Linear-scaling density-functional theory with Gaussian orbitals and periodic boundary conditions: Efficient evaluation of energy and forces via the fast multipole method

We report methodological and computational details of our Kohn-Sham density-functional method with Gaussian orbitals for systems with periodic boundary conditions. Our approach for the Coulomb problem is based on the direct space fast multipole method, which achieves not only linear scaling of computational time with system size but also very high accuracy in all infinite summations. The latter is pivotal for avoiding numerical instabilities that have previously plagued calculations with large bases, especially those containing diffuse functions. Our program also makes extensive use of other linear-scaling techniques recently developed for large clusters. Using these theoretical tools, we have implemented computational programs for energy and analytic energy gradients (forces) that make it possible to optimize geometries of periodic systems with great efficiency and accuracy. Vibrational frequencies are then accurately obtained from finite differences of forces. We demonstrate the capabilities of our methods with benchmark: calculations on polyacetylene, polyphenylenevinylene, and a (5,0) carbon nanotube, employing basis sets of double zeta plus polarization quality, in conjunction with the generalized gradient approximation and kinetic-energy density-dependent functionals. The Largest calculation reported in this paper contains 244 atoms and 1344 contracted Gaussians in the unit fell.