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Thermophysical properties and collision-induced lightscattering as a probe for gaseous helium interatomicpotentialsM.S.A. El-Kader aa Department of Engineering Mathematics and Physics, Faculty of Engineering , CairoUniversity , Giza, 12211 , EgyptAccepted author version posted online: 03 Sep 2012.Published online: 28 Sep 2012.

To cite this article: M.S.A. El-Kader (2013) Thermophysical properties and collision-induced light scattering as a probe forgaseous helium interatomic potentials, Molecular Physics: An International Journal at the Interface Between Chemistry andPhysics, 111:2, 309-322, DOI: 10.1080/00268976.2012.723144

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Molecular Physics, 2013Vol. 111, No. 2, 309–322, http://dx.doi.org/10.1080/00268976.2012.723144

RESEARCH ARTICLE

Thermophysical properties and collision-induced light scattering as a probe forgaseous helium interatomic potentials

M.S.A. El-Kader*

Department of Engineering Mathematics and Physics, Faculty of Engineering,Cairo University, Giza, 12211, Egypt

(Received 1 June 2012; final version received 15 August 2012)

Isotropic and anisotropic collision-induced light scattering spectra of helium gas at room temperature 294.5Kand at 99.6K with the second pressure virial coefficients, second acoustic virial coefficients, viscosity and thermalconductivity have been used for deriving the empirical models of the pair-polarizability trace and anisotropy andthe interaction potential. Theoretical zeroth and second moments of the binary spectra using various models forthe pair-polarizabilities and interatomic potential are compared with the experimental values performed by LeDuff’s group. In addition, third pressure virial coefficients, isotopic thermal factors, self diffusion coefficients,second virial dielectric constants and second Kerr coefficients calculated for these models are compared withexperimental ones. The results show that these models are the most accurate models reported to date for thissystem.

Keywords: collision-induced light scattering spectra; potential; helium

1. Introduction

Collision-induced light scattering (CILS) has been

widely used in the past to study intermolecular

properties, especially the invariant of the interaction-

induced polarizability tensor of collisional pairs of

atoms or molecules [1–4]. Accurate Raman spectra of

the rare-gases could be recorded at low enough

densities so that the spectra are due to binary interac-

tions, virtually unaffected by many-body interactions.

Empirical models of the rare-gas diatom polarizabil-

ities were obtained by comparing observed spectro-

scopic features with calculations based on advanced

interatomic potentials and suitable models of the

diatom polarizability invariants.Empirical diatom polarizability models are typi-

cally based on the classical long-range dipole-induced

dipole (DID) mechanism [5] to which a short-range

term is added to simulate the quantal exchange and

overlap contributions [3,6]. For example, very success-

ful models of the trace and anisotropy of the diatom

polarizability tensor are given by the sum of the DID

contributions (often up to second order) and an added

short-range term, which falls off exponentially with

increasing interatomic separation r [3,7]. Such models

typically contain two adjustable parameters that are

chosen to obtain a close fit to the measured momentsof the isotropic and anisotropic Raman spectra and,some times, of other experimental data [3,7]. Theseparameters are the amplitude and range parameters ofthe exponential short-range term in both models of thepolarizabilities.

Also, the analytical model for the interaction-induced polarizability anisotropy of rare-gas atompairs [8] was obtained which includes the effects ofelectrostatic and exchange forces. This model containsonly one free parameter and is based on a theory of thepolarizability of a pair of hydrogen atoms in the tripletstate that treats the interatomic potential as aperturbation.

However, we note that these previous analyticalmodels all suffer from the fact that the differentexperimental data such as lineshapes, integrated inten-sities and second dielectric constant are not fitted well.Some modifications have been made on the trace andanisotropy models such as the computation of thecoefficient A8 of (1/r

8) [9] to reproduce both measure-ments, light scattering spectra, and virial coefficient.A precise determination of a model of the inducedtrace and anisotropy is thought to be important notonly for its own sake, for comparison with ab initiocalculations of the same quantities, but also for the

*Email: [email protected]

[29] D. Henderson and D. Boda, Phys. Chem. Chem. Phys.11, 3822 (2009).

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[32] M. Hahn, O. Barbieri, F.P. Campana, R. Kotz andR. Gallay, Appl. Phys. A 82, 633 (2006).

[33] M. Hahn, O. Barbieri, R. Gallay and R. Kotz, Carbon

44, 2523 (2006).[34] Y. Bar-Cohen Editor, Electroactive Polymer Actuators as

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[36] J.O. Besenhard, M. Winter, J. Yang, W. Biberacher and

J. Power, Sources 54, 228 (1995).[37] S-K. Jeong, M. Inaba, T. Abe and Z. Ogumi,

J. Electrochem. Soc. 148, A989 (2001).

[38] K. Kiyohara and K. Asaka, J. Phys. Chem. C 111,15903 (2007).

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Lett. 104, 097802 (2010).

[41] G. Pastore, P.V. Giaquinta, J.S. Thakur and M.P. Tosi,J. Chem. Phys. 84, 1827 (1986).

[42] J. Lekner, Physica A 176, 485 (1991).[43] R. Sperb, Mol. Simul. 20, 179 (1998).

[44] G.M. Torrie, J.P. Valleau and G.N. Patey, J. Chem.Phys. 76, 4615 (1982).

[45] Y.S. Jho, G. Park, C.S. Chang, P.A. Pincus and

M.W. Kim, Phys. Rev. E 76, 011920 (2007).[46] M.S. Loth, B. Skinner and B.I. Shklovskii, Phys. Rev. E

82, 056102 (2010).

[47] J.I. Siepmann and M. Sprik, J. Chem. Phys. 102,511 (1995).

[48] S.K. Reed, O.J. Lanning and P.A. Madden, J. Chem.Phys. 126, 084704 (2007).

[49] J. Vatamanu, O. Borodin and G.D. Smith, Phys. Chem.Chem. Phys. 12, 170 (2010).

[50] J.H. Irving and J.G. Kirkwood, J. Chem. Phys. 18,817 (1950).

[51] R. Mysyk, E. Raymundo-Pinero and F. Beguin,

Electrochem. Comm. 11, 554 (2009).[52] Y. Liu, S. Liu, J. Lin, D. Wang, V. Jain, R. Montazami,

J.R. Heflin, J. Li, L. Madsen and Q.M. Zhang, Appl.

Phys. Lett. 96, 223503 (2010).[53] K. Kiyohara, T. Sugino, I. Takeuchi, K. Mukai, and

K. Asaka, J. Appl. Phys. 105, 063506 (2009); Erratum:105, 11902 (2009).

[54] E.J.W. Verway and J.Th.G. Overbeek, Theory of theStability of Lyophobic Colloids (Elsevier, Amsterdam,1948: Kluwer Academic/Plenum Publishers, New York,

1999).[55] E.R.A Lima, D. Horinek, R.R. Netz, E.C. Biscaia,

F.W. Tavares, W. Kunz and M. Bostrom, J. Phys.

Chem. B 112, 1580 (2008).[56] C.T. Chan, W.A. Kamitakahara and K.M. Ho, Phys.

Rev. Lett. 58, 1528 (1987).

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310 M.S.A. El-Kader

investigation of collision-induced light scattering

(CILS) phenomena at high density which are believed

to be critically shaped by the pair-polarizability.No adequate potential with the parameters fitted

well with the different thermophysical and transport

properties at different temperatures is available to study

the gas phase of helium. We calculate the interatomic

potential for the helium interaction using mostly the

methods outlined in a previous paper [10]. Since the

details of the methods are given there and the references

therein, 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 light scattering data in addition to the data

on pressure virial coefficients, viscosity, diffusion, and

thermal conductivity coefficients at higher range of

temperatures to fit the simple functional form of the

interatomic potentials for He–He interactions.The CILS and the thermophysical properties used

in the fitting are complementary ones for that purpose.

For these pairs of gas atoms, the measured CILS at

different temperatures used is most sensitive to the

attractive potential from rm which is the separation at

the minimum of the interatomic potential out to the

asymptotic long-range region, and the rainbow and

supernumerary oscillations give detailed information

about that part of the potential [11]. Pressure virial

coefficients reflect the size of rm and the volume of the

attractive well [12], while the viscosity, thermal con-

ductivity and diffusion data are most sensitive to the

wall of the potential from rm inward to a point where

the potential is repulsive [13].In this paper we present a new analysis of the

isotropic and anisotropic light scattering spectra of

helium gas at T¼ 294.5K and at T¼ 99.6K, based on

fitting the spectral profiles and their moments of the

measurements. Spectral profiles are calculated numer-

ically with the help of a quantal computer program, and

compared with the measured spectra. The comparison

of calculated and measured spectra provides valuable

clues concerning the quality of existing models of both

the interaction-induced pair-polarizabilities and the

interatomic potential. The interatomic potential and

calculation of the different properties are presented in

Section 2. The diatom polarizability forms adopted are

presented in Section 3. The theoretical method for

calculating the lineshape is briefly given in Section 4,

together with the computational implementation.

Analysis of CILS spectral moments to determine the

parameters of the trace and anisotropy models of the

pair-polarizabilities is given in Section 5. Results are

presented and discussed in Sec. 6 and the concluding

remarks are given in Section 7.

2. The intermolecular potential models andmulti-property analysis

In order to calculate the line profiles of scattering andtheir associated moments, the interatomic potential isneeded. Results with different potentials can be com-pared with experiment to assess the quality of thepotential.

The interatomic potentials we provide here areobtained through the analysis of the second and thirdpressure virial coefficients [14–33], second acousticvirial coefficients [34] and set of gaseous transportproperties [31,32,35–72].

For the analysis of all these experimental data weconsider the empirical Exp-6 [73], Hartree–Fock dis-persion types (HFD-B) [74], Morse-Spline van derWaals (MSV) [75], Barker-Watts-Lee-Schafer-Lee(BWLSL) [76], four-parameter diatomic potential(FPDP) [77], Exponential Spline-Morse-Spline-vander Waals (ESMSV) [78,79] and Tang-Toennies (TT)potential models [80]. In addition to these empiricalpotentials the recent ab initio potential of Przybytek[81] was considered. The parameters of our empiricalpotentials are shown in Table 1.

2.1. Analysis of the second and third pressure virialcoefficients

An effective means for checking the validity of thedifferent potential models can be made using second(B), third (C) pressure virial coefficients and secondacoustic virial coefficients (A) data [14–34] at differenttemperatures.

A quantum-mechanical treatment of the elasticscattering is needed to obtain very accurate values forthe thermophysical properties of helium. For thispurpose, the relative phase shifts �l have to beevaluated as asymptotic limiting values of the relativephases of the perturbed and unperturbed radial factorwave functions. To obtain the relative phase shifts�l(k), the Schrodinger equation is solved by numericalintegration for many values of the wave numberk ¼

ffiffiffiffiffiffiffiffiffiffi2mE

p=�h, where E is the energy of the incoming

wave and m is the reduced mass. The procedures ofcalculating the second pressure virial coefficient B(T)are described in details in Ref. [82].

For the evaluation of the third pressure virialcoefficient C(T), naturally occuring helium is againassumed to be a pure gas composed of atoms with thesame mass. Further, C(T) is calculated as a sum of thethree contributions [83,84], one term for the pairwiseadditivity of the two-body interatomic potential Cadd,an extra genuine term Cnon-add for non-additivityDV3ðr12, r13, r23Þ of the three-body interaction potential

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Molecular Physics 311

V3ðr12, r13, r23Þ, and a first-order correction term forthe quantum effects Cqm,1:

C Tð Þ ¼ CaddðTÞ þ Cnon�addðTÞ þ �Cqm,1ðTÞ ð1Þ

The formulas for the three contributions havealready been given in Ref. [82]. The non-additivitycontribution DV3ðr12, r13, r23Þ to the three-body poten-tial is again approximated by the Axilrod–Teller triple-dipole potential term [85,86], in which thenon-additivity coefficient of the triple-dipole termcalculated for helium by Meath [87], C9 ¼ 1:472 a.u.,is used.

The calculated B, C and A were compared with theexperimental results [14–34] and more recent calculatedones [88–93]. The comparison in the case of the secondpressure virial coefficient is shown in Figure 1.

2.2. Analysis of traditional transport properties

An additional check on the proposed potentials consistof the calculation of the transport properties, i.e.viscosity (�), thermal conductivity (�), self diffusion

coefficient (D) and isotopic thermal factors (S) atdifferent temperatures of helium, obtained via theformulas of Hurly–Mehl [89,90], Bich et al. [82] andHurly–Moldover [30] and their comparison to theaccurate experimental results [31,32,35–72].Determination of these properties of a quantum gasrequires two major computational steps. The first stageis the determination of the quantum phase shifts forHe–He atomic scattering. The second is the use of the phaseshifts to compute the quantum cross-sections, then thetemperature-dependent collision integrals, and finally,the transport coefficients. The compuatational methodsdescribed in detail by Hurly andMehl [89,90] were used.

3. Diatom polarizability

The polarizability of a cluster of inert atoms dependingon the interatomic forces is a key element for both thequalitative and the quantitative understanding ofseveral macroscopic properties of a dense fluid(second-and higher-order virial coefficients of thedielectric constant, refractivity, optical Kerr constant).

Table 1. Parameters of the trial potentials of gaseous helium and the associated values of �a.

Potential parameters Exp-6 HFD-B BWLSL FPDP ESMSV MSV TT

"ðKÞ=kB 11.3 11.25 9.76 11.48 11.15 10.985 10.974rmðnmÞ 0.295 0.2958 0.297 0.2985 0.2958 0.29636 0.29645

�ðA�1Þ – – 9.5 – 6.175 2.14942 3.82416� 15.75 – – – – – –A – 11.85 1.1464 5.75 0.5684 – 3053500.0B – �0.997 �3.52108 – – – 2.0L – – – 4.45 – – –D – 1.3 – – – – –B0 – – 1.1464 – 13.53 – –B1 – – �3.52108 – – – –B2 – – 10.5 – – – –B3 – – 15.0 – – – –B4 – – �50.0 – – – –B5 – – �50.0 – – – –Del – – 0.01 – – – –�B 0.97 0.46 0.69 0.84 0.64 0.57 0.18�C 0.96 0.84 0.94 0.95 0.93 0.88 0.67�A 0.95 0.73 0.89 0.93 0.87 0.79 0.64�� 0.79 0.45 0.75 0.76 0.72 0.49 0.15�D 0.84 0.41 0.73 0.81 0.69 0.45 0.29�� 0.91 0.66 0.86 0.87 0.81 0.68 0.13�S 0.88 0.59 0.79 0.79 0.74 0.63 0.22�t 0.91 0.61 0.81 0.85 0.78 0.66 0.39

�a is defined by �a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=NÞ

PNj¼1 ð1=nJ

Pnjj¼1 D

�2ji ðPji�pjiÞ2ÞÞ

q, where Pji and pji are, respectively, the calculated and experimental

values of property j at point i and Dji is the experimental uncertainty of property j at point i. The subscripts B, C, A, �, D, �, S andt refer, respectively, to the pressure second virial coefficient, pressure third virial coefficient, acoustic virial coefficient, viscosity,diffusion coefficient, thermal conductivity, thermal isotopic factor and total.

investigation of collision-induced light scattering

(CILS) phenomena at high density which are believed

to be critically shaped by the pair-polarizability.No adequate potential with the parameters fitted

well with the different thermophysical and transport

properties at different temperatures is available to study

the gas phase of helium. We calculate the interatomic

potential for the helium interaction using mostly the

methods outlined in a previous paper [10]. Since the

details of the methods are given there and the references

therein, 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 light scattering data in addition to the data

on pressure virial coefficients, viscosity, diffusion, and

thermal conductivity coefficients at higher range of

temperatures to fit the simple functional form of the

interatomic potentials for He–He interactions.The CILS and the thermophysical properties used

in the fitting are complementary ones for that purpose.

For these pairs of gas atoms, the measured CILS at

different temperatures used is most sensitive to the

attractive potential from rm which is the separation at

the minimum of the interatomic potential out to the

asymptotic long-range region, and the rainbow and

supernumerary oscillations give detailed information

about that part of the potential [11]. Pressure virial

coefficients reflect the size of rm and the volume of the

attractive well [12], while the viscosity, thermal con-

ductivity and diffusion data are most sensitive to the

wall of the potential from rm inward to a point where

the potential is repulsive [13].In this paper we present a new analysis of the

isotropic and anisotropic light scattering spectra of

helium gas at T¼ 294.5K and at T¼ 99.6K, based on

fitting the spectral profiles and their moments of the

measurements. Spectral profiles are calculated numer-

ically with the help of a quantal computer program, and

compared with the measured spectra. The comparison

of calculated and measured spectra provides valuable

clues concerning the quality of existing models of both

the interaction-induced pair-polarizabilities and the

interatomic potential. The interatomic potential and

calculation of the different properties are presented in

Section 2. The diatom polarizability forms adopted are

presented in Section 3. The theoretical method for

calculating the lineshape is briefly given in Section 4,

together with the computational implementation.

Analysis of CILS spectral moments to determine the

parameters of the trace and anisotropy models of the

pair-polarizabilities is given in Section 5. Results are

presented and discussed in Sec. 6 and the concluding

remarks are given in Section 7.

2. The intermolecular potential models andmulti-property analysis

In order to calculate the line profiles of scattering andtheir associated moments, the interatomic potential isneeded. Results with different potentials can be com-pared with experiment to assess the quality of thepotential.

The interatomic potentials we provide here areobtained through the analysis of the second and thirdpressure virial coefficients [14–33], second acousticvirial coefficients [34] and set of gaseous transportproperties [31,32,35–72].

For the analysis of all these experimental data weconsider the empirical Exp-6 [73], Hartree–Fock dis-persion types (HFD-B) [74], Morse-Spline van derWaals (MSV) [75], Barker-Watts-Lee-Schafer-Lee(BWLSL) [76], four-parameter diatomic potential(FPDP) [77], Exponential Spline-Morse-Spline-vander Waals (ESMSV) [78,79] and Tang-Toennies (TT)potential models [80]. In addition to these empiricalpotentials the recent ab initio potential of Przybytek[81] was considered. The parameters of our empiricalpotentials are shown in Table 1.

2.1. Analysis of the second and third pressure virialcoefficients

An effective means for checking the validity of thedifferent potential models can be made using second(B), third (C) pressure virial coefficients and secondacoustic virial coefficients (A) data [14–34] at differenttemperatures.

A quantum-mechanical treatment of the elasticscattering is needed to obtain very accurate values forthe thermophysical properties of helium. For thispurpose, the relative phase shifts �l have to beevaluated as asymptotic limiting values of the relativephases of the perturbed and unperturbed radial factorwave functions. To obtain the relative phase shifts�l(k), the Schrodinger equation is solved by numericalintegration for many values of the wave numberk ¼

ffiffiffiffiffiffiffiffiffiffi2mE

p=�h, where E is the energy of the incoming

wave and m is the reduced mass. The procedures ofcalculating the second pressure virial coefficient B(T)are described in details in Ref. [82].

For the evaluation of the third pressure virialcoefficient C(T), naturally occuring helium is againassumed to be a pure gas composed of atoms with thesame mass. Further, C(T) is calculated as a sum of thethree contributions [83,84], one term for the pairwiseadditivity of the two-body interatomic potential Cadd,an extra genuine term Cnon-add for non-additivityDV3ðr12, r13, r23Þ of the three-body interaction potential

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312 M.S.A. El-Kader

The diatom polarizability is defined as the excesspolarizability due to the interaction. It is given by thepolarizability of two interacting atoms minus the sumof polarizabilities of two unperturbed atoms. The tensorcomponents are functions of separation r, and vanish asr approaches infinity. The invariants are trace,�ðrÞ ¼ ð�jj þ 2�?Þ=3 and anisotropy �ðrÞ ¼ �jj � �?,where �jj and �? designate the diatom polarizabilitycomponents in the molecular frame, with external fieldsparallel and perpendicular to the diatom axis.

For the sake of comparison and discussion, for thepresent calculations we considered five models of thepair-polarizability trace and anisotropy which are

(a) The two-term dipole-induced dipole models(DID) [94]:

�ðrÞ ¼ 4�3or6

þ 4�4or9

ð2Þ

�ðrÞ ¼ 6�2or3

þ 6�3or6

ð3Þ

where �o is the polarizability of the individual isolatedatoms.

(b) The empirical Proffitt–Keto–Frommhold(PKF) model, fitting their experimental polar-ized and depolarized spectra [95]:

�ðrÞ ¼ A6

r6� to exp � r� �

rt

� �ð4Þ

�ðrÞ ¼ 6�2or3

� go exp � r� �ro

� �ð5Þ

where the values of the parameters have been deter-

mined as

to ¼ 0:1333661262=1000 A3, rt ¼ 0:333382 A

for the trace and go ¼ 2:222769=1000 A3,

ro ¼ 0:37042 A for the anisotropy:

(c) The polarizability models of Certain and

Fortune [96] denoted CF, where distortions of

the 6�3o coefficient in the 1/r6 DID classical

term were evaluated in terms of a coefficient

accounting for the London long-distance dis-

persion effects.(d) The MHWA ab initio model of Moszynski

et al. [97] was obtained by means of symmetry-adapted perturbation theory (SAPT) in a

many-body formulation especially adapted to

the intermolecular interactions. The DF

ab initio model of Dacre and Frommhold [98]

was constructed from a large-scale ab initio

computation based on a mixed self-consistent

field (SCF) and configuration-interaction (CI)

approach. The FC ab initio model proposed byFortune and Certain [99] was obtained with a

coupled self-consistent field (SCF) calculation

based on a finite field method. The more recent

ab initio model of Cencek et al. [100] was used

large expansions in basis sets of explicitly

correlated Gaussian functions and the varia-

tion-perturbation technique.

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

2

4

6

8

10

12

14Calculated virial coefficients using TT

Calculated virial coefficients using Exp-6

Calculated virial coefficients using HFD-B

Calculated virial coefficients using FPDP

Calculated virial coefficients using BWSLS

Calculated virial coefficients using ESMSV

Calculated virial coefficients using MSV

Calculated virial coefficients using ab initio pot.[81]

Experimental virial coefficients [30]

T(K)

B(c

m3 /m

ol)

Figure 1. He-He second pressure virial coefficients in cm3/mole vs. temperature in K using different intermolecular potentials.The experimental data [30].

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(e) The analytical models of Buckingham et al. [9],

which it will been seen to provide a useful basis

to describe diatom polarizabilities should be of

the forms:

�ðrÞ ¼ A6

r6þ 20�2oC

r8� to exp � r

rt

� �ð6Þ

�ðrÞ ¼ 6�2or3

þ A

r6þ 24�2oC

r8� go exp � r

ro

� �ð7Þ

with [4,101,102]

A6 ¼ 4�3o þ5

9�C6=�o

� �ð8Þ

and

A ¼ 6�3o þ �C6=3�o� �

ð9ÞHere, �o and � designate atomic polarizability and

hyperpolarizability respectively, C6 is the dispersion

force coefficient and C is the quadrupole polarizability.

The values of the parameters used here are as follows

[103–105]:

�o ¼ 1:3222 a:u, � ¼ 36:0 a:u,

C ¼ 2:315892 a:u and

C6=104 ¼ 1:01304251KA

6:

We shall use (6) and (7) below to see if the trace and

anisotropy can be approximated by such simple models

which will be seen to provide a useful empirical basis to

describe diatom polarizabilities.

4. The spectral lineshape and its moments

The quantum theory is applied for the accurate com-

putation of the CILS absolute intensities of the helium

pairs. Numerically, this is done by means of the

propagative two-point Fox–Goodwin integrator

[106,107], where the ratio of the wavefunction, defined

at adjacent points on a spatial grid, is built step-by-step.As regards our problem, binary isotropic and

anisotropic spectra are computed quantum-mechani-

cally, as a function of �, at temperature T by using the

expressions [108,109]:

Iisoð�Þ ¼ hc�3k4sXJmax

J¼o,J even

ð2Jþ 1Þ

�Z Emax

0

E0,J0 �j j E,J

� �� 2expð�E=kBTÞdE ð10Þ

Ianisoð�Þ ¼2

15hc�3k4s

XJmax

J¼o,J even

gJbJ0

J ð2Jþ1Þ

�Z Emax

0

E0,J0 �� E,J

� �� 2expð�E=kBTÞdE ð11Þ

The symbol ks stands for the Stokes wave number

of the scattered light, h is Planck’s constant and c is the

speed of the light. Constant � account for the thermal

de Broglie wavelength, � ¼ h=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2��kBT

p, with � the

reduced mass of He2 and kB Boltzmann’s constant.

Symbol E,J designates the scattering wave function

and Emax the maximum value of the energy that is

required to obtain convergence of the integrals.In these expressions, � ¼ �ðrÞ and � ¼ �ðrÞ denote

the trace and anisotropy of the quasimolecule, gJ the

nuclear statistical weight; for a rare gas, one has gJ ¼ 0

for J odd and gJ ¼ 2 for J even. Finally, bJ0

J are intensity

factors involving the rotational quantum numbers J and

J0 of the initial and final states, respectively.With the spectral intensities Iisoð�Þ and Ianisoð�Þ in

cm6 as input and through the following analytical

expressions we are able to deduce the isotropic moments

M2nð Þ ¼ �o2�

� �4Z 1

�1ð2�c�Þ2nIisoð�Þd� ð12Þ

and anisotropic moments

M2nð Þ ¼ 15

2

�o2�

� �4Z 1

�1ð2�c�Þ2nIanisoð�Þd� ð13Þ

where �o denotes the laser wavelength and n is a non-

negative integer.An effective means for checking the validity of the

different models of induced trace �ðrÞ is to compare the

first two even moments of the isotropic spectrum which

are deduced from the experiment to those determined

from the sum rules with L¼ 0 [110]:

M0 ¼ 4�

Z 1

0

ð�LðrÞÞ2gðrÞr2dr, ð14Þ

M1 ¼4��h

2m

Z 1

0

��d�LðrÞdr

�2

þ LðLþ 1Þ��LðrÞ

r

�2�gðrÞr2dr, ð15Þ

M2 ¼4��h2

m2

Z 1

0

� �LðrÞ4

d4�LðrÞdr4

� ��

� d�LðrÞ2dr

� �d3�LðrÞdr3

� �þ LðLþ 1Þ 1

2r2d�LðrÞdr

� �2

� LðLþ 1Þ �LðrÞr3

� �d�LðrÞdr

� �

þL2ðLþ 1Þ2 ð�LðrÞÞ2

4r4

�gðrÞr2dr

þ 4�

m

Z 1

0

ð�LðrÞdV

dr

� �d�LðrÞdr

� �

The diatom polarizability is defined as the excesspolarizability due to the interaction. It is given by thepolarizability of two interacting atoms minus the sumof polarizabilities of two unperturbed atoms. The tensorcomponents are functions of separation r, and vanish asr approaches infinity. The invariants are trace,�ðrÞ ¼ ð�jj þ 2�?Þ=3 and anisotropy �ðrÞ ¼ �jj � �?,where �jj and �? designate the diatom polarizabilitycomponents in the molecular frame, with external fieldsparallel and perpendicular to the diatom axis.

For the sake of comparison and discussion, for thepresent calculations we considered five models of thepair-polarizability trace and anisotropy which are

(a) The two-term dipole-induced dipole models(DID) [94]:

�ðrÞ ¼ 4�3or6

þ 4�4or9

ð2Þ

�ðrÞ ¼ 6�2or3

þ 6�3or6

ð3Þ

where �o is the polarizability of the individual isolatedatoms.

(b) The empirical Proffitt–Keto–Frommhold(PKF) model, fitting their experimental polar-ized and depolarized spectra [95]:

�ðrÞ ¼ A6

r6� to exp � r� �

rt

� �ð4Þ

�ðrÞ ¼ 6�2or3

� go exp � r� �ro

� �ð5Þ

where the values of the parameters have been deter-

mined as

to ¼ 0:1333661262=1000 A3, rt ¼ 0:333382 A

for the trace and go ¼ 2:222769=1000 A3,

ro ¼ 0:37042 A for the anisotropy:

(c) The polarizability models of Certain and

Fortune [96] denoted CF, where distortions of

the 6�3o coefficient in the 1/r6 DID classical

term were evaluated in terms of a coefficient

accounting for the London long-distance dis-

persion effects.(d) The MHWA ab initio model of Moszynski

et al. [97] was obtained by means of symmetry-adapted perturbation theory (SAPT) in a

many-body formulation especially adapted to

the intermolecular interactions. The DF

ab initio model of Dacre and Frommhold [98]

was constructed from a large-scale ab initio

computation based on a mixed self-consistent

field (SCF) and configuration-interaction (CI)

approach. The FC ab initio model proposed byFortune and Certain [99] was obtained with a

coupled self-consistent field (SCF) calculation

based on a finite field method. The more recent

ab initio model of Cencek et al. [100] was used

large expansions in basis sets of explicitly

correlated Gaussian functions and the varia-

tion-perturbation technique.

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

2

4

6

8

10

12

14Calculated virial coefficients using TT

Calculated virial coefficients using Exp-6

Calculated virial coefficients using HFD-B

Calculated virial coefficients using FPDP

Calculated virial coefficients using BWSLS

Calculated virial coefficients using ESMSV

Calculated virial coefficients using MSV

Calculated virial coefficients using ab initio pot.[81]

Experimental virial coefficients [30]

T(K)

B(c

m3 /m

ol)

Figure 1. He-He second pressure virial coefficients in cm3/mole vs. temperature in K using different intermolecular potentials.The experimental data [30].

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314 M.S.A. El-Kader

þ 2�LðrÞd2�LðrÞdr2

� �VðrÞÞ gðrÞr2dr

� 8�

m

Z 1

0

�LðrÞd2�LðrÞdr2

� �geðrÞr2dr

þ 4�

m2

Z 1

0

�LðrÞr2


� �� LðrÞ

r3d�LðrÞdr

� ��

þLðLþ 1Þ ð�LðrÞÞ2

2r4

�gmðrÞr2dr

ð16Þ

The quantum mechanical expressions for the pair

distribution functions g(r), ge(r) and gm(r) are indicated

in Ref. [110].Although the spectroscopic method used in this

work is a quite powerful tool because we compare the

shapes of the spectral distributions on an absolute

intensity scale, the straightforward measurement of the

second dielectric virial coefficient B" is a complemen-

tary tool. By simply comparing two numbers, that is

measured [111] and a computed B", this quantity can

serve as a sensitive probe for checking the quality of an

incremental trace model. This is done by using the

following formula [112]:

B" ¼8�2

3N2

A

Z 1

0

�ðrÞgðrÞr2dr ð17Þ

Also, for checking the validity of the different

models of induced anisotropy �ðrÞ we compare the first

three moments of the anisotropic spectrum which are

deduced from the experiment with the theoretical ones

through the sum rules with L¼ 2 [110].The quantum CILS lineshape calculations have

been extensively used by Frommhold and co- workers

[113,114] for the determination of induced polarizabil-

ities of noble gas diatoms. The method consists of

adopting a pair potential and computing lineshapes for

various forms of the induced anisotropy, using a

computer code based on (8). Close agreement between

calculated and experimental lineshapes is possible and

can be utilized to define an empirical induced

anisotropy.Let us first consider the lineshape that we have

calculated for the helium pair at T¼ 294.5K using our

empirical potential and the different models of the pair

polarizabilities. Table 2 shows the comparison between

the moments of the isotropic and anisotropic spectra

calculated using Equations (12) and (13) with the

experimental ones.In an attempt to fit the isotropic and anisotropic

spectra even more closely, we use the trace and

anisotropy models, as in Equations (6) and (7), we

found that the parameters for these polarizabilities are

rt ¼ 0:3485 A and to ¼ 2:55 A3

for the trace and.

ro ¼ 0:3055 A and go ¼ 8:265 A3

for the anisotropy.

Table 2. Comparison between theoretical and experimental isotropic and anisotropic moments of helium using the present traceand anisotropy polarizabilities and TT interatomic potential.

ðM0Þ10�4 A9ðM2Þ1023 A

9=s2

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{Isotropic moments

ðM0Þ10�2 A9ðM2Þ1024 A

9=s2

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{Anisotropic moments

ExperimentalRachet et al.[115]C.Guillot-Noel et al.[116,117] 2.60a –b 3.2� 1.3a –b 1.70� 0.19c 1.46d 3.18� 0.48c 0.935d

CalculatedDID [94] 0.01541 0.00725 0.01849 0.003093 2.35 1.779 6.48 1.81CF [96] 0.27 – 0.23 – 2.75 1.82 7.03 1.847PKF [95] 3.95 – 6.32 – 2.04 1.49 3.22 0.951DF [98] 2.56 – 3.01 – 2.44 1.81 3.39 1.037FC [99] 3.6 – 3.73 – 1.67 1.22 2.52 0.768MHWA [97] 2.43 – 3.09 – 1.97 1.45 2.57 0.882Ab initio [100] 2.367 0.855 4.13 0.9579 1.9974 1.765 3.147 0.9963Ab initio [118] 2.388 0.8625 4.164 0.9605 1.995 1.763 3.155 0.9887Present workThe empirical trace and anisoptropy 2.50 1.044 3.365 0.74624 1.789 1.54 3.18 1.015

Notes: aValues calculated from (12) from the lineshape Iisoð�Þ at T¼ 294.5K,bValues calculated from (12) from the lineshape Iisoð�Þ at T¼ 99.6K,cValues calculated from (13) from the lineshape Ianisoð�Þ at T¼ 294.5K,dValues calculated from (13) from the lineshape Ianisoð�Þ at T¼ 99.6K.

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Table 2 reports the values of the moments calculated

for these models of the trace and anisotropy and shows

that the discrepancies between these moments and

its experimental are eliminated for all models of �ðrÞand �ðrÞ.

5. Analysis of CILS spectral moments to determineaðrÞ and bðrÞ

The method of detailed analysis of the first three even

moments of the depolarized light scattering spectrum

(CILS) has been used by Barocchi–Zoppi [119] and

Chrysos–Dixneuf [120] for the determination of the

extra-dipole-induced dipole (DID) contribution to the

pair-polarizability anisotropy of argon. This consists

of establishing an appropriate parameterized model

form for anisotropy and then searching by means of a

computer for the sets of parameters that are consistent

with the experimental values of the moments.In order to make the presentation of our results to

be comparable with those given by other authors, it is

convenient to rewrite �ðrÞ and �ðrÞ of the Equations (6)and (7) in terms of the reduced variable x ¼ r=rm where

rm is the separation at the minimum of the interatomic

potential V(r). In this case one has:

�ðxÞ ¼ A�6x

�6 þ B�1x

�8 � t�o exp � x

xt

� �� ð18Þ

�ðxÞ ¼ 6�2or3m

x�3 þ A�x�6 þ B�2x

�8 � g�o exp � x

xo

� ��

ð19Þ

where

A�6 ¼ A6=r

6m; B�

1 ¼ 20�2oC=r8m; t�o ¼ to;

xt ¼ rt=rm; A� ¼ A=6�2or3m; B�

2 ¼ 4C=r5m;

g�o ¼ gor3m=6�

2o and xo ¼ ro=rm:

The substitutions of (18) into the moment expres-

sions (14), (15) and (16) with L¼ 0 for the isotropic

spectrum and of (19) into the same expressions with

L¼ 2 for the anisotropic spectrum make it possible to

rewrite them in the form of quadratic equations for the

unknown t�o and g�o with coefficients which are para-

metric functions of xt and xo. The equations those one

obtains from the moments of the isotropic and

anisotropic spectra respectively are of the form:

Xiþ1 ¼ Xi �D�11 xiFðXiÞ�� ð20Þ

Yiþ1 ¼ Yi �D�12 yiFðYiÞ�� ð21Þ

where the Jacobian matrices are given by

D1ðxÞ ¼

@Miso0

@B1

@Miso0

@to

@Miso0

@rt@Miso

1

@B1

@Miso1

@to

@Miso1

@rt@Miso

2

@B1

@Miso2

@to

@Miso2

@rt

0BBBBBB@

1CCCCCCA

ð22Þ

D2ðYÞ ¼

@Maniso0

@B2

@Maniso0

@go

@Maniso0

@ro@Maniso

1

@B2

@Maniso1

@go

@Maniso1

@ro@Maniso

2

@B2

@Maniso2

@go

@Maniso2

@ro

0BBBBBB@

1CCCCCCA

ð23Þ

and the two column vectors X and F(X) for isotropicmoments and Y and F(Y) for anisotropic moments aredefined as

X ¼B1

tort

0@

1A, ð24Þ

FðXÞ ¼DMiso

0

DMiso1

DMiso2

0@

1A ð25Þ

Y ¼B2

goro

0@

1A, ð26Þ

FðYÞ ¼DManiso

0

DManiso1

DManiso2

0@

1A ð27Þ

where DMn are the difference between theoretical andexperimental moments of the isotropic and anisotropiclight scattering spectra.

Once convergences are obtained, the column vec-tors solutions X and Y have to satisfy FðXÞ¼ 0 for theisotropic moments and FðYÞ ¼ 0 for the anisotropicmoments. Each element of the matrices D1 and D2

reads

D1ð Þab¼ 4�

Z 1

0

aabðXÞ expð�VðrÞ=kBTÞr2dr ð28Þ

D2ð Þab¼ 4�

Z 1

0

babðYÞ expð�VðrÞ=kBTÞr2dr ð29Þ

where a,b¼ 1, 2 and 3 stand for the line and columnnumbers of the matrices D1 and D2, respectively. Forthe specific functional forms of �ðrÞ and �ðrÞ given byEquations (6) and (7), the nine elements aab(X) andbab(Y) are given in the Appendix.

The great advantages of this method for calculatingthe parameters of the models for �ðrÞ and �ðrÞ are thespeed of computation and that the trial-and-error

þ 2�LðrÞd2�LðrÞdr2

� �VðrÞÞ gðrÞr2dr

� 8�

m

Z 1

0

�LðrÞd2�LðrÞdr2

� �geðrÞr2dr

þ 4�

m2

Z 1

0

�LðrÞr2


� �� LðrÞ

r3d�LðrÞdr

� ��

þLðLþ 1Þ ð�LðrÞÞ2

2r4

�gmðrÞr2dr

ð16Þ

The quantum mechanical expressions for the pair

distribution functions g(r), ge(r) and gm(r) are indicated

in Ref. [110].Although the spectroscopic method used in this

work is a quite powerful tool because we compare the

shapes of the spectral distributions on an absolute

intensity scale, the straightforward measurement of the

second dielectric virial coefficient B" is a complemen-

tary tool. By simply comparing two numbers, that is

measured [111] and a computed B", this quantity can

serve as a sensitive probe for checking the quality of an

incremental trace model. This is done by using the

following formula [112]:

B" ¼8�2

3N2

A

Z 1

0

�ðrÞgðrÞr2dr ð17Þ

Also, for checking the validity of the different

models of induced anisotropy �ðrÞ we compare the first

three moments of the anisotropic spectrum which are

deduced from the experiment with the theoretical ones

through the sum rules with L¼ 2 [110].The quantum CILS lineshape calculations have

been extensively used by Frommhold and co- workers

[113,114] for the determination of induced polarizabil-

ities of noble gas diatoms. The method consists of

adopting a pair potential and computing lineshapes for

various forms of the induced anisotropy, using a

computer code based on (8). Close agreement between

calculated and experimental lineshapes is possible and

can be utilized to define an empirical induced

anisotropy.Let us first consider the lineshape that we have

calculated for the helium pair at T¼ 294.5K using our

empirical potential and the different models of the pair

polarizabilities. Table 2 shows the comparison between

the moments of the isotropic and anisotropic spectra

calculated using Equations (12) and (13) with the

experimental ones.In an attempt to fit the isotropic and anisotropic

spectra even more closely, we use the trace and

anisotropy models, as in Equations (6) and (7), we

found that the parameters for these polarizabilities are

rt ¼ 0:3485 A and to ¼ 2:55 A3

for the trace and.

ro ¼ 0:3055 A and go ¼ 8:265 A3

for the anisotropy.

Table 2. Comparison between theoretical and experimental isotropic and anisotropic moments of helium using the present traceand anisotropy polarizabilities and TT interatomic potential.

ðM0Þ10�4 A9ðM2Þ1023 A

9=s2

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{Isotropic moments

ðM0Þ10�2 A9ðM2Þ1024 A

9=s2

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{Anisotropic moments

ExperimentalRachet et al.[115]C.Guillot-Noel et al.[116,117] 2.60a –b 3.2� 1.3a –b 1.70� 0.19c 1.46d 3.18� 0.48c 0.935d

CalculatedDID [94] 0.01541 0.00725 0.01849 0.003093 2.35 1.779 6.48 1.81CF [96] 0.27 – 0.23 – 2.75 1.82 7.03 1.847PKF [95] 3.95 – 6.32 – 2.04 1.49 3.22 0.951DF [98] 2.56 – 3.01 – 2.44 1.81 3.39 1.037FC [99] 3.6 – 3.73 – 1.67 1.22 2.52 0.768MHWA [97] 2.43 – 3.09 – 1.97 1.45 2.57 0.882Ab initio [100] 2.367 0.855 4.13 0.9579 1.9974 1.765 3.147 0.9963Ab initio [118] 2.388 0.8625 4.164 0.9605 1.995 1.763 3.155 0.9887Present workThe empirical trace and anisoptropy 2.50 1.044 3.365 0.74624 1.789 1.54 3.18 1.015

Notes: aValues calculated from (12) from the lineshape Iisoð�Þ at T¼ 294.5K,bValues calculated from (12) from the lineshape Iisoð�Þ at T¼ 99.6K,cValues calculated from (13) from the lineshape Ianisoð�Þ at T¼ 294.5K,dValues calculated from (13) from the lineshape Ianisoð�Þ at T¼ 99.6K.

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approach is avoided. The method of moments analysishas been applied to the collision-induced light scatter-ing (CILS) spectra of Ne, Ar, Kr, Xe and CH4 [121].Because the correction to the first-order DID expres-sion is quite small over most of the interatomicseparations probed by the atomic motions, and thetwo correction terms to a large extent cancel out theeffects of each other, the numerical values derived forthe parameters to of the trace and go of the anisotropy,are quite sensitive to the input values of the moments.

For the calculation of the theoretical values of themoments with the empirical models of the trace andanisotropy polarizabilities the empirical pair potentialTT labelled 7 in Table 1, was used.

Tables 3 and 4 give the numerical results of thesearch while Figures 2 and 3 show the different regionsof the polarizability behavior which are consistent withthe experimental results. Here, also, the first order(DID), PKF and ab initio models of the trace andanisotropy are reported.

Although the numerical values obtained for theseparameters in Tables 3 and 4 are very different, thecorresponding functions �ðrÞ and �ðrÞ are really verysimilar, as are evident in Figures 2 and 3. Importantdeviations from the two-term DID models occur onlyat separations r� 1.607 rm for the trace and r� 1.2495rm for the anisotropy and the models differ from each

other significantly only at separations which are notprobed by the molecules anyway.

The moment analysis program was easily modifiedto calculate the moments of the isotropic and aniso-tropic spectra as functions of to and go using theexpressions (18) and (19). The calculation requiresabout 15 min. of CPU time on the Pentium 4 computerfor 1500 values of both to and go.

The experimentally determined values of themoments, with error limits, now each define a rangeof acceptable values for the parameters. This approachto the construction of an empirical models for �ðrÞ and�ðrÞ is acceptable if all moments define a commonrange of, to and go, rt and ro values. As one can seefrom Tables 3 and 4, the agreement between theexperimental values and the theoretical ones using ourempirical potential is excellent and we have verifiedthat it remains acceptable for to ¼ 2:5625� 0:2375 A

3

and go ¼ 8:8� 3:7 A3

in the case of the trace andanisotropy models respectively.

6. Results

In the following the results of the analysis for heliumwill be described and the empirical models for both�ðrÞ and �ðrÞ are compared with the DID, PKF, FC,

Table 4. Pair-polarizability anisotropy models and calculated moments of the helium anisotropic light scattering spectrumobtained by using the present TT potential model at T¼ 294.5K.

Model ro go

M0ð10�2 A9Þ

Experimental1.7� 0.19 [116]Calculated

M1ð1010 A9=sÞ

Experimental-

Calculated

M2ð1024 A9=s2Þ

Experimental3.18� 0.48 [116]

Calculated

BKð10�35 C2m8J�2ÞCalculated 2.3341* [123]

Calculated

1 0.32 5.10 1.837 4.217 3.548 2.1592 0.315 7.0 1.805 3.961 3.30 2.1133 0.305 8.25 1.789 3.836 3.18 2.0894 0.30 9.85 1.768 3.693 3.047 2.0595 0.295 12.5 1.72 3.427 2.813 1.993

Note: *At T¼ 407.6K.

Table 3. Pair-polarizability trace models and calculated moments of the helium isotropic light scattering spectrum obtained byusing the present TT potential model at T¼ 294.5K.

Model rt to

M0ð10�4 A9Þ

Experimental2.6 [115]Calculated

M1ð109 A9=sÞ

Experimental3.05

Calculated

M2ð1023 A9=s2Þ

Experimental3.2� 1.3[115]Calculated

B"ðcm6=mole2ÞExperimental

�0.07 *� 0.01 [122]Calculated

1 0.358 2.325 2.858 3.5273 3.611 �0.08942 0.355 2.40 2.710 3.412 3.503 �0.08583 0.350 2.545 2.50 3.260 3.365 �0.08044 0.345 2.675 2.24 3.03 3.141 �0.0745 0.340 2.80 1.964 2.764 2.881 �0.067

Note: *At T¼ 323.15K.

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CF, DF and the ab initio results of MHWA. [97] andthe more recent ab initio models [100]. Good agreementbetween the ab initio models of MHWA for the traceand PKF for the anisotropy with our empirical ones isgenerally observed and the calculated profiles byour models of the polarizabilities agree with themeasurements to within the uncertainty of the latter.

Figures 4–7 display the quantum profiles of theisotropic and anisotropic collision-induced light scat-tering spectra at T¼ 294.5K and 99.6K using models 3in Tables 3 and 4 for both the trace and anisotropycompared with experiment [115–117].

That the present trace and anisotropy polarizabil-ities are better than those previously available, this is

Figure 3. The empirical model of the pair polarizability anisotropy �ðrÞ with the different literature models.

Figure 2. The empirical model of the pair polarizability trace �ðrÞ with the different literature models.

approach is avoided. The method of moments analysishas been applied to the collision-induced light scatter-ing (CILS) spectra of Ne, Ar, Kr, Xe and CH4 [121].Because the correction to the first-order DID expres-sion is quite small over most of the interatomicseparations probed by the atomic motions, and thetwo correction terms to a large extent cancel out theeffects of each other, the numerical values derived forthe parameters to of the trace and go of the anisotropy,are quite sensitive to the input values of the moments.

For the calculation of the theoretical values of themoments with the empirical models of the trace andanisotropy polarizabilities the empirical pair potentialTT labelled 7 in Table 1, was used.

Tables 3 and 4 give the numerical results of thesearch while Figures 2 and 3 show the different regionsof the polarizability behavior which are consistent withthe experimental results. Here, also, the first order(DID), PKF and ab initio models of the trace andanisotropy are reported.

Although the numerical values obtained for theseparameters in Tables 3 and 4 are very different, thecorresponding functions �ðrÞ and �ðrÞ are really verysimilar, as are evident in Figures 2 and 3. Importantdeviations from the two-term DID models occur onlyat separations r� 1.607 rm for the trace and r� 1.2495rm for the anisotropy and the models differ from each

other significantly only at separations which are notprobed by the molecules anyway.

The moment analysis program was easily modifiedto calculate the moments of the isotropic and aniso-tropic spectra as functions of to and go using theexpressions (18) and (19). The calculation requiresabout 15 min. of CPU time on the Pentium 4 computerfor 1500 values of both to and go.

The experimentally determined values of themoments, with error limits, now each define a rangeof acceptable values for the parameters. This approachto the construction of an empirical models for �ðrÞ and�ðrÞ is acceptable if all moments define a commonrange of, to and go, rt and ro values. As one can seefrom Tables 3 and 4, the agreement between theexperimental values and the theoretical ones using ourempirical potential is excellent and we have verifiedthat it remains acceptable for to ¼ 2:5625� 0:2375 A

3

and go ¼ 8:8� 3:7 A3

in the case of the trace andanisotropy models respectively.

6. Results

In the following the results of the analysis for heliumwill be described and the empirical models for both�ðrÞ and �ðrÞ are compared with the DID, PKF, FC,

Table 4. Pair-polarizability anisotropy models and calculated moments of the helium anisotropic light scattering spectrumobtained by using the present TT potential model at T¼ 294.5K.

Model ro go

M0ð10�2 A9Þ

Experimental1.7� 0.19 [116]Calculated

M1ð1010 A9=sÞ

Experimental-

Calculated

M2ð1024 A9=s2Þ

Experimental3.18� 0.48 [116]

Calculated

BKð10�35 C2m8J�2ÞCalculated 2.3341* [123]

Calculated

1 0.32 5.10 1.837 4.217 3.548 2.1592 0.315 7.0 1.805 3.961 3.30 2.1133 0.305 8.25 1.789 3.836 3.18 2.0894 0.30 9.85 1.768 3.693 3.047 2.0595 0.295 12.5 1.72 3.427 2.813 1.993

Note: *At T¼ 407.6K.

Table 3. Pair-polarizability trace models and calculated moments of the helium isotropic light scattering spectrum obtained byusing the present TT potential model at T¼ 294.5K.

Model rt to

M0ð10�4 A9Þ

Experimental2.6 [115]Calculated

M1ð109 A9=sÞ

Experimental3.05

Calculated

M2ð1023 A9=s2Þ

Experimental3.2� 1.3[115]Calculated

B"ðcm6=mole2ÞExperimental

�0.07 *� 0.01 [122]Calculated

1 0.358 2.325 2.858 3.5273 3.611 �0.08942 0.355 2.40 2.710 3.412 3.503 �0.08583 0.350 2.545 2.50 3.260 3.365 �0.08044 0.345 2.675 2.24 3.03 3.141 �0.0745 0.340 2.80 1.964 2.764 2.881 �0.067

Note: *At T¼ 323.15K.

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immediately apparent in Figures 4–7 when wecompared the calculated spectral profiles using thedifferent models of polarizabilities with the experimen-tal values.

The result of our analysis is therefore that heliumpairs develop an incremental polarizability trace andanisotropy during collisions, besides the DID one,which contributes substantially at intermediate-rangedistances and can be ascribed to other mechanisms of

electron cloud distortion, such as overlap and electron-correlation effects.

7. Conclusion

The present study further demonstrate that the TTpotential model is a very good representation of theinteratomic potential of helium gas. The treatments

100 200 300 400 500 600 7000.00001

0.0001

0.001

0.01

0.1

1

10

Theoretical anisotropic spectrum using DID

Theoretical anisotropic spectrum using CF

Theoretical anisotropic spectrum using FC

Theoretical anisotropic spectrum using PKF

Theoretical anisotropic spectrum using DF

Theoretical anisotropic spectrum using MHWA

Theoretical anisotropic spectrum using CCSD(T)

Theoretical anisotropic spectrum using the empirical anisotropy

Experimental anisotropic spectrum[116]

)(cm 1-ν

I ani

so (

10-5

7 cm

6 )

Figure 4. The anisotropic light scattering spectrum of He-He at T¼ 294.5K using TT potential given in Table 1.

Figure 5. The isotropic light scattering spectrum of He-He at T¼ 294.5K using TT potential given in Table 1.

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proposed in this study may represent an improvementof the model from the thermodynamics and transportproperties over a wider temperature range. Also, wehave adopted a models for the pair-polarizability trace�ðrÞ and anisotropy �ðrÞ with adjustable parameters,which we determine by fitting to the first few evenmoments of the measured isotropic and anisotropiclight scattering spectra at T¼ 294.5K and 99.6K usingmethods of classical mechanics.

The different models derived for �ðrÞ and �ðrÞ forhelium gas produce isotropic and anisotropic

lineshapes in good agreement with experiments. Also,

the calculations of the second dielectric constant using

our empirical models of the trace polarizabilities agree

very well with the experimental values.

References

[1] G.C. Tabisz, Specialist Periodical Rep. Chem. Soc. 6, 136

(1979).[2] L. Frommhold, Adv. Chem. Phys. 46, 1 (1981).

50 100 150 200 250 3000.001

0.01

0.1

1

10Theoretical anisotropic spectrum using DID








Experimental anisotropic light scattering [117]

)(cm 1-ν

I ani

so (

10-5

7 cm

6 )


100 200 300 4000.001

0.01

0.1

1

10Theoretical isotropic spectrum using DID trace

Theoretical isotropic spectrum using CF trace

Theoretical isotropic spectrum using FC trace

Theoretical isotropic spectrum using PKF trace

Theoretical isotropic spectrum using DF trace

Theoretical isotropic spectrum using MHWA trace

Theoretical isotropic spectrum using CCSD(T) trace

Theoretical isotropic spectrum using the empirical trace

Experimental isotropic light scattering [117]

)(cm 1-ν

I iso

(10-5

8 cm

6 )


immediately apparent in Figures 4–7 when wecompared the calculated spectral profiles using thedifferent models of polarizabilities with the experimen-tal values.

The result of our analysis is therefore that heliumpairs develop an incremental polarizability trace andanisotropy during collisions, besides the DID one,which contributes substantially at intermediate-rangedistances and can be ascribed to other mechanisms of

electron cloud distortion, such as overlap and electron-correlation effects.

7. Conclusion

The present study further demonstrate that the TTpotential model is a very good representation of theinteratomic potential of helium gas. The treatments

100 200 300 400 500 600 7000.00001

0.0001

0.001

0.01

0.1

1

10

Theoretical anisotropic spectrum using DID








Experimental anisotropic spectrum[116]

)(cm 1-ν

I ani

so (

10-5

7 cm

6 )



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320 M.S.A. El-Kader

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703 (1976).

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56, 4119 (1972).[38] G.P. Flynn, R.V. Hanks, N.A. Lemaire and J. Ross,

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322 M.S.A. El-Kader

Appendix

For the specific functional form of the induced trace andanisotropy given by Equations (28) and (29), the ninefunctions aij(X) and bij(Y) are

a11ðB1, to, rtÞ ¼ �2� expð�r=rtÞa12ðB1, to, rtÞ ¼ 2�tor expð�r=rtÞ

a13ðB1, to, rtÞ ¼ 2�=r8

a21ðB1, to, rtÞ ¼�h

�rt�0 expð�r=rtÞ


�ð�0ð1� r=rtÞÞto expð�r=rtÞ

a23ðB1, to, rtÞ ¼ �8�h

�

�0

r9

� �


�

� �2 �

4

1

rt

� �4

þ �000

4

� �

� �0

2r3t� �

000

2rt

�þ 1

�

� ��

V0

rt� 2V

r2t

� ��

� ð�0V0 þ 2�00VÞÞ þ 2

�

� ��

r2tþ �00

� �

� 1

�2

� ��

r2r2tþ �

00

r2þ �

r3rt� �

0

r3

� ��expð�r=rtÞ


�

� �2 � �4

r

rt� 4

� �tor3t

� �

� �000 0rto4

� �þ �

0to2r2t

r

rt� 3

� �þ �

000to2

� �r

rt� 1

� �!

� 1

�

� �ð� V0r

rt�V0 � 2Vr

r2tþ 4V

rt

� �to

þ ð�0V0 þ 2�00VÞrtoÞ

� 2

�

� ��

r

rt� 2

� �tort

� �þ �00tor

� �

þ 1

�2

� ��

r2r

rt� 2

� �tort

� �þ �

00

rto

�

þ �

r3r

rt� 1

� �to �

�0

r2to

��expð�r=rtÞ


�

� �2

�180 � 11�r12

� �000 0

4r8þ 360�0

r11

� �

þ 4�000

r9

!� 1

�

� ��

8V0

r9� 144V

r10

� ��

� ðV0�0 þ 2�00VÞ

r8

�

� 2

�

72�

r10þ �

00

r8

� ��

þ 1

�2

� �80�

r12þ �00

r10� �0

r11

� �

b11ðB2, go, roÞ ¼ �2� expð�r=roÞb12ðB2, go, roÞ ¼ 2�gor expð�r=roÞb13ðB2, go, roÞ ¼ 2�=r8

b21ðB2, go, roÞ ¼�h

�

�0

ro� 6�

r2

� �expð�r=roÞ


��0 1� r

ro

� �þ 6�

r

� �go expð�r=roÞ


�� 8�0

r9þ 6�

r10

� �


�

� �2 �

4

1

ro

� �4

þ �000 0

4

� �� 0

2r3o

� �000

2roþ 6

�0

r2ro� 6

�

r3roþ 6

�0

r3� 18

�

r4

�

þ 1

�

� ��

V0

ro� 2V

r2o

� �� ð�0V0 þ 2�00VÞ

� �

þ 2

�

� ��

r2oþ �00

� �� 1

�2

� ��

r2r2oþ �

00

r2

�

þ �

r3ro� �

0

r3þ 6

�

r4

!!exp�� r=ro

!

b32ðB2, go, roÞ ¼

�h

�

� �2 � �

4

r

ro� 4

� �gor3t

� �000 0rgo4

� �

þ�0go2r2o

r

ro� 3

� �þ �

000go2

� �r

ro� 1

� �!

� 6�0gor2

r

ro� 1

� �þ 6�

r

ro� 1

� �gor3

� 6�0gor2

þ 18�gor3

� 1

�

� �ð�ðV0go

r

ro� 1

� �

� 2Vgo�r

ro� 2

� �!� ðV0�0 þ 2�00VÞ gor

!

� 2

�

� � �

r

ro� 2

� �goro

� �þ �00gor

!

þ 1

�2

� ��

r2r

ro� 2

� �goro

� �þ �

00

rgo

�

þ �

r3go

r

ro� 1

� ��

0

r2go

þ6go�

r3

��expð�r=roÞ


�

� �2

� 7854�

r12� �

000

4r8

�

þ 306�0

r11þ 4�

000

r9

�� 1

�

� ��

8V0

r9� 144V

r10

� ��

� ðV0�0 þ 2�00VÞr8

�

� 2

�

� �72�

r10þ �

00

r8

� ��

þ 1

�2

� �86�

r12þ �00

r10� �0

r11

� �

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thermophysical properties and collision-induced light scattering as a probe for gaseous helium...

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