Desenvolvimento de uma nova proposta matemática e computacional para a metodologia da matriz densidade tight-binding ordem-N

Abstract

ln this work, a mathematical-computational basis for the electronic structure calculation methodology known as Electronic Density Matrix Method Tight Binding {DMTB) is developed in a comprehensive context. The adopted approach evidences that the final mathematical formulation of this method is completely dependent on the computational strategies chosen for its implementation. ln this way, the DMTB is established as a mathem.atical-computational model of variable final formulation. Our formulation uses a computational strategy based on Nonlinear Conjugate Gradients {NLCG) algorithms. The final model obtained is slightly different from the DMTB that was originally presented in the literature by Vanderbilt group. lt is in accordance with the version presented by Millam and Scuseria, in 1997. Our development considered nonstandard themes of both Mathematics and Computation such as functions with domains in matrix spaces and the development of differential calculus techniques in these spaces and modifications in the structure of NLCG. The approach which was used develops the m.athematical aspects airrúng at effective computational implementation of the methodology. Thus, given that technological advances and scientific knowledge must be closely related, the present dissertation intends to fulfill a gap in the specialized literature, since there are no texts with pedagogical objectives or that are associated, in a natural way ,with m.athematical and computational modeling aspects ofthe DMTB.