From 2f40ff5776183eab249c238765007182b03cde48 Mon Sep 17 00:00:00 2001
From: AbdAmmar <59544987+AbdAmmar@users.noreply.github.com>
Date: Tue, 16 Jan 2024 01:13:44 +0100
Subject: [PATCH] Update README.md
---
plugins/local/jastrow/README.md | 57 ++++++++++++++++++++-------------
1 file changed, 34 insertions(+), 23 deletions(-)
diff --git a/plugins/local/jastrow/README.md b/plugins/local/jastrow/README.md
index 33ed177b..0b74b6c6 100644
--- a/plugins/local/jastrow/README.md
+++ b/plugins/local/jastrow/README.md
@@ -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, \[ u(\mathbf{r}_1, \mathbf{r}_2) = u(r_{12}) = \frac{r_{12}}{2} \left[ 1 - \text{erf}(\mu \, r_{12}) \right] - \frac{\exp\left[- (\mu \, r_{12})^2\right]}{2 \sqrt{\pi} \mu} \]
+
+ )
+
+ with,
+
+ =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)
+
## 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 \(v\):
+
+ %5C,v(%5Cmathbf%7Br%7D_i)%5C,v(%5Cmathbf%7Br%7D_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 **prod-gauss**:
+
+ =%5Cprod_%7BA%7D%5Cleft(1-e%5E%7B-%5Calpha_A(%5Cmathbf%7Br%7D-%5Cmathbf%7BR%7D_A)%5E2%7D%5Cright))
+
-- 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**:
+
+ =1-%5Csum_%7BA%7Dc_A%20e%5E%7B-%5Calpha_A(%5Cmathbf%7Br%7D-%5Cmathbf%7BR%7D_A)%5E2%7D)
+
+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:
+
+ )
+
- 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
+
+ =%5Csum_A%5Csum_%7Bp_A%7Dc_%7Bp_A%7De%5E%7B-%5Calpha_%7Bp_A%7D(%5Cmathbf%7Br%7D-%5Cmathbf%7BR%7D_A)%5E2%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
+
+ =-%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))
+