  We report on the impact of magnetoelastic coupling on the magnetocaloric
properties of LaFe$_{11.4}$Si$_{1.6}$H$_{1.6}$ in terms of the vibrational
density of states, which we determined with $^{57}$Fe nuclear resonant
inelastic X-ray scattering measurements and with density-functional-theory
based first-principles calculations in the ferromagnetic low-temperature and
paramagnetic high-temperature phase. In experiments and calculations, we
observe pronounced differences in the shape of the Fe-partial VDOS between
non-hydrogenated and hydrogenated samples. This shows that hydrogen does not
only shift the temperature of the first-order phase transition, but also
affects the elastic response of the Fe-subsystem significantly. In turn, the
anomalous redshift of the Fe VDOS, observed by going to the low-volume PM
phase, survives hydrogenation. As a consequence, the change in the Fe specific
vibrational entropy $\Delta S_\mathrm{lat}$ across the phase transition has the
same sign as the magnetic and electronic contribution. DFT calculations show
that the same mechanism, which is a consequence of the itinerant electron
metamagnetism associated with the Fe subsystem, is effective in both the
hydrogenated and he hydrogen-free compounds. Although reduced by 50 % as
compared to the hydrogen-free system, the measured change $\Delta
S_\mathrm{lat}$ of 3.2\pm1.9 J/kgK across the FM to PM transition contributes
with 35 % significantly and cooperatively to the total isothermal entropy
change $\Delta S_\mathrm{iso}$. Hydrogenation is observed to induce an overall
blueshift of the Fe-VDOS with respect to the H-free compound; this effect,
together with the enhanced Debye temperature observed, is a fingerprint of the
hardening of the Fe sublattice by hydrogen incorporation. In addition, the mean
Debye velocity of sound of LaFe$_{11.4}$Si$_{1.6}$H$_{1.6}$ was determined from
the NRIXS and the DFT data.
