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A new empirical potential energy function for Ar2

Philip T. Myatt, Ashok K. Dham, Pragna Chandrasekhar, Frederick R. W.McCourt & Robert J. Le Roy

To cite this article: Philip T. Myatt, Ashok K. Dham, Pragna Chandrasekhar, Frederick R. W.McCourt & Robert J. Le Roy (2018) A new empirical potential energy function for Ar2, MolecularPhysics, 116:12, 1598-1623, DOI: 10.1080/00268976.2018.1437932

To link to this article: https://doi.org/10.1080/00268976.2018.1437932

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MOLECULAR PHYSICS, VOL. , NO. , –https://doi.org/./..

RESEARCH ARTICLE

A new empirical potential energy function for Ar

Philip T. Myatta, Ashok K. Dhama,b, Pragna Chandrasekhara, Frederick R. W. McCourta and Robert J. Le Roya

aDepartment of Chemistry, University of Waterloo, Waterloo, Ontario, Canada; bDepartment of Physics, Punjabi University, Patiala, India

ARTICLE HISTORYReceived July Accepted January

KEYWORDSAr potential curve; Ar virialcoefficients; Ar transportcoefficients; empiricalanalysis

ABSTRACTA critical re-analysis of all available spectroscopic and virial coefficient data for Ar2 has been used todetermine an improvedempirical analytic potential energy function that has been ‘tuned’to optimiseits agreement with viscosity, diffusion and thermal diffusion data, and whose short-range behaviouris in reasonably good agreement with the most recent ab initio calculations for this system. The rec-ommended Morse/long-range potential function is smooth and differentiable at all distances, andincorporates both the correct theoretically predicted long-range behaviour and the correct limitingshort-range functional behaviour. The resulting value of the well depth isDe = 99.49 (±0.04) cm−1

and the associated equilibrium distance is re = 3.766 (±0.002) A, while the 40Ar s-wave scatteringlength is −714 A.

1. Introduction

Diatomic molecules in their ground electronic states maybe assigned fairly readily to one of two distinct classes,one being chemically bound species with deep poten-tial energy wells that support numerous excited vibra-tional and rotational energy levels, the other being physi-cally bound (Van derWaals) species which generally havequite shallow potential energy wells that support rela-tively few vibrational and rotational levels. The formerclass is typified by diatomic molecules formed by pairs ofopen-shell atoms, while the latter is typified by diatomicmolecules formed by pairs of closed-shell atoms, in par-ticular, noble gas and alkaline earth atoms. Diatomicmolecules formed from pairs of open s-shell atoms, suchas hydrogen or alkali atoms, have both chemically boundground singlet electronic states andweakly bound excited

CONTACT Robert J. Le Roy leroy@uwaterloo.caSupplementary material for this article can be accessed at https://doi.org/./...

triplet ‘Van der Waals’ electronic states sharing the samedissociation asymptote. By and large, the developmentof characteristic functional forms to represent interac-tion energies between atom pairs of these two classes ofspecies followed separate paths. This ultimately unneces-sary distinction between intramolecular and intermolec-ular interactions for (diatomic) molecules was reinforcedby the fact that copious amounts of vibration–rotationspectroscopic data tended to be available for ‘chemicallybound’ species, while relatively fewer data were availablefor ‘physically bound’ species.

Representation of the interaction energy between apair of closed-shell atoms has a long history, begin-ning with the rather simplistic (and unrealistic) fifth-power repulsive force law introduced by Maxwell [1]in the nineteenth century, followed by introduction of

