  The collective electronic excitation in planar sodium clusters is studied by
time-dependent density functional theory calculations. The formation and
development of the resonances in photoabsorption spectra are investigated in
terms of the shape and size of the 2-dimensional (2-D) systems. The nature of
these resonances is revealed by the frequency-resolved induced charge densities
present on a real-space grid. For long double chains, the excitation is similar
to that in long single atomic chains, showing longitudinal modes, end and
central transverse modes. However, for 2-D planes consisting of ($n \times n$)
atoms with $n$ up to 16, new 2-D characteristic modes emerge regardless of the
symmetries considered. For a kick parallel to the plane, besides the equivalent
end mode, mixed modes with contrary polarity occur, while for an impulse
perpendicular to the plane there will be corner, side center, bulk center, and
circuit modes. Our calculation reveals the importance of dimensionality for
plasmon excitation and how it evolves from 1-D to 2-D.
