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

1. Introduction

In a semiclassical fluid, the second virial coefficient can be expressed as [1,2,3,4,5,6,7,8]

$$B_2\left(T\right)=B^{\left(0\right)}\left(T\right)+{\displaystyle \frac{{\hslash }^2}{m}}{B}^{\left(1\right)}\left(T\right)+\mathcal{O}\left(\frac{{\hslash }^4}{m^2}\right),$$

(1)

where

$$B^{\left(0\right)}\left(T\right)={\displaystyle \frac{{\Omega }_d}{2}}{\int }_0^{\infty }\mathrm{d}r{r}^{d-1}\left[1-{\mathrm{e}}^{-\beta \varphi \left(r\right)}\right],{\Omega }_d={\displaystyle \frac{2{\pi }^{\frac{d}{2}}}{\Gamma \left(\frac{d}{2}\right)}},$$

(2)

is the classical contribution, and

$$B^{\left(1\right)}\left(T\right)={\displaystyle \frac{{\Omega }_d{\beta }^3}{24}}{\int }_0^{\infty }\mathrm{d}r{r}^{d-1}{\mathrm{e}}^{-\beta \varphi \left(r\right)}{\left[{\displaystyle \frac{\mathrm{d}\varphi \left(r\right)}{\mathrm{d}r}}\right]}^2$$

(3)

represents the first quantum correction. In Eqs. (2) and (3), $\varphi \left(r\right)$ denotes the pair potential, $\beta \equiv {\left(k_{B}T\right)}^{-1}$, and ${\Omega }_d$ is the total solid angle in d dimensions.

The prototypical pair potential in liquid-state theory is the Lennard-Jones (LJ) potential,

$$\varphi \left(r\right)=4ϵ\left[{\left({\displaystyle \frac{\sigma }{r}}\right)}^{2n}-{\left({\displaystyle \frac{\sigma }{r}}\right)}^n\right],$$

(4)

where $ϵ$ and $\sigma$ set the energy and length scales, respectively. The standard LJ (sLJ) fluid corresponds to $d=3$ and $n=6$, while the generalized Lennard-Jones (gLJ) model allows arbitrary dimensionality d and stiffness parameter $n>d$. In the limit $n\to \infty$, the gLJ potential of Eq. (4) approaches the hard-sphere potential.

By introducing the reduced (dimensionless) coefficients

$$B_c\equiv {\displaystyle \frac{2d}{{\Omega }_d{\sigma }^d}}{B}^{\left(0\right)},B_q\equiv {\displaystyle \frac{24ϵ}{{\Omega }_d{\sigma }^{d-2}}}{B}^{\left(1\right)},$$

(5)

we obtain

$$B_c\left(T^{*}\right)=d{\int }_0^{\infty }\mathrm{d}x{x}^{d-1}\left[1-{\mathrm{e}}^{-4{\beta }^{*}(x^{-2n}-x^{-n})}\right],$$

(6a)

$$B_q\left(T^{*}\right)={{\beta }^{*}}^3{\left(4n\right)}^2{\int }_0^{\infty }\mathrm{d}x{x}^{-(2n+3-d)}{\mathrm{e}}^{-4{\beta }^{*}(x^{-2n}-x^{-n})}\left(1-4x^{-n}+4x^{-2n}\right),$$

(6b)

where $T^{*}\equiv 1/{\beta }^{*}=k_{B}T/ϵ$ is the reduced temperature.

Several equivalent representations of $B_c$ for the sLJ fluid can be found in the literature (see, for instance, Ref. [8] and references therein). Perhaps the most compact expression—valid for the gLJ fluid—is [9, Sec. 3.7]

$$B_c\left(T^{*}\right)=\Gamma (1-\frac{d}{n}){\left(8{\beta }^{*}\right)}^{\frac{d}{2n}}{\mathrm{e}}^{{\beta }^{*}/2}{\mathrm{D}}_{{\displaystyle \frac{d}{n}}}\left(-\sqrt{2{\beta }^{*}}\right),$$

(7)

where

$${\mathrm{D}}_a\left(z\right)={\displaystyle \frac{{\mathrm{e}}^{-z^2/4}}{\Gamma (-a)}}{\int }_0^{\infty }\mathrm{d}t{t}^{-a-1}\left[{\mathrm{e}}^{-t^2/2-zt}-\Theta \left(a\right)\right],a<1,$$

(8)

is the parabolic cylinder function [10]. In Eq. (8), $\Theta (·)$ denotes the Heaviside step function.

