The structural, electronic, magnetic, and vibrational properties of LaFeSiH$_x$ for $x$ between 0 and 1 are investigated using density functional calculations. We find that the electronic and magnetic properties are strongly controlled by the hydrogen concentration $x$ in LaFeSiH$_x$. While fully hydrogenated LaFeSiH has a striped antiferromagnetic ground state, the underdoped LaFeSiH$_x$ for $xleq0.75$ is not magnetic within the virtual crystal approximation or with explicit doping of supercells. The antiferromagnetic configuration breaks the symmetry of Fe $d$ orbitals and increases electron-phonon coupling up to $50%$, especially for modes in the 20-50 meV range that are associated with Fe atomic movement. We find competing nearest and next-nearest neighbor exchange interactions and significant spin-phonon coupling, qualitatively similar but smaller in magnitude compared those found in LaOFeAs superconductors. The superconducting $T_c$ for antiferromagnetic LaFeSiH$_x$, assuming conventional superconductivity via McMillans equation, therefore is computed to be 2-10 K, in contrast to $T_capprox0$ for the nonmagnetic material. We also predict that the LaFeSiH$_x$ could be a good proton conductor due to phase stability with a wide range of hydrogen concentration $x < 1$.
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