  In purple bacteria, the fundamental charge-separation step that drives the
conversion of radiation energy into chemical energy proceeds along one branch -
the A branch - of a heterodimeric pigment-protein complex, the reaction center.
Here, we use first principles time-dependent density functional theory (TDDFT)
with an optimally-tuned range-separated hybrid functional to investigate the
electronic and excited-state structure of the primary six pigments in the
reaction center of \textit{Rhodobacter sphaeroides}. By explicitly including
amino-acid residues surrounding these six pigments in our TDDFT calculations,
we systematically study the effect of the protein environment on energy and
charge-transfer excitations. Our calculations show that a forward charge
transfer into the A branch is significantly lower in energy than the first
charge transfer into the B branch, in agreement with the unidirectional charge
transfer observed experimentally. We further show that inclusion of the protein
environment redshifts this excitation significantly, allowing for energy
transfer from the coupled $Q_x$ excitations. Through analysis of transition and
difference densities, we demonstrate that most of the $Q$-band excitations are
strongly delocalized over several pigments and that both their spatial
delocalization and charge-transfer character determine how strongly affected
they are by thermally-activated molecular vibrations. Our results suggest a
mechanism for charge-transfer in this bacterial reaction center and pave the
way for further first-principles investigations of the interplay between
delocalized excited states, vibronic coupling, and the role of the protein
environment of this and other complex light-harvesting systems.
