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Jmu added
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@ -154,14 +154,14 @@ with the same name suffixed by ~_im~.
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** Periodic boundary calculations (pbc group)
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A single k-point per TREXIO file can be stored. The k-point is
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A single $k$-point per TREXIO file can be stored. The $k$-point is
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defined in this group.
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#+NAME: pbc
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| Variable | Type | Dimensions | Description |
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|------------+---------+------------+-------------------------|
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| ~periodic~ | ~int~ | | ~1~: true or ~0~: false |
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| ~k_point~ | ~float~ | ~(3)~ | k-point sampling |
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| ~k_point~ | ~float~ | ~(3)~ | $k$-point sampling |
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#+CALL: json(data=pbc, title="pbc")
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@ -285,7 +285,7 @@ with the same name suffixed by ~_im~.
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\chi_j(r) = \exp \left( -i \mathbf{k}_j \mathbf{r} \right)
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\]
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The basis set is defined as the array of k-points in the
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The basis set is defined as the array of $k$-points in the
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reciprocal space, defined in the ~pbc~ group. The kinetic energy
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cutoff ~e_cut~ is the only input data relevant to plane waves.
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@ -612,7 +612,8 @@ power = [
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\chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r})
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\]
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where $i$ is the atomic orbital index, $P$ refers to either the
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where $i$ is the atomic orbital index,
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$P$ encodes for either the
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polynomials or the spherical harmonics, $\theta(i)$ returns the
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shell on which the AO is expanded, and $\eta(i)$ denotes which
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angular function is chosen.
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@ -1129,7 +1130,7 @@ power = [
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* Correlation factors
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** Jastrow factor (jastrow group)
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The Jastrow factor is an N-electron function to which the CI
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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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@ -1189,7 +1190,7 @@ power = [
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The terms $J_{\text{ee}}^\infty$ and $J_{\text{eN}}^\infty$ are shifts to ensure that
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$J_{\text{ee}}$ and $J_{\text{eN}}$ have an asymptotic value of zero.
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$f$ and $g$ are scaling functions defined as
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$f$ and $g$ are scaling function defined as
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\[
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f(r) = \frac{1-e^{-\kappa\, r}}{\kappa} \text{ and }
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@ -1198,34 +1199,66 @@ power = [
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*** Mu
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The "mu" Jastrow factor has only a single parameter $\mu$ for the
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[[https://doi.org/10.1063/5.0044683][electron-electron term]]:
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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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\[
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J_{\text{eeN}}(\mathbf{r}) =
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\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
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\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
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\]
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where the electron-electron function $u$ is given by the symetric function
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\[
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u\left(\mu, r\right) = \frac{r}{2} \, \left[ 1 - \text{erf}(\mu\, r) \right] - \frac{1}{2 \, \mu \, \sqrt{\pi}} \exp \left[ -(\mu \, r)^2 \right].
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\]
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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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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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\[
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E_\alpha(R) = 1 - \exp\left( - \gamma_{\alpha} \, R^2 \right).
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\]
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In particular, if the parameters $\gamma$ tends to zero, the Mu-Jastrow factor becomes
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\[
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J_{\text{ee}}(\mathbf{r}) =
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\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} r_{ij}
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\left( 1 - \text{erf}(\mu\, r_{ij})\right) - \frac{1}{\mu\sqrt{\pi}}
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e^{-(\mu\,r_{ij})^2}
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\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right)
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\]
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*** Mu with frozen core
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and for large $\gamma$ 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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\[
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J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}} \,
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\left[ \exp\left( \kappa_{\alpha} R_{i \alpha}^2 \right) - 1\right].
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\]
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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~, ~Mu~ or ~MuFC~ |
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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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| 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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#+CALL: json(data=jastrow, title="jastrow")
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