Bound state vibrational spectrum of the 3–9 atom-surface interaction

The potential $\text{V(z) =}\text{C}{}_9\text{z}{}^9\text{?}\text{C}{}_3\text{z}{}^3$ is a reasonable parametrization of the atom-surface interaction. We evaluate the discrete spectrum E_n of bound states for this potential with arbitrary coefficients C₉ and C₃. The resulting form in the WKB approximation is $\text{E}{}_{\mathrm{n}}\text{= ?D 1 ?}\text{(n +}\text{l}\text{2}\text{)}\text{L}\text{]}{}^6$, where L depends on the mass and D is the well depth. We find that the exact solution of the Schrödinger equation can be written in the same form, with n shifted slightly by an amount δ_n, which we calculate. The results are applied to the case of He near a NaF surface, in which the calculated eigenvalue spectrum agrees well with experimental values.