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Corrections on Jastrow mu
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@ -1132,11 +1132,6 @@ power = [
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The Jastrow factor is an $N$-electron function to which the CI
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expansion is multiplied: $\Psi = \Phi \times \exp(J)$,
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where
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\[
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J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R})
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\]
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In the following, we use the notations $r_{ij} = |\mathbf{r}_i - \mathbf{r}_j|$ and
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$R_{i\alpha} = |\mathbf{r}_i - \mathbf{R}_\alpha|$, where indices
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@ -1152,7 +1147,12 @@ power = [
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*** CHAMP
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The first form of Jastrow factor is the one used in
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the [[https://trex-coe.eu/trex-quantum-chemistry-codes/champ][CHAMP]] program.
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the [[https://trex-coe.eu/trex-quantum-chemistry-codes/champ][CHAMP]] program:
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\[
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J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R})
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\]
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$J_{\text{eN}}$ contains electron-nucleus terms:
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@ -1199,66 +1199,85 @@ power = [
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*** Mu
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[[https://aip.scitation.org/doi/10.1063/5.0044683][Mu-Jastrow]] is based on a one-parameter correlation factor that has been introduced in the context of transcorrelated methods.
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This correlation factor imposes the electron-electron cusp and it is built such that the leading order in $1/r_{12}$ of the
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effective two-electron potential reproduces the long-range interaction of the range-separated density functional theory.
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Its analytical expression reads as
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[[https://aip.scitation.org/doi/10.1063/5.0044683][Mu-Jastrow]] is based on a one-parameter correlation factor that has
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been introduced in the context of transcorrelated methods. This
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correlation factor imposes the electron-electron cusp and it is
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built such that the leading order in $1/r_{12}$ of the effective
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two-electron potential reproduces the long-range interaction of the
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range-separated density functional theory. Its analytical
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expression reads
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\[
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J_{\text{eeN}}(\mathbf{r}) =
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J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) +
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J_{\text{eN}}(\mathbf{r}, \mathbf{R})
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\].
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The electron-electron cusp is incorporated in the three-body term.
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\[
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J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
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\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
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\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
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\]
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where the electron-electron function $u$ is given by the symetric function
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$u$ is an electron-electron function is given by the symetric function
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\[
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u\left(\mu, r\right) = \frac{r}{2} \, \left[ 1 - \text{erf}(\mu\, r) \right] - \frac{1}{2 \, \mu \, \sqrt{\pi}} \exp \left[ -(\mu \, r)^2 \right].
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\]
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This electron-electron term is tunned by the parameter $\mu$ which controls the depth and the range of the coulomb hole between electrons.
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This electron-electron term is tuned by the parameter $\mu$ which
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controls the depth and the range of the Coulomb hole between
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electrons.
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An enveloppe function has been introduced to cancel out the Jastrow effects between two-electrons when they are both close
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to a nucleus (to perform a frozen-core calculation). The envelop function is given by
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An envelope function has been introduced to cancel out the Jastrow
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effects between two-electrons when they are both close to a nucleus
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(to perform a frozen-core calculation). The envelope function is
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given by
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\[
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E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right).
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\]
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In particular, if the parameters $\gamma$ tends to zero, the Mu-Jastrow factor becomes
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In particular, if the parameters $\gamma_\alpha$ tend to zero, the
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Mu-Jastrow factor becomes a two-body Jastrow factor:
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\[
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J_{\text{ee}}(\mathbf{r}) =
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\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
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\]
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and for large $\gamma$ it becomes zero.
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and for large $\gamma_\alpha$ it becomes zero.
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To increase the flexibility of the Jastrow and improve the electronic density we add the following electron-nucleus term
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To increase the flexibility of the Jastrow and improve the
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electron density the following electron-nucleus term is added
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\[
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J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}} \,
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\left[ \exp\left( \kappa_{\alpha} R_{i \alpha}^2 \right) - 1\right].
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\left[ \exp\left( a_{\alpha} R_{i \alpha}^2 \right) - 1\right].
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\]
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The parameter $\mu$ is stored in the ~ee~ array, the parameters
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$\gamma_\alpha$ are stored in the ~een~ array, and the parameters
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$a_\alpha$ are stored in the ~en~ array.
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*** Table of values
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#+name: jastrow
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| Variable | Type | Dimensions | Description |
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|----------------+----------+---------------------+-----------------------------------------------------------------------|
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| ~type~ | ~string~ | | Type of Jastrow factor: ~CHAMP~ or ~Mu~ |
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| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
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| ~en_num~ | ~dim~ | | Number of Electron-nucleus parameters |
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| ~een_num~ | ~dim~ | | Number of Electron-electron-nucleus parameters |
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| ~ee~ | ~float~ | ~(jastrow.ee_num)~ | Electron-electron parameters |
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| ~en~ | ~float~ | ~(jastrow.en_num)~ | Electron-nucleus parameters |
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| ~een~ | ~float~ | ~(jastrow.een_num)~ | Electron-electron-nucleus parameters |
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| ~en_nucleus~ | ~index~ | ~(jastrow.en_num)~ | Nucleus relative to the eN parameter |
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| ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter |
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| ~ee_scaling~ | ~float~ | | $\kappa$ value in CHAMP Jastrow for electron-electron distances |
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| ~en_scaling~ | ~float~ | ~(nucleus.num)~ | $\kappa$ value in CHAMP and Mu Jastrow for electron-nucleus distances |
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| ~ee_hole~ | ~float~ | | $\mu$ value in Mu Jastrow |
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| ~en_enveloppe~ | ~float~ | ~(nucleus.num)~ | $\gamma$ value in Mu Jastrow |
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| Variable | Type | Dimensions | Description |
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|---------------+----------+---------------------+-----------------------------------------------------------------|
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| ~type~ | ~string~ | | Type of Jastrow factor: ~CHAMP~ or ~Mu~ |
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| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
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| ~en_num~ | ~dim~ | | Number of Electron-nucleus parameters |
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| ~een_num~ | ~dim~ | | Number of Electron-electron-nucleus parameters |
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| ~ee~ | ~float~ | ~(jastrow.ee_num)~ | Electron-electron parameters |
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| ~en~ | ~float~ | ~(jastrow.en_num)~ | Electron-nucleus parameters |
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| ~een~ | ~float~ | ~(jastrow.een_num)~ | Electron-electron-nucleus parameters |
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| ~en_nucleus~ | ~index~ | ~(jastrow.en_num)~ | Nucleus relative to the eN parameter |
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| ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter |
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| ~ee_scaling~ | ~float~ | | $\kappa$ value in CHAMP Jastrow for electron-electron distances |
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| ~en_scaling~ | ~float~ | ~(nucleus.num)~ | $\kappa$ value in CHAMP Jastrow for electron-nucleus distances |
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#+CALL: json(data=jastrow, title="jastrow")
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