# Exact Analytic Second Virial Coefficient for the Lennard-Jones Fluid

## TL;DR

This paper derives an exact analytic expression for the second virial coefficient of the Lennard-Jones fluid valid across all temperatures, using a variable transformation that simplifies the Hamiltonian to a harmonic oscillator form.

## Contribution

It provides the first exact analytic formula for the second virial coefficient of the Lennard-Jones fluid applicable at all temperatures, expressed via special functions.

## Key findings

- Exact formula valid for entire temperature range
- Derived limiting behaviors at low and high temperatures
- Expressed in terms of parabolic cylinder and hypergeometric functions

## Abstract

An exact analytic form for the second virial coefficient, valid for the entire range of temperature, is presented for the Lennard-Jones fluid in this paper. It is derived by making variable transformation that gives rise to the Hamiltonian mimicking a harmonic oscillator-like dynamics for negative energy. It is given in terms of parabolic cylinder functions or confluent hypergeometric functions. Exact limiting laws for the second virial coefficient in the limits of low and high temperatuers are also deduced for the Lennard-Jones fluid.