A first-order phase transition from a high-density liquid to a low-density liquid has been proposed to explain the various thermodynamic anomies of water. It also has been proposed that such liquid-liquid phase transition would exist in supercooled silicon. Computer simulation studies show that, across the transition, the diffusivity drops roughly 2 orders of magnitude, and the structures exhibit considerable tetrahedral ordering. The resulting phase is a highly viscous, low-density liquid silicon. Investigations on the atomic diffusion of such a novel form of liquid silicon are of high interest. Here we report such diffusion results from molecular dynamics simulations using the classical Stillinger-Weber (SW) potential of silicon. We show that the atomic diffusion of the low-density liquid is highly correlated with local tetrahedral geometries. We also show that atoms diffuse through hopping processes within short ranges, which gradually accumulate to an overall random motion for long ranges as in normal liquids. There is a close relationship between dynamical heterogeneity and hopping process. We point out that the above diffusion mechanism is closely related to the strong directional bonding nature of the distorted tetrahedral network. Our work offers new insights into the complex behavior of the highly viscous low density liquid silicon, suggesting similar diffusion behaviors in other tetrahedral coordinated liquids that exhibit liquid-liquid phase transition such as carbon and germanium.