  We present an extension of TRAPHIC, the method for radiative transfer of
ionising radiation in smoothed particle hydrodynamics simulations that we
introduced in Pawlik & Schaye (2008). The new version keeps all advantages of
the original implementation: photons are transported at the speed of light, in
a photon-conserving manner, directly on the spatially adaptive, unstructured
grid traced out by the particles, in a computation time that is independent of
the number of radiation sources, and in parallel on distributed memory
machines. We extend the method to include multiple frequencies, both hydrogen
and helium, and to model the coupled evolution of the temperature and
ionisation balance. We test our methods by performing a set of simulations of
increasing complexity and including a small cosmological reionisation run. The
results are in excellent agreement with exact solutions, where available, and
also with results obtained with other codes if we make similar assumptions and
account for differences in the atomic rates used. We use the new implementation
to illustrate the differences between simulations that compute photoheating in
the grey approximation and those that use multiple frequency bins. We show that
close to ionising sources the grey approximation asymptotes to the
multi-frequency result if photoheating rates are computed in the optically thin
limit, but that the grey approximation breaks down everywhere if, as is often
done, the optically thick limit is assumed.
