  The first-principle study on double perovskite compound Ba$_2$PrVO$_6$ has
been presented in this paper. By analysing band structures and integrated
density of states, it is found that Ba$_2$PrVO$_6$ is ferromagnetic metallic
within LSDA and compensated half metallic whithin LSDA+U. According to the
total and partial density of states, the anti-ferromagnetism of Ba$_2$PrVO$_6$
is originated from the spin down state Pr$^{4+}$ ($4f_{-2}$) and the spin up
hybridized state between the partially filled $t_{2g}$ state of V and the
partially filled triple-degeneration state $4f_{-1}$, $4f_{+1}$, $4f_{+3}$ of
Pr. We have investigated the magnetic evaluation of Ba$_2$PrVO$_6$ through
fixed spin moment calculations, and the results indicate that the compensated
half-metallic state is the ground state. The thermodynamic properties of
Ba$_2$PrVO$_6$ are also studied by employing the quasi-harmonic Debye model.
