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The Hartree-Fock (HF) approximation has been an important tool forquantum-chemical calculations since its earliest appearance in the late 1920s,and remains the starting point of most single-reference methods in use today.Intuition suggests that the HF kinetic energy should not exceed the exactkinetic energy, but no proof of this conjecture exists, despite a near centuryof development. Beginning from a generalized virial theorem derived fromscaling considerations, we derive a general expression for the kinetic energydifference that applies to all systems. For any atom or ion this triviallyreduces to the well-known result that the total energy is the negative of thekinetic energy and since correlation energies are never positive, proves theconjecture in this case. Similar considerations apply to molecules at theirequilibrium bond lengths. We use highly precise calculations on Hooke's atom(two electrons in a parabolic well) to test the conjecture in a non-trivialcase, and to parameterize the difference between density-functional and HFquantities, but find no violations of the conjecture.

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