Necessary Conditions on Radial Distribution Functions

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Abstract
Certain necessary conditions must be met by any function introduced to serve as a radial distribution function of a uniform N‐particle system. One large class of necessary conditions is based on the statement that the expectation value of a potential energy (for an arbitrary potential function between pairs of particles) cannot fall below the classical minimum potential energy of the system. To convert this statement into a family of useful inequalities, we have evaluated the classical potential energies of close‐packed crystals for a linear combination of Yukawa and Coulomb two‐particle interactions. The numerical evaluation of lattice sums is performed by two procedures: (1) direct summation over the lattice (suitable for short‐range potentials), and (2) an adaptation of the Ewald summation procedure suitable for long‐range potentials. Results are given for the Coulomb and Yukawa potentials and also for the potentials 1/r(r + a) and 1/r(r + a)2. Two simple approximate forms are developed both giving close lower bounds on the classical potential energy of interacting particles forming a regular crystalline lattice.

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