“Lagrange Functions”: A Family of Powerful Basis Sets for Real-Space Order-N Electronic Structure Calculations

Plane waves have unparalleled simplicity and systematic convergence by a single monotonic parameter, the energy cutoff, but they are limited to speriodic systems and require Fourier transforms that scale as N²logN, where N is the number of atoms. Real-space methods for order-N scaling are computationally complex and convergence depends on several variables. Here we introduce and demonstrate “Lagrange functions” as a family of analytical, complete, and orthonormal basis sets that are suitable for efficient, accurate, real-space, order-N electronic structure calculations. Convergence is controlled by a single monotonic parameter, the dimension of the basis set, and computational complexity is lower than that of plane waves.