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@ -12,46 +12,57 @@ The main keywords are:
1. **none:** No 2e-Jastrow is used.
2. **rs-dft:** 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, \[ u(\mathbf{r}_1, \mathbf{r}_2) = u(r_{12}) = \frac{r_{12}}{2} \left[ 1 - \text{erf}(\mu \, r_{12}) \right] - \frac{\exp\left[- (\mu \, r_{12})^2\right]}{2 \sqrt{\pi} \mu} \]
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7D%5Ctau=%5Cfrac%7B1%7D%7B2%7D%5Csum_%7Bi,j%5Cneq%20i%7Du(%5Cmathbf%7Br%7D_i,%5Cmathbf%7Br%7D_j)">
</p>
with,
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7D%20u(%5Cmathbf%7Br%7D_1,%5Cmathbf%7Br%7D_2)=u(r_%7B12%7D)=%5Cfrac%7Br_%7B12%7D%7D%7B2%7D%5Cleft%5B1-%5Ctext%7Berf%7D(%5Cmu%20r_%7B12%7D)%5Cright%5D-%5Cfrac%7B%5Cexp%5B-(%5Cmu%20r_%7B12%7D)%5E2%5D%7D%7B2%5Csqrt%7B%5Cpi%7D%5Cmu%7D">
</p>
## env_type Options
The Jastrow used is multiplied by an envelope \(v\):
\begin{equation}
\tau = \frac{1}{2} \sum_{i,j \neq i} u(\mathbf{r}_i, \mathbf{r}_j) \, v(\mathbf{r}_i) \, v(\mathbf{r}_j)
\end{equation}
The 2-electron Jastrow is multiplied by an envelope \(v\):
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7D%5Ctau=%5Cfrac%7B1%7D%7B2%7D%5Csum_%7Bi,j%5Cneq%20i%7Du(%5Cmathbf%7Br%7D_i,%5Cmathbf%7Br%7D_j)%5C,v(%5Cmathbf%7Br%7D_i)%5C,v(%5Cmathbf%7Br%7D_j)">
</p>
- 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 **prod-gauss**:
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7D%20v(%5Cmathbf%7Br%7D)=%5Cprod_%7BA%7D%5Cleft(1-e%5E%7B-%5Calpha_A(%5Cmathbf%7Br%7D-%5Cmathbf%7BR%7D_A)%5E2%7D%5Cright)">
</p>
- if `env_type` is **sum-gauss**: \(v(\mathbf{r}) = 1 - \sum_{a} \left(1 - c_a e^{-\alpha_a (\mathbf{r} - \mathbf{R}_a)^2 } \right)\)
Here, \(A\) designates the nuclei, and the coefficients and exponents are defined in the tables `enc_coef` and `env_expo` respectively.
- if `env_type` is **sum-gauss**:
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7D%20v(%5Cmathbf%7Br%7D)=1-%5Csum_%7BA%7Dc_A%20e%5E%7B-%5Calpha_A(%5Cmathbf%7Br%7D-%5Cmathbf%7BR%7D_A)%5E2%7D">
</p>
Here, \(A\) designates the nuclei, and the coefficients and exponents are defined in the tables `env_coef` and `env_expo` respectively.
## j1e_type Options
The Jastrow used is:
\begin{equation}
\tau = \sum_i u_{1e}(\mathbf{r}_i)
\end{equation}
The 1-electron Jastrow used is:
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7D%5Ctau=%5Csum_i%20u_%7B1e%7D(%5Cmathbf%7Br%7D_i)">
</p>
- if `j1e_type` is **none**: No one-electron Jastrow is used.
- if `j1e_type` is **gauss**: We use \(u_{1e}(\mathbf{r}) = \sum_A \sum_{p_A} c_{p_A} e^{-\alpha_{p_A} (\mathbf{r} - \mathbf{R}_A)^2}\), where the \(c_p\) and \(\alpha_p\) are defined by the tables `j1e_coef` and `j1e_expo`, respectively.
- if `j1e_type` is **gauss**: We use
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7Du_%7B1e%7D(%5Cmathbf%7Br%7D)=%5Csum_A%5Csum_%7Bp_A%7Dc_%7Bp_A%7De%5E%7B-%5Calpha_%7Bp_A%7D(%5Cmathbf%7Br%7D-%5Cmathbf%7BR%7D_A)%5E2%7D">
</p>
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7D%20c_%7Bp_A%7D%5C,%5Ctext%7Band%7D%5C,%5Calpha_%7Bp_A%7D">
- if `j1e_type` is **charge-harmonizer**: The one-electron Jastrow factor depends on the two-electron Jastrow factor \(u_{2e}\) such that the one-electron term is added to compensate for the unfavorable effect of altering the charge density caused by the two-electron factor:
\begin{equation}
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),
\end{equation}
Feel free to review and let me know if any further adjustments are needed.
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
<p align="center">
<img src="https://latex.codecogs.com/png.image?%5Cinline%20%5Clarge%20%5Cdpi%7B200%7D%5Cbg%7Bwhite%7Du_%7B1e%7D(%5Cmathbf%7Br%7D_1)=-%5Cfrac%7BN-1%7D%7B2N%7D%5C,%5Csum_%7B%5Csigma%7D%5C,%5Cint%20d%5Cmathbf%7Br%7D_2%5C,%5Crho%5E%7B%5Csigma%7D(%5Cmathbf%7Br%7D_2)%5C,u_%7B2e%7D(%5Cmathbf%7Br%7D_1,%5Cmathbf%7Br%7D_2)">
</p>