Cálculos teóricos e computacionais de estrutura eletrônica

Abstract

In the present dissertation, some methods used in electronic structure calculations are discussed together with examples of applications. In Chapter 1, we start with the Hartree-Fock-Roothaan method deducing the equations to be solved from the energy of an electronic state and solvingthese equations for the one-dimensional helium atom. Keeping the study of two-electron systems in Chapter 2, it is presented the Hylleraass method, published in a series of three articles between the years 1928 and 1930, together with a simplification of the method presented byChandrasekhar and Herzberg in 1955. In chapter 3, the cusp conditions are deduced for the first time in the Hylleraas coordinates and these are applied in some functions found in the literature in order to analyze if these conditions are satisfied. Also in this chapter, constraints for thewavefunctions are obtained to guarantee that the cusp conditions will be satisfied. In Chapter 4 we introduce a new model used in electronic structure calculations developed simultaneously by Thomas and Fermi in the years 1927 and 1928, respectively, in which the electronic densityis used to obtain the energy of the system, thus giving rise to the Density Functional Theory, considering that this density is uniform. The equations to be solved are deduced in this chapter, considering only the classical effects as in the original works of Thomas and Fermi, in which allnecessary mathematical passages are presented and the results are discussed. In Chapter 5 it is presented a continuation in the development of the Thomas-Fermi model in which the quantum exchange effect, as a functional of the density, was introduced by Dirac in 1930. The deductiondoes not follow the original work of Dirac, but all necessary concepts and all the mathematical passages are presented throughout the development of this chapter . In Chapter 6 are introduced the Khon and Sham model, proposed in 1965, which uses the functionals of density to perform the calculations, but obtained great success in the results so that this model is still used in the present day to perform electronic structure calculations. In this chapter the correlation functional is introduced and the improvement in the results when considering this effect becomesevident when an example is solved after deduction of the necessary equations. In Chapter 7 a software is used to study systems of many electrons using more complex functionals than those presented in Chapters 5 and 6. The systems chosen for the study are the gold clusters containing two to six atoms. The results are compared with available experimental and theoretical data. The population analysis of the orbitals was also carried out to determine the charges in each atom of the clusters, and these results are used to relate the charge distributions in clusters with the geometries of interaction between clusters and a electron receptor, or donor, ligand.