Thereby n is a non-integer charge of the system and a can be 1 or -1 depending on the situation of adding or re-moving an electron. The resulting deviations ∆ a (n) are shown in Fig. for four sets of parameters with the smallest deviations from linearity. Note that the deviations from the piecewise linearity of E(n) in Eq. ( ) for the tuned functional are very small and the values are at least two orders of magnitude smaller than for standard density functionals. As can be seen, the global minimum (0.00; 0.14) performs best also in this fractional-electron test. Changing the parameter pair to the next higher α value (0.05; 0.13) gives a similarly good description of the ionized system (n -N ≤ 0).