From f6184620f1c1e5563fbd27afbee811b9509d528e Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Fri, 6 Jan 2023 19:58:54 +0100 Subject: [PATCH] Corrections on Jastrow mu --- trex.org | 87 ++++++++++++++++++++++++++++++++++---------------------- 1 file changed, 53 insertions(+), 34 deletions(-) diff --git a/trex.org b/trex.org index d2233e6..011e042 100644 --- a/trex.org +++ b/trex.org @@ -1132,12 +1132,7 @@ 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 $i$ and $j$ correspond to electrons and $\alpha$ to nuclei. @@ -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: @@ -1199,66 +1199,85 @@ 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 | - | ~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 | + | 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 Jastrow for electron-nucleus distances | #+CALL: json(data=jastrow, title="jastrow")