  The equation of state of liquid metallic hydrogen is solved numerically.
Investigations are carried out at temperatures, which correspond both to the
experimental conditions under which metallic hydrogen is produced on earth and
the conditions in the cores of giant planets of the solar system such as
Jupiter and Saturn. It is assumed that hydrogen is in an atomic state and all
its electrons are collectivized. Perturbation theory in the electron and proton
interaction is applied to determine the thermodynamic potentials of metallic
hydrogen. The electron subsystem is considered in the randomphase approximation
with regard to the exchange interaction and the correlation of electrons in the
local field approximation. The interproton interaction is taken into account in
the hard spheres approximation. The thermodynamic characteristics of metallic
hydrogen are calculated with regard to the zero-, second-, and thirdorder
perturbation theory terms. The third-order term proves to be rather essential
at moderately high temperatures and densities, although it is much smaller than
the second-order term. The thermodynamic potentials of metallic hydrogen are
monotonically increasing functions of density and temperature. The values of
pressure for the temperatures and pressures that are characteristic of the
conditions under which metallic hydrogen is produced on earth coincide with the
corresponding values reported by the discoverers of metallic hydrogen to a high
degree of accuracy. The temperature and density ranges are found in which there
exists a liquid phase of metallic hydrogen.
