In the primary process of photosynthesis, energy of the electronic excited state, A*, in the core antenna is trapped by the reaction center into a state, B, of charge separation between pigments therein. This energy-trapping process is mediated by the excited state, P*, of the special pair of chlorophylls or bacteriochlorophylls in the reaction center. Energy transfer from A* to P* has been reported to be uphill with an excitation-energy difference, for example, amounting to about 200 and 350-400 cm(-1) in Rhodobacter (Rb.) sphaeroides and Rhodopseudomonas (Rps.) viridis, respectively, in purple bacteria. The rate constant of this energy-trapping process was calculated with special attention to these species, reproducing observed data in the whole temperature range. At physiological temperatures, this process proceeds by the ordinary sequential mechanism, in which the second step from P* to B takes place after thermalization of phonons at P*, in both species. The uphill excitation-energy difference in the first step from A* to P* is blurred out by thermal activation with nearly uniform rate constants of about (50 ps)(-1) for the energy trapping at room temperature. In Rb. sphaeroides, P* is thermally (i.e., in the free energy) lower than A* by about 50 cm(-1) because of reorganizing relaxation of the medium in association with excitation of pigments. Correspondingly therein, the energy-trapping process is determined by the ordinary sequential mechanism at all temperatures. In Rps. viridis, P* is thermally still higher than A* by about 150 cm(-1). Correspondingly, below about 50 K therein, the energy-trapping process changes to the superexchange mechanism in which mediation at P* takes place as a quantum-mechanical virtual process without real population at P*. This mechanism resolves a serious difficulty that the observed data and the expectation from the ordinary sequential mechanism deviate to an astronomical extent at low temperatures in Rps. viridis.