How harmonic is dipole resonance of metal clusters?

We discuss the degree of anharmonicity of dipole plasmon resonances in metal clusters. We employ the time-dependent variational principle and show that the relative shift of the second phonon scales as $N^{-4/3}$ in energy, $N$ being the number of particles. This scaling property coincides with that for nuclear giant resonances. Contrary to the previous study based on the boson-expansion method, the deviation from the harmonic limit is found to be almost negligible for Na clusters, the result being consistent with the recent experimental observation.