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The structural, elastic properties and electronic structure of FeTi and its hydrides, and hydrogen diffusion behavior in FeTi alloy were investigated by the first-principles calculations within the frame work of density functional theory (DFT).The calculated lattice constants for FeTi were in good agreement with the experimental values, and the lattice distortion of hydrides increases with the rise of hydrogen content.The calculation results of formation enthalpy, fusion heat of hydrogen and octahedron interstice size showed that hydrogen atom is more likely to exist in the central site of octahedron with the frame of four Ti atoms and two Fe atoms.The independent elastic constants as well as polycrystalline elastic parameters (bulk modulus B, shear modulus G, Youngs modulus E, Poissons ratio v and anisotropy value A) were calculated, and then the ductility and elastic anisotropy of FeTi and its hydrides were discussed in details.In addition, the density of states, electron density difference and population analysis for FeTi and its hydrides were also calculated, which reveals the underlying mechanism of structural stability and chemical bonding.Finally, the activation energy of next nearest neighbor diffusion for H between octahedron interstices in FeTi alloy was calculated to be 2.92 eV, clarifying the atomic mechanism of hydrogen diffusion.