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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.

Structure through pair correlations. For a homogeneous isotropic monatomic liquid of number density $n=N/V$, the radial distribution function $g(r)$ is defined byIt compares the conditional density around a tagged particle with the bulk density. A repulsive core makes close neighbours unlikely; a first peak identifies a preferred coordination shell, and damped peaks at larger radii describe short-range order. Far away, $g(r)\to 1$ in the bulk limit. Unlike a crystal, an ordinary liquid has no indefinitely repeated positional order. An ideal uncorrelated gas has $g(r)=1$ away from the tagged particle. The shell factor $4\pi r^2$ must not be confused with $g(r)$ itself.

The same pair structure enters the static structure factor measured through scattering. In the bulk isotropic convention,Thus real-space coordination and reciprocal-space scattering peaks describe the same correlations.

Energy from the structure. If interactions are sums of a central pair potential $v(r)$, the equipartition theorem gives the classical kinetic energy $3N{k}_{B}T/2$. Count the interaction of a tagged particle with its neighbours and divide by two to avoid counting each pair twice. The pair-correlation expression for internal energy isThis makes structure thermodynamically useful: short-distance repulsion and the attractive region contribute with different weights. Additional internal molecular modes or many-body interactions require extra terms. The formula is a classical bulk pair-potential result, not a universal energy formula for every liquid.

Pressure and the dilute expansion. The mechanical virial theorem relates the pair forces to pressure. The virial pressure from pair correlations isAt low density, $g(r)\simeq e^{-\beta v(r)}$, giving the virial expansionThe second virial coefficient has units of volume per particle; higher coefficients encode additional cluster correlations. A hard-sphere exclusion diameter $d$ gives $B_2=2\pi d^3/3>0$. Attractions contribute negative terms and can make $B_2$ negative. Although the pair-correlation formulas can describe a dense liquid when its actual $g(r)$ is known, truncating the virial series after a few terms is normally a dilute-fluid approximation. It should not be silently extrapolated through condensation or into a dense liquid.

For a differentiable central pair potential in a classical homogeneous three-dimensional fluid, the mechanical virial theorem gives the displayed pressure. The radial distribution function accounts for the probability of each pair separation. In the dilute limit, $g(r)=e^{-v(r)/(k_{B}T)}+O(n)$, and integration by parts yields $B_2(T)=-2\pi {\int }_0^{\infty }[e^{-v(r)/(k_{B}T)}-1]r^{2}dr$. Thus the second virial coefficient controls $p/(n{k}_{B}T)=1+B_{2}n+\cdots$. A few virial terms are a low-density approximation, not generally a converged description of a dense liquid. A discontinuous hard core needs its contact form or a limiting argument.