Generalization of equivalent crystal theory to include angular dependence

The original equivalent crystal theory is a semi-empirical method for calculating the configurational energy of atomic systems. Each atomic site in the real crystal with defects is assigned an equivalent lattice constant, in general different from the ground state value. This parameter corresponds to a local compression or expansion of the perfect lattice. The basic method considers these volumetric transformations and, in addition, introduces the possibility that the reference lattice is anisotropically distorted. These distortions however, were introduced ad-hoc. In this work, we generalize the original Equivalent Crystal Theory by introducing site-dependent directional distortions of the lattice which account for the dependence of the energy on anisotropic local density variations. This is done in the spirit of the original framework of ECT, but includes a gradient in the density. This approach is introduced to ECT and may apply to other semi-empirical methods by making use of readily available first-principles results to fix parameters. We develop here the basic framework, and apply it to the calculation of Fe(1 1 0) and Fe(1 1 1) surface energy formation, and to the vacancy energy curve. The results, compared with first-principles calculations, show an improvement over previous semi-empirical approaches.