From 1adaec0da012865d53f9e6d33bcd01a82c36582a Mon Sep 17 00:00:00 2001
From: Anthony Scemama <scemama@irsamc.ups-tlse.fr>
Date: Tue, 28 Mar 2023 10:40:49 +0200
Subject: [PATCH] Update pseudo documentation

---
 trex.org | 26 ++++++++++++++------------
 1 file changed, 14 insertions(+), 12 deletions(-)

diff --git a/trex.org b/trex.org
index f233fb97..ba901a00 100644
--- a/trex.org
+++ b/trex.org
@@ -386,20 +386,22 @@ prim_factor =
    \[
    V_A^{\text{ECP}} =
    V_{A \ell_{\max}+1} +
-   \sum_{\ell=0}^{\ell_{\max}} V_{A \ell}
-   \sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle  \langle Y_{\ell m} |
+   \sum_{\ell=0}^{\ell_{\max}}
+   \delta V_{A \ell}\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \langle Y_{\ell m} |
    \]
 
-   The first term in the equation above is sometimes attributed to the local channel,
-   while the remaining terms correspond to the non-local channel projections.
-
-   All the functions $V_{A\ell}$ are parameterized as:
-   \[
-   V_{A \ell}(\mathbf{r}) =
-   \sum_{q=1}^{N_{q \ell}}
-   \beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
-   e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 }
-   \].
+   The first term in this equation is attributed to the local channel, while
+   the remaining terms correspond to non-local channel projections.  $\ell_{\max}$
+   refers to the maximum angular momentum in the non-local component of the ECP.
+   The functions \(\delta V_{A \ell}\) and \(V_{A \ell_{\max}+1}\) are parameterized as:
+   \begin{eqnarray}
+      \delta V_{A \ell}(\mathbf{r}) &=&
+      \sum_{q=1}^{N_{q \ell}}
+      \beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
+      e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 } \nonumber\\
+      V_{A \ell_{\max}+1}(\mathbf{r}) &=& -\frac{Z_\text{eff}}{|\mathbf{r}-\mathbf{R}_{A}|}+\delta V_{A \ell_{\max}+1}(\mathbf{r})
+   \end{eqnarray}
+   where $Z_\text{eff}$ is the effective nuclear charge of the center.
 
    See http://dx.doi.org/10.1063/1.4984046 or https://doi.org/10.1063/1.5121006 for more info.