appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law.
They are characteristic of the interaction potential between the particles and in general depend on the temperature.
depends only on the pair interaction between the particles, the third (
The first step in obtaining a closed expression for virial coefficients is a cluster expansion[1] of the grand canonical partition function Here
λ = exp [ μ
is the canonical partition function of a subsystem of
is the Hamiltonian (energy operator) of a subsystem of
The Hamiltonian is a sum of the kinetic energies of the particles and the total
The grand partition function
can be expanded in a sum of contributions from one-body, two-body, etc.
In this manner one derives These are quantum-statistical expressions containing kinetic energies.
Note that the one-particle partition function
the kinetic energy operators commute with the potential operators and the kinetic energies in numerator and denominator cancel mutually.
The trace (tr) becomes an integral over the configuration space.
It follows that classical virial coefficients depend on the interactions between the particles only and are given as integrals over the particle coordinates.
virial coefficients becomes quickly a complex combinatorial problem.
Making the classical approximation and neglecting non-additive interactions (if present), the combinatorics can be handled graphically as first shown by Joseph E. Mayer and Maria Goeppert-Mayer.
[2] They introduced what is now known as the Mayer function: and wrote the cluster expansion in terms of these functions.
are related to the irreducible Mayer cluster integrals
through The latter are concisely defined in terms of graphs.
The rule for turning these graphs into integrals is as follows: The first two cluster integrals are The expression of the second virial coefficient is thus: where particle 2 was assumed to define the origin (
This classical expression for the second virial coefficient was first derived by Leonard Ornstein in his 1908 Leiden University Ph.D. thesis.