The behavior of low-energy electrons near the Fermi level is of vital importance to functional properties of materials and to novel materials discovery and design. Using theoretical modeling and numerical simulation methods, we investigated how to approach the properties of low-energy electrons in solid compound materials and illustrated how they affect the physical and chemical properties of materials; in this dissertation, we mainly focused on the topological, the optical, and the defect properties of materials. Firstly, by first-principles calculations, we studied the topological properties in wurtzite and litharge III-V materials and suggested experimentally accessible methods to realize nontrivial topology in the class of material. Secondly, we calculated the optical conductivity of a topological semimetal, CoSi, and connected the main features in the conductivity spectrum to the topological nature of CoSi. The strategies we used to analyze the optical conductivity can be further generalized to other complex topological materials and to higher-order optical properties. Finally, we studied the mechanical and the electronic properties in the WS2 monolayer. We demonstrated that defect states, which comprise the non-bonding states arising from the vacancy, could strongly affect the photo-excited electrons and holes and further the band-bending properties in the WS2 monolayer.