Non‐local relation between kinetic and exchange energy densities in Hartree–Fock theory

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Abstract
A non‐local generalization K(r, r') of the kinetic energy t(r) such that t(r) = ∫K(r, r') dr' is defined using the idempotency property of the Hartree–Fock first‐order density matrix. This is, in turn, related by means of an explicit differential equation to the non‐local exchange energy density X(r, r'). The relationship is illustrated for a couple of examples: with the Fermi‐hole in a uniform electron gas, of importance in the local density version of density functional theory, and with inhomogeneous electron systems.