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Corrections on Jastrow mu

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Anthony Scemama 2023-01-06 19:58:54 +01:00
parent 664fcbbb95
commit f6184620f1
1 changed files with 53 additions and 34 deletions
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@ -1132,11 +1132,6 @@ power = [
The Jastrow factor is an $N$-electron function to which the CI
expansion is multiplied: $\Psi = \Phi \times \exp(J)$,
where
\[
J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R})
\]
In the following, we use the notations $r_{ij} = |\mathbf{r}_i - \mathbf{r}_j|$ and
$R_{i\alpha} = |\mathbf{r}_i - \mathbf{R}_\alpha|$, where indices
@ -1152,7 +1147,12 @@ power = [
*** CHAMP
The first form of Jastrow factor is the one used in
the [[https://trex-coe.eu/trex-quantum-chemistry-codes/champ][CHAMP]] program.
the [[https://trex-coe.eu/trex-quantum-chemistry-codes/champ][CHAMP]] program:
\[
J(\mathbf{r},\mathbf{R}) = J_{\text{eN}}(\mathbf{r},\mathbf{R}) + J_{\text{ee}}(\mathbf{r}) + J_{\text{eeN}}(\mathbf{r},\mathbf{R})
\]
$J_{\text{eN}}$ contains electron-nucleus terms:
@ -1199,53 +1199,74 @@ power = [
*** Mu
[[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.
This correlation factor imposes the electron-electron cusp and it is built such that the leading order in $1/r_{12}$ of the
effective two-electron potential reproduces the long-range interaction of the range-separated density functional theory.
Its analytical expression reads as
[[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. This
correlation factor imposes the electron-electron cusp and it is
built such that the leading order in $1/r_{12}$ of the effective
two-electron potential reproduces the long-range interaction of the
range-separated density functional theory. Its analytical
expression reads
\[
J_{\text{eeN}}(\mathbf{r}) =
J(\mathbf{r}, \mathbf{R}) = J_{\text{eeN}}(\mathbf{r}, \mathbf{R}) +
J_{\text{eN}}(\mathbf{r}, \mathbf{R})
\].
The electron-electron cusp is incorporated in the three-body term.
\[
J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
\]
where the electron-electron function $u$ is given by the symetric function
$u$ is an electron-electron function is given by the symetric function
\[
u\left(\mu, r\right) = \frac{r}{2} \, \left[ 1 - \text{erf}(\mu\, r) \right] - \frac{1}{2 \, \mu \, \sqrt{\pi}} \exp \left[ -(\mu \, r)^2 \right].
\]
This electron-electron term is tunned by the parameter $\mu$ which controls the depth and the range of the coulomb hole between electrons.
This electron-electron term is tuned by the parameter $\mu$ which
controls the depth and the range of the Coulomb hole between
electrons.
An enveloppe function has been introduced to cancel out the Jastrow effects between two-electrons when they are both close
to a nucleus (to perform a frozen-core calculation). The envelop function is given by
An envelope function has been introduced to cancel out the Jastrow
effects between two-electrons when they are both close to a nucleus
(to perform a frozen-core calculation). The envelope function is
given by
\[
E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right).
\]
In particular, if the parameters $\gamma$ tends to zero, the Mu-Jastrow factor becomes
In particular, if the parameters $\gamma_\alpha$ tend to zero, the
Mu-Jastrow factor becomes a two-body Jastrow factor:
\[
J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
\]
and for large $\gamma$ it becomes zero.
and for large $\gamma_\alpha$ it becomes zero.
To increase the flexibility of the Jastrow and improve the electronic density we add the following electron-nucleus term
To increase the flexibility of the Jastrow and improve the
electron density the following electron-nucleus term is added
\[
J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}} \,
\left[ \exp\left( \kappa_{\alpha} R_{i \alpha}^2 \right) - 1\right].
\left[ \exp\left( a_{\alpha} R_{i \alpha}^2 \right) - 1\right].
\]
The parameter $\mu$ is stored in the ~ee~ array, the parameters
$\gamma_\alpha$ are stored in the ~een~ array, and the parameters
$a_\alpha$ are stored in the ~en~ array.
*** Table of values
#+name: jastrow
| Variable | Type | Dimensions | Description |
|----------------+----------+---------------------+-----------------------------------------------------------------------|
|---------------+----------+---------------------+-----------------------------------------------------------------|
| ~type~ | ~string~ | | Type of Jastrow factor: ~CHAMP~ or ~Mu~ |
| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
| ~en_num~ | ~dim~ | | Number of Electron-nucleus parameters |
@ -1256,9 +1277,7 @@ power = [
| ~en_nucleus~ | ~index~ | ~(jastrow.en_num)~ | Nucleus relative to the eN parameter |
| ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter |
| ~ee_scaling~ | ~float~ | | $\kappa$ value in CHAMP Jastrow for electron-electron distances |
| ~en_scaling~ | ~float~ | ~(nucleus.num)~ | $\kappa$ value in CHAMP and Mu Jastrow for electron-nucleus distances |
| ~ee_hole~ | ~float~ | | $\mu$ value in Mu Jastrow |
| ~en_enveloppe~ | ~float~ | ~(nucleus.num)~ | $\gamma$ value in Mu Jastrow |
| ~en_scaling~ | ~float~ | ~(nucleus.num)~ | $\kappa$ value in CHAMP Jastrow for electron-nucleus distances |
#+CALL: json(data=jastrow, title="jastrow")