Kinetic energy density of nearly free electrons. I. Response functionals of the external potential

Free electrons have a uniform kinetic energy density (KED), which evolves into a spatially varying quantity as the electrons respond to the gradual imposition of an external potential. In this paper and a companion paper, we examine two sets of functionals for describing the local, non-negative KED that emerges after such a perturbation. In this paper, we emphasize potential functionals, deriving the first- and second-order deviations from the free-electron KED as functionals of the perturbing potential, also reconsidering the analogous functionals for the local density of states and the electron density. (In the second paper, we use these results to re-express the KED response in terms of functionals of the induced electron density.) We develop reciprocal-space formulations of the response kernels to complement previously known real-space forms. The first-order function is straightforward to obtain, but the second-order function requires considerable effort. To manage the derivations, we relate the KED response to that of the one-electron Green function, and then examine the latter in detail. Finally, we provide extensive validation of the derived response functions based on asymptotic analysis of an integral representation, numerical integration of the same generating integral, and application to the linear potential model.