Validity of the extended electron-electron cusp condition

The extended cusp condition asserts that h(u)-a₀dh(u)/du≥0, where a₀ is the Bohr radius, u is the interelectronic spacing, and h(u) is the angle-averaged pair density in the ground state. We prove that this inequality is obeyed by Hooke’s atom for any value of the spring constant. However, we also show that this condition is violated by the uniform electron gas of high density. We explain the qualitative difference between these two systems by subtracting a long-range contribution from h(u), leaving a short-range contribution which is amenable to a local density approximation. Thus the extended cusp condition is not a universal property of the ground state of inhomogeneous electronic systems.