For accurately reflecting the practical features of turbulence, the modified atmospheric spectrum model should be applyed in the simulation of optical wave propagation. To do this, the method of generating high precision turbulent phase screen is proposed for the modified atmospheric spectrum, which is on the bases of optimized phase screen model. By extending the low frequency region and changing the sampling Settings of the model, the maximum relative error in low frequency region was reduced to 1%. As a comparison, the maximum relative error in low frequency region was 6.75% for Optimization-based method before improvement, 22.99% for original FFT method and 16.81% for subharmonic method. By using this method, the simulation of Gaussian beam propagating in turbulence has been done and the second-order statistical properties include beam spread and beam wander have been estimated. The results show that, in the case of weak fluctuation level, the degree of compliance with the analytical approximations is good. However, under the condition of strong fluctuation level, the deviation between the simulation results and the theoretical results is increasing with distance. The deviation of beam spread is up to 6 cm, while the deviation of beam wander is up to 1 cm, which can be explained by the fact that the theoretical model cannot predict the beam wander saturation. In comparison with the simulation results of the Von-Karman spectrum, the beam spread estimated by modified atmospheric spectrum is slightly larger than that estimated by Von-Karman spectrum, and beam wander predicted by modified atmospheric spectrum shows a faster rate to reach saturation than that predicted by the Von Karman spectrum, it is precisely induced by the "bump" of the modified atmosphere spectrum. So we conclude that the phase screen generated by the method presented here can characterize the refractive index perturbation of the actual atmosphere effectively.