© Informa UK Limited, trading as Taylor & Francis Group

MOLECULAR PHYSICS 1599

simple two-term repulsive-attractive representations ofthe interaction energy introduced during the first halfof the twentieth century, as exemplified by the two-parameter (Lennard-)Jones (LJ) [2,3] and exponential-six (exp-6) [4,5] models, both of which incorporatedthe form of the leading inverse-power dispersion energyterm associated with pairs of closed-shell atoms. Themost common LJ potential energy function (PEF), char-acterised by a minimum at energy −De and radial posi-tion re, is the sum of a repulsive term that varies withatomic separation r as r−12 and an attractive term thatvaries as r−6. However, it was realised soon after theintroduction of the LJ model in 1924 that, while such asimple two-term inverse-power form was adequate fordescribing the relatively imprecise experimental virialand transport coefficients available at the time, it waswholly inadequate for describing the much more precisemeasurements of spectroscopic transition frequencies ofchemically bound diatomic molecules. This led to peo-ple thinking of intramolecular and intermolecular inter-action energies in qualitatively, rather than only quan-titatively, different manners. Indeed, in 1929, Morse [6]proposed the two-term three-parameter functional formnow known as the Morse potential, in which exponen-tial functions were employed to represent both repulsiveand attractive branches of the interaction energy betweenpairs of (open-shell) atoms, and showed that it providedan adequate description of most of the then-availablediatomic molecular vibrational spectroscopic data.

Although early second virial and viscosity coefficientswere available over reasonably extensive temperatureranges, they could still adequately be described in termsof simple two-parameter models of the LJ form becausetheir values could only be determined within experimen-tal uncertainties of the order of a few per cent. Onceexperimental uncertainties were reduced to fractions of1%, such overly simple forms were no longer adequate,and more sophisticated functional forms were required.For example, the two-parameter LJ model was modi-fied to give a three-parameter extended LJ model [7]in which the inverse power of the long-range attractiveterm was fixed at six while the inverse power of therepulsive term was replaced by a constrained linear func-tion of the interatomic separation, and this model suf-ficed to provide a consistent rendering of the temperaturedependences of the available second (pressure) virialcoefficient and transport coefficients. More realistic rep-resentations of both the repulsive and attractive compo-nents of the interaction energy based upon theoreticalconstraints were also introduced around the same time.Of particular note were the multiparameter Hartree–Fock plus damped dispersion (HFD) [8,9] and exchange-Coulomb (XC) [10,11] models. Although themost recent

multiparameter versions of these functional forms for thepotential energy may readily be tuned to give good rep-resentations of the temperature dependences of both sec-ond virial and bulk transport properties, it proved to beconsiderably more challenging to obtain unique accuratefunctional representation of all properties.

It should also be pointed out that even though quan-tum chemical computational methods have recentlybecome able to provide rather accurate ab initiopair inter-action energies, for all but the smallest chemically boundspecies (e.g. H2 and H+

2 ) they are still not sufficientlyaccurate to be able to predict spectroscopic transitionswith a precision that matches experiment. Consequently,empirical potential energy functional forms remain themainstay for the determination of highly accurate inter-action energies for chemically bound diatomic species.The most accurate available ab initio interaction ener-gies for the ground electronic states of noble gas pairs,specifically He2 [12,13], Ne2 [14] and Ar2 [15,16], giveexcellent agreement with experimental values of the virialand transport coefficients of these bulk gases, and givegood agreement with vibrational level spacings deducedfrom spectroscopic data. However, it has not yet beendemonstrated clearly that the best published ab initioAr2(X 1�+

g ) PEFs [15,16] are sufficiently accurate to givedirectly computed spectroscopic transition energies thatlie within the reported experimental uncertainties (see,however, Section 5.1).

It has been well established that the shapes andradial positions of PEFs for chemically bound diatomicmolecules in their ground electronic states aremost accu-rately determined from analyses of the transition ener-gies associated with their allowed spectroscopic transi-tions amongst the plethora of molecular states possessedby stable molecules. A number of different representa-tions of the PEF for a diatomic molecule have been pro-posed over the years, arguably beginning with the MorsePEF [6] in 1929. The Morse PEF was followed in 1932by Dunham’s introduction of a PEF written as a polyno-mial in powers of a dimensionless radial variable [17],together with analytic expressions relating the lower-order coefficients of the polynomial PEF to the coeffi-cients of empirically determined polynomial expressionsrepresenting the molecular energy levels as double powerseries in the vibrational and rotational quantum num-bers. For much of the next half century, spectroscopicdata analyses focussed upon performing empirical fits tothe Dunham double-power-series expansion and, fromthe mid 1960s onward, generating sets of turning pointsfrom the resulting functions using the semiclassicalRydberg–Klein–Rees [18] (RKR) procedure. However,relatively little attention was paid to finding more appro-priate analytic forms for the underlying PEFs.

