  By means of quantum tensor network calculations, we investigate the large
Purcell effect experienced by an organic molecule placed in the vicinity of a
plasmonic nanostructure. In particular, we consider a donor-{\pi}
bridge-acceptor dye at the gap of two Ag nanospheres. Our theoretical approach
allows for a realistic description of the continua of both molecular vibrations
and optical nanocavity modes. We analyze both the exciton dynamics and the
corresponding emission spectrum, showing that these magnitudes are not
accurately represented by the simplified models used up to date. By
disentangling the molecule coupling to radiative and non-radiative plasmonic
modes, we also shed light into the quenching phenomenology taking place in the
system.
