The differential virial theorem of March and Young for N fermions moving in a common one-dimensional potential energy V(x) is here combined with the Euler equation of density functional theory expressing the constancy of the chemical potential throughout the entire inhomogeneous particle density. The functional derivative of the single-particle kinetic energy is thereby expressed directly in terms of the kinetic energy density; a line integral being involved in establishing the connection. This result is then used to establish a formula measuring departures from simple homogeneity of the kinetic energy functional: a matter of current interest in density functional theory. Finally, the general theory of the functional derivative of the single-particle kinetic energy with respect to the particle density is exemplified for the case of harmonic confinement of fermions in one dimension.
Explicit kinetic-energy density functionals for noninteracting electron systems that are based on reproducing the linear response function in the homogeneous limit are the most accurate approximations to the exact Kohn-Sham functional. In this work we study several such models for one-dimensional systems. In this case, the geometrical simplicity allows the calculation of the static linear response resulting from these functionals for inhomogeneous systems. Likewise, the discussion of the energies and density profiles obtained is easier and clearer than that for more realistic problems. Special attention is paid to the application of explicit kinetic energy functionals to systems with odd and noninteger number of particles.
For first-principles density functional theory of a many fermion system, the determination of the kinetic energy functional is important. We consider
N independent fermions with spatially varying effective mass in two dimensions, we derive the corresponding kinetic energy density using the $\ensuremath{\hbar}$ semiclassical approach. Our result reduces, as expected, to the one obtained in the literature for a constant effective mass. We examine the analytical expressions of the position dependent effective mass terms in the kinetic energy density functional with respect to the dimensionality
d=1,2,3 of the space.