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In Density Functional Theory (DFT), the Hohenberg-Kohn theorem allows us to map the interacting system to a fictitious non-interacting Kohn-Sham system, with an effective one-body potential, that reproduces the groundstate density of the interacting system. Formally, this mapping is exact, which suggests that studying the non-interacting system may provide useful insight into the interacting system. We use the Bohr atom, with non-interacting electrons in hydrogenic orbitals, as a model to find the asymptotic expansion of its exchange energy in a charge-neutral scaling as atomic number Z approaches infinity. We show that, like in the interacting atom, the Generalized Gradient Approximations B88 and PBE are very close to having the correct asymptotic form, while the Gradient Expansion Approximation GE2 is off by a factor of 2. We adjust the parameter in GE2 to create a modified functional with the correct asymptotic form. We have also developed a new piecewise smooth model for
the density of the Bohr atom that improves upon the Thomas-Fermi density.
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