Leading correction to the local density approximation of the kinetic energy in one dimension

A mathematical framework is constructed for the sum of the lowest N eigenvalues of a potential. Exactness is illustrated on several one-dimensional systems (harmonic oscillator, particle in a box, and Poschl-Teller well). Semiclassical expansion yields the leading corrections for finite systems, identifying the error in common gradient expansions in density functional theory. Some singularities can be avoided when evaluating the correction to the leading term. Correcting the error in the gradient expansion greatly improves accuracy. The relevance to practical density functional calculations is discussed.