From 664fcbbb95b5494641f0fb8ecdf4298f47b90055 Mon Sep 17 00:00:00 2001 From: Abdallah Ammar Date: Fri, 6 Jan 2023 18:48:07 +0100 Subject: [PATCH] Jmu added --- trex.org | 83 +++++++++++++++++++++++++++++++++++++++----------------- 1 file changed, 58 insertions(+), 25 deletions(-) diff --git a/trex.org b/trex.org index 8504b6b..d2233e6 100644 --- a/trex.org +++ b/trex.org @@ -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")