Moment analysis in collision-induced absorption: Determination of a single parameter empirical model for the induced dipole moment of He-Ar gas mixtures

Abstract

We present a method for the construction of a one-adjustable-parameter empirical model for the induced dipole moment. The method is based on classical physics principles and relies on the first three spectral moments of the collision-induced absorption spectra at various temperatures and new interaction potentials. In this work it is applied to the spectra of He-Ar mixtures. Our values are in good agreement with the available ab initio data. The profiles calculated with these models at various temperatures are in excellent agreement with experiment.

Keywords

Induced dipole moment interatomic potentials He-Ar mixtures

1. Introduction

In a previous paper, we calculated the collision-induced light scattering (CILS) spectra of mercury (Hg) vapor at various temperatures using an empirical induced trace and anisotropy polarizability. The derivation of these quantities is based on classical physics and relies on the first three even moments of scattering [1]. Due to the observed agreement for scattering spectra of mercury, we extend this treatment to absorption by He-Ar pairs. We use the available thermophysical properties for this mixture to construct the parameters of the relevant induced dipole moment and interaction potential.

Mixtures of noble gases show an absorption band in the far infrared region of the spectrum [2, 3]. This absorption is due to the induced dipole moment arising from the deformation of the electronic clouds during the collision of two unlike atoms. As the induced dipole moment depends on the distance between the colliding atoms, the translational state of the system can change owing to the interaction of the induced dipole with the electromagnetic field, resulting in a translational absorption band. From the early observation of this phenomenon in spectra of mixtures of noble gases, substantial effort has been dedicated to their study, because the parameters involved in the construction of the interatomic potential and the induced dipole moment may be deduced from measurements of the translational band [4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

Measurements of collision-induced absorption (CIA) spectra give reliable information for the interatomic interactions. Specifically, spectral lineshapes and intensities reflect certain details of the induced dipole as function of the interatomic separation and the collision dynamics (i.e. the interatomic potential).

No accurate potential is available for He-Ar. We calculated an approximate interatomic potential, using mostly the methods outlined in a previous paper [14]. As the relevant details and references were given therein [14], we will only restate the equations when it is necessary, for the sake of continuity. To reiterate, the basic strategy in this paper is to include collision induced absorption data in addition to second pressure virial coefficients data, mixtures viscosity, diffusion, thermal conductivity coefficients and isotopic thermal factors at various temperatures, to fit the simple functional form of Barker, Fisher and Watts (BFW), modified Tang-Toennies model (MTT) and modified Morse-Spline van der Waals (MMSV) interatomic potential for the He-Ar interaction.

The CIA and thermophysical properties used in the fitting are complementary. For this mixture, the measured CIA at various temperatures is most sensitive to the repulsive part of the potential [15], while the second pressure virial coefficients reflect the size of $r_{m}$ and the volume of the attractive well [16]. The viscosity, thermal conductivity, isotopic thermal factor and diffusion data are most sensitive to the wall of the potential from $r_{m}$ inward to a point where the potential is repulsive [17].

In this paper we present a new analysis of the translational band of collision-induced absorption spectra (CIA) of He-Ar at different temperatures, based on fitting the moments of the spectral profiles of the measurements. Spectral profiles are calculated numerically with the help of a quantal computer program and compared with the measured spectra. The comparison of calculated and measured spectra provides valuable insights concerning the quality of existing models of both the induced dipole moments and the interatomic potential. The induced dipole moments forms adopted are presented in Section 2. The calculations of the different properties using the present interatomic potentials are presented in Section 3. Analysis of CIA spectral moments to determine the parameters of the induced dipole moments is given in Section 4. The theoretical method for calculating the lineshapes is briefly given in Section 5, together with the computational implementation. Results are presented and discussed in Section 6 and the concluding remarks are given in Section 7.

2. Induced dipole moment

In order to calculate the spectral line profiles of absorption and the associated zeroth, first and second moments one needs the induced dipole moment. Results with different models can be compared with experiment to assess the quality of the induced dipole moment. In this paper and for the sake of comparison and discussion, we considered seven models of the induced dipole moment which are the dispersion type [18, 19], the exponential function models [13, 20, 21], the analytical dipole moment models [11, 22], higher-order polynomial model [23], the empirical dipole model [19], the ab initio MP2 and CCSD models [24, 25].

We shall use the analytical dipole moment model below to see if the induced dipole moment can be approximated by such a simple model. Particularly in the case of the He-Ar gas mixture, for which the fundamental theory is at present of limited value, this model will be seen to provide a useful empirical basis for the description interaction induced dipole.

Over a broader range of separations [11, 22], it has been argued that an analytical dipole model like

$\displaystyle\mu(r)=\mu_{{0}}\exp\left(-\frac{(r-\sigma)}{r_{0}}\right)-\frac{% {D}_{{7}}}{{r}^{{7}}}$ (1)

should be expected to approximate the dipole moment more closely.

Suitable values for the coefficients $\mu_{{0}}$ , ${D}_{{7}}$ and ${r}_{{0}}$ were obtained via an economical least-squares routine fitting the lowest three spectral moments of absorption at different temperatures to the values computed from the theoretical moments [26]. This solution was then checked against the lineshapes.

3. The interatomic potential and multi-property analysis

The interatomic potential we provide here is obtained through the analysis of the pressure second virial coefficients [27, 28, 29, 30, 31] and a set of gaseous transport properties [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].

For the analysis of all these experimental data we consider the following potentials.

A. BFW potential

The empirical Barker, Fisher and Watts (BFW) interatomic potential [44] is represented by the following formula

$\displaystyle V(r)=\varepsilon\left(\sum\limits_{{i}={0}}^{{5}}{{A}_{{i}}{(x-1% )}^{{i}}\exp(\xi{(1-x))}}-{}\sum\limits_{{n}={3}}^{5}{\frac{{C}_{{2n}}}{{(x}^{% {2n}}+{\textit{del}})}}\right)$ (2)

where $x=r/r_{m}$ is the reduced distance, $\varepsilon$ is the potential depth, $r_{m}$ is the distance at the minimum potential and the rest are fitting parameters.

B. MTT potential

The modified Tang-Toennies model (MTT) which in the whole range of interactions can be represented by the formula [45]

$\displaystyle V(r)=Ar^{(\frac{{7}}{{2\textit{Bet}}}{-1)}}{e}^{{-2\textit{Bet }% r}}-\sum\limits_{{n}={3}}^{5}{\frac{{C}_{{2n}}{f}_{{2n}}{(br)}}{{r}^{{2n}}}}$ (3)

where ${f}_{{2n}}{(br)}$ is the appropriate damping functions, given by the expression derived by Tang and Toennies [46]

$\displaystyle{f}_{{2n}}{(br)}{=1}-{e}^{{-br}}\sum\limits_{{k}={0}}^{{2n}}{% \frac{{(br)}^{{k}}}{{k!}}}$ (4)

and

$\displaystyle b=2\textit{Bet}-\frac{\frac{{7}}{{2\textit{Bet}}}-1}{{r}}$ (5)

C. MMSV potential

In this paper we consider the updated empirical Modified-Morse-Spline-van der Waals (MMSV) potential. In region one it is represented by Modified Morse potential [47]

$\displaystyle V_{1}(r)=\varepsilon\left\{{\left(\frac{\exp(A)-\lambda}{\exp(% \textit{Ar/rm})-\lambda}\right)^{2}-2\left(\frac{\exp(A)-\lambda}{\exp(\textit% {Ar/rm})-\lambda}\right)}\right\}$ (6)

The potential $V$ is represented in region two as a single cubic Hermite spline

$\displaystyle{V}_{{2}}={\beta}_{{1}}+(x-{x}_{{1}})\{\beta_{{2}}+(x-{x}_{{2}})[% \beta_{{3}}+(x-{x}_{{1}})\beta_{{4}}]\}$ (7)

and in region three by the asymptotic London or van der Waals expansion

$\displaystyle{V}_{{3}}=-{C}_{{6}}{x}^{{-6}}-{C}_{{8}}{x}^{{-8}}-{C}_{{10}}{x}^% {{-10}}$ (8)

Given ${V}_{{1}}$ and $V_{{3}}$ , $V_{{2}}$ is completely determined by the requirements of continuity of the potential and its slope.

Even at the present (BFW) level, there are really thirteen parameters ( $\varepsilon$ , $r_{{m}}$ , ${A}_{{0}}$ , $A_{{1}}$ , $A_{{2}}$ , $A_{{3}}$ , ${A}_{{4}}$ , $A_{{5}}$ , $\xi$ , del, $C_{{6}}$ , ${C}_{{8}}$ , $C_{{10}}$ ), in the (MTT) level, there are seven free parameters ( $\varepsilon$ , $r_{{m}}$ , $A$ , Bet, ${C}_{{6}}$ , $C_{{8}}$ , $C_{{10}}$ ) and in the (MMSV) level, there are nine parameters ( $\varepsilon$ , $r_{{m}}$ , $A$ , $\lambda$ , $r_{{1}}$ , $r_{{2}}$ , $C_{{6}}$ , $C_{{8}}$ , $C_{{10}}$ ) which are far too many to determine from the present data. Accordingly we proceeded as follows: the coefficients ${A}_{{0}}$ , $A_{{1}}$ , $A$ and Bet are determined from the conditions of continuity. The dispersion coefficients ${C}_{{6}}$ , ${C}_{{8}}$ , $C_{{10}}$ were taken from theoretical calculations of Jiang [48]. leaving rest parameters in these models that were varied to fit the second pressure virial coefficients $\delta_{{B}}$ . This minimization is further supported by calculating $\delta_{\eta}$ , $\delta_{\lambda}$ , $\delta_{{D}}$ and $\delta_{\alpha}$ , the rms deviations calculated from mixture viscosity, thermal conductivity, diffusion coefficients and thermal diffusion factor, respectively. This decision leads to the potentials parameters in Table 1 as our best estimate of He-Ar interatomic potential.

Table 1

Parameters of the isotropic interatomic potentials and the associated values of $\delta$

Pots.	$\varepsilon/{k}_{{B}}$ $(K)$	${r}_{{m}}$ (Å)	$\sigma$ (Å)	$\xi$	del	${A}_{{0}}$	${A}_{{1}}$	${A}_{{2}}$	${A}_{{3}}$	${A}_{{4}}$	${A}_{{5}}$	$A$	Bet	$\lambda$	${r}_{{1}}$ ( Å)	${r}_{{2}}$ (Å)
BFW	29.8	3.492	3.0985	10.3	0.01	0.952	$-$ 3.74	$-$ 3.88	$-$ 36	229	$-$ 145	–	–	–	–	–
MTT	29.5	3.48	3.10374	–	–	–	–	–	–	–	–	13.13	1.18626	–	–	–
MMSV	29.6	3.438	3.083	–	–	–	–	–	–	–	–	6.65	–	5.0	4.97	5.68
$\delta_{{B}_{{12}}}$	$\delta_{\eta}$	$\delta_{\lambda}$	$\delta_{{D}}$	$\delta_{\textit{iso}}$	$\delta_{{o}}$
0.28	0.65	0.74	0.93	1.08	0.78
0.48	0.58	0.67	0.89	0.97	0.74
0.42	0.53	0.61	0.85	0.91	0.69

$\delta^{{a}}$ is defined by $\delta^{{a}}=\sqrt{(1/N)\sum\limits_{{j}={1}}^{{N}}\left(1/n_{{J}}\sum\limits_% {{j}={1}}^{{n}_{{j}}}\Delta_{{ji}}^{{-2}}({P}_{{ji}}-p_{{ji}})^{2}\right)}$ , where ${P}_{{ji}}$ and $p_{{ji}}$ are, respectively, the calculated and experimental values of property $j$ at point $i$ and $\Delta_{{ji}}$ is the experimental uncertainty of property $j$ at point $i$ . The subscripts ${B}_{{12}}$ , $\eta$ , $\lambda$ , $D$ , Iso and $o$ refer, respectively, to the second pressure virial coefficient, the viscosity, the thermal conductivity, the diffusion coefficient, the isotopic thermal factor and total.

In addition to the present potentials, we also considered some older empirical Lennard-Jones (12-6) [32, 49, 50], Exp-6 [37], Morse-6 Hybrid [51], ESMSV [52], Arora (m, 6, 8) [53], HFD1 [54], SPFD [54] interatomic potentials as well as the potential of Rizzo et al. [25].

3.1 Analysis of pressure second virial coefficients

An effective means for checking the validity of the different potentials parameters can be made using the second pressure virial coefficient data [27, 28, 29, 30, 31] at different temperatures. The interaction second pressure virial coefficient ${B}_{12}$ at temperature $T$ was calculated classically with the first three quantum corrections from [55]:

$\displaystyle{B}_{{12}}{(T)}={B}_{{cl}}{(T)}+\lambda{B}_{{qm,1}}{(T)}+\lambda^% {2}{B}_{{qm,2}}{(T)}+\lambda^{3}{B}_{{qm,3}}{(T)}$ (9)

where

$\displaystyle{B}_{{cl}}{(T)}=2\pi{N}_{{o}}\int\limits_{0}^{\infty}{{[1-\exp(-V% (r)/k}_{{B}}{T)]r}^{{2}}{dr}}$ (10)

and the first three quantum corrections ${B}_{{qm,1}}{(T),B}_{{qm,2}}(T)$ and $B_{{qm,3}}{(T)}$ are given in Ref. [55], with ${\lambda}=\hslash^{2}{/(24}m{k}_{{B}}{T)}$ , $\hslash=h{/2}\pi$ , $m$ and $N_{o}$ are the reduced mass and Avogadro’s number. The calculated ${B}_{{12}}$ including the first three quantum corrections were compared with the experimental results [27, 28, 29, 30, 31] shown in Fig. 1.

Figure 1.

Temperature dependence of the He-Ar gas interaction pressure second virial coefficients $B_{12}$ in cm ${}^{{3}}$ mol ${}^{{-1}}$ versus temperature in K. Comparison is made with previously available experimental results [27, 28, 29, 30, 31]. The Calculations were performed using different interatomic potentials.

3.2 Analysis of traditional transport properties

An additional check on the proposed potentials consists of the calculation of the transport properties i.e. viscosity ( $\eta_{\text{mix}}$ ), thermal conductivity ( $\lambda_{\text{mix}}$ ), diffusion coefficient ( $D_{\text{mix}}$ ), and thermal diffusion factor (Iso) at different temperatures of He-Ar mixtures. These are obtained via the formulae of Monchick et al. [56] and their comparison to the accurate experimental and theoretical results [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43] which are clear in Figs 2–5 by calculating the associated values of $\delta a$ as shown in Table 1. The agreements for this system under consideration are excellent in the whole temperature range.

Figure 2.

Deviations in per cent as a function of temperature of the mixture viscosity ( $\eta_{\text{mix}}$ ) of He-Ar gas using the present BFW potential.

Figure 3.

Deviations in per cent as a function of temperature of the mixture thermal conductivity ( $\lambda_{\text{mix}}$ ) of He-Ar using the present BFW potential.

Figure 4.

Deviations in per cent as a function of temperature of the mixture diffusion coefficients ( $D_{\text{mix}}$ ) of He-Ar using the present BFW potential.

Figure 5.

Deviations in per cent as a function of temperature of the mixture thermal diffusion factor (Iso) of He-Ar using the present BFW potential.

In this respect, according to the kinetic theory of gases at low density and the Chapman-Enskog solution of the Boltzmann equation, the transport properties can be expressed with the help of a series of collision integrals that depend on the interatomic potential energy, and are defined as [57]

$\displaystyle\theta=\pi{-2b}\int\limits_{{r}_{{0}}}^{\infty}{\frac{{dr}}{{r}^{% {2}}{(1-(b/r)}^{{2}}{-(V(r)/E))}^{{1/2}}}}$ (11) $\displaystyle Q^{(l)}(E)=2\pi\left({{1-}\frac{{1}+{(-1)}^{l}}{2(1+l)}}\right)^% {{-1}}\int\limits_{{0}}^{\infty}{(1-{\text{Cos}}^{l}\theta{)bdb}}$ (12) $\displaystyle\Omega^{(l,s)}(T)=\left({{(s}+{1)!(k}_{{B}}{T)}^{{s}+{2}}}\right)% ^{{-1}}\int\limits_{{0}}^{\infty}{Q^{(l)}{(E)\exp(-E/k}_{{B}}{T)E}^{{s}+{1}}{% dE}}$ (13)

where $\theta$ is the scattering angle, $Q^{(l)}(E)$ the transport collision integral, $b$ the impact parameter, $E$ the relative kinetic energy of colliding atoms and $r_{0}$ the closest approach of two atoms. Thus, three successive numerical integrations are required to obtain a collision integral. The reduced collision integral is defined by

$\displaystyle\Omega^{\ast(l,s)}(T)=\frac{\Omega^{(l,s)}(T)}{\pi\sigma^{2}}$ (14)

where $\sigma$ is the length scaling factor, such that $V(\sigma)=0$ .

The potential energy would serve as the input information required in calculating the collision integrals, and consequently the transport properties. Kinetic-theory expressions for the transport properties (viscosity, thermal conductivity, diffusion coefficient and thermal diffusion factor) in terms of the collision integrals for the binary gas mixtures are given by the following equations [32]:

$\displaystyle\frac{1}{\eta_{\text{mix}}}=\frac{{X}_{\eta}+{Y}_{\eta}}{1+Z_{% \eta}}$ (15)

where $\eta_{\text{mix}}$ is mixture viscosity and

$\displaystyle{X}_{\eta}=\frac{x_{1}^{2}}{\eta_{1}}+\frac{2x_{1}x_{2}}{\eta_{12% }}+\frac{x_{2}^{2}}{\eta_{2}}$ (16) $\displaystyle{Y}_{\eta}=\frac{{3}}{{5}}{A}_{{12}}^{{\ast}}\left(\frac{x_{1}^{2% }}{\eta_{1}}\left(\frac{M{}_{1}}{M_{2}}\right)+\frac{2x_{1}x_{2}}{\eta_{12}}% \left(\frac{(M{}_{1}+M_{2})^{2}}{4M_{1}M_{2}}\right)\left(\frac{\eta_{12}^{2}}% {\eta_{1}\eta_{2}}\right)+\frac{x_{2}^{2}}{\eta_{2}}\left(\frac{M{}_{2}}{M_{1}% }\right)\right)$ (17) $\displaystyle{Z}_{\eta}=\frac{{3}}{{5}}{A}_{{12}}^{{\ast}}\left(x_{1}^{2}\left% (\frac{M{}_{1}}{M_{2}}\right)+2x_{1}x_{2}\left(\frac{(M{}_{1}+M_{2})^{2}}{4M_{% 1}M_{2}}\left(\frac{\eta_{12}}{\eta_{1}}+\frac{\eta_{12}}{\eta_{2}}\right)-1% \right)+x_{2}^{2}\left(\frac{M{}_{2}}{M_{1}}\right)\right)$ (18)

where $x_{i}$ , $M_{i}$ , $\eta_{i}$ and ${T}^{{\ast}}$ are mole fractions, molecular weights, viscosity at the mixture temperature of species $i$ ( $i=$ 1, 2) and reduced temperature, respectively.

In the above expressions the interaction viscosity $\eta_{12}$ is given by

$\displaystyle\eta_{12}=\frac{{5}}{{16}}\left(\frac{{2M}_{{1}}{M}_{{2}}{k}_{{B}% }{T}}{\pi({M}_{{1}}+{M}_{{2}})}\right)^{1l2}\frac{1}{\sigma^{2}\Omega^{\ast(2,% 2)}(T^{\ast})}$ (19) $\displaystyle{A}_{{12}}^{{\ast}}=\frac{\Omega^{\ast(2,2)}({T}^{\ast})}{\Omega^% {\ast(1,1)({T}^{\ast})}}$ (20)

In addition, the binary thermal conductivity, diffusion coefficient and mixture thermal diffusion factor may be written as

$\displaystyle\lambda_{12}=\frac{{75}}{{64}}\left(\frac{{(M}_{{1}}+{M}_{{2}}{)k% }_{{B}}^{{3}}{T}}{2\pi{M}_{{1}}{M}_{{2}}}\right)^{1l2}\frac{1}{\sigma^{2}% \Omega^{\ast(2,2)}(T^{\ast})}$ (21) $\displaystyle D_{\text{mix}}=\frac{{3}}{{8}}\left(\frac{{(M}_{{1}}+{M}_{{2}}{)% k}_{{B}}^{{3}}{T}^{{3}}}{2\pi{M}_{{1}}{M}_{{2}}}\right)^{1l2}\frac{(1+\Delta)}% {P\sigma^{2}\Omega^{\ast(1,1)}(T^{\ast})}$ (22) $\displaystyle\text{Iso}={(6C}_{{12}}^{{\ast}}-5)\left(\frac{{x}_{{1}}{S}_{{1}}% -{x}_{{2}}{S}_{{2}}}{{x}_{{1}}^{{2}}{Q}_{{1}}+{x}_{{2}}^{{2}}{Q}_{{2}}+{x}_{{1% }}{x}_{{2}}{Q}_{{12}}}\right)(1+K_{2})$ (23)

with the pressure $P$ and the temperature $T$ . The expressions for the second-order correction $\Delta$ as well as the values of ${C}_{{12}}^{{\ast}}$ , $S$ and $Q$ are listed in Appendix C of Ref. [32]. The correction term $K_{2}$ is small in magnitude and may be negligible in the present calculations.

4. Theory of lineshapes of collision-induced absorption

In this section the quantum mechanical calculations for collision-induced absorption (CIA) are described. The atomic wavefunctions, which enter the computation of the matrix elements, are obtained by numerical integration of the radial Schrödinger equation [58] using the energy density normalization.

Collision-induced absorption spectra (CIA) can be computed from quantum mechanical theory if the interaction potential is known along with a suitable model of the collision-induced dipole moment [59]. The absorption coefficient $\alpha(\omega)$ is related to the product of volume $v$ and the so-called spectral function, $G(\omega)$ , according to [59]

$\displaystyle\alpha{(}\omega{)=}\frac{{4}\pi^{{2}}}{3\hslash c}{n}_{{1}}{n}_{{% 2}}\omega(1-\exp(-\hslash\omega/{k}_{{B}}{T})){vG}(\omega)$ (24)

Here, $\omega$ designates angular frequency; $\hslash$ is the Planck’s constant; $c$ is the speed of light in vacuum; $n_{1}$ and $n_{2}$ are the number densities of the mixed gases; and $vG(\omega)$ is the spectral density, which may be calculated if the induced dipole $\mu(r)$ and potential functions are given. The spectral moments are defined as

$\displaystyle{M}_{{n}}{=}\int\limits_{{-}\infty}^{\infty}{\omega^{{n}}{vG}(% \omega){d}\omega}$ (25)

for $n=$ 0, 1, 2, …. These moments can be compared with values calculated directly from the sum rules [26], where quantum corrections were made for the pair distribution functions $g(r)$ , $g_{e}(r)$ and $g_{m}(r)$ of the first three spectral moments of absorption [60].

It is often inconvenient to use tabular data in spectral moments and line shape computations. Therefore, we have obtained an analytical model of the exchange or overlap dipole in the range of interest (near 5.98 bohr) by a least mean squares fit. It is of the form Eq. (1).

Range and strength are the parameters determined by the analysis, with the results 0.0039 $\leqslant\mu_{0}\leqslant$ 0.0048 a.u., and 0.0346 $\leqslant{r}_{{0}}\leqslant$ 0.0359 nm for He-Ar gas mixture.

As a first step and for inert gas mixtures, we used the exponential dipole, Eq. (1) with $D_{7}=$ 0 and the parameters given above. The star curve in Fig. 1. Gives the calculated line shape for comparison with the measurement [5] for He-Ar mixtures. The agreement is less than perfect. Near the peak of absorption, the calculated absorption is too weak and the high frequency wing is not well represented at all, in spite of the fact that the lowest three spectral moments of the measurements agree very closely with those of the computed line shape. Therefore, we add a dispersion part $D_{7}>$ 0 to the exponential dipole Eq. (1).

Figure 6.

Comparison between the calculated translational collision-induced absorption spectra of He-Ar at T $=$ 295 K using the present BFW interatomic potential with the parameters given in Table 1 and different models of the induced dipole moment with the experimental one.

Since an accurate determination of these spectral integrals requires knowledge of the absorption coefficient $\alpha(\omega)$ at low and high frequencies, which is not available, it is best to approximate the spectral function $vG(\omega)$ by an analytical model profile with three parameters, the so-called $K_{0}$ model [61], which was chosen to provide a remarkably close representation of virtually all line shapes arising from exchange and dispersion force induction. These parameters have been determined by fitting the experimental spectrum, using a least mean squares procedure. From the fit the three lowest spectral moments of the measurements at different temperatures for the considered mixture are readily obtained with 0.0051 $\leqslant\mu_{0}\leqslant$ 0.0064 a.u., 0.0345 $\leqslant{r}_{{0}}\leqslant$ 0.0366 nm and $-$ 268 $\leqslant{D}_{{7}}\leqslant-$ 315 a.u.

5. Analysis of CIA spectral moments to determine

{\mu}(r)

The method of detailed analysis of the first three even moments of the polarized and depolarized light scattering spectrum (CILS) has been used by Meinander et al. [62] and El-Kader [1, 63, 64, 65, 66, 67, 68] for the determination of the extra-dipole-induced dipole (DID) contribution to the pair-polarizability trace and anisotropy of $\text{CH}_{{4}}$ , inert gases and mercury. This consists of establishing an appropriate parameterized model form for the trace and the anisotropy, subsequently then searching for the sets of parameters that are consistent with the experimental values of the polarized and depolarized spectral moments.

In order to proceed, it is convenient to rewrite the induced dipole moment ${\mu(r)}$ of the Eq. (1) in terms of the reduced variable ${x}={r/r}_{{m}}$ where ${r}_{{m}}$ is the separation at the minimum of the interatomic potential $V(r)$ . In this case one has:

$\displaystyle\mu{(x)=}\mu_{{0}}\exp\left(-\left(\frac{{x-}\sigma^{{\ast}}}{{x}% _{{0}}}\right)\right)-\frac{{D}_{{7}}^{{\ast}}}{{x}^{{6}}}$ (26)

where $\sigma^{\ast}=\sigma/{r}_{{m}}$ ; ${x}_{{0}}={r}_{{0}}/{r}_{{m}}$ and ${D}_{{7}}^{{\ast}}={D}_{{7}}/{r}_{{m}}^{{7}}$ .

Substituting Eq. (26) into the moment expressions of absorption spectrum [26] make it feasible to rewrite them in the form of quadratic equations for the unknown $\mu_{0}$ and $D_{{7}}$ with coefficients as parametric functions of ${r}_{{0}}$ . The equations obtained from these moments are of the form:

$\displaystyle D{(}X{)}=\left|{\begin{array}[]{l}\frac{\partial{M}_{{0}}}{% \partial\mu_{{0}}}{}\frac{\partial{M}_{{0}}}{\partial{r}_{{0}}}{}\frac{% \partial{M}_{{0}}}{\partial{D}_{{7}}}\\ \\ \frac{\partial{M}_{{1}}}{\partial\mu_{{0}}}{}\frac{\partial{M}_{{1}}}{\partial% {r}_{{0}}}{}\frac{\partial{M}_{{1}}}{\partial{D}_{{7}}}\\ \\ \frac{\partial{M}_{{2}}}{\partial\mu_{{0}}}{}\frac{\partial{M}_{{2}}}{\partial% {r}_{{0}}}{}\frac{\partial{M}_{{2}}}{\partial{D}_{{7}}}\\ \end{array}}\right|$ (27) $\displaystyle X=\left({\begin{array}[]{l}\mu_{{0}}\\ {r}_{{0}}\\ {D}_{{7}}\\ \end{array}}\right)$ (28) $\displaystyle F{(}X{)}=\left({\begin{array}[]{l}\Delta{M}_{{0}}\\ \Delta{M}_{{1}}\\ \Delta{M}_{{2}}\\ \end{array}}\right)$ (29)

where $\Delta{M}_{{n}}={M}_{{n}}-{M}_{{n,\exp}}(n=0,1,2)$ , and $M_{{n}}$ , ${M}_{{n,\exp}}$ are the calculated and experimental spectral moments, respectively.

$\displaystyle X_{{i}+{1}}=X_{{i}}-D^{-1}\left|{{}_{{x}_{{i}}}F{(}X_{i}{)}}\right.$ (30)

at any step $i$ of the iteration procedure. Once convergence is obtained, the column vector solution $X$ has to satisfy $F(X)=0$ . For the specific functional form of $\mu{(r)}$ given by Eq. (1), the nine elements of the matrix $D$ are determined and given in the Appendix.

Table 2

Comparison of experimental and theoretical spectral moments

Mixture	T (K)	$M_{{0}}$ *10 ${}^{{62}}$ (erg cm ${}^{{6}}$ )		$M_{{1}}$ *10 ${}^{{49}}$ (erg cm ${}^{{6}}$ /s)		$M_{{2}}$ *10 ${}^{{35}}$ (erg cm ${}^{{6}}$ /s ${}^{{2}}$ )
		Cal.	Exp.	Cal.	Exp.	Cal.	Exp.
He-Ar	140	(0.99) ${}^{\text{a}}$ , (1.153) ${}^{\text{b}}$ , (0.858) ${}^{\text{c}}$ , (1.09) ${}^{\text{d}}$ , (1.136) ${}^{\text{e}}$ , (0.939) ${}^{\text{f}}$ , (0.99) ${}^{\text{g}}$	0.94 [10]	(0.715) ${}^{\text{a}}$ , (0.880) ${}^{\text{b}}$ , (0.762) ${}^{\text{c}}$ , (0.839) ${}^{\text{d}}$ , (0.872) ${}^{\text{e}}$ , (0.745) ${}^{\text{f}}$ , (0.765) ${}^{\text{g}}$	0.74 [10]	(0.361) ${}^{\text{a}}$ , (0.447) ${}^{\text{b}}$ , (0.401) ${}^{\text{c}}$ , (0.423) ${}^{\text{d}}$ , (0.473) ${}^{\text{e}}$ , (0.373) ${}^{\text{f}}$ , (0.39) ${}^{\text{g}}$	–
	165	(1.113) ${}^{\text{a}}$ , (1.293) ${}^{\text{b}}$ , (0.988) ${}^{\text{c}}$ , (1.24) ${}^{\text{d}}$ , (1.245) ${}^{\text{e}}$ , (1.07) ${}^{\text{f}}$ , (1.12) ${}^{\text{g}}$	1.18 [10]	(0.805) ${}^{\text{a}}$ , (0.975) ${}^{\text{b}}$ , (0.854) ${}^{\text{c}}$ , (0.946) ${}^{\text{d}}$ , (0.952) ${}^{\text{e}}$ , (0.837) ${}^{\text{f}}$ , (0.857) ${}^{\text{g}}$	0.88 [10]	(0.459) ${}^{\text{a}}$ , (0.557) ${}^{\text{b}}$ , (0.502) ${}^{\text{c}}$ , (0.538) ${}^{\text{d}}$ , (0.577) ${}^{\text{e}}$ , (0.472) ${}^{\text{f}}$ , (0.492) ${}^{\text{g}}$	–
	200	(1.29) ${}^{\text{a}}$ , (1.49) ${}^{\text{b}}$ , (1.174) ${}^{\text{c}}$ , (1.45) ${}^{\text{d}}$ , (1.396) ${}^{\text{e}}$ , (1.26) ${}^{\text{f}}$ , (1.295) ${}^{\text{g}}$	1.3 [10]	(0.934) ${}^{\text{a}}$ , (1.108) ${}^{\text{b}}$ (0.984) ${}^{\text{c}}$ , (1.1) ${}^{\text{d}}$ , (1.063) ${}^{\text{e}}$ , (0.967) ${}^{\text{f}}$ , (0.987) ${}^{\text{g}}$	1.02 [10]	(0.619) ${}^{\text{a}}$ , (0.731) ${}^{\text{b}}$ , (0.619) ${}^{\text{c}}$ , (0.726) ${}^{\text{d}}$ (0.7397) ${}^{\text{e}}$ , (0.632) ${}^{\text{f}}$ , (0.657) ${}^{\text{g}}$	–
	240	(1.499) ${}^{\text{a}}$ , (1.72) ${}^{\text{b}}$ , (1.392) ${}^{\text{c}}$ , (1.7) ${}^{\text{d}}$ , (1.565) ${}^{\text{e}}$ , (1.481) ${}^{\text{f}}$ , (1.504) ${}^{\text{g}}$	1.67 [10]	(1.09) ${}^{\text{a}}$ , (1.26) ${}^{\text{b}}$ , (1.125) ${}^{\text{c}}$ , (1.28) ${}^{\text{d}}$ , (1.19) ${}^{\text{e}}$ , (1.12) ${}^{\text{f}}$ , (1.14) ${}^{\text{g}}$	1.18 [10]	(0.834) ${}^{\text{a}}$ , (0.959) ${}^{\text{b}}$ , (0.869) ${}^{\text{c}}$ , (0.978) ${}^{\text{d}}$ , (0.948) ${}^{\text{e}}$ , (0.846) ${}^{\text{f}}$ , (0.876) ${}^{\text{g}}$	–
	295	(1.79) ${}^{\text{a}}$ , (2.03) ${}^{\text{b}}$ , (1.695) ${}^{\text{c}}$ , (2.048) ${}^{\text{d}}$ , (1.792) ${}^{\text{e}}$ , (1.794) ${}^{\text{f}}$ , (1.795) ${}^{\text{g}}$	1.7 [5]	(1.299) ${}^{\text{a}}$ , (1.46) ${}^{\text{b}}$ , (1.31) ${}^{\text{c}}$ , (1.53) ${}^{\text{d}}$ , (1.352) ${}^{\text{e}}$ , (1.327) ${}^{\text{f}}$ , (1.349) ${}^{\text{g}}$	1.32 [5]	(1.185) ${}^{\text{a}}$ , (1.32) ${}^{\text{b}}$ , (1.193) ${}^{\text{c}}$ , (1.39) ${}^{\text{d}}$ , (1.273) ${}^{\text{e}}$ , (1.19) ${}^{\text{f}}$ , (1.23) ${}^{\text{g}}$	1.17 [5]
	480	(2.815) ${}^{\text{a}}$ , (3.08) ${}^{\text{b}}$ , (2.712) ${}^{\text{c}}$ , (3.29) ${}^{\text{d}}$ , (2.513) ${}^{\text{e}}$ , (2.886) ${}^{\text{f}}$ , (2.8) ${}^{\text{g}}$	3.5 [11]	(2.046) ${}^{\text{a}}$ , (2.13) ${}^{\text{b}}$ , (1.87) ${}^{\text{c}}$ , (2.42) ${}^{\text{d}}$ , (1.877) ${}^{\text{e}}$ , (2.03) ${}^{\text{f}}$ , (2.07) ${}^{\text{g}}$	4.31 [11]	(2.859) ${}^{\text{a}}$ , (2.95) ${}^{\text{b}}$ , (2.567) ${}^{\text{c}}$ , (3.37) ${}^{\text{d}}$ , (2.67) ${}^{\text{e}}$ , (2.80) ${}^{\text{f}}$ , (2.89) ${}^{\text{g}}$	–

${}^{\text{a}}$ Values calculated using the present MTT interatomic potential and induced dipole moment Eq. (1) with $D_{7}=$ 0. ${}^{\text{b}}$ Values calculated using SPFD interatomic potential [54] and induced dipole moment Eq. (1). ${}^{\text{c}}$ Values calculated using M3SV interatomic potential [69] and induced dipole moment Eq. (1). ${}^{\text{d}}$ Values calculated using CCSD(T) interatomic potential [25] and induced dipole moment Eq. (1). ${}^{\text{e}}$ Values calculated using the present BFW interatomic potential and induced dipole moment Eq. (1). ${}^{\text{f}}$ Values calculated using the present MTT interatomic potential and induced dipole moment Eq. (1). ${}^{\text{g}}$ Values calculated using the present MMSV interatomic potential and induced dipole moment Eq. (1).

Figure 7.

Empirical induced moment $\mu(r)$ of He-Ar with Ab initio models [24, 25] and the theoretical one [12].

The advantage of this method to obtain $\mu{(r)}$ is the speed of computation and the avoidance of the trial-and-error approach. The method of moments analysis has been applied to the collision-induced light scattering (CILS) spectra of inert gases, mercury and methane [1, 62, 63, 64, 65, 66, 67, 68]. Because the correction to the first-order DID expressions is quite small over most of the interatomic separations probed by the atomic motions, and the two correction terms cancel out to a large extent the effects of each other, the numerical values derived for the parameters $\mu_{{0}}$ and ${D}_{{7}}$ of the induced dipole moment, are quite sensitive to the input values of the moments.

The empirical pair potentials BFW, MTT and MMSV with the different parameters are given in Table 1 for the considered mixture. Table 2 gives the numerical results of the search while Fig. 7 shows the different regions of the dipole moment behavior which are consistent with the experimental results. Here the remaining theoretical and the ab initio induced dipole models [24, 25] are reported as well.

The moment analysis program was easily modified to calculate the moments of the absorption spectra as functions of $\mu_{{0}}$ and ${D}_{{7}}$ using the expression (1). The experimentally determined values of the moments, with error limits, now each define a range of acceptable values for the parameters. This approach to the construction of an empirical models for $\mu{(r)}$ is acceptable if all moments define a common range of $\mu_{{0}}$ , ${D}_{{7}}$ and ${r}_{{0}}$ values. As one can see from Table 2 and Fig. 7, the agreement between the experimental values and the theoretical ones using our empirical potential is excellent and we have verified that it remains acceptable for $\mu_{0}=$ 0.006 a.u., ${r}_{{0}}=$ 0.03576 nm and ${D}_{{7}}=-$ 290 a.u.

6. Results

In this section, we present the results of the analysis for noble gas mixtures and the empirical models for $\mu{(r)}$ are compared to the ab initio results of Maroulis et al. [24] and Cacheiro et al. [25]. Good agreement between the ab initio models and our empirical ones is generally observed. The profiles and relevant spectral moments obtained via our models for the dipole moment agree with the measurements. Figure 6 and Table 2 display the quantum profiles of the collision-induced light absorption spectra and their moments at different temperatures using the empirical models of the induced dipole moment compared with experiments [5, 10, 11].

Summarizing our analysis we find that He-Ar inert gas mixtures develop an incremental induced dipole moment during collisions, besides the exponential one, which contributes substantially at intermediate-range distances and can be ascribed to other mechanisms of electron cloud distortion, such as overlap and electron-correlation effects.

7. Conclusion

We have developed a model for the induced dipole moment $\mu{(r)}$ with adjustable parameters fitted to the spectral profiles of collision-induced absorption at various temperatures using quantum mechanics and to the first few spectral moments of the measured spectra using classic mechanics.

It is clear that the two methods are complementary and could be used jointly to reduce the computational cost and allow the maximum of information to be extracted from measurements.

The present study further demonstrate that the present BFW, MTT and MMSV models yield reliable approximations of the interatomic potential of He-Ar gas mixtures. The treatment proposed in this study represents an improvement on the model from thermodynamical and transport properties over a wider temperature range as in the cases of Ar-Kr [70] and Ne-Kr [71] mixtures. Also, it is interesting to note that the empirical model derived for the induced dipole agrees reasonably well with the ab initio results of Cacheiro et al. and Maroulis et al. for this mixture and produces lineshapes in good agreement with experiment.

Footnotes

Acknowledgments

We are grateful to Dr. J. Borysow, Dr. L. Frommhold and Dr. G. Birnbaum for making available their published Fortran code with the different results of the collision induced absorption (CIA) for various systems.

Appendix

For the specific functional form of the induced dipole moment $\mu{(r)}$ given by Eq. (1), the nine functions ${D}_{{ij}}(X)$ are

$\displaystyle D_{{11}}{(}\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=8\pi\int\limits_{0}^% {\infty}{\mu{(r)\exp(-(r-}\sigma{)/r}_{{0}})}{g(r)r}^{{2}}{dr}$ $\displaystyle D_{{12}}{(}\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=8\pi\int\limits_{0}^% {\infty}{\frac{\mu_{{0}}}{{r}_{{0}}^{{2}}}\mu{(r)\exp(-(r-}\sigma{)/r}_{{0}}){% g(r)r}^{{3}}{dr}}$ $\displaystyle D_{{13}}{(}\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=-8\pi\int\limits_{{0% }}^{\infty}{\frac{{1}}{{r}^{{5}}}\mu{(r)}}{g(r)dr}$ $\displaystyle D_{{21}}{(}\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=\frac{-4\pi\hslash}{% {m}}\int\limits_{0}^{\infty}{\left({\frac{\mu{{}^{\prime}(r)}}{{r}_{{0}}}-% \frac{2\mu{(r)}}{{r}^{{2}}}}\right){\exp(-(r-}\sigma{)/r}_{{0}})}{g(r)r}^{{2}}% {dr}$ $\displaystyle D_{{22}}{(}\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=\frac{4\pi\hslash}{{% m}{r}_{{0}}^{{2}}}\mu_{0}\int\limits_{0}^{\infty}{\left({-\left(\frac{{r}}{{r}% _{{0}}}-{1}\right)\mu{{}^{\prime}(r)}+\frac{2\mu{(r)}}{{r}}}\right){\exp(-(r-}% \sigma{)/r}_{{0}})}{g(r)r}^{{2}}{dr}$ $\displaystyle D_{{23}}{(}\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=\frac{2\pi\hslash}{{% m}}\int\limits_{0}^{\infty}\left(\mu^{\prime}(r)\left(\left(\frac{{14}}{{r}^{{% 8}}}\right)+4\frac{\sigma^{2}}{{r}^{10}}\right)-\frac{4\mu{(r)}}{{r}^{{9}}}% \right){g(r)r}^{{2}}{dr}$ $\displaystyle D_{{31}}{(}\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=\frac{4\pi\hslash^{2% }}{{m}^{{2}}}\int\limits_{0}^{\infty}\left(-\frac{\mu^{\prime\prime\prime% \prime}{(r)}}{{4}}+\frac{\mu^{\prime\prime\prime}{(r)}}{{2r}_{{0}}}+\mu^{% \prime}{(r)}\left(\frac{1}{{2r}_{{0}}^{{3}}}-\frac{2}{{r}_{{0}}{r}^{{2}}}-% \frac{2}{{r}^{{3}}}\right)\right.$ $\displaystyle\quad{}+\mu{(r)}\left.\left(\frac{-1}{4{r}_{{0}}^{{4}}}+\frac{2}{% {r}_{{0}}{r}^{{3}}}+\frac{2}{{r}^{{4}}}\right)\right)\exp(-(r-\sigma)/r_{{0}})% g(r)r^{{2}}{dr}$ $\displaystyle\quad{}+\frac{4\pi}{{m}}\int\limits_{{0}}^{\infty}\left(2\mu^{% \prime\prime}{(r)V(r)}+\mu^{\prime}(r)V^{\prime}(r)+\frac{\mu{(r)}}{{r}_{{0}}}% \left(-{V^{\prime}(r)}+\frac{2{V(r)}}{{r}_{{0}}}\right)\right)$ $\displaystyle\quad{}\exp(-(r-\sigma)/r_{{0}})g(r)r^{{2}}{dr}-\frac{8\pi}{{m}}% \int\limits_{0}^{\infty}\left(\mu^{\prime\prime}{(r)}+\frac{\mu{(r)}}{{r}_{{0}% }^{{2}}}\right)\exp(-(r-\sigma{)/r}_{{0}}){g}_{{e}}{(r)r}^{{2}}{dr}$ $\displaystyle\quad{}+\frac{{4}\pi}{{m}^{{2}}}\int\limits_{0}^{\infty}\left(% \frac{\mu^{\prime\prime}{(r)}}{{r}^{{2}}}-\frac{\mu^{\prime}{(r)}}{{r}^{{3}}}+% \frac{\mu{(r)}}{{r}^{{2}}}\left(\frac{2}{{r}^{{2}}}+\frac{1}{{r}_{{0}}^{{2}}}+% \frac{1}{{rr}_{{0}}}\right)\right)\exp(-(r-\sigma)/r_{{0}}){g}_{{m}}{(r)r}^{{2% }}{dr}$ $\displaystyle D_{{32}}(\mu_{{0}}{,r}_{{0}}{,D}_{{7}}{)}=\frac{4\pi\hslash^{2}}% {{m}^{{2}}}\int\limits_{0}^{\infty}\left(-\frac{{r}\mu_{0}\mu^{\prime\prime% \prime\prime}{(r)}}{{4r}_{{0}}^{{2}}}+\frac{\mu_{0}\mu^{\prime\prime\prime}{(r% )}}{{2r}_{{0}}^{{2}}}\left(\frac{{r}}{{r}_{{0}}}{-1}\right)+\frac{\mu_{0}\mu^{% \prime}{(r)}}{{r}_{{0}}^{{2}}}\right.$ $\displaystyle\quad\left.\left(\frac{{r}}{{2r}_{{0}}^{{3}}}-\frac{3}{{2r}_{{0}}% ^{{2}}}-\frac{2}{{rr}_{{0}}}\right)+\frac{\mu_{0}\mu{(r)}}{{r}_{{0}}^{{3}}}% \left(\frac{-{r}}{4{r}_{{0}}^{{3}}}+\frac{1}{{r}_{{0}}^{{2}}}+\frac{2}{{r}^{{2% }}}\right)\right)\exp(-(r-\sigma{)/r}_{{0}}){g(r)r}^{{2}}{dr}$ $\displaystyle\quad{}+\frac{4\pi}{{m}}\int\limits_{{0}}^{\infty}\left(\frac{2% \mu_{0}\mu^{\prime\prime}(r)V(r)r}{{r}_{{0}}^{{2}}}+\mu_{0}\mu^{\prime}(r)V^{% \prime}(r)\frac{{r}}{{r}_{{0}}^{{2}}}+\frac{\mu_{0}\mu{(r)}}{{r}_{{0}}^{{2}}}% \left(-\frac{{r}}{{r}_{{0}}}V^{\prime}(r)+V^{\prime}(r)+\frac{2rV(r)}{{r}_{{0}% }^{{2}}}\right.\right.$ $\displaystyle\quad{}-4\left.\left.\frac{{V(r)}}{{r}_{{0}}}\right)\right)\exp(-% (r-\sigma)/r_{{0}})g(r)r^{{2}}{dr}-\frac{8\pi}{{m}}\int\limits_{0}^{\infty}% \left(\frac{{r}}{{r}_{{0}}^{{2}}}\mu_{0}\mu^{\prime\prime}(r)+\frac{\mu_{0}\mu% {(r)}}{{r}_{{0}}^{{3}}}\left(-2+\frac{{r}}{{r}_{{0}}}\right)\right)$ $\displaystyle\quad\exp(-(r-\sigma)/r_{{0}}){g}_{{e}}(r)r^{{2}}{dr}+\frac{{4}% \pi}{{m}^{{2}}}\int\limits_{0}^{\infty}\frac{\mu_{0}\mu^{\prime\prime}{(r)}}{{% rr}_{{0}}^{{2}}}-\frac{\mu_{0}\mu^{\prime}(r)}{{r}^{{2}}{r}_{{0}}^{{2}}}+\frac% {\mu_{0}\mu{(r)}}{{rr}_{{0}}^{{2}}}\left(\frac{1}{{r}_{{0}}^{{2}}}-\frac{1}{{% rr}_{{0}}}+\frac{1}{{r}^{{2}}}\right)$ $\displaystyle\quad{}\exp(-(r-\sigma)/r_{{0}})g_{{m}}(r)r^{{2}}{dr}$

and

$\displaystyle D_{{33}}(\mu_{{0}},r_{{0}},D_{{7}})=\frac{4\pi\hslash^{2}}{{m}^{% {2}}}\int\limits_{0}^{\infty}\left(\frac{\mu^{\prime\prime\prime\prime}{(r)}}{% {4r}^{{7}}}-\frac{\mu^{\prime\prime\prime}{(r)}}{{2}}\left(\frac{7}{{r}^{{8}}}% +\frac{2\sigma^{2}}{{r}^{10}}\right)-\frac{\mu^{\prime}{(r)}}{{r}^{{10}}}\right.$ $\displaystyle\quad{}\left.\left(236+\frac{239\sigma^{2}}{{r}^{2}}+\frac{120% \sigma^{4}}{{r}^{4}}\right)+\frac{\mu{(r)}}{{r}^{{11}}}\left(1244+\frac{{1706}% \sigma^{{2}}}{{r}^{2}}+\frac{{1326}\sigma^{{4}}}{{r}^{{4}}}\right)\right)g(r)r% ^{{2}}{dr}$ $\displaystyle\quad{}+\frac{4\pi}{{m}}\int\limits_{{0}}^{\infty}\left(-\frac{2% \mu{}^{\prime\prime}(r)V(r)}{{r}^{{7}}}-\frac{\mu^{\prime}(r)V^{\prime}(r)}{{r% }^{{7}}}+\frac{\mu(r)}{{r}^{{8}}}\left(7V^{\prime}(r)+\frac{{2}\sigma^{2}}{{r}% ^{{2}}}V^{\prime}(r)-\frac{112{V(r)}}{{r}}\right.\right.$ $\displaystyle\quad{}\left.\left.-\frac{{4}\sigma^{2}V(r)}{{r}^{{3}}}\left(17+4% \left(\frac{\sigma}{{r}}\right)^{2}\right)\right)\right)g(r)r^{{2}}dr-\frac{8% \pi}{{m}}\int\limits_{0}^{\infty}\left(-\frac{\mu^{\prime\prime}{(r)}}{{r}^{{7% }}}-\frac{2\mu(r)}{{r}^{{9}}}\right.$ $\displaystyle\quad{}\left.\left(28+\frac{\sigma^{2}}{{r}^{{2}}}\left(17+4\left% (\frac{\sigma}{{r}}\right)^{2}\right)\right)\right){g}_{{e}}(r)r^{{2}}dr+\frac% {{4}\pi}{{m}^{{2}}}\int\limits_{0}^{\infty}\left(-\frac{\mu^{\prime\prime}{(r)% }}{{r}^{{9}}}+\frac{\mu^{\prime}{(r)}}{{r}^{{10}}}-\frac{\mu{(r)}}{{r}^{{11}}}\right.$ $\displaystyle\quad{}\left.\left(65+\frac{2\sigma^{2}}{{r}^{{2}}}\left(18+4% \left(\frac{\sigma}{{r}}\right)^{2}\right)\right)\right){g}_{{m}}(r)r^{{2}}dr$

References

El-Kader

M.S.A.

and Phys

, D: Applied Physics, 32 (1999), 869.

Kiss

Z.J.

and Welsh

H.L.

, Phys.Rev.Lett., 2 (1959), 166.

Buontempo

Cunsolo

Dore

and Maselli

, Intermolecular spectroscopy and dynamical properties of dense system, LXXV Corso, Soc., Italiana di Fisica, Bologna, Italy, 1980, 211.

Marteau

P.H.

Obriot

and Fondere

, Can.J.Phys., 64 (1986), 822.

Bosomworth

D.R.

and Gush

H.P.

, Can.J.Phys., 43 (1965), 751.

Quazza

, JQSRT, 16 (1976), 491.

Birnbaum

Krauss

and Frommhold

, J.Chem.Phys., 80 (1984), 2669.

Birnbaum

Brown

M.S.

and Frommhold

, Can.J.Phys., 59 (1981), 1544.

Frommhold

and Tonkov

M.V.

, Phys.Rev.A, 54 (1996), 5438.

10.

Bukhtoyarova

V.J.

and Tonkov

M.V.

, OS, 43 (1977), 27.

11.

Bar-Ziv

and Weiss

, J.Chem.Phys., 64 (1976), 2412.

12.

Birnbaum

Brown

M.S.

and Frommhold

, Can.J.Phys., 59 (1981), 1544.

13.

Dore

Finzi

Nucara

Postorino

and Rovere

, Mol.Phys., 84 (1995), 1065.

14.

El-Kader

M.S.A.

and Bancewicz

, Mol.Phys., 109 (2011), 457.

15.

Futrelle

R.P.

, Phys.Rev.Lett., 9 (1967), 479.

16.

Maitland

G.C.

Rigby

Smith

E.B.

and Wakeham

W.A.

, Intermolecular Forces-Their Origin and Determination, (Clarendon, Oxford), 1981.

17.

Maitland

G.C.

and Wakeham

W.A.

, Direct determination of intermolecular potentials from gaseous transport coefficients alone, Mol.Phys. 35 (1978), 1443.

18.

A.D. Buckingham Intermolecular Forces, Adv.Chem.Phys., 12 (1967), 107.

19.

Wishnant

D.M.

and Browns

W.B.

, Mol.Phys., 26 (1973), 1105.

20.

Matcha

R.L.

and Nesbet

R.K.

, Phys.Rev., 160 (1967), 72.

21.

Lacey

A.J.

and Brown

W.B.

, Mol.Phys., 27 (1974), 1013.

22.

Dacre

P.D.

and McWeeny

R.M.

, Proc.R.Soc.A, 317 (1970), 435.

23.

Meyer

and Frommhold

, Phys.Rev.A, 33 (1986), 3807.

24.

Maroulis

Haskopoulos

Glaz

Bancewicz

and Godet

J.L.

, Chem.Phys.Lett., 428 (2006), 28.

25.

Cacheiro

J.L.

Fernándoz

Marchesan

Coriani

Hättig

and Rizzo

, Mol.Phys., 102 (2004), 101.

26.

Moraldi

Borysow

and Frommhold

, Chem.Phys., 86 (1984), 339.

27.

Dymond

J.H.

Marsh

K.N.

and Wilhoit

R.C.

, Virial coefficients of pure gases and mixtures, Edited by M. Fenkel and K.N. Marsh, Landolt-Bornstein – Group IV Physical Chemistry, 21, Springer-Verlag, Heidelberg, 2003.

28.

Martin

M.L.

Trengove

R.D.

Harris

K.R.

and Dunlop

P.J.

, Aus.J.Chem., 35 (1982), 1525.

29.

Blancett

A.L.

Hall

K.R.

and Canfield

F.B.

, Physica, 47 (1970), 75.

30.

Brewer

and Vaughn

G.W.

, J.Chem.Phys., 50 (1969), 2960.

31.

Kalfoglou

N.K.

and Miller

J.G.

, J.Phys.Chem., 71 (1967), 1256.

32.

Kestin

Knierim

Mason

E.A.

Najafi

S.T.

and Waldman

, J.Phys.Chem.Ref.Data, 13(1) (1984), 229.

33.

Thornton

and Baker

W.A.D.

, Viscosity and thermal conductivity of binary gas mixtures: Argon-Neon, Argon-Helium, and Neon-Helium, Proceedings of the Physical Society 80 (1962), 1171–1175.

34.

Kalelkar

A.S.

and Kestin

, J.Chem.Phys., 52 (1970), 4248.

35.

Felix Sharipov, and Victor J. Benites, J.Chem.Phys., 143 (2015), 154104.

36.

Taylor

W.L.

and Cain

, J.Chem.Phys., 78 (1983), 6220.

37.

Srivastava

K.P.

, J.Chem.Phys., 28 (1958), 543.

38.

Vandael

and Cauwenbergh

, Physica, 40 (1968), 173.

39.

Liner

J.C.

and Stanley Weissman, J.Chem.Phys., 56 (1972), 2288.

40.

Hogervorst

, Physica, 51 (1971), 59.

41.

Taylor

W.L.

, 72 (1980), 4973.

42.

Trengove

R.D.

and Dunlop

P.J.

, Ber. Bunsenges. Phys. Chem., 87 (1983), 874.

43.

Trengove

R.D.

Harris

K.R.

Robjohns

H.L.

and Dunlop

P.J.

, Physica 131 A, 1985, 506.

44.

Barker

J.A.

Fisher

R.A.

and Watts

R.O.

, Mol.Phys., 21 (1971), 657.

45.

Peng

J.F.

Ren

Qiao

L.W.

and Tang

K.T.

, J.At.Mol.Sci., 2 (2011), 289.

46.

Tang

K.T.

and Toennies

J.P.

, J. Chem. Phys., 80 (1984), 3726.

47.

Lim

T.C.

, J.Math.Chem., 47 (2010), 984.

48.

Jiang

Mitroy

Cheng

and Bromley

M.W.J.

, At. Data Nucl. Data Tables, 101 (2015), 158.

49.

Vandael

and Cauwenbergh

, Physica, 40 (1968), 173.

50.

Liner

J.C.

and Stanley Weissman, J.Chem.Phys., 56 (1972), 2288.

51.

Konowalow

D.D.

and Zakheim

D.S.

, J.Chem.Phys., 57 (1972), 4375.

52.

Chen

C.H.

Siska

P.E.

and Lee

Y.T.

, J.Chem.Phys., 59 (1973), 601.

53.

Arora

P.S.

Robjohns

H.L.

and Dunlop

P.J.

, Physica 95 A, 1979, 561.

54.

Aziz

R.A.

Riley

P.W.

Buck

Maneke

Schleusener

Scoles

and Valbusa

, J.Chem.Phys., 71 (1979), 2637.

55.

Bich

Hellmann

and Vogel

, Mol. Phys., 106 (2008), 1107.

56.

Monchick

Yun

K.S.

and Mason

E.A.

, J.Chem.Phys., 39 (1963), 654.

57.

Chapman

and Cowling

T.G.

, The Mathematical Theory of Non-Uniform Gases.; (Cambridge University Press, New York), 1952.

58.

Frommhold

, Collision-induced scattering of light and the diatom polarizabilities, Adv.Chem.Phys. 46 (1981), 1.

59.

Guillot

Bratos

and Birnbaum

, Mol.Phys., 44 (1981), 1021.

60.

Nienhuis

, J.Math.Phys., 11 (1970), 239.

61.

Frommhold

, Collision-induced absorption in Gases (Cambridge University Press, Cambridge), 1993.

62.

Meinander

Tabisz

G.C.

and Zoppi

, J.Chem.Phys., 84 (1986), 3005.

63.

El-Kader

M.S.A.

, Z.Phys.Chem., 218 (2004), 293.

64.

El-Kader

M.S.A.

, Z.Phys.Chem., 219 (2005), 1169.

65.

El-Kader

M.S.A.

, Phys.Letters A373, 2009, 243.

66.

El-Kader

M.S.A.

, Mol.Phys., 109 (2011), 863.

67.

El-Kader

M.S.A.

, Mol.Phys., 111 (2013), 307.

68.

El-Kader

M.S.A.

, Phys.Letters A373, 2009, 2054.

69.

El-Kader

M.S.A.

and Quant

, Spectrosc. Rad. Transf., 112 (2011), 1533.

70.

Fakhardji

Szabó

El-Kader

M.S.A.

Haskopoulos

Maroulis

and Gustafsson

, J.Chem.Phys., 151 (2019), 144303.

71.

Fakhardji

Szabó

El-Kader

M.S.A.

and Gustafsson

, J.Chem.Phys., 152 (2020), 234302.