Assessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theory

We present a detailed study of different exchange-correlation (xc) functionals in describing the dynamical properties of finite systems. For that purpose, we calculated the static polarizabilities, ionization potentials, and optical absorption spectrum of four small clusters, ${\mathrm{Na}}_{\mathrm{2}},$ ${\mathrm{Na}}_{\mathrm{4}},$ ${\mathrm{SiH}}_{\mathrm{4}},$ and ${\mathrm{Si}}_{\mathrm{2}}{\mathrm{H}}_{\mathrm{6}},$ using a real-space, real-time technique. The computed static polarizabilities and ionization potentials seem to be in rather good agreement with the available experimental data, once the proper asymptotics of the potential are taken into account. The same conclusion holds for the absorption spectra, although the xc kernels in use do not provide a sufficiently strong attractive interaction between electrons and holes, leading to spectra slightly shifted towards higher energies. This deficiency is traced back to the insufficient description of dynamical effects in the correlation functional. Furthermore, it is shown that the xc potential used to obtain the ground state is the key factor to get reasonable spectra, whereas the choice of the xc kernel just amounts to small, although important, quantitative changes.