A Mathematical and Computational Proposal for the Development of Tight-Binding Order-N Density Matrix Methodology

Abstract

The aim of this paper is to present, in a broad context, a proposal of mathematical foundation that serves as framework for the development of the electronic structure calculation methodology called “DMTB-Densit Matrix Tight Binding method.” ), which was originally presented in the literature by the group led by physicist David Vanderbilt in 1993 (\cite{vanderbilt}), as well as its relationship to the computational modeling that can be chosen for its implementation. The approach adopted makes it clear that the final mathematical formulation of this methodology is completely dependent on the computational strategies chosen for its effective implementation. Thus, we put the DMTB as a mathematical-computational model of variable final formulation. Finally, we propose an implementation based on the nonlinear conjugate gradient method. The final model obtained is slightly different from the DMTB that was originally presented in 1993, in agreement with the version presented by Millam and Scuseria in 1997 (\cite{scuseria}). The approach used develops the mathematical aspects, aiming at the effective computational implementation of the methodology.

Keywords

• Tight-Binding Density Matrix
• Mathematical and Computational Modeling
• Lagrange multipliers