  The radiative decay of excited states of the negative ion of rhodium, Rh$^-$,
has been investigated experimentally and theoretically. The experiments were
conducted at the Double ElectroStatic Ion Ring Experiment (DESIREE) facility at
Stockholm University using selective photodetachment from a stored ion beam to
monitor the time evolution of the excited state populations. The lifetimes of
the Rh$^-$ $^3F_{3}$ and $^3F_{2}$ fine structure levels were measured to be
3.2(6)~s and 21(4)~s, respectively. An additional, previously unreported,
higher-lying bound state of mixed $^1D_2+^3P_2+(4d^95s)^1D_2+^3F_2$ composition
was observed and found to have a lifetime of 10.9(8)s. The binding energy of
this state was determined to be in the interval $0.1584(2) $ eV $ < E_b <
0.2669(2)$ eV, using laser photodetachment threshold (LPT) spectroscopy. An
autodetaching state with a lifetime of 480(10) microseconds was also observed.
Theoretical calculations of the excited-state compositions, energies, and
magnetic-dipole transition lifetimes were performed using the
multiconfiguration Dirac-Hartree-Fock and relativistic configuration
interaction methods. The calculated lifetimes of the $^3F_{3}$ and $^3F_{2}$
fine structure levels are in excellent agreement with the measured values. The
present study should provide valuable insights into electron correlation
effects in negative ions and forbidden radiative transitions.
