  {\em Ab initio} techniques based on density functional theory in the
projector-augmented-wave implementation are used to calculate the free energy
and a range of other thermodynamic properties of liquid iron at high pressures
and temperatures relevant to the Earth's core. The {\em ab initio} free energy
is obtained by using thermodynamic integration to calculate the change of free
energy on going from a simple reference system to the {\em ab initio} system,
with thermal averages computed by {\em ab initio} molecular dynamics
simulation. The reference system consists of the inverse-power pair-potential
model used in previous work. The liquid-state free energy is combined with the
free energy of hexagonal close packed Fe calculated earlier using identical
{\em ab initio} techniques to obtain the melting curve and volume and entropy
of melting. Comparisons of the calculated melting properties with experimental
measurement and with other recent {\em ab initio} predictions are presented.
Experiment-theory comparisons are also presented for the pressures at which the
solid and liquid Hugoniot curves cross the melting line, and the sound speed
and Gr\"{u}neisen parameter along the Hugoniot. Additional comparisons are made
with a commonly used equation of state for high-pressure/high-temperature Fe
based on experimental data.
