Lithium-sulfur (Li-S) batteries are a promising alternative to the Li-ion technology due to their high theoretical capacity and low cost. However, unlike intercalation compounds that are currently used in commercial lithium ion batteries, the sulfur cathode undergoes a series of complex electrochemical reactions with substantial structural and morphological changes during charge and discharge. Various lithium polysulfides are formed that may shuttle between the electrodes, resulting in poor Coulombic efficiency. In this work, first principles calculations based on the density function theory are performed to study the structural evolution of alpha-sulfur during discharging. First, the surface energies of a few low index surfaces are obtained, and the lithium diffusion barriers into those surfaces are calculated. While clear driving force is observed for both the (001) and the (010) surfaces, a diffusion barrier exists for the (100) surface. Then, the structural evolution is studied with ab initio molecular dynamics by adding lithium ions at a specific rate. The effects of electrolytes are also studied, and the interface diffusion barrier of lithium from the electrolyte into the cathode are calculated. Finally, the elementary process that leads to the formation of soluable polysulfide is also simulated. These calculations help to reveal an atomistic understanding of the complex electrochemical reactions that take place in the sulfur cathode.