A quantitative study of the astrophysically important sub-barrier fusion of $^{12}$C+$^{12}$C is presented. Low-energy collisions are described in the body-fixed reference frame using wave-packet dynamics within a nuclear molecular picture. A collective Hamiltonian drives the time propagation of the wave-packet through the collective potential-energy landscape. The fusion imaginary potential for specific dinuclear configurations is crucial for understanding the appearance of resonances in the fusion cross section. The theoretical sub-barrier fusion cross sections explain some observed resonant structures in the astrophysical S-factor. These cross sections monotonically decline towards stellar energies. The structures in the data that are not explained are possibly due to cluster effects in the nuclear molecule, which are to be included in the present approach.
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