Can the Hartree-Fock kinetic energy exceed the true kinetic energy?

Hartree-Fock (HF) theory has remained an important tool for quantum chemical calculations since its earliest appearances in the late 1920s. Despite a near century of development, the sign of the difference between the true and Hartree-Fock kinetic energies remains unclear for most systems. Intuition suggests that the true kinetic energy should always be larger than the HF value, but we know of no proof of this. For atoms or ions, the virial theorem shows that the true kinetic energy is always larger than its HF counterpart. Beginning from a generalized virial theorem derived from density scaling considerations, we derive a general expression for the kinetic energy difference, that applies to all systems. Calculations on Hooke’s atom illustrate this relation and show that the difference of the kinetic energies always remains positive, thereby not providing a counterexample.