Troullier-Martins Pseudopotentials

The Kerker method was generalized by Troullier and Martins [96] to polynomials of higher order. The rational behind this was to use the additional parameters (the coefficients of the higher terms in the polynomial) to construct smoother pseudopotentials. The Troullier-Martins wavefunctions has the following form

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

with

$$p(r) = c_0 + c_2 r^2 + c_4 r^4 + c_6 r^6 + c_8 r^8 + c_{10} r^{10} + c_{12} r^{12}$$

and the coefficients $c_n$ are determined from

• norm-conservation
• For $n = 0 \ldots 4$
  $$\left. {d^n \Phi \over dr^n} \right\vert _{r=r_c} = \left. {d^n \Psi \over dr^n} \right\vert _{r=r_c}$$

• $$\left. {d \Phi \over dr} \right\vert _{r=0} = 0$$