On the realizability of pair correlation functions

The pair correlation function g2(r) provides a basic geometric descriptor for many-particle systems. It must obey two necessary conditions: (i) non-negativity for all distances r, and (ii) non-negativity of its associated structure factor S(k) for all k. Here we utilize an improved stochastic construction algorithm for particle configurations to establish conditions in which (i) and (ii) are also sufficient, i.e., g2(r) is in fact realizable. Two types of target pair correlation functions have been investigated in one, two, and three dimensions for hard-core particles, specifically a unit step function, and a contact δ plus step pair correlation function. Results indicate that the former target function is realizable up to a terminal density set by necessary condition (ii), at which the particle core packing fraction equals 2-d in d dimensions. Furthermore, results are consistent with the proposition that for d>1 the contact δ plus step function is realizable up to a terminal density due to condition (ii) at which the packing fraction of cores is (d+2)/2d+1 [Torquato and Stillinger, J. Phys. Chem. B 106 (2000) 8354, 11406].