Upper bound to the gradient-based kinetic energy density of noninteracting electrons in an external potential

We examine a simple upper bound to the gradient-based kinetic energy density (KED) of noninteracting electrons in an external potential: t(r) ≤ [μ-v(r)]n(r)+14λ2n(r), where t(r) is the gradient-based (non-negative) KED, μ is the Fermi energy, v(r) is the external potential, and n(r) is the electron density. The bound emerges naturally from a well-known expression for t(r), leading to an intuitive physical interpretation. For example, t(r) approaches the upper bound in regions where the electron density consists mainly of contributions from states with energies close to the Fermi energy. This upper bound complements the orbital-free lower bound provided by the gradient form of the von Weizsäcker (vW) KED, which is also non-negative. Both bounds yield t(r) exactly for single-orbital systems, and accordingly, they merge in single-orbital regions of more general systems. We demonstrate the universality of the two bounds over a wide range of test systems, including model potentials, atoms, diatomic molecules, and pseudopotential approximations of crystals. We also show that the exact t(r) frequently exceeds the sum of the vW and Thomas-Fermi KEDs, rendering that sum unsuitable as a strict upper bound to the gradient-based KED.