Kerker Pseudopotentials

Also in this approach [95] pseudopotentials with the HSC properties are constructed. But instead of using cutoff functions ($f_1$ , $f_2$ , $f_3$ ) the pseudo wavefunctions are directly constructed from the all-electron wavefunctions by replacing the all-electron wavefunction inside some cutoff radius by a smooth analytic function that is matched to the all-electron wavefunction at the cutoff radius. The HSC properties then translate into a set of equations for the parameters of the analytic form. After having determined the pseudo wavefunction the Schrödinger equation is inverted and the resulting potential unscreened. Note that the cutoff radius of this type of pseudopotential construction scheme is considerably larger than the one used in the HSC scheme. Typically the cutoff radius is chosen slightly smaller than $R_{\rm max}$ , the outermost maximum of the all-electron wavefunction. The analytic form proposed by Kerker is

$$\Phi_l(r) = r^{l+1} e^{p(r)} \qquad r < r_c$$

with

$$p(r) = \alpha r^4 + \beta r^3 + \gamma r^2 + \delta \enspace .$$

The term linear in r is missing to avoid a singularity of the potential at $r=0$ . The HSC conditions can be translated into a set of equations for the parameters $\alpha, \beta, \gamma, \delta$ .