Analysis of exchange and exchange-correlation potentials with non-zero constant asymptotics

Using a constructive approach, it is shown that the asymptotics of the exchange and exchange-correlation potentials and energy densities of finite systems must decay asymptotically to zero as the inverse of the distance. By the traditional inversion approach there appears an undetermined constant in obtaining those potentials, which must be fixed to zero, otherwise it causes a shift in the electronic energy proportional to the number of electrons. It is shown that the requirement of non-zero constant asymptotics for newly developed exchange and exchange-correlation potentials also leads to the unphysical result that the exchange and exchange-correlation energy-density is a non-zero constant in the zero electron density limit