Linear-response functions in spin-density-functional theory

A model system is studied for which exact results are known in order to exhibit directly the limits of density-functional theory when applied to excitation properties. As one might intuitively expect, the ground-state theory accurately describes properties reflecting certain static limits, the paramagnetic susceptibility, for example. The ground-state theory does not, however, accurately describe physical properties involving the density of low-lying excitations, spin-wave, and plasmon dispersion, for example. To demonstrate these limitations we have developed a spin-density-functional theory of the frequency and wave-vector-dependent linear response of the inhomogeneous magnetic electron gas. It is shown that in the local-exchange approximation of Gaspar-Kohn-Sham, the static long-wavelength limits of these functions are identical to those obtained from the time-dependent Hartree-Fock theory [random-phase approximation (RPA) with exchange contributions] of the uniform magnetic electron system in the same limits; however, the plasmon dispersion relation ${\omega }_{\mathrm{pl}}^2\mathrm{}\left(q\right)\mathrm{}\cong {\omega }_{\mathrm{pl}}^2\mathrm{}\left(0\right)[1+A(\zeta \left){\left(\frac{q}{k_F}\right)}^2\right]$, and the spin-wave dispersion ${\omega }_{\mathrm{sw}}\mathrm{}\left(q\right)\mathrm{}\cong D\left(\zeta \right){\left(\frac{q}{k_F}\right)}^2$ for long wavelengths $\frac{q}{k_F}\ll 1$, do not correspond with the RPA results. In fact, we find, $A_{\mathrm{LD}}\left(\zeta \right)<\mathrm{}{A}_{\mathrm{RPA}}\left(\zeta \right)$ and $D_{\mathrm{LD}}\left(\zeta \right)>\mathrm{}{D}_{\mathrm{RPA}}\left(\zeta \right)$, where $\zeta$ is the magnetization of the system. The discrepancies found in the dynamical limit are a reflection of the fact that the lowlying excitations close to the ground state are not described by the local scheme. Some comments are made as to the modifications needed in the time-dependent spin-density-functional theory, which may rectify these discrepancies at least in the low-frequency long-wavelength limit.