We present quantum chemical molecular dynamics (MD) simulations for a model formation process of fullerene molecules. Trajectories of up to 24-ps lengths were computed for (5,5), (7,3), (8,0), (9,0), (10,0), and (10,5) open-ended single-walled carbon nanotubes for a temperature range between 2000 and 4000 K at various tube lengths, using density functional based tight-binding (DFTB) molecular dynamics. DFTB was selected because geometries and energies obtained are found to qualitatively agree with B3LYP/6-31G(d) results at much smaller cost of computer time. Extremely fast cage formation was observed with simulation times as short as 3 ps, and most simulations at 3000 and 4000 K led to the formation of fullerene structures within less than 14-ps simulation times. Key structural features for the transformation of tubes to fullerenes are identified, such as the overwhelming presence of acetylenic "wobbling C-2 units", which form spontaneously in great abundance at the open ends of the tubes. A comparison of DFTB simulations is made with corresponding semiclassical reactive bond-order force field MD trajectory calculations, which exhibit much slower structural transformations without the "wobbling" C-2 units. We also compare DFTB energetics of optimized MD snapshot structures with B3LYP energies.