1600 P. T. MYATT ET AL.

A key subsequent development in analytic functionalforms for chemically bound molecules was due to Coxonand Hajigeorgiou [19], who showed that a Morse-typePEF with an exponent coefficient that varied with dis-tance was an extremely flexible form that required sub-stantially fewer parameters to represent a given spec-troscopic data-set accurately than did an equivalentDunham expansion. Replacing its radial distance vari-able by a dimensionless radial variable [20,21] that mapsthe semi-infinite domain [0, �) onto the finite domain[−1,+1], and introduction of a dimensionless prefactor tothe exponential term [22] that enables the PEF to have anyspecified long-range tail behaviour [23–25] then yieldedthe ‘Morse/long-range’ (MLR) PEF form [25,26].

MLR PEFs have been employed successfully to obtainexcellent representations of large, and often diverse spec-troscopic data-sets for a variety of ground and excitedstate species, such as N2 [27], MgH(D) [23,28], Li2 [24],hydrogen halides [29] and CH+ [30]. Moreover, suchstudies have also provided excellent representations ofboth ‘conventionally shallow’ Van der Waals potentialssuch as the a 3�+

u state of Li2 [31], and the ground statesof the semi-chemically bound Van der Waals species Be2[32] andMg2 [33]. This functional formhas been selectedfor the present redetermination of the PEF for the X 1�+

gstate of Ar2.

It was not until 1970, when Tanaka and Yoshino [34]reported their first measurements of electronic transi-tions in argon, that discrete vibrational spectroscopic databecame available for a noble gas diatomic species. How-ever, their vibrational band-head measurements sufficedonly to determine the energies of six of the nine [35]vibrational levels of ground-state Ar2. A few years later,a similar study employing a higher-resolution spectro-graph that delineated the rotational structure of thosesame six vibrational levels of Ar2 allowed Colbourn andDouglas [36] to obtain a RKR potential for ground-stateAr2 , to which they attached an analytic long-range tailconsisting of a sum of inverse-power terms. A decadelater, Herman et al. [37] utilised high-resolution VUVemission from three excited electronic (excimer) states ofAr2 to obtain improved values of the band constants andrelative ro-vibrational level energies for v = 0,… , 5 of theelectronic ground state plus a number of vibrational levelsfor three excited electronic states.

Gaseous argon has played an important role as oneof the key substances utilised for establishing bench-mark standards for equation-of-state and transport prop-erty measurements. For this reason, there has been astrong interest in obtaining a highly accurate pair inter-action PEF for its ground electronic (X 1�+

g ) state. Thefirst serious attempt to attain this goal was carriedout by Barker and coworkers [38–40], having shown

conclusively [41,42] that both the popular LJ and exp-6model functional forms were inadequate. As new micro-scopic [34,36,43,44] and bulk [45–47] experimental data,plus more extensive and/or accurate [48–54] experi-mental transport data became available for argon, moresophisticated functional forms, especially of the HFD-type [8,9], began to be employed rather successfully [55–57] to represent the dependence of the potential energy,V(r), on the separation, r, between a pair of ground-electronic-state Ar atoms. However, much of that workconsisted of manual (trial-and-error) fits, and none of itcombined fully quantum simulations of the actual spec-troscopic data together with high-level simulations ofthe bulk property measurements in a unified non-linearleast-squares fit analysis. The present work describes suchan analysis and determines an improved empirical ana-lytic PEF for ground-state Ar2.

