Dense sphere packings from optimized correlation functions
  Elementary smooth functions (beyond contact) are employed to construct pair correlation functions that mimic jammed disordered sphere packings. Using the g2-invariant optimization method of Torquato and Stillinger [J. Phys. Chem. B 106, 8354, 2002], parameters in these functions are optimized under necessary realizability conditions to maximize the packing fraction phi and average number of contacts per sphere Z. A pair correlation function that incorporates the salient features of a disordered packing and that is smooth beyond contact is shown to permit a phi of 0.6850: this value represents a 45% reduction in the difference between the maximum for congruent hard spheres in three dimensions, pi/sqrt{18} ~ 0.7405, and 0.64, the approximate fraction associated with maximally random jammed (MRJ) packings in three dimensions. We show that, surprisingly, the continued addition of elementary functions consisting of smooth sinusoids decaying as r^{-4} permits packing fractions approaching pi/sqrt{18}. A translational order metric is used to discriminate between degrees of order in the packings presented. We find that to achieve higher packing fractions, the degree of order must increase, which is consistent with the results of a previous study [Torquato et al., Phys. Rev. Lett. 84, 2064, 2000]. 
Adam B. Hopkins, Frank H. Stillinger, Salvatore Torquato