  We present a systematic investigation of the effect of H, B, C, and N
interstitials on the electronic, lattice and magnetic properties of
La(Fe,Si)$_{13}$ using density functional theory. The parent LaSiFe$_{12}$
alloy has a shallow, double-well free energy function that is the basis of
itinerant metamagnetism. On increasing the dopant concentration, the resulting
lattice expansion causes an initial increase in magnetisation for all
interstitials that is only maintained at higher levels of doping in the case of
hydrogen. Strong s-p band hybridisation occurs at high B,C and N
concentrations. We thus find that the electronic effects of hydrogen doping are
much less pronounced than those of other interstitials, and result in the
double-well structure of the free energy function being least sensitive to the
amount of hydrogen. This microscopic picture accounts for the vanishing first
order nature of the transition by B,C, and N dopants as observed
experimentally. We use our calculated electronic density of states for
LaSiFe$_{12}$ and the hydrogenated alloy to infer changes in magneto-elastic
coupling and in phonon entropy on heating through $T_C$ by calculating the
fermionic entropy due to the itinerant electrons. Lastly, we predict the
electron thermopower in a spin-mixing, high temperature limit and compare our
findings to recent literature data.
