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Jmu added

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This commit is contained in:
Abdallah Ammar 2023-01-06 18:48:07 +01:00
parent 352ed6fb53
commit 664fcbbb95
1 changed files with 58 additions and 25 deletions
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@ -154,14 +154,14 @@ with the same name suffixed by ~_im~.
** Periodic boundary calculations (pbc group)
A single k-point per TREXIO file can be stored. The k-point is
A single $k$-point per TREXIO file can be stored. The $k$-point is
defined in this group.
#+NAME: pbc
| Variable | Type | Dimensions | Description |
|------------+---------+------------+-------------------------|
| ~periodic~ | ~int~ | | ~1~: true or ~0~: false |
| ~k_point~ | ~float~ | ~(3)~ | k-point sampling |
| ~k_point~ | ~float~ | ~(3)~ | $k$-point sampling |
#+CALL: json(data=pbc, title="pbc")
@ -285,7 +285,7 @@ with the same name suffixed by ~_im~.
\chi_j(r) = \exp \left( -i \mathbf{k}_j \mathbf{r} \right)
\]
The basis set is defined as the array of k-points in the
The basis set is defined as the array of $k$-points in the
reciprocal space, defined in the ~pbc~ group. The kinetic energy
cutoff ~e_cut~ is the only input data relevant to plane waves.
@ -612,7 +612,8 @@ power = [
\chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r})
\]
where $i$ is the atomic orbital index, $P$ refers to either the
where $i$ is the atomic orbital index,
$P$ encodes for either the
polynomials or the spherical harmonics, $\theta(i)$ returns the
shell on which the AO is expanded, and $\eta(i)$ denotes which
angular function is chosen.
@ -1129,7 +1130,7 @@ power = [
* Correlation factors
** Jastrow factor (jastrow group)
The Jastrow factor is an N-electron function to which the CI
The Jastrow factor is an $N$-electron function to which the CI
expansion is multiplied: $\Psi = \Phi \times \exp(J)$,
where
@ -1189,7 +1190,7 @@ power = [
The terms $J_{\text{ee}}^\infty$ and $J_{\text{eN}}^\infty$ are shifts to ensure that
$J_{\text{ee}}$ and $J_{\text{eN}}$ have an asymptotic value of zero.
$f$ and $g$ are scaling functions defined as
$f$ and $g$ are scaling function defined as
\[
f(r) = \frac{1-e^{-\kappa\, r}}{\kappa} \text{ and }
@ -1198,34 +1199,66 @@ power = [
*** Mu
The "mu" Jastrow factor has only a single parameter $\mu$ for the
[[https://doi.org/10.1063/5.0044683][electron-electron term]]:
[[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
\[
J_{\text{eeN}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
\]
where the electron-electron function $u$ is given by the symetric function
\[
u\left(\mu, r\right) = \frac{r}{2} \, \left[ 1 - \text{erf}(\mu\, r) \right] - \frac{1}{2 \, \mu \, \sqrt{\pi}} \exp \left[ -(\mu \, r)^2 \right].
\]
This electron-electron term is tunned by the parameter $\mu$ which controls the depth and the range of the coulomb hole between electrons.
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
\[
E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right).
\]
In particular, if the parameters $\gamma$ tends to zero, the Mu-Jastrow factor becomes
\[
J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} r_{ij}
\left( 1 - \text{erf}(\mu\, r_{ij})\right) - \frac{1}{\mu\sqrt{\pi}}
e^{-(\mu\,r_{ij})^2}
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
\]
*** Mu with frozen core
and for large $\gamma$ it becomes zero.
To increase the flexibility of the Jastrow and improve the electronic density we add the following electron-nucleus term
\[
J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}} \,
\left[ \exp\left( \kappa_{\alpha} R_{i \alpha}^2 \right) - 1\right].
\]
*** Table of values
#+name: jastrow
| Variable | Type | Dimensions | Description |
|---------------+----------+---------------------+-----------------------------------------------------------------|
| ~type~ | ~string~ | | Type of Jastrow factor: ~CHAMP~, ~Mu~ or ~MuFC~ |
| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
| ~en_num~ | ~dim~ | | Number of Electron-nucleus parameters |
| ~een_num~ | ~dim~ | | Number of Electron-electron-nucleus parameters |
| ~ee~ | ~float~ | ~(jastrow.ee_num)~ | Electron-electron parameters |
| ~en~ | ~float~ | ~(jastrow.en_num)~ | Electron-nucleus parameters |
| ~een~ | ~float~ | ~(jastrow.een_num)~ | Electron-electron-nucleus parameters |
| ~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 Jastrow for electron-nucleus distances |
| 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 |
| ~een_num~ | ~dim~ | | Number of Electron-electron-nucleus parameters |
| ~ee~ | ~float~ | ~(jastrow.ee_num)~ | Electron-electron parameters |
| ~en~ | ~float~ | ~(jastrow.en_num)~ | Electron-nucleus parameters |
| ~een~ | ~float~ | ~(jastrow.een_num)~ | Electron-electron-nucleus parameters |
| ~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 |
#+CALL: json(data=jastrow, title="jastrow")