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Analytical formulae are derived for the zero-point vibrational energy andanharmonicity corrections of the cohesive energy and the mode Gr\"{u}neisenparameter within the Einstein model for the cubic lattices (sc, bcc and fcc)and for the hexagonal close-packed structure. This extends the work done byLennard Jones and Ingham in 1924, Corner in 1939 and Wallace in 1965. Theformulae are based on the description of two-body energy contributions by aninverse power expansion (extended Lennard-Jones potential). These make use ofthree-dimensional lattice sums, which can be transformed to fast convergingseries and accurately determined by various expansion techniques. We applythese new lattice sum expressions to the rare gas solids and discuss associatedcritical points. The derived formulae give qualitative but nevertheless deepinsight into vibrational effects in solids from the lightest (helium) to theheaviest rare gas element (oganesson), both presenting special cases because ofstrong quantum effects for the former and strong relativistic effects for thelatter.

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