First principles calculations can be used to study many material properties from a fundamental point of view. This talk will cover the calculations of natural vibration frequencies (local vibrational modes) of impurities and defects in crystals. These vibration frequencies can be probed experimentally by infrared spectroscopy techniques. After a brief review of the computation techniques and details, two different cases of local vibrational modes will be explained. The first case covers the vibration frequencies of hydrogen atoms that form strong bonds with N or O in GaN and ZnO. In these cases, the vibration mode is very distinct from the crystal phonon modes. Therefore, the reduce mass approach can be used and the calculations can be done by decoupling the vibration from the rest of the system. However, special treatment of the anharmonicity is needed in these particular cases due to the large vibration amplitude. The second case covers the calculation of defect systems that have vibrational modes closer to that of crystal phonon modes. For the second case, a full dynamic matrix calculation is needed.