Naturally, the situation is more involved for the quantum contribution $B_q$. In a recent work, Zhao et al. [8] derived a linear, second-order homogeneous ordinary differential equation for the sLJ coefficient $B_q$. From its solution, they obtained

$$B_q\left(T^{*}\right)={\displaystyle \frac{1}{3\times 2^{\frac{11}{6}}}}\left[\Gamma \left(\frac{5}{12}\right)F\left(T^{*}\right)-\Gamma (-\frac{1}{12})G\left(T^{*}\right)\right],$$

(9)

where

$$F\left(T^{*}\right)={{\beta }^{*}}^{\frac{19}{12}}\left[721{\mathrm{F}}_1(\frac{5}{12};\frac{1}{2};{\beta }^{*})-221{\mathrm{F}}_1(\frac{5}{12};\frac{3}{2};{\beta }^{*})\right],$$

(10a)

$$G\left(T^{*}\right)={{\beta }^{*}}^{\frac{13}{12}}\left[12{\beta }^{*}1{\mathrm{F}}_1(\frac{11}{12};\frac{3}{2};{\beta }^{*})+111{\mathrm{F}}_1(-\frac{1}{12};\frac{1}{2};{\beta }^{*})\right].$$

(10b)

Here, $1{\mathrm{F}}_1\left(a;b;z\right)$ denotes the Kummer confluent hypergeometric function [10]. The result given by Eqs. (9) and (10) was first obtained by Michels [4].

2. First Quantum Correction to the Second Virial Coefficient

Our goal is to derive an alternative, more compact expression for $B_q$ in the broader case of the gLJ fluid. We begin by stating the final result:

$$B_q\left(T^{*}\right)={\displaystyle \frac{\Gamma (2-\frac{d-2}{n})n}{8}}{\left(8{\beta }^{*}\right)}^{{\displaystyle \frac{d-2}{2n}}+1}{\mathrm{e}}^{{\beta }^{*}/2}\left[{\mathrm{D}}_{{\displaystyle \frac{d-2}{n}}}\left(-\sqrt{2{\beta }^{*}}\right)+{\mathrm{D}}_{{\displaystyle \frac{d-2}{n}}-2}\left(-\sqrt{2{\beta }^{*}}\right)\right].$$

(11)

Before proving Eq. (11), we list several useful properties of the parabolic cylinder function [10]:

$${\mathrm{D}}_a\left(z\right)=z{\mathrm{D}}_{a-1}\left(z\right)+(1-a){\mathrm{D}}_{a-2}\left(z\right),$$

(12a)

$${\displaystyle \frac{\partial {\mathrm{D}}_a\left(z\right)}{\partial z}}=a{\mathrm{D}}_{a-1}\left(z\right)-{\displaystyle \frac{z}{2}}{\mathrm{D}}_a\left(z\right),$$

(12b)

$$\underset{z\to 0}{lim}{\mathrm{D}}_a\left(z\right)={\displaystyle \frac{\sqrt{\pi }{2}^{\frac{a}{2}}}{\Gamma \left(\frac{1-a}{2}\right)}},\underset{z\to \infty }{lim}{\mathrm{D}}_a(-z)={\displaystyle \frac{\sqrt{2\pi }}{\Gamma (-a)}}{\mathrm{e}}^{z^2/4}{z}^{-a-1},$$

(12c)

$${\mathrm{D}}_0\left(z\right)={\mathrm{e}}^{-z^2/4},{\mathrm{D}}_{-2}\left(z\right)={\mathrm{e}}^{-z^2/4}-\sqrt{{\displaystyle \frac{\pi }{2}}}{\mathrm{e}}^{z^2/4}z\mathrm{erfc}\left({\displaystyle \frac{z}{\sqrt{2}}}\right),$$

(12d)

$${\mathrm{D}}_a\left(\sqrt{2}z\right)=2^{-\frac{a}{2}}{\mathrm{e}}^{-z^2/2}{\mathrm{H}}_a\left(z\right).$$

(12e)

Equation (12e) defines the generalized Hermite functions ${\mathrm{H}}_a\left(z\right)$ for arbitrary (noninteger) degree $a<1$ [10].

