Oxygen rich combustion in diesel engines produces nitrogen oxides, NOx (NO, NO2), which are pollutants causing smog and acid rain, and must be removed to meet emission regulations. Cu-SSZ-13, a microporous zeolite exchanged with Cu metal, is the leading choice for selective catalytic reduction (SCR) of NOx to harmless N2 and H2O, using NH3 as the reductant. The Cu site in Cu-SSZ-13 is the active site for SCR reaction, as demonstrated by many previous literatures. However, the understanding of standard SCR mechanisms from a molecular level is not as well developed. In this dissertation I used different molecular modeling tools to study the dynamics and mobilities of the Cu sites under simulated reaction conditions, developed a correlation to accurately calculate the free energies of adsorption on Cu sites, and compared sulfur poisoning species between the two different types of Cu sites.Chapter 2 includes a general description of the computational techniques used in this dissertation, these techniques applied to solve specifc problems are described more in detail in the methods section of each chapter. Chapter 3 contains detailed study of the dynamic nature of the Cu sites in Cu-SSZ-13 under different gas exposure conditions. Cu metals are solvated by H2O ligands when the samples are exposed to air, and are solvated NH3 ligands when the samples are exposed to SCR gas conditions. The Cu centers have limited mobility when they have no ligand, and their mobility increased dramatically after H2O or NH3 solvation. Chapter 4 contains simulation of UV-Visible absorption spectra of the Cu sites. We found it essential to perform more than 100 ps of ab initio molecular dynamics (AIMD) simulations at a relevant temperature to extract a collection of geometries at those conditions for spectrum calculation. The final spectrum averaged from the 400 AIMD snapshots agrees closely with experimental spectra. We also studied eight different Cu dimers with a variety of geometric isomers, each giving peaks at different absorption energies in the simulated UV-Visible spectra, but hard to be assigned to contributions in the observed spectra. Chapter 5 introduced a method to accurately calculate the free energies of adsorption of small molecules onto Cu-SSZ-13, using constrained molecular dynamics sampling technique and the potential of mean force method. The simple correlations between the entropy of adsorption and the ideal gas entropy of adsorbate allow us to quickly compute the free energy of adsorption under experimental gas pressures and temperatures. Chapter 6 compares the sulfur poisoning species on the two Cu sites. (Bi)sulfates are shown to be the stable poisoning species under simulated standard SCR conditions on both types of Cu sites, agreeing well with X-ray experimental evidence of copper and sulfur oxidation states.