Quasi-two-dimensional approach to states of an adsorbed atom

A method is developed for solving the Schrödinger equation to obtain the eigenfunctions $\psi (z,\overrightarrow{\mathrm{R}})$ and eigenvalues of an adsorbed atom. By solving a one-dimensional equation for a given position $\overrightarrow{\mathrm{R}}$ on the surface, one generates an effective potential $\epsilon \left(\overrightarrow{\mathrm{R}}\right)$ for the problem of lateral motion. The leading correction to this Born-Oppenheimer-like approach is expressed in analytic form. A numerical calculation for the case of He on graphite illustrates the simplicity and accuracy of the method. The mean distance of a ground-state ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom is found to agree with an experimental result of Carneiro, Passell, Thomlinson, and Taub.