The static and dynamic properties of interstitial H2, HD and D2 molecules in crystalline silicon are obtained from ab initio molecular-dynamics simulations with atomic-like basis sets. The static (T = 0) calculations agree with those of most other authors: the centre of mass (CM) of H2 is at the tetrahedral interstitial (T) site, the molecule is a nearly-free rotator, and the activation energy for diffusion is 0.90 eV. However, these results fail to explain a number of experimental observations, such as why H2 is infrared (IR) active, why the expected ortho/para splitting is not present, why the symmetry is C1, why the piezospectroscopic tensors of H2 and D2 are identical or why the exposure to an H/D mix results in a single HD line which is not only at the wrong place but also much weaker than expected. In the present work, we extend the static calculations to include the constant-temperature dynamics for H2 in Si. At T > 0 K, the CM of the molecule no longer remains at the T site. Instead, H2 'bounces' off the walls of its tetrahedral cage and exchanges energy with the host crystal. The average position of the CM is away from the T site along <100>. Under uniaxial stress, the CM shifts off that axis and the molecule has C1 symmetry. The H-H stretch frequency calculated from the Fourier transform of the v-v autocorrelation function is close to the measured one. Since the potential energy experienced by H2 in Si near the T site is very flat, we argue that H2 should be a nearly free quantum mechanical rotator. Up to room temperature, only the j = 0 and j = 1 rotational states are occupied, H2 resembles a sphere rather than a dumbbell, the symmetry is determined by the position of the CM and HD is equivalent to DH in any symmetry. The rapid motion of the CM implies that an ortho-to-para transition will occur if a large magnetic moment is nearby. Several candidates are proposed. Since nuclear quantum effects are not included in our calculations, we cannot address the possibility that the observed vibrational spectrum of H2 results from a tunnelling excitation as proposed by Stoneham.