 Three-ion axial normal modes

Three identical repelling charges in a one-dimensional harmonic trap have a translation mode $(1,1,1)$, a breathing mode $(1,0,-1)$, and a symmetric middle-against-outer mode $(1,-2,1)$. Their frequencies are $\omega$, $\sqrt{3}\omega$, and $\sqrt{29/5}\omega$. The translation frequency is unchanged by internal Coulomb forces. Linearizing each pair potential produces a positive spring coefficient $2q^2/(4\pi {ϵ}_0{d}^3)$.