Accurate quantum-mechanical evaluation of the electric polarization of periodic solids using a multi-step method

The electric polarization of a periodic solid can be expressed as a Berry phase, and computed from the occupied Bloch state overlap matrix between neighboring wavevectors on a grid sampling the Brillouin zone. The usual discretized expression for the electric polarization leads to slow convergence with respect to the number of wavevectors on the grid. In this work, we improve the integration scheme, and obtain a better accuracy without the need for a finer sampling. Our technique uses multi-step overlap matrices instead of only nearest-neighbor overlap matrices, and results in higher precision for almost no additional computation cost. We test the method on AlAs, confirm the theoretical convergence rate, and observe that medium-order multi-step formulas provide superior accuracy without fluctuation problems