Ab initio local energy and local stress: application to tilt and twist grain boundaries in Cu and Al

The energy-density and stress-density schemes (Shiihara et al 2010 Phys. Rev. B 81 075441) within the projector augmented wave (PAW) method based on the generalized gradient approximation (GGA) have been applied to tilt and twist grain boundaries (GBs) and single vacancies in Cu and Al. Local energy and local stress at GBs and defects are obtained by integrating the energy and stress densities in each local region by the Bader integration using a recent algorithm (Yu et al 2011 J. Chem. Phys. 134 064111) as well as by the layer-by-layer integration so as to settle the gauge-dependent problem in the kinetic terms. Results are compared with those by the fuzzy-Voronoi integration and by the embedded atom method (EAM). The features of local energy and local stress at GBs and vacancies depend on the bonding nature of each material. Valence electrons in Al mainly located in the interatomic regions show remarkable response to structural disorder as significant valence charge redistribution or bond reconstruction, often leading to long-range variations of charges, energies and stresses, quite differently from d electrons in Cu mainly located near nuclei. All these features can be well represented by our local energy and local stress. The EAM potential for Al does not reproduce correct local energy or local stress, while the EAM potential for Cu provides satisfactory results.