Brillouin zone treatment in total energy calculations of Peierls distorted chains

Abstract

Most band structure calculations approximate the integral over the Brillouin zone of momentum (i.e., wave vector) dependent properties with an appropriately weighted sum over a discrete set of points in the Brillouin zone. The best choice for such a set of points has long been a point of discussion in crystalline band structure calculations. For one-dimensionally periodic systems, however, the usual choice of points has been evenly spaced points in the one-dimensional Brillouin zone with equal weights. We have analyzed the exact error for the integral over the π band of a tight-binding model of trans-polyacetylene as a function of bond alternation. We find that the error in π band energy decreases in magnitude as q−2, where q is the total number of points treated in the Brillouin zone, for the metallic polyacetylene system with equal bond lengths. As bond alternation increases, however, we find that the error in π band energy decreases in magnitude roughly exponentially as a function of bond alternation for any given value of q. We find that this systematic change in error as a function of bond alternation can lead to either apparent overestimation or underestimation of the equilibrium dimerization and stabilization energy of Peierls distorted systems using first-principles total energy calculations.