Cluster Expansion for Superlattice Physisorption: Helium Adsorbed upon Graphite

The ground-state energy of the $``X_g=\frac{1}{3}"$ and $``X_g=\frac{1}{4}"$ superlattice states of ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ adsorbed on graphite is calculated via a variational wave function incorporating Jastrow factors. The single-particle factors in the wave function are constructed as an expansion in an orthogonal basis set and the Jastrow terms incorporate only lateral correlations. The ground-state energy is calculated using the Nosanow cluster expansion. The stability of the superlattice states is examined and both are found to be unstable at zero pressure. Estimates of the stability of these states to "melting" indicate the $``X_g=\frac{1}{3}"$ state is stable and the $``X_g=\frac{1}{4}"$ state is less stable or possibly unstable. The sensitivity of the results to reasonable substrate-induced many-body corrections to the adatom-adatom potential is found to be significant.