Theory of helium submonolayers: Formulation of the problem and solution in the zero coverage limit

Abstract

We are concerned here with the physical adsorption of helium atoms on a crystal surface. A general formulation of the problem is first given, which takes into account both the static periodic potential of a model substrate and He-He interactions. This formulation leads in a natural way to expressions of surface phonon states obtained recently by Jackson. As the first step of carrying out our formalism, we extend an earlier single-particle calculation by Lennard-Jones and Devonshire. A set of basis functions consisting of products of Mathieu functions and eigenfunctions of a Morse Hamiltonian are defined, and a low-order perturbative calculation is performed. While the procedure provides a qualitative description of the energy bands, it appears to be quantitatively inadequate. To facilitate a basis for future work, which will include the effects of correlations between the helium atoms, we carry out variational calculations in this zero-coverage limit, using various combinations of Gaussian and Morse functions. A reasonably single combination results in a ground-state energy which is just 1% higher than that of Ricca, Pisani, and Garrone, who performed elaborate computations to solve a secular equation obtained with harmonic oscillator basis functions.