  tructural transformation, pressure dependent elasticity behaviors, phonon,
and thermodynamic properties of the equiatomic TiZr alloy are investigated by
using first-principles density-functional theory. Our calculated lattice
parameters and equation of state for $\alpha$ and $\omega$ phases as well as
the phase transition sequence of
$\alpha$$\mathtt{\rightarrow}$$\omega$$\mathtt{\rightarrow}$$\beta$ are
consistent well with experiments. Elastic constants of $\alpha$ and $\omega$
phases indicate that they are mechanically stable. For cubic $\beta$ phase,
however, it is mechanically unstable at zero pressure and the critical pressure
for its mechanical stability is predicted to equal to 2.19 GPa. We find that
the moduli, elastic sound velocities, and Debye temperature all increase with
pressure for three phases of TiZr alloy. The relatively large $B/G$ values
illustrate that the TiZr alloy is rather ductile and its ductility is more
predominant than that of element Zr, especially in $\beta$ phase. Elastic wave
velocities and Debye temperature have abrupt increase behaviors upon the
$\alpha$$\mathtt{\rightarrow}$$\omega$ transition at around 10 GPa and exhibit
abrupt decrease feature upon the $\omega$$\mathtt{\rightarrow}$$\beta$
transition at higher pressure. Through Mulliken population analysis, we
illustrate that the increase of the \emph{d}-band occupancy will stabilize the
cubic $\beta$ phase. Phonon dispersions for three phases of TiZr alloy are
firstly presented and the $\beta$ phase phonons clearly indicate its
dynamically unstable nature under ambient condition. Thermodynamics of Gibbs
free energy, entropy, and heat capacity are obtained by quasiharmonic
approximation and Debye model.
