Jastrow

Information related to the Jastrow factor in trans-correlated calculations.

The main keywords are: - j2e_type - j1e_type - env_type

j2e_type Options

None: No 2e-Jastrow is used.
Mu: 2e-Jastrow inspired by Range Separated Density Functional Theory. It has the following shape:

$\tau=\frac{1}{2}\sum_{i,j\neq i}u(\mathbf{r}_i,\mathbf{r}_j)$

with,

$u(\mathbf{r}_1,\mathbf{r}_2)=u(r_{12})=\frac{r_{12}}{2}\left[1-\text{erf}(\mu r_{12})\right]-\frac{\exp[-(\mu r_{12})^2]}{2\sqrt{\pi}\mu}$
Mu_Nu: A valence and a core correlation terms are used

$u(\mathbf{r}_1,\mathbf{r}_2)=u(\mu;r_{12})\,v(\mathbf{r}_1)\,v(\mathbf{r}_2)+u(\nu;r_{12})[1-v(\mathbf{r}_1)\,v(\mathbf{r}_2)]$

with envelop (v).

The 2-electron Jastrow is multiplied by an envelope (v):

$\tau=\frac{1}{2}\sum_{i,j\neq i}u(\mathbf{r}_i,\mathbf{r}_j)\,v(\mathbf{r}_i)\,v(\mathbf{r}_j)$

if env_type is None: No envelope is used.
if env_type is Prod_Gauss:

$v(\mathbf{r})=\prod_{A}\left(1-e^{-\alpha_A(\mathbf{r}-\mathbf{R}_A)^2}\right)$
if env_type is Sum_Gauss:

$v(\mathbf{r})=1-\sum_{A}c_A e^{-\alpha_A(\mathbf{r}-\mathbf{R}_A)^2}$

Here, (A) designates the nuclei, and the coefficients and exponents are defined in the tables env_coef and env_expo respectively.

The 1-electron Jastrow used is:

$\tau=\sum_i u_{1e}(\mathbf{r}_i)$

if j1e_type is None: No one-electron Jastrow is used.
if j1e_type is Gauss: We use

$u_{1e}(\mathbf{r})=\sum_A\sum_{p_A}c_{p_A}e^{-\alpha_{p_A}(\mathbf{r}-\mathbf{R}_A)^2}$

$c_{p_A}\,\text{and}\,\alpha_{p_A}$

are defined by the tables j1e_coef and j1e_expo, respectively.

if j1e_type is Charge_Harmonizer: The one-electron Jastrow factor aims to offset the adverse impact of modifying the charge density induced by the two-electron factor
$u_{1e}(\mathbf{r}_1)=-\frac{N-1}{2N}\,\sum_{\sigma}\,\int d\mathbf{r}_2\,\rho^{\sigma}(\mathbf{r}_2)\,u_{2e}(\mathbf{r}_1,\mathbf{r}_2)$
if j1e_type is Charge_Harmonizer_AO: The one-electron Jastrow factor Charge_Harmonizer is fitted by the product of atomic orbitals:
$u_{1e}(\mathbf{r})=\sum_{\alpha,\beta}C_{\alpha,\beta}\chi_{\alpha}(\mathbf{r})\chi_{\beta}(\mathbf{r})$