Leading corrections to local approximations

For the kinetic energy of one-dimensional model finite systems the leading corrections to local approximations as a functional of the potential are derived using semiclassical methods. The corrections are simple, nonlocal functionals of the potential. Turning points produce quantum oscillations leading to energy corrections, which are completely different from the gradient corrections that occur in bulk systems with slowly varying densities. Approximations that include quantum corrections are typically much more accurate than their local analogs. The consequences for density functional theory are discussed.