## Pseudopotential The pseudopotential is an effective potential constructed to replace the atomic all-electron potential, such that core states are eliminated, and the valence electrons are described by pseudo-wavefunctions with lower mode, it has the form of $\left\{ \begin{aligned} &\text{local:} \quad V_{\text{loc}}^{\text{PP}}(r) = -\frac{Z_{\text{eff}}}{r} \, \text{erf}\left(\alpha^{\text{PP}} r\right) + \sum_{i=1}^4 C_i^{\text{PP}} \left( \sqrt{2} \alpha^{\text{PP}} r \right)^{2i-2} \exp\left[-\left( \alpha^{\text{PP}} r \right)^2\right] \\ &\text{non-local:} \quad V_{\text{nl}}^{\text{PP}}(\mathbf{r}, \mathbf{r}') = \sum_{lm} \sum_{ij} \langle \mathbf{r} \, | \, p_i^{lm} \rangle \, h_{ij}^l \, \langle p_j^{lm} \, | \, \mathbf{r}' \rangle \end{aligned} \right.$ and in the local potential expression, we have a long range effect (the one with effective ionic charges $Z_{\text{eff}}$) plus a short range effect (the one with summation of Gaussians). $\alpha^{\text{PP}}=\frac{1}{\sqrt2 r_{\text{loc}}^{\text{PP}}}$ This $r_{\text{loc}}$ is the range of Gaussian ionic charge distribution. In the non-local part, $\langle \mathbf{r} \, | \, p_i^{lm} \rangle = N_i^l \, Y^{lm}(\hat{r}) \, r^{l+2i-2} \, \exp\left[-\frac{1}{2} \left( \frac{r}{r_l} \right)^2\right]$. $N_i^l$ is the normalization constant, $Y^{lm}(\hat{r})$ is spherical harmonics, and $r_{l}$ is the radius. The $h_{ij}^{l}$ is a coefficient. ![[Drawing 2024-08-30 00.35.52.excalidraw.svg]] The basic idea is: inside our cut-off $r_{c}$, the wave function and the potential are oscillatory, but in many cases they are not very critical. Therefore, we replace the complex expression with something more smooth, namely pseudopotentials. Outside the cut-off, we have smooth pseudopotentials and the real potentials being the same. Approaches to follow is: ![[Drawing 2024-08-30 00.53.11.excalidraw.svg]] >[!Info] >See the original paper: [Pseudopotentials for H to Kr optimized for gradient-corrected exchange-correlation functionals](https://link.springer.com/article/10.1007/s00214-005-0655-y). Use https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/31572/214_2005_Article_655.pdf?sequence=2 as an alternative if access is limited.