Electronic structure methods based on the Kohn-Sham (KS) formalism of density-functional theory are ubiquitously and highly successfully used in physics, chemistry, and materials science. Whether the KS formalism is universally applicable is, however, an open question because it is not known whether a KS model system exists for all physical electron systems. This is the question of whether electron densities of real physical systems are always noninteracting $v$-representable. Here we give indications that this is not always the case. To that end, we calculated highly accurate electron densities of first and second row atoms with a partially filled $p$ shell by high-level quantum chemistry methods and by full configuration interaction and then tried to determine the effective KS potential by KS inversion. Except for nitrogen and phosphorus this was only possible for an occupation pattern violating the Aufbau principle. This means that the resulting wave function is not a ground state of the KS Hamiltonian operator and therefore not a valid KS wave function, which indicates that the KS formalism is not always applicable. Strategies to avoid the presented $v$-representability problem are discussed.

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• Density functional theory
• First-principles calculations
• Quantum chemistry methods