The time evolution of the rotational distribution of diatomic molecules induced by a train of broadband optical pulses can be modeled as a class of continuous-time quantum walk (CTQW) on a half line. As the transition moments asymptotically decrease to zero like Gaussian distribution, the rotational distribution cannot reach infinity far from the initial state and exhibits a localization-like behavior where the upper limit of the distribution is not trivial. We investigated a time evolution of the density distribution with the transition moments of the Gaussian distribution peaked at the boundary of the half line for the initial state of J=0. Even though the time evolution exhibited oscillatory motion inside a certain region repeatedly, we observed a clear time-averaged distribution that is well fitted to the stretched exponential distribution with parameters close to the Gaussian distribution by numerical simulation. By varying the coefficients of the Gaussian distribution of the transition moments, the coefficients of the stretched exponential distribution can be determined by regression.