Both unit cell parameters and fractional atomic coordinates of β-Si3N4 have been fully relaxed using Crysta106 program that adopts all-electron gauss-type basis sets and the density functional theory (PW91 and B3LYP). The geometrically optimized results are in excellent agreement with experiment. The band structure and density of states are also calculated. Based on the optimized geometries, the vibration frequencies have been calculated at Γ point using PW91 and B3LYP method with symmetry assignment. The 11 Raman mode frequencies are compared with experiment and other theoretical results, and the missing Ag mode frequency is assigned to 459cm-1. The present calculation shows that the results of B3LYP are better than those of PW91 In the intermediate and low-frequency region, the largest differences from experimental results are 2 and 17cm-1, while in the high-frequency region the largest differences are 16　and 35cm-1 respectively. For IR frequencies, the symmetries of the experimental data are assigned by comparing them with the present theoretical results.