  We have investigated the thermal equation of state of tantalum from first
principles using the Linearized Augmented Plane Wave (LAPW) and pseudopotential
methods for pressures up to 300 GPa and temperatures up to 10000 K. The
equation of state at zero temperature was computed using LAPW. For finite
temperatures, mixed basis pseudopotential computations were performed for 54
atom supercells. The vibrational contributions were obtained by computing the
partition function using the particle in a cell model, and the the finite
temperature electronic free energy was obtained from the LAPW band structures.
We discuss the behavior of thermal equation of state parameters such as the
Gr\"uneisen parameter $\gamma$, $q$, the thermal expansivity $\alpha$, the
Anderson-Gr\"uneisen parameter $\delta_T$ as functions of pressure and
temperature. The calculated Hugoniot shows excellent agreement with shock-wave
experiments. An electronic topological transition was found at approximately
200 GPa.
