Necessary Conditions on Radial Distribution Functions

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.