Within Kirkwood theory, we study the translational diffusion coefficient of a single polymer chain in dilute solution, and focus on the small difference between the short-time Kirkwood value D-(K) and the asymptotic long-time value D. We calculate this correction term by highly accurate large-scale Brownian dynamics simulations, and show that it is in perfect agreement with the rigorous variational result DD-(K); the older data are shown to have insufficient statistical accuracy to resolve this question. We then discuss the Green-Kubo integrand in some detail. This function behaves very differently for pre-averaged versus fluctuating hydrodynamics, as shown for the initial value by analytical considerations corroborated by numerical results. We also present further numerical data on the chain's statics and dynamics. (C) 2003 American Institute of Physics.