However, for "fc", S 1 and S 2 do not capture this population transition. Moreover, in Fig. we may hypothesize that the fc method fails to describe how the active orbital switches locality around ϵ d = -0.05. This hypothesis is confirmed in Fig. (a), where we plot the impurity population on the pc and fc active orbitals. For the pc method, the HOMO is not always localized on the impurity (true for ϵ d < -0.05) and in fact becomes a bath orbital for ϵ d > -0.05; however, for fc, the HOMO is always on the impurity. Such a change in frontier orbital localization can also be identified by the small tip/dip in Fig. , where we plot the eigenvalues of the impurity projection operator, Pimp = 2(d †