Bachelet-Hamann-Schlüter (BHS) form

Bachelet et al. [94] proposed an analytic fit to the pseudopotentials generated by the HSC recipe of the following form

$$V_{ps}(r) = V_{core}(r) + \sum_L \Delta V_L^{ion}(r)$$

$$V_{core}(r) = -{Z_v \over r} \left[ \sum_{i=1}^2 c_i^{core} \mbox{erf} \left( \sqrt{\alpha_i^{core}} r \right) \right]$$

$$\Delta V_L^{ion}(r) = \sum_{i=1}^3 \left( A_i + r^2 A_{i+3} \right) \exp(-\alpha_i r^2)$$

The cutoff functions were slightly modified to be $f_1(x) = f_2(x) = f_3(x) = \exp(-x^{3.5})$ . They generated pseudopotentials for almost the entire periodic table (for the local density approximation), where generalizations of the original scheme to include spin-orbit effects for heavy atoms were made. Useful is also their list of atomic reference states.

BHS did not tabulate the $A_i$ coefficients as they are often very big numbers but another set of numbers $C_i$ , where

$$C_i = - \sum_{l=1}^6 A_l Q_{il}$$

and

$$A_i = - \sum_{l=1}^6 C_l Q_{il}^{-1}$$

with

$$Q_{il} = \left\{ \begin{array}{cl} 0 & \mbox{for} \enspace i > l... ... Q_{ki} Q_{kl} \right]^{1/2} & \mbox{for}\enspace i < l \end{array} \right.$$

where $S_{il} = \int_0^{\infty} r^2 \varphi_i(r) \varphi_l(r) dr$ , and

$$\varphi_i(r) = \left\{ \begin{array}{cl} e^{-\alpha_i r^2} & \mb... ...^{-\alpha_i r^2} & \mbox{for}\enspace i=4,5,6 \end{array} \right. \enspace .$$