By introducing the change of variable $x\to t=\sqrt{8{\beta }^{*}}{x}^{-n}$ in Eq. (6b), we obtain

$$B_q\left(T^{*}\right)={\displaystyle \frac{n}{8}}{\left(8{\beta }^{*}\right)}^{\frac{a}{2}+1}{\int }_0^{\infty }\mathrm{d}t{t}^{-a+1}{\mathrm{e}}^{-t^2/2-zt}\left(z^2+2zt+t^2\right),$$

(13)

where we have introduced the shorthand notation $a\equiv \frac{d-2}{n}$, $z\equiv -\sqrt{2{\beta }^{*}}$. Using the definition of the parabolic cylinder function, Eq. (8), Eq. (13) can be rewritten as

$$\begin{array}{ccc}\hfill B_q\left(T^{*}\right)=& {\displaystyle \frac{\Gamma (2-a)n}{8}}{\left(8{\beta }^{*}\right)}^{\frac{a}{2}+1}{\mathrm{e}}^{z^2/4}\left[z^2{\mathrm{D}}_{a-2}\left(z\right)+2(2-a)z{\mathrm{D}}_{a-3}\left(z\right)\right.\hfill & \hfill \left.+(3-a)(2-a){\mathrm{D}}_{a-4}\left(z\right)\right].\end{array}$$

This expression is already quite compact, but it can be further simplified. Iterative application of Eq. (12a) yields

$$\begin{array}{ccc}\hfill {\mathrm{D}}_a\left(z\right)=& z\left[z{\mathrm{D}}_{a-2}\left(z\right)+(2-a){\mathrm{D}}_{a-3}\left(z\right)\right]+(1-a){\mathrm{D}}_{a-2}\left(z\right)=\hfill & \hfill z^2{\mathrm{D}}_{a-2}\left(z\right)+(2-a)z{\mathrm{D}}_{a-3}\left(z\right)+(2-a){\mathrm{D}}_{a-2}\left(z\right)-{\mathrm{D}}_{a-2}\left(z\right).\end{array}$$

Next, we apply Eq. (12a) to the term $(2-a){\mathrm{D}}_{a-2}\left(z\right)$, which gives

$${\mathrm{D}}_a\left(z\right)=z^2{\mathrm{D}}_{a-2}\left(z\right)+2(2-a)z{\mathrm{D}}_{a-3}\left(z\right)+(3-a)(2-a){\mathrm{D}}_{a-4}\left(z\right)-{\mathrm{D}}_{a-2}\left(z\right).$$

(14)

Substituting this identity into Eq. (14), and returning to the physical variables $a\to \frac{d-2}{n}$ and $z\to -\sqrt{2{\beta }^{*}}$, we recover Eq. (11). In terms of the generalized Hermite functions, Eq. (11) can be rewritten as

$$B_q\left(T^{*}\right)={\displaystyle \frac{\Gamma (2-\frac{d-2}{n})n}{4}}{\left(4{\beta }^{*}\right)}^{{\displaystyle \frac{d-2}{2n}}+1}\left[{\mathrm{H}}_{{\displaystyle \frac{d-2}{n}}}\left(-\sqrt{{\beta }^{*}}\right)+2{\mathrm{H}}_{{\displaystyle \frac{d-2}{n}}-2}\left(-\sqrt{{\beta }^{*}}\right)\right].$$

(15)

For the particular case of the sLJ model ($d=3$, $n=6$),

$$\begin{array}{ccc}\hfill B_q\left(T^{*}\right)=& {\displaystyle \frac{3\Gamma \left(\frac{11}{6}\right)}{4}}{\left(8{\beta }^{*}\right)}^{{\displaystyle \frac{13}{12}}}{\mathrm{e}}^{{\beta }^{*}/2}\left[{\mathrm{D}}_{{\displaystyle \frac{1}{6}}}\left(-\sqrt{2{\beta }^{*}}\right)+{\mathrm{D}}_{-{\displaystyle \frac{11}{6}}}\left(-\sqrt{2{\beta }^{*}}\right)\right]=\hfill & \hfill {\displaystyle \frac{3\Gamma \left(\frac{11}{6}\right)}{2}}{\left(4{\beta }^{*}\right)}^{{\displaystyle \frac{13}{12}}}\left[{\mathrm{H}}_{{\displaystyle \frac{1}{6}}}\left(-\sqrt{{\beta }^{*}}\right)+2{\mathrm{H}}_{-{\displaystyle \frac{11}{6}}}\left(-\sqrt{{\beta }^{*}}\right)\right].\end{array}$$

It can be verified that Eq. (18) is equivalent to the combination of Eqs. (9) and (10).

The limits given by Eq. (12c) allow us to determine the low- and high-temperature behaviors of $B_q\left(T^{*}\right)$ for the gLJ fluid:

$$\underset{T^{*}\to 0}{lim}{B}_q\left(T^{*}\right)=\sqrt{\pi }{2}^{{\displaystyle \frac{d-2}{n}}+1}n{\mathrm{e}}^{1/T^{*}}{T^{*}}^{-{\displaystyle \frac{3}{2}}},\underset{T^{*}\to \infty }{lim}{B}_q\left(T^{*}\right)={\displaystyle \frac{n\Gamma (2-\frac{d-2}{2n})}{2}}{\left({\displaystyle \frac{4}{T^{*}}}\right)}^{{\displaystyle \frac{d-2}{2n}}+1}.$$

(16)

In the second equality of Eq. (19), use has been made of the identity $\Gamma \left(2x\right)/\Gamma \left(x\right)=2^{2x-1}\Gamma (x+\frac{1}{2})/\sqrt{\pi }$.

