On the First Quantum Correction to the Second Virial Coefficient of a Generalized Lennard-Jones Fluid

TL;DR

This paper derives a compact, explicit formula for the first quantum correction to the second virial coefficient of generalized Lennard-Jones fluids, enabling analysis of quantum effects in gases and extending to higher-order corrections.

Contribution

It provides a new, simplified analytic expression for the quantum correction, incorporating dimensionality and stiffness, and applicable to various Lennard-Jones fluids.

Findings

Explicit formula for quantum correction derived

Analysis of quantum effects on Boyle temperature

Application to noble gases like helium, neon, and argon

Abstract

We derive an explicit analytic expression for the first quantum correction to the second virial coefficient of a $d$-dimensional fluid whose particles interact via the generalized Lennard-Jones $(2n,n)$ potential. By introducing an appropriate change of variable, the correction term is reduced to a single integral that can be evaluated in closed form in terms of parabolic cylinder or generalized Hermite functions. The resulting expression compactly incorporates both dimensionality and stiffness, providing direct access to the low- and high-temperature asymptotic regimes. In the special case of the standard Lennard-Jones fluid ($d=3$, $n=6$), the formula obtained is considerably more compact than previously reported representations based on hypergeometric functions. The knowledge of this correction allows us to determine the first quantum contribution to the Boyle temperature, whose dependence on dimensionality and stiffness is explicitly analyzed, and enables quantitative assessment of quantum effects in noble gases such as helium, neon, and argon. Moreover, the same methodology can be systematically extended to obtain higher-order quantum corrections.