  We systematically calculate the structure, formation enthalpy, formation free
energy, elastic constants and electronic structure of Ti$_{0.98}$X$_{0.02}$
system by density functional theory (DFT) simulations to explore the effect of
transition metal X (X=Ag, Cd, Co, Cr, Cu, Fe, Mn, Mo, Nb, Ni, Pd, Rh, Ru, Tc,
and Zn) on the stability mechanism of $\beta$-titanium. Based on our
calculations, the results of formation enthalpy and free energy show that
adding trace X is beneficial to the thermodynamic stability of
$\beta$-titanium. This behavior is well explained by the density of state
(DOS). However, the tetragonal shear moduli of Ti$_{0.98}$X$_{0.02}$ systems
are negative, indicating that $\beta$-titanium doping with a low concentration
of X is still elastically unstable at 0 K. Therefore, we theoretically explain
that $\beta$-titanium doping with trace transition metal X is unstable in the
ground state.
