The ground-state structural optimization and the terahertz spectrum calculation of an organic electro-optical crystal of 4-N, N-dimethylamino-4'-N'-methyl-stilbazolium tosylate (DAST) are performed using dispersion-corrected density functional theory (DFT-D2). DAST consists of an organic pyridinium salt (cation), one of the most efficient non-linear optical active chromophores and a sulfonate (anion) for enhancing the stability of the noncentrosymmetric macroscopic crystal. Such an organic crystalline salt DAST exhibits highly electro-optical and nonlinear optical coefficients, and it is an efficient emitter of THz pulses. The steady ground-state structure of DAST is obtained by a step-by-step optimization method with gradually increasing the convergence accuracy. The calculated terahertz spectra in 0-4 THz are in good agreement with experimental measurements, implying the reasonability of DFT-D2 method. Moreover, the vibration displacement vector diagrams for DAST molecular structure are obtained using Cambridge sequential total energy package animation simulation function. The results indicate that the phonon modes of DAST crystal at 1.12 THz are attributed to the optical phonon modes of the anion and cation, and DAST cation (organic pyridinium salt) and anion (sulfonate) undergo translational vibrations in their respective (benzene ring) plane. In contrast the vibrations at 1.46 THz and 1.54 THz are mainly related to the vibration of the sulfonate, among which 1.46 THz vibration is caused by the rotation of the sulfonate along the a axis, while 1.54 THz is due to the motion of the whole sulfonate along the c axis. And the vibrations at 2.63 THz and 3.16 THz originate from the torsional vibrations of cations and the rotation of anions, respectively. The results presented in this work clearly illustrate the contributions of the anion and cation of DAST in the THz responses. The mode assignments provide important reference and guidance for further synthesis of new DAST derivatives with larger electro-optical coefficients. In particular, our results suggest that DFT method is a powerful theoretical tool for studying the THz photonics and it is helpful not only for better understanding the mechanisms of the THz responses of organic electro-optic crystals, but also for controlling their performances.