The complete diagonalized Hamiltonian matrixes of order 120 of 3d³ /3d⁷ ions configuration in the trigonal symmetry have been established by irreducible representation method and group theory,taking spin-orbit interaction,spin-spin interaction and spin-other-orbit interaction into account. Ground-state energy levels and zero-field splitting parameter were calculated by the matrixes,and the effects of spin doublets on the ground-state energy levels were studied. The calculated values were conformed with the experimental values. The results show that the contributions of doublets cannot be omitted. On these bases,the effects of spin-spin interaction and spin-other-orbit interaction on the fine structures of spectra and zero-field splitting parameter of emerald crystal were studied further. It was found that the effects of spin-spin interaction and spin-other-orbit interaction on the ground-state energy levels and the zero-field splitting parameter cannot be omitted. By comparing the values of theoretical calculation with the experimental values,the existence of Jahn-Teller effects in emerald crystal is confirmed and this can give more reasonable explain on the splitting of fine structures of spectra.