The geometrical structure, stability, vibrant frequency and optical spectrum of Mgn(2≤n≤15) clusters are studied by the B3LYP method of the density functional theory (DFT) at 3-21g and lanl2dz levels to search for the stable structures and the magic clusters. The spectroscopic properties of the structures are investigated in detail. The results indicate that the consequences computed by the two sets at ground state are identical to a great degree and the slight difference is discovered only in second-order energy difference (D2E). In addition, the bond length of Mgn clusters calculated at lanl2dz level is longer than that at 3- 21g level. The clusters including Mg4, Mg10, and Mg15 are more stable than others, and Mg4 and Mg10 are the magic clusters. All of the clusters do not only have infrared activity but also have Raman activity except Mg2 cluster. Through the optical spectrum analysis of Mg4, Mg5, Mg7, Mg9, Mg10, Mg15 clusters, it is found that the Mg10 cluster has the most infrared spectral peaks and the Mg15 cluster has the most intensive spectral peaks.