Fast Fourier methods for systems with reduced periodicity: a case study in electronic structure methods.
Shankar Sundararaman (Physics, Cornell)

Spectral methods based on the Fast Fourier Transform exhibit exponential basis convergence for partial differential equations with smooth solutions, but they are limited to special geometries such as periodic boundary conditions. We use truncated Green's functions to extend the applicability of these methods to open boundary conditions, and demonstrate $\cal{O}(N \log N)$ spectral methods for the three dimensional Poisson equation with arbitrary combinations of free-space and periodic directions. Using this technique, we extend the exponential basis convergence of plane-wave electronic structure methods for crystalline solids to molecules, nanowires and thin films. Finally, we show that non-periodic Poisson solutions naturally arise even in periodic systems and are required to fix the poor basis convergence of the Fock exchange term in the electronic energy.