  There is longstanding fundamental interest in 6-fold coordinated $d^6$
($t_{2g}^6$) transition metal complexes such as [Ru(bpy)$_3$]$^{2+}$ and
Ir(ppy)$_3$, particularly their phosphorescence. This interest has increased
with the growing realisation that many of these complexes have potential uses
in applications including photovoltaics, imaging, sensing, and light-emitting
diodes. In order to design new complexes with properties tailored for specific
applications a detailed understanding of the low-energy excited states,
particularly the lowest energy triplet state, $T_1$, is required. Here we
describe a model of pseudo-octahedral complexes based on a pseudo-angular
momentum representation and show that the predictions of this model are in
excellent agreement with experiment - even when the deviations from octahedral
symmetry are large. This model gives a natural explanation of zero-field
splitting of $T_1$ and of the relative radiative rates of the three sublevels
in terms of the conservation of time-reversal parity and total angular momentum
modulo two. We show that the broad parameter regime consistent with the
experimental data implies significant localization of the excited state.
