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.