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2nd screening of Sec IV done

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Pierre-Francois Loos 2020-08-18 10:42:39 +02:00
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Manuscript/rsdft-cipsi-qmc.tex
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@ -495,7 +495,7 @@ The first question we would like to address is the quality of the
nodes of the wave function $\Psi^{\mu}$ obtained for intermediate values of the nodes of the wave function $\Psi^{\mu}$ obtained for intermediate values of the
range separation parameter (\ie, $0 < \mu < +\infty$). range separation parameter (\ie, $0 < \mu < +\infty$).
For this purpose, we consider a weakly correlated molecular system, namely the water For this purpose, we consider a weakly correlated molecular system, namely the water
molecule \titou{at its experimental geometry. \cite{Caffarel_2016}} molecule at its experimental geometry. \cite{Caffarel_2016}
We then generate trial wave functions $\Psi^\mu$ for multiple values of We then generate trial wave functions $\Psi^\mu$ for multiple values of
$\mu$, and compute the associated FN-DMC energy keeping fixed all the $\mu$, and compute the associated FN-DMC energy keeping fixed all the
parameters impacting the nodal surface, such as the CI coefficients and the molecular orbitals. parameters impacting the nodal surface, such as the CI coefficients and the molecular orbitals.
@ -583,7 +583,7 @@ To do so, we have made the following numerical experiment.
First, we extract the 200 determinants with the largest weights in the FCI wave First, we extract the 200 determinants with the largest weights in the FCI wave
function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants, function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants,
we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}] we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
for different values of $\mu$ \titou{using the srPBE functional}. This gives the CI expansions $\Psi^\mu$. for different values of $\mu$ using the srPBE functional. This gives the CI expansions $\Psi^\mu$.
Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of
a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and
\begin{subequations} \begin{subequations}
@ -601,7 +601,7 @@ where the sum over $i < j$ loops over all unique electron pairs.
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 $j$. 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 $j$.
The parameters $a=1/2$ The parameters $a=1/2$
and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$ and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$
were obtained by energy minimization of a single \titou{HF?} determinant. were obtained by energy minimization of a single determinant.
The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements 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 of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the
basis of Jastrow-correlated determinants $e^J D_i$: basis of Jastrow-correlated determinants $e^J D_i$: