A local tensor that unifies kinetic energy density and vorticity in density functional theory

We present a kinetic energy tensor that unifies a scalar kinetic energy density commonly used in meta-generalized gradient approximation functionals and the vorticity density that appears in paramagnetic current-density-functional theory. Both types of functionals can thus be subsumed as special cases of a novel functional form that is naturally placed on the third rung of Jacob's ladder. Moreover, the kinetic energy tensor is related to the exchange hole curvature, is gauge invariant, and has very clearcut N-representability conditions. The latter conditions enable the definition of an effective number of non-negligible orbitals. Whereas quantities such as the electron localization function can discriminate effective one-orbital regions from other regions, the present kinetic energy tensor can discriminate between one-, two-, three-, and four-or-more orbital regions.