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 $t\left(\mathbf{r}\right)$ in terms of the ground-state density $\rho \left(\mathbf{r}\right).\mathrm{}$ Of course, the direct route to total kinetic energy is from the momentum density $n\left(\mathbf{p}\right),$ which in turn is directly related by Fourier transform to the first-order density matrix $\gamma (\mathbf{r},{\mathbf{r}}^{\prime }).\mathrm{}$ Here, an alternative route to calculate the total kinetic energy is explored, via the Fourier transform $ñ\left(\mathbf{r}\right)$ of the momentum density $n\left(\mathbf{p}\right).\mathrm{}$ It is shown that $ñ\left(\mathbf{r}\right)$ is related to the density matrix γ through its contracted form $\int \gamma ({\mathbf{r}}^{\prime }-\mathbf{r},{\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }=ñ\left(\mathbf{r}\right).\mathrm{}$ As examples, bare Coulomb field and harmonic confinement for arbitrary numbers of closed shells are treated. Finally, a localized potential $V\left(\mathbf{r}\right)$ embedded in an initially uniform electron gas is considered, but now to low order in a perturbation series in $V\left(\mathbf{r}\right).\mathrm{}$