Generalization of Equivalent Crystal Theory to Include Angular Dependence

In the original Equivalent Crystal Theory, each atomic site in the real crystal is assigned an equivalent lattice constant, in general different from the ground state one. This parameter corresponds to a local compression or expansion of the 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 systematically introducing site-dependent directional distortions of the lattice, whose corresponding distortions account for the dependence of the energy on anisotropic local density variations. This is done in the spirit of the original framework, but including a gradient term in the density. This approach is introduced to correct a deficiency in the original Equivalent Crystal Theory and other semiempirical methods in quantitatively obtaining the correct ratios of the surface energies of low index planes of cubic metals (100), (110), and (111). We develop here the basic framework, and apply it to the calculation of Fe (110) and Fe (111) surface energy formation. The results, compared with first principles calculations, show an improvement over previous semiempirical approaches.