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a lowering of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm
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0.3$~m\hartree{} at the triple-zeta level are obtained.
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0.3$~m\hartree{} at the triple-$\zeta$ level are obtained.
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Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
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Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
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intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
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intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
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a smooth behaviour is obtained:
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a smooth behaviour is obtained:
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starting from $\mu=0$ (\textit{i.e.} the KS determinant),
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starting from $\mu=0$ (\ie, the KS determinant),
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the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,
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the FN-DMC error is reduced continuously until it reaches a minimum
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and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).
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for an optimal value of $\mu$ (which is obviously basis set and functional dependent),
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For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set,
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and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
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with the optimal value of $\mu$, one can obtain a lowering of the
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For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$,
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FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}, with an optimal value at
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one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
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$\mu=1.75$~bohr$^{-1}$.
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with an optimal value of $\mu=1.75$~bohr$^{-1}$.
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When the basis set is increased, the gain in FN-DMC energy with
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When the basis set is increased, the gain in FN-DMC energy with
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respect to the FCI trial wave function is reduced, and the optimal
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respect to the FCI trial wave function is reduced, and the optimal
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value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes
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value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes
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of the wave functions $\Psi^\mu$ obtained with the short-range
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of the wave functions $\Psi^\mu$ obtained with the srLDA
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LDA exchange-correlation functional give very similar FN-DMC energies with respect
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exchange-correlation functional give very similar FN-DMC energies with respect
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to those obtained with the short-range PBE functional, even if the
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to those obtained with the srPBE functional, even if the
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RS-DFT energies obtained with these two functionals differ by several
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RS-DFT energies obtained with these two functionals differ by several
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tens of m\hartree{}.
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tens of m\hartree{}.
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\titou{The key fact here is that, 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 here is that RS-DFT trial wave functions yield a lower fixed-node energy with more compact multideterminant 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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\label{sec:rsdft-j}
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\label{sec:rsdft-j}
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@ -548,9 +552,9 @@ Then, within the same set of determinants we optimize the CI coefficients in the
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a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$.
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a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$.
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Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
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Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
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on the same Slater determinant basis.
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on the same Slater determinant basis.
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In figure~\ref{fig:overlap}, we plot the overlaps
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In Fig.~\ref{fig:overlap}, we plot the overlaps
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$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,
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$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,
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and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions
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and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions
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$\Psi^\mu$ together with that of $\Psi^J$.
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$\Psi^\mu$ together with that of $\Psi^J$.
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%%% FIG 3 %%%
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%%% FIG 3 %%%
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@ -663,7 +667,7 @@ $\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
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and the overlap curve.
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and the overlap curve.
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These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
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These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
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and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
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and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
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Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}),
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Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}),
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one can notice that the differences with respect to the usual Hamiltonian come
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one can notice that the differences with respect to the usual Hamiltonian come
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from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
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from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
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@ -694,7 +698,7 @@ As a conclusion of the first part of this study, we can notice that:
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\item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$,
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\item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$,
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one-body density, on-top pair density, and FN-DMC energy) indicate
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one-body density, on-top pair density, and FN-DMC energy) indicate
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that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
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that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
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\textit{i.e.} mimicking short-range correlation effects. The latter
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\ie, mimicking short-range correlation effects. The latter
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statement can be qualitatively understood by noticing that both RS-DFT
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statement can be qualitatively understood by noticing that both RS-DFT
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and transcorrelated approaches deal with an effective non-divergent
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and transcorrelated approaches deal with an effective non-divergent
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electron-electron interaction, while keeping the density constant.
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electron-electron interaction, while keeping the density constant.
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@ -764,7 +768,7 @@ one-electron, two-electron and one-nucleus-two-electron terms.
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The problematic part is the two-electron term, whose simplest form can
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The problematic part is the two-electron term, whose simplest form can
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be expressed as
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be expressed as
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\begin{equation}
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\begin{equation}
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J_\text{ee} = \sum_i \sum_{j<i} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
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J_\text{ee} = \sum_{i<j} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
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\end{equation}
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\end{equation}
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The parameter
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The parameter
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$a$ is determined by cusp conditions, and $b$ is obtained by energy
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$a$ is determined by cusp conditions, and $b$ is obtained by energy
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@ -997,7 +1001,7 @@ when the trial wave function is expanded on more than a few million
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determinants.
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determinants.
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At the RS-DFT-CIPSI level, we can remark that with the triple-zeta
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At the RS-DFT-CIPSI level, we can remark that with the triple-zeta
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basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the
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basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the
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FCI. For the largest systems, as shown in figure~\ref{fig:g2-ndet}
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FCI. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet}
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there are many systems which did not reach the threshold
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there are many systems which did not reach the threshold
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$\EPT<1$~m\hartree{}, and the number of determinants exceeded
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$\EPT<1$~m\hartree{}, and the number of determinants exceeded
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10~million so the calculation stopped. In this regime, there is a
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10~million so the calculation stopped. In this regime, there is a
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@ -1024,7 +1028,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
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Searching for the optimal value of $\mu$ may be too costly, so we have
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Searching for the optimal value of $\mu$ may be too costly, so we have
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computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
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computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
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are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
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are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
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in table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
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in Table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
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0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
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0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
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the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
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the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
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3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values
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3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values
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