Recent experiments with self-phoretic particles at low concentrations show a pronounced dynamic clustering [I. Theurkauff et al., Phys. Rev. Lett. 108, 268303 (2012)]. We model this situation by taking into account the translational and rotational diffusiophoretic motion, which the active particles perform in their self-generated chemical field. Our Brownian dynamics simulations show pronounced dynamic clustering only when these two phoretic contributions give rise to competing attractive and repulsive interactions, respectively. We identify two dynamic clustering states and characterize them by power-law-exponential distributions. In case of mere attraction a chemotactic collapse occurs directly from the gaslike into the collapsed state, which we also predict by mapping our Langevin dynamics on the Keller-Segel model for bacterial chemotaxis.