Interestingly, Eq. (11) simplifies considerably in the case of a two-dimensional fluid ($d=2$). Using Eq. (12d), we obtain

$$B_q\left(T^{*}\right)=n{\beta }^{*}\left[2+\sqrt{\pi {\beta }^{*}}{\mathrm{e}}^{{\beta }^{*}}\mathrm{erfc}(-\sqrt{{\beta }^{*}})\right].$$

(17)

In this case, the ratio $B_q/n$ is independent of the stiffness parameter n.

Figure 1 illustrates the temperature dependence of $B_q(T^{)}$ for the two- and three-dimensional gLJ fluids with $n=4$, 5, 6, 7, 8, and 12. As can be seen, for a given $T^{*}$, the reduced first quantum correction $B_q$ increases as the potential becomes stiffer. Moreover, the influence of n is more pronounced at high temperatures than at low temperatures, consistent with the limiting behaviors described by Eq. (19). For the same values of $T^{*}$ and n, $B_q$ is larger in the two dimensions than in three.

3. First Quantum Correction to the Boyle Temperature

From Eq. (1), the reduced second virial coefficient of the gLJ fluid can be written as

$${\displaystyle \frac{2d}{{\Omega }_d{\sigma }^d}}{B}_2\left(T^{*}\right)=B_c\left(T^{*}\right)+{\displaystyle \frac{d}{12}}q{B}_q\left(T^{*}\right)+\mathcal{O}\left(q^2\right),q\equiv {\displaystyle \frac{{\hslash }^2}{m{\sigma }^2ϵ}}.$$

(18)

The Boyle temperature, $T_{\mathrm{B}}^{*}$, is defined by the condition $B_2\left(T_{\mathrm{B}}^{*}\right)=0$. It marks the balance between the attractive and repulsive contributions to the intermolecular potential: the attractive interactions dominate for $T^{*}<T_{\mathrm{B}}^{*}$, whereas the repulsive ones dominate for $T^{*}>T_{\mathrm{B}}^{*}$. In the semiclassical regime, the Boyle temperature can be expanded as

$$T_{\mathrm{B}}^{*}=T_0^{*}-q{T}_1^{*}+\mathcal{O}\left(q^2\right),$$

(19)

where $T_0^{*}$ is the classical Boyle temperature, i.e., the solution of $B_c\left(T_0^{*}\right)=0$, or equivalently, ${\mathrm{D}}_{d/n}(-\sqrt{2/T_0^{*}})=0$. By inserting Eq. (22) into Eq. (21), one obtains the first quantum correction to the Boyle temperature,

$$T_1^{*}={\displaystyle \frac{d}{12}}{\displaystyle \frac{B_q\left(T_0^{*}\right)}{{\left.\partial B_c\left(T^{*}\right)/\partial T^{*}\right|}_{T_0^{*}}}}.$$

(20)

Note that $\partial B_c/\partial T^{*}=-{T^{*}}^{-2}\partial B_c/\partial {\beta }^{*}$, where, from Eqs. (7) and (12b),

$${\displaystyle \frac{\partial B_c}{\partial {\beta }^{*}}}={\displaystyle \frac{d}{n}}\left[{\displaystyle \frac{B_c}{2{\beta }^{*}}}-{\displaystyle \frac{\Gamma (1-\frac{d}{n})}{\sqrt{2{\beta }^{*}}}}{\left(8{\beta }^{*}\right)}^{{\displaystyle \frac{d}{2n}}}{\mathrm{e}}^{{\beta }^{*}/2}{\mathrm{D}}_{{\displaystyle \frac{d}{n}}-1}\left(-\sqrt{2{\beta }^{*}}\right)\right].$$

(21)

Thus, one finally obtains

$$T_1^{*}={\displaystyle \frac{n^2\Gamma (2-\frac{d-2}{n})}{3\Gamma (1-\frac{d}{n})}}{\left({\displaystyle \frac{T_0^{*}}{8}}\right)}^{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{n}}}{\displaystyle \frac{{\mathrm{D}}_{\frac{d-2}{n}}(-\sqrt{2/T_0^{*}})+{\mathrm{D}}_{\frac{d-2}{n}-2}(-\sqrt{2/T_0^{*}})}{{\mathrm{D}}_{\frac{d}{n}-1}(-\sqrt{2/T_0^{*}})}}.$$

(22)

Figure 2 shows $T_0^{*}$ and $T_1^{*}$ as functions of n for $d=2$ and $d=3$. While the classical Boyle temperature $T_0^{*}$ decreases as the potential becomes stiffer, the first quantum correction $T_1^{*}$ increases with n. As a result, quantum effects amplify the decrease of the Boyle temperature with increasing stiffness, as illustrated by the curves representing $T_0^{*}-q{T}_1^{*}$ with $q=5\times {10}^{-3}$. This effect is more pronounced in two-dimensional fluids than in three-dimensional ones.

