Using the time-dependent local density functional (TDLDF) theory and analytically continuing the resultant response function into the lower half frequency plane, the authors calculate the dispersion relations and Landau damping of the lightly damped surface plasmon modes of a bare jellium aluminium surface, free of foreign overlayers. For the correctly self-consistent zeroth-order surface electron density profile, they find at least one such mode in addition to the familiar 'monopole' surface plasmon.
Dobson, J., & Harris, G. (1988). An additional surface plasmon mode of a bare jellium aluminium surface from self-consistent microscopic calculations. Journal of Physics Condensed Matter, 21(21), L729-L734. https://doi.org/10.1088/0022-3719/21/21/001