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

TL;DR

This paper derives a compact analytic expression for the first quantum correction to the second virial coefficient $B_q$ in a $d$-dimensional fluid with generalized Lennard-Jones interactions, expressing the result in terms of parabolic cylinder functions $D_a(z)$ and, equivalently, generalized Hermite functions $H_a(z)$. By introducing suitable dimensionless variables, the authors reveal how $B_q$ depends on dimensionality through $a=(d-2)/n$ and demonstrate a unified treatment of dimensionality and stiffness, with explicit low- and high-temperature asymptotics. They also obtain the first quantum correction to the Boyle temperature $T_B^*$, showing how quantum effects alter the classical balance between attraction and repulsion, and apply the theory to noble gases He, Ne, and Ar to illustrate the relative importance of quantum corrections across species. The methodology is systematically extendable to higher-order quantum corrections and provides both theoretical insight and practical tools for assessing quantum effects in real fluids, including pedagogical value due to the explicit closed-form expressions.

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.

Figures (3)

• Figure S1: Reduced first quantum correction to the second virial coefficient, as given by Equation \ref{['9']}, for the gLJ fluid with (a) $d=2$ and (b) $d=3$. The curves correspond, from bottom to top, to stiffness parameters $n=4$, $5$, $6$, $7$, $8$, and $12$.
• Figure S2: Classical Boyle temperature $T_0^*$ (solid lines) and its first quantum correction $T_1^*$ (dashed lines) as functions of the stiffness parameter $n$ for the gLJ fluid with (a) $d=2$ and (b) $d=3$. The dotted lines represent the quantum-corrected Boyle temperature $T_{\text{B}}^*$, obtained from Equation \ref{['22']} with $q=5\times 10^{-3}$.
• Figure S3: Absolute value of the relative deviation, $|\delta B_2^*(T)|$, for helium, neon, and argon, obtained from Equation \ref{['deltaB']} to first order in $q$.