The ground state energy of an atom of nuclear charge Ze and in a magnetic field B is evaluated exactly in the asymptotic regime Z → ∞. We present the results of a rigorous analysis that reveals the existence of 5 regions as Z → ∞: B ≪Z4/3, B ≈Z4/3, Z4/3 ≪B ≪Z3, B ≈Z3, B ≫Z3. Different regions have different physics and different asymptotic theories. Regions 1, 2, 3, 5 are described exactly by a simple density functional theory, but only in regions 1, 2, 3 is it of the semiclassical Thomas-Fermi form. Region 4 cannot be described exactly by any simple density functional theory; surprisingly, it can be described by a simple density matrix functional theory.
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