  The surface plasmon in simple metal clusters is red-shifted from the Mie
frequency, the energy shift being significantly larger than the usual spill-out
correction. Here we develop a variational approach to the RPA collective
excitations. Using a simple trial form, we obtain analytic expressions for the
energy shift beyond the spill-out contribution. We find that the additional red
shift is proportional to the spill-out correction and can have the same order
of magnitude.
