The Lennard Jones Potential Revisited -- Analytical Expressions for Vibrational Effects in Cubic and Hexagonal Close-Packed Lattices

Analytical formulae are derived for the zero-point vibrational energy and anharmonicity corrections of the cohesive energy and the mode Gr\"{u}neisen parameter within the Einstein model for the cubic lattices (sc, bcc and fcc) and for the hexagonal close-packed structure. This extends the work done by Lennard Jones and Ingham in 1924, Corner in 1939 and Wallace in 1965. The formulae are based on the description of two-body energy contributions by an inverse power expansion (extended Lennard-Jones potential). These make use of three-dimensional lattice sums, which can be transformed to fast converging series and accurately determined by various expansion techniques. We apply these new lattice sum expressions to the rare gas solids and discuss associated critical points. The derived formulae give qualitative but nevertheless deep insight into vibrational effects in solids from the lightest (helium) to the heaviest rare gas element (oganesson), both presenting special cases because of strong quantum effects for the former and strong relativistic effects for the latter.