Analytical algorithm for Equivalent Crystal Theory

Equivalent crystal theory (ECT) is a semi-empirical technique used for the calculation of defect energetics in metals and semiconductors. The implementation of the method involves the solution of transcendental equations. Although this is not a problem for simple defects, for complex defects, Monte Carlo and molecular dynamics calculations in large systems, it could be the speed-determining limitation in a calculation. In this paper we propose a procedure for bypassing this step and obtaining the desired result directly. The form of the particular transcendental equations suggest a different approach, namely, the equation can be cast in the form of the Lambert function that can be readily evaluated from standard routines. We test this scheme by evaluating the surface energies for a variety of metallic elements and by standard numerical approaches, and demonstrate that they agree to within a few ppm.