In cases where optimal (α, ω) pairs correspond to similar J * values it is customary to employ an additional criterion for the selection of optimal values. For example, according to Janak's theorem, the energy E(N) for an exact exchange-correlation functional must vary linearly for fractional electron numbers between integer N values. In reality, this condition is not fulfilled for many functionals pointing to a (de)localization error. We have tested the performance of the present parameters along minimum valley of J * (α, ω) in Figure (white line). For all points of (α; ω) in the valley, we have calculated the deviation of the curvatures of the E(N)-dependence for fractional charges from being piecewise linear according to