Extension of the Lennard-Jones potential: Theoretical investigations into rare-gas clusters and crystal lattices of He, Ne, Ar, and Kr using many-body interaction expansions

The many-body expansion ${V}_{\mathrm{int}}={\ensuremath{\sum}}_{ilj}{V}^{(2)}({r}_{ij})+{\ensuremath{\sum}}_{iljlk}{V}^{(3)}({r}_{ij},{r}_{ik},{r}_{jk})+\ensuremath{\cdots},$ in terms of interaction potentials between rare-gas atoms converges fast at distances $rg{r}_{\mathit{HS}}$, with ${r}_{\mathit{HS}}$ being the hard-sphere radius at the start of the repulsive wall of the interaction potential. Hence, for the solid state where the minimum distance is always above ${r}_{\mathit{HS}}$, a reasonable accuracy is already obtained for the lattice parameters and cohesive energies of the rare-gas elements using precise two-body terms. All tested two-body potentials show a preference of the hcp over the fcc structure. We demonstrate that this is always the case for the Lennard-Jones potential. We extend the Lennard-Jones potential to obtain analytical expressions for the lattice parameters, cohesive energy, and bulk modulus using the solid-state parameters of Lennard-Jones and Ingham [Proc. R. Soc. London, Ser. A 107, 636 (1925)], which we evaluate up to computer precision for the cubic lattices and hcp. The inclusion of three-body terms does not change the preference of hcp over fcc, and zero-point vibrational effects are responsible for the transition from hcp to fcc as shown recently by Rosciszewski et al. [Phys. Rev. B 62, 5482 (2000)]. More precisely, we show that it is the coupling between the harmonic modes which leads to the preference of fcc over hcp, as the simple Einstein approximation of moving an atom in the static field of all other atoms fails to describe this difference accurately. Anharmonicity corrections to the crystal stability are found to be small for argon and krypton. We show that at pressures higher than $15\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ three-body effects become very important for argon and good agreement is reached with experimental high-pressure density measurements up to $30\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, where higher than three-body effects become important. At high pressures we find that fcc is preferred over the hcp structure. Zero-point vibrational effects for the solid can be successfully estimated from an extrapolation of the cluster zero-point vibrational energies with increasing cluster size $N$. For He, the harmonic zero-point vibrational energy is predicted to be always above the potential energy contribution for all cluster sizes up to the solid state at structures obtained from the two-body force. Here anharmonicity effects are very large which is typical for a quantum solid.