An effective Frenkel-exciton Hamiltonian for the LH2 photosynthetic complex from Rhodospirillum molischianum is calculated using the collective electronic oscillator (CEO) approach combined with the crystal structure. The absorption spectra of the various bacteriochlorophyll aggregates forming the complex are computed using the CEO. Each electronic transition is further analyzed in terms of its characteristic electron-hole motions in real space. Using a two-dimensional representation of the underlying transition density matrices, we identify localized and delocalized electronic transitions, test the applicability of the exciton model, and compute interchromophore electronic couplings. Förster energy-transfer hopping time scales within B800 and from the B800 to the B850 system, obtained using the computed coupling constants, are in excellent agreement with experiment.