Momentum density and its Fourier transform: Relation to the first-order density matrix and some scaling properties

Density-functional theory requires knowledge of the kinetic-energy density t(r) in terms of the ground-state density ρ(r). Of course, the direct route to total kinetic energy is from the momentum density n(p), which in turn is directly related by Fourier transform to the first-order density matrix γ(r,r^′). Here, an alternative route to calculate the total kinetic energy is explored, via the Fourier transform ñ(r) of the momentum density n(p). It is shown that ñ(r) is related to the density matrix γ through its contracted form ∫γ(r^′-r,r^′)dr^′=ñ(r). As examples, bare Coulomb field and harmonic confinement for arbitrary numbers of closed shells are treated. Finally, a localized potential V(r) embedded in an initially uniform electron gas is considered, but now to low order in a perturbation series in V(r).