The Lennard-Jones Potential Revisited: Analytical Solutions for the Solid State from Lattice Sums and Epstein Zeta Functions

Abstract:  The Lennard-Jones potential is the most widely used interaction potential between atoms with widespread applications in physical, chemical and biological sciences. This simple potential also has the advantage that the cohesive energy, pressure and the bulk modulus of a simple solid (simple cubic, body-centered cubic, and face-centered cubic) can be expressed analytically as a function of volume using lattice sums in three dimensions, originally developed by Hund, Born and Lennard-Jones. In a similar procedure to Lennard-Jones we derive analytical expressions for the zero-point vibrational energy and first-order anharmonicity corrections for these crystals by an inverse power expansion in terms of the internuclear distance, which we call the Extended Lennard-Jones potential. These new expressions are applied to the Lennard-Jones systems of rare gas solids from helium down to superheavy element oganessian (Z=118). We will show how to deal with slow converging lattice sums by expansion techniques such as the Epstein zeta function or Van der Hoff-Benson expansions in terms of Bessel functions. We also give an analytical solution for the hcp lattice. By doing so we can solve some old problems, e.g. why a simple Lennard-Jones potential always prefers hcp over fcc contrary to what is known from experiment. Moreover, through many-body expansions using computer intensive relativistic coupled cluster methods we can get the cohesive energy for solid Argon accurate to within 1 J/mol and within experimental accuracy.