saving work
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@ -272,7 +272,7 @@ improvements on the path from HF to FCI:
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i) increasing the maximum degree of excitation of CI methods (CISD, CISDT,
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i) increasing the maximum degree of excitation of CI methods (CISD, CISDT,
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CISDTQ, \ldots), or ii) expanding the size of a complete active space
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CISDTQ, \ldots), or ii) expanding the size of a complete active space
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(CAS) wave function until all the orbitals are in the active space.
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(CAS) wave function until all the orbitals are in the active space.
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Selected CI methods take a shortcut between the HF
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SCI methods take a shortcut between the HF
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determinant and the FCI wave function by increasing iteratively the
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determinant and the FCI wave function by increasing iteratively the
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number of determinants on which the wave function is expanded,
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number of determinants on which the wave function is expanded,
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selecting the determinants which are expected to contribute the most
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selecting the determinants which are expected to contribute the most
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@ -281,7 +281,7 @@ extracted from the CI matrix expressed in the determinant subspace,
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and the FCI energy can be estimated by adding up to the variational energy
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and the FCI energy can be estimated by adding up to the variational energy
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a second-order perturbative correction (PT2), $\EPT$.
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a second-order perturbative correction (PT2), $\EPT$.
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The magnitude of $\EPT$ is a measure of the distance to the FCI energy
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The magnitude of $\EPT$ is a measure of the distance to the FCI energy
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and a diagnostic of the the quality of the wave function.
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and a diagnostic of the quality of the wave function.
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Within the CIPSI algorithm originally developed by Huron \textit{et al.} in Ref.~\onlinecite{Huron_1973} and efficiently implemented as described in Ref.~\onlinecite{Garniron_2019}, the PT2
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Within the CIPSI algorithm originally developed by Huron \textit{et al.} in Ref.~\onlinecite{Huron_1973} and efficiently implemented as described in Ref.~\onlinecite{Garniron_2019}, the PT2
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correction is computed simultaneously to the determinant selection at no extra cost.
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correction is computed simultaneously to the determinant selection at no extra cost.
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$\EPT$ is then the sole parameter of the CIPSI algorithm and is chosen to be its convergence criterion.
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$\EPT$ is then the sole parameter of the CIPSI algorithm and is chosen to be its convergence criterion.
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@ -598,10 +598,12 @@ were obtained by energy minimization with 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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\begin{eqnarray}
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\begin{subequations}
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H_{ij} & = & \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J} } \\
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\begin{gather}
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S_{ij} & = & \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} }
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H_{ij} = \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J} } \\
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\end{eqnarray}
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S_{ij} = \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} }
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\end{gather}
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\end{subequations}
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and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}
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and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}
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We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
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We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
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@ -1064,7 +1066,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
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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=\textwidth]{g2-dmc.pdf}
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\includegraphics[width=\textwidth]{g2-dmc.pdf}
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\caption{Errors in the FN-DMC atomization energies with the different
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\caption{Errors in the FN-DMC atomization energies with various
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trial wave functions. Each dot corresponds to an atomization
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trial wave functions. Each dot corresponds to an atomization
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energy.
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energy.
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The boxes contain the data between first and third quartiles, and
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The boxes contain the data between first and third quartiles, and
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@ -1090,7 +1092,7 @@ cancellations of errors.
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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=\textwidth]{g2-ndet.pdf}
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\includegraphics[width=\textwidth]{g2-ndet.pdf}
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\caption{Number of determinants in the different trial wave
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\caption{Number of determinants for various trial wave
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functions. Each dot corresponds to an atomization energy.
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functions. Each dot corresponds to an atomization energy.
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The boxes contain the data between first and third quartiles, and
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The boxes contain the data between first and third quartiles, and
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the line in the box represents the median. The outliers are shown
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the line in the box represents the median. The outliers are shown
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@ -1132,8 +1134,8 @@ solution would have been the PBE single determinant.
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%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%
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In the present work, we have shown that introducing short-range correlation via
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In the present work, we have shown that introducing short-range correlation via
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a range-separated Hamiltonian in a full CI expansion yields improved
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a range-separated Hamiltonian in a FCI expansion yields improved
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nodal surfaces, especially with small basis sets. The effect of sr-DFT
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nodal surfaces, especially with small basis sets. The effect of short-range DFT
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on the determinant expansion is similar to the effect of re-optimizing
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on the determinant expansion is similar to the effect of re-optimizing
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the CI coefficients in the presence of a Jastrow factor, but without
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the CI coefficients in the presence of a Jastrow factor, but without
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the burden of performing a stochastic optimization.
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the burden of performing a stochastic optimization.
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