4. Conclusions

In this paper, we have derived an explicit and compact expression, Eq. (11), for the first quantum correction to the second virial coefficient of a d-dimensional fluid composed of particles interacting through the gLJ $(2n,n)$ potential defined in Eq. (4). As in the classical case, Eq. (8), the first quantum correction has been conveniently expressed in terms of parabolic cylinder functions. For the particular case of the sLJ fluid ($d=3$, $n=6$), the expression obtained here for $B_q$ [see Eq. (18)] is considerably more concise than the combination of Eqs. (9) and (10) reported previously [4,8].

An additional asset of the present results is that they allow one to explore the combined influence of dimensionality and stiffness on the quantum correction $B_q$. From Eq. (11), it follows that the ratio $B_q/n$ depends on d and n only through the combination $(d-2)/n$. This implies that, at a given temperature, the value of $B_q/n$ for a three-dimensional fluid with stiffness n is identical to that of a d-dimensional fluid ($d>3$) with an effective stiffness $n_d=(d-2)n$. In contrast, for two-dimensional fluids, $B_q/n$ is independent of n and is given by the particularly simple expression of Eq. (17).

The knowledge of $B_q$ has enabled us to derive the first quantum correction to the Boyle temperature [see Eq. (22)]. As illustrated by Figure 1 and Figure 2, the general trend is that the quantum corrections to both the second virial coefficient and the Boyle temperature become more significant as the potential stiffness increases and the system dimensionality decreases.

Although in this work we have focused on the first quantum correction to $B_2$, the same methodology can be extended to higher-order terms. For example, the general expression for the second-order correction reads [1,2]

$$\begin{array}{ccc}\hfill B^{\left(2\right)}\left(T\right)=& -{\displaystyle \frac{{\Omega }_d{\beta }^4}{24}}{\int }_0^{\infty }\mathrm{d}r{r}^{d-1}{\mathrm{e}}^{-\beta \varphi \left(r\right)}\left\{{\displaystyle \frac{1}{10}}{\left[{\displaystyle \frac{{\mathrm{d}}^2\varphi \left(r\right)}{\mathrm{d}{r}^2}}\right]}^2+{\displaystyle \frac{1}{5r^2}}{\left[{\displaystyle \frac{\mathrm{d}\varphi \left(r\right)}{\mathrm{d}r}}\right]}^2\right.\hfill & \hfill \left.+{\displaystyle \frac{\beta }{9r}}{\left[{\displaystyle \frac{\mathrm{d}\varphi \left(r\right)}{\mathrm{d}r}}\right]}^3-{\displaystyle \frac{{\beta }^2}{72}}{\left[{\displaystyle \frac{\mathrm{d}\varphi \left(r\right)}{\mathrm{d}r}}\right]}^4\right\}.\end{array}$$

Specializing to the gLJ potential, Eq. (4), and introducing the change of variable $r\to t=\sqrt{8{\beta }^{*}}{(r/\sigma )}^{-n}$, one can express $B^{\left(2\right)}$ in terms of the parabolic cylinder functions ${\mathrm{D}}_{a-2}(-\sqrt{2{\beta }^{*}})$, ${\mathrm{D}}_{a-3}(-\sqrt{2{\beta }^{*}})$, …, ${\mathrm{D}}_{a-8}(-\sqrt{2{\beta }^{*}})$, with $a={\displaystyle \frac{d-4}{n}}$. This expression can be further simplified through repeated application of Eq. (12a).

In summary, we have obtained a compact and fully explicit expression for the first quantum correction to the second virial coefficient of a d-dimensional gLJ fluid, expressed in terms of parabolic cylinder or generalized Hermite functions. The formulation unifies the treatment of dimensionality and stiffness, provides analytic access to the limiting behaviors, and naturally yields the quantum correction to the Boyle temperature. Beyond its intrinsic theoretical interest, the approach presented here provides a systematic framework for deriving higher-order quantum corrections of relevance in quantum and semiclassical fluid theory