Convergence acceleration scheme for self-consistent orthogonal-basis-set electronic structure methods

A new self-consistent convergence acceleration scheme that is a variant of the Newton-Raphson algorithm for non-linear systems of equations is presented. With this scheme, which is designed for use with minimal orthogonal basis set electronic structure methods, the conventional Newton-Raphson scaling with respect to the number of atoms is enhanced from quartic to cubic. The scheme is demonstrated using a self-consistent environment-dependent tight binding model for hydrocarbons that allows an efficient and reasonably precise simulation of charge density distortions due to external electric fields, finite system sizes, and surface effects. In the case of a metallic system, self-consistency convergence starts at a high fictitious temperature, typically 1500 K. As the electron density approaches the self-consistent configuration the temperature is decreased. Typically, seven to nine iterations are required to achieve self-consistency in metallic systems to a final temperature of 300 K. For systems with a finite band gap the convergence may start at the target temperature so that temperature reduction is unnecessary, and typically two iterations are needed to achieve self-consistency. The convergence algorithm can handle extremely high applied fields and is very robust with respect to initial electron densities.