The main goal of this research was to apply first-principles electronic structure calculations to investigate atomic motions in several condensed materials. This thesis consists of five separate but related topics that are classified into two main categories: structure of materials under pressure and Li ion dynamics in lithium battery materials. The atomic structure of liquid gallium was investigated in order to resolve a controversy about an anomalous structural feature observed in the x-ray and neutron scattering patterns. We explored the pressure effect when modifying the liquid structure close to the solid-liquid melting line. The atomic trajectories obtained from first-principles molecular dynamics (FPMD) calculations were examined. The results clarified the local structure of liquid gallium and explained the origin of a peculiar feature observed in the measured static structure factor. We also studied the structure of a recently discovered phase-IV of solid hydrogen over a broad pressure range near room temperature. The results revealed novel structural dynamics of hydrogen under extreme pressure. Unprecedented large amplitude fluxional atomic dynamics were observed. The results helped to elucidate the complex vibrational spectra of this highly-compressed solid. The atomic dynamics of Li ions in cathode, anode, and electrolyte materials - the three main components of a lithium ion battery - were also studied. On LiFePO4, a promising cathode material, we found that in addition to the commonly accepted one-dimensional diffusion along the Li channels in the crystal structure, a second but less obvious multi-step Li migration through the formation of Li-Fe antisites was identified. This discovery confirms the two-dimensional Li diffusion model reported in several Li conductivity measurements and illustrates the importance of the distribution of intrinsic defects in the enhancement of Li transport ability. The possibility of using type-II clathrate Si136 as an anode material was investigated. It was found that lithiated Si-clathrates are intrinsic metals and their crystal structures are very stable. Calculations revealed the charge and discharge voltages are very low and almost independent of the Li concentrations, an ideal property for an anode material. Significantly, migration pathways for Li ions diffusing through the cavities of the clathrate structures were found to be rather complex. Finally, the feasibility of a family of Li3PS4 crystalline and nanoporous cluster phases were studied for application as solid electrolytes. It was found that the ionic conductivity in the nanocluster is much higher than in crystalline phases. It is anticipated that the knowledge gained in the study of battery materials will assist in future design of new materials with improved battery charge and discharge performance.