2nd screening of Sec IV done
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@ -495,7 +495,7 @@ The first question we would like to address is the quality of the
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nodes of the wave function $\Psi^{\mu}$ obtained for intermediate values of the
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nodes of the wave function $\Psi^{\mu}$ obtained for intermediate values of the
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range separation parameter (\ie, $0 < \mu < +\infty$).
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range separation parameter (\ie, $0 < \mu < +\infty$).
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For this purpose, we consider a weakly correlated molecular system, namely the water
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For this purpose, we consider a weakly correlated molecular system, namely the water
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molecule \titou{at its experimental geometry. \cite{Caffarel_2016}}
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molecule at its experimental geometry. \cite{Caffarel_2016}
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We then generate trial wave functions $\Psi^\mu$ for multiple values of
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We then generate trial wave functions $\Psi^\mu$ for multiple values of
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$\mu$, and compute the associated FN-DMC energy keeping fixed all the
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$\mu$, and compute the associated FN-DMC energy keeping fixed all the
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parameters impacting the nodal surface, such as the CI coefficients and the molecular orbitals.
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parameters impacting the nodal surface, such as the CI coefficients and the molecular orbitals.
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@ -583,7 +583,7 @@ To do so, we have made the following numerical experiment.
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First, we extract the 200 determinants with the largest weights in the FCI wave
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First, we extract the 200 determinants with the largest weights in the FCI wave
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function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants,
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function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants,
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we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
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we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
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for different values of $\mu$ \titou{using the srPBE functional}. This gives the CI expansions $\Psi^\mu$.
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for different values of $\mu$ using the srPBE functional. This gives the CI expansions $\Psi^\mu$.
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Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of
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Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of
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a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and
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a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and
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\begin{subequations}
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\begin{subequations}
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@ -601,7 +601,7 @@ where the sum over $i < j$ loops over all unique electron pairs.
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In Eqs.~\eqref{eq:jast-eN} and \eqref{eq:jast-ee}, $r_{iA}$ is the distance between the $i$th electron and the $A$th nucleus while $r_{ij}$ is the interlectronic distance between electrons $i$ and $j$.
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In Eqs.~\eqref{eq:jast-eN} and \eqref{eq:jast-ee}, $r_{iA}$ is the distance between the $i$th electron and the $A$th nucleus while $r_{ij}$ is the interlectronic distance between electrons $i$ and $j$.
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The parameters $a=1/2$
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The parameters $a=1/2$
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and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$
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and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$
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were obtained by energy minimization of a single \titou{HF?} determinant.
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were obtained by energy minimization of a single determinant.
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The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements
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The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements
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of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the
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of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the
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basis of Jastrow-correlated determinants $e^J D_i$:
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basis of Jastrow-correlated determinants $e^J D_i$:
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