Representative atomic and molecular systems, including various inorganic and organic molecules with covalent and ionic bonds, have been studied by using density functional theory. The calculations were done with the commonly used exchange-correlation functional B3LYP followed by a comprehensive analysis of the calculated highest-occupied and lowest-unoccupied Kohn-Sham orbital (HOMO and LUMO) energies. The basis set dependence of the DFT results shows that the economical 6-31+G* basis set is generally sufficient for calculating the HOMO and LUMO energies (if the calculated LUMO energies are negative) for use in correlating with molecular properties. The directly calculated ionization potential (IP), electron affinity (EA), electronegativity (chi), hardness (eta), and first electron excitation energy (iota) are all in good agreement with the available experimental data. A generally applicable linear correlation relationship exists between the calculated HOMO energies and the experimental/calculated IPs. We have also found satisfactory linear correlation relationships between the calculated LUMO energies and experimental/calculated EAs (for the bound anionic states), between the calculated average HOMO/LUMO energies and chi values, between the calculated HOMO-LUMO energy gaps and eta values, and between the calculated HOMO-LUMO energy gaps and experimental/calculated first excitation energies. By using these linear correlation relationships, the calculated HOMO and LUMO energies can be employed to semiquantitatively estimate ionization potential, electron affinity, electronegativity, hardness, and first excitation energy.