Pressure formulae for liquid metals and plasmas based on the density-functional theory

First, pressure formulae for electrons under the external potential produce d by fixed nuclei are derived both in surface integral and volume integral forms for an arbitrary volume chosen in the system; the surface integral fo rm is based on a pressure tensor which is the sum of the kinetic and exchan ge-correlation parts in the density-functional theory, and the volume integ ral form represents the virial theorem with subtraction of the nuclear viri al term. Secondly, on the basis of these formulae, the thermodynamical pres sure of liquid metals and plasmas is represented in the forms of a surface integral and a volume integral including the nuclear contribution. From the se results, we obtain a virial pressure formula for liquid metals, which is more accurate and simpler than the standard representation. On the basis o f our formulation, some comments are made on pressure formulae derived prev iously and on a definition of pressure used widely.