Density-functional theory of elastic moduli: Hard-sphere and Lennard-Jones crystals

We propose a density-functional method for calculating elastic moduli of crystalline solids. The method is based on the second-order Ramakrishnan-Yussouff (RY) expansion of the variational grand-canonical potential around a uniform liquid state. The densities of the strained and unstrained crystal are represented as sums of narrow Gaussians. We express the crystal moduli in terms of the liquid structure factor its first and second derivatives evaluated at the reciprocal-lattice points of the crystal. We evaluate the elastic moduli for fcc hard-sphere and Lennard-Jones crystals using the Percus-Yevick and computer-simulation liquid structure factors, respectively. An indirect comparison with available experimental and theoretical values shows that although our calculated moduli are accurate to an order of magnitude, higher-order terms in the RY expansion might be significant. We find important contributions from density equilibration within the strained unit cell.