Pair-correlation expression for internal energy ($U/N=3k_{B}T/2+2\pi n{\int }_0^{\infty }{r}^{2}v(r)g(r)dr$)

For a classical monatomic homogeneous fluid with pairwise central pair potential $v(r)$, the mean interaction energy is one half the sum over neighbours of a tagged particle. Using the radial distribution function gives the displayed formula; the factor one half prevents counting each pair twice. The kinetic term follows from the equipartition theorem. Its use requires the relevant integral to exist and a thermodynamic state specified by number density and temperature. Quantum kinetic contributions, many-body forces and internal molecular modes require modifications.