I have found this resource quite good for a brief summary of pseudopotentials, of which I am trying to learn more about.
The author shows that, once a functional form for the pseudo-wavefunctions has been found, the 'screened' pseudopotential is given by:
$$ V^{ion}_l = \epsilon_l - V_H^{valence} - V_{XC}^{valence} -\frac{l(l+1)}{2r^2} + \frac{1}{2\psi_l^{PS}} + \frac{1}{2\psi_l^{PS}} \frac{d^2}{dr^2} \psi_l^{PS} $$
For $r<r_c$ (cutoff radius), where $\psi_l^{PS}$ is the pseudo-wavefunction, $V_H$ is the Hartree potential and $V_{XC}$ is the exchange-correlation potential. This expression is just the screened potential (from inverting the Schrodinger equation), with the $V_H$ and $V_{XC}$ contributions of the valence electrons subtracted off. For $r>r_c$, the potential is just the all-electron potential.
The author then states: "Based on these atomic pseudopotentials, the pseudopotential for the entire system takes the form:"
$$ V_{psp} = \sum_{l=0}^{l_{max}} \sum_{m=-l}^{m=l} Y_{lm}(\hat{\textbf{r}}) \ V^{ion}_l(|\textbf{r}|) \ \delta\big(|\textbf{r}| - |\textbf{r}'|\big) \ Y^*_{lm}(\hat{\textbf{r}}')$$
I would like to know where this expression has come from.
