This dissertation is focused on material property exploration and analysis using computational quantum mechanics methods. Theoretical calculations were performed on the recently discovered hexahydride materials A2SiH6 (A=Rb, K) to calculate the lattice dynamics of the systems in order to check for structural stability, verify the experimental Raman and infrared spectrospcopy results, and obtain the theoretical free energies of formation. The electronic structure of the systems was calculated and the bonding and ionic properties of the systems were analyzed. The novel hexahydrides were compared to the important hydrogen storage material KSiH3. This showed that the hypervalent nature of the SiH62- ions reduced the Si-H bonding strength considerably. These hydrogen rich compounds could have promising energy applications as they link to alternative hydrogen fuel technology. The carbide systems Li-C (A=Li,Ca,Mg) were studied using \emph{ab initio} and evolutionary algorithms at high pressures. At ambient pressure Li2C2 and CaC2 are known to contain C22- dumbbell anions and CaC2 is polymorphic. At elevated pressure both CaC2 and Li2C2 display polymorphism. At ambient pressure the Mg-C system contains several experimentally known phases, however, all known phases are shown to be metastable with respect to the pure elements Mg and C. First principle investigation of the configurational space of these compounds via evolutionary algorithms results in a variety of metastable and unique structures. The binary compounds ZnSb and ZnAs are II-V electron-poor semiconductors with interesting thermoelectric properties. They contain rhomboid rings composed of Zn2Sb2 (Zn2As2) with multi-centered covalent bonds which are in turn covalently bonded to other rings via two-centered, two-electron bonds. Ionicity was explored via Bader charge analysis and it appears that the low ionicity that these materials display is a necessary condition of their multicentered bonding. Both compounds were found to have narrow, indirect band gaps with multi-valley valence and conduction bands; which are important characteristics for high thermopower in thermoelectric materials. Future work is needed to analyze the lattice properties of the II-V CdSb-type systems, especially in order to find the origin of the extremely low thermal conductivity that these systems display.