Kinetic Energy Optimized Pseudopotentials

This scheme is based on the observation that the total energy and the kinetic energy have similar convergence properties when expanded in plane waves. Therefore, the kinetic energy expansion is used as an optimization criteria in the construction of the pseudopotentials. Also this type [97] uses an analytic representation of the pseudo wavefunction within $r_c$

$$\Phi_l(r) = \sum_{i=1}^n a_i j_l(q_i r) \qquad r < r_c$$

where $j_l(q r)$ are spherical Bessel functions with $i-1$ zeros at positions smaller than $r_c$ . The values of $q_i$ are fixed such that

$${j^{\prime}(q_i r_c) \over j(q_i r_c)} = {\Psi^{\prime}_l(r_c) \over \Psi_l(r_c)} \enspace .$$

The conditions that are used to determine the values of $a_i$ are:

• $\Phi_l$ is normalized
• First and second derivatives of $\Phi_l$ are continuous at $r_c$
• $\Delta E_{\rm K}(\{ a_i \}, q_c)$ is minimal
  $$\Delta E_{\rm K} = - \int d^3r \Phi^\star_l \nabla^2 \Phi_l - \int_0^{q_c} dq q^2 \mid \Phi_l(q) \mid^2$$
  $\Delta E_{\rm K}$ is the kinetic energy contribution above a target cutoff value $q_c$ . The value of $q_c$ is an additional parameter (as for example $r_c$ ) that has to be chosen at a reasonable value. In practice $q_c$ is changed until it is possible to minimize $\Delta E_{\rm K}$ to a small enough value.