Quantum Crystallography: X‐Ray Orbitals Are Kohn–Sham Orbitals

It is suggested that, under restrictive ideal circumstances, the interacting electronic energy of a system can be obtained directly from experimentally accessible x‐ray orbitals, defined here as orbitals obtained from direct fitting of the one‐body density matrix to measured x‐ray structure factors, through inverse Kohn–Sham equation functional integration. The argument rests on the identity, within a unitary transformation, between experimentally fitted x‐ray orbitals and theoretically calculated Kohn–Sham orbitals, both of which—in ideal circumstances—yield the same electron density. Using x‐ray orbitals, one may evaluate both the Kohn–Sham kinetic energy and the interacting potential energy by functional integration of the exact Kohn–Sham potential, taken directly from the Kohn–Sham equations, expressed in atomic units (a.u.), . This approach allows the total interacting Kohn–Sham energy: , where E KS is the interacting electronic energy, is the non‐interacting kinetic‐energy, and is the interacting electronic potential‐energy obtained from the Kohn–Sham potential , to be reconstructed entirely from orbital data measurable through high‐resolution x‐ray diffraction. A theorem is enunciated stating that for a non‐degenerate ground state, absent magnetic fields, in a real finite one‐particle basis, if the experimentally accessible constraints uniquely determine an idempotent rank‐ N x‐ray projector P XR , and if a Kohn–Sham projector P KS satisfies the same set of constraints, then the two projectors must be identical ( P XR = P KS ). Any remaining freedom corresponds only to unitary rotations within the occupied subspace and does not affect the projector itself. A mathematical development and an illustrative application to the beryllium atom are presented. Since this procedure reconstructs Kohn–Sham energies from diffraction‐constrained electronic structure, it offers an experimentally anchored route for benchmarking and evaluating approximate DFT exchange–correlation functionals via their associated potentials and energy decompositions.