  We investigate the origin of the volume change and magnetoelastic interaction
observed at the magnetic first-order transition in the magnetocaloric system
La(Fe$_{1-x}$Si$_x$)$_{13}$ by means of first-principles calculations combined
with the fixed-spin moment approach. We find that the volume of the system
varies with the square of the average local Fe moment, which is significantly
smaller in the spin disordered configurations compared to the ferromagnetic
ground state. The vibrational density of states obtained for a hypothetical
ferromagnetic state with artificially reduced spin-moments compared to a
nuclear inelastic X-ray scattering measurement directly above the phase
transition reveals that the anomalous softening at the transition essentially
depends on the same moment-volume coupling mechanism. In the same spirit, the
dependence of average local Fe moment on the Si content can account for the
occurence of first- and second-order transitions in the system.
