  Collective excitations in simple metal systems can be described successfully
in terms of a local one-body excitation operator Q, due to the long range
nature of the coulomb interaction. For the plasmon modes of a simple-metal
slab, momentum expansions of Q are calculated using a variational procedure,
equivalent to a restricted RPA calculation. The dispersion relation and the
density fluctuation for each mode are found in the sudden approximation using
the proper Q operator and the RPA sum-rule formalism. The contribution of the
exchange and correlation energy is estimated using a local density functional.
The positive background is described within a jellium model while the
ground-state electronic density is approximated by a double step profile. The
density fluctuation of the plasmon modes above the plasma frequency form
standing waves across the slab. The spectra below the plasma frequency is
qualitatively different to that of local optics calculations, due to the
appearance of two multipole plasmon modes that shift down the origin of the
$\omega_{+}$ plasmon. The dependence of the results on the width of the slab,
the density of the simple-metal and the surface diffuseness is discussed.
Throughout, we compare our results with previous RPA and TDLDA calculations.
