$O ( N log N )$ scaling method to evaluate the ion–electron potential of crystalline solids

We propose a simple $O ( N log N )$ scaling expression in reciprocal space for evaluating the ion–electron potential of crystalline solids. The expression replaces the long-range ion–electron potential with an equivalent localized charge distribution and corresponding boundary conditions on the unit cell. Given that no quadratic scaling structure factor is required—as used in traditional methods—the expression shows the inherent $O ( N log N )$ behavior, and is well suited to simulating large-scale systems within orbital-free density functional theory. The scheme is implemented in the ATLAS software package and benchmarked by using a solid Mg body-centered cubic lattice containing tens of thousands of atoms in the unit cell. The test results show that the method can efficiently simulate large scale crystals with high computational accuracy.