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