Over the past decade or so, significant improve-ments in both computer hardware and quantum chem-ical methodology (reviewed extensively in Refs. [15] and[16]) have enabled increasingly accurate ab initio com-putations of V(r) for the Ar2 dimer [58–65], culminatingin the essentially converged results reported in Refs. [15]and [16], both of which were fitted to (different) multi-parametermodifications of the TT version [9] of theHFDfunctional form. In contrast, the present work will pro-vide what will hopefully be the final and ‘best’ empiricalpotential, against which such calculations may be tested.

2. Empirical determination of diatomicpotential energy functions

Chemically bound diatomic molecules typically supportlarge numbers of vibrational and rotational levels in theirelectronic ground states, which in principle allows awealth of pure rotational and ro-vibration data, as well aselectronic ro-vibrational data, to be obtained and utilisedfor the determination of accurate PEFs. Since the turn ofthe present century this process has increasingly involvedthe application of robust numerical Schrödinger-solvercomputer programs to preliminary, but realistic, param-eterised analytic model functions V(r; {pj}) to gener-ate estimates of the energies E(v ′J′) and E(v ′′, J′′), ofthe upper, respectively, lower energy levels (expressed incm−1 units) associatedwith each experimentally reportedspectroscopic transition. This process yields a set of sim-ulated transition energies

νcalc(v′, J′; v ′′, J′′) = E(v ′, J′) − E(v ′′, J′′), (1)

for comparison with the corresponding experimentaldata (traditionally given as a set of wave-number val-ues). An optimised analytic PEFmay then be obtained by

MOLECULAR PHYSICS 1601

carrying out a sequential fitting procedure that involvesa systematic variation of the set of potential param-eters {pj} in order to minimise the root-mean-squaredeviation (RMSD) between simulated and experimentalresults. This procedure is facilitated by the fact that theSchrödinger solver that yields an eigenvalue E(v , J) nor-mally also yields the associated eigenfunction ψv , J(r),which in turn allows the Hellmann-Feynman theoremexpression

∂E(v, J)∂ p j

=⟨ψv,J

∣∣∣∣∂V (r; {p j})∂ p j

∣∣∣∣ ψv,J

⟩to be used for efficient calculation of the partial deriva-tives with respect to the potential function param-eters {pj} required for the least-squares parameter-optimisation procedure.

Because it remains a well-entrenched tradition in boththe spectroscopic and PEF/surface research communitiesto report spectroscopic transitions and potential energiesin terms of wave-numbers and cm−1 units and to reportwavelengths and separations in Angstrom (A) units, wehave chosen to conform with this tradition.

Physically bound diatomic molecules support manyfewer bound vibrational and rotational levels than dochemically bound diatomicmolecules, with a consequentreduction in the number of spectroscopic transitionsavailable for observation and analysis. As dipolemomentsand dipole-moment-derivatives for such molecules tendto be small, pure rotational and rovibrational transitionsare relatively weak. Moreover, their typically rather smallbinding energies result in their thermal populations beingquite low in most environments. Hence, the amount ofhighly accurate spectroscopic data available is often insuf-ficient to enable a fully reliable empirical PEF for theatom–atom potential energy well to be obtained by fit-ting to them alone. In any case, such eigenvalue differ-ences would only be directly sensitive to the PEF at ener-gies below the dissociation asymptote, so that interactionenergies for separations smaller than the location, σ , ofthe (finite-separation) zero of V(r) cannot be determineddirectly from discrete spectroscopic data.

As short-range interaction energies between pairsof rare gas atoms play significant roles in determin-ing bulk properties of these species, it is important toincorporate experimental data that are influenced by thenature of the repulsive wall of the PEF into the fit-ting procedure. Fortunately, bulk-gas phenomena, suchas equation-of-state and acoustic virial coefficients mayreadily be treated together with spectroscopic data insuch a ‘direct-potential-fit’ (DPF) analysis. Moreover,experimental data for transport properties, such as shearviscosity, thermal conductivity and, to a more limited

extent, diffusion and thermal diffusion, may also beutilised for refining PEFs for species bound by Van derWaals interactions.

Equation-of-state virial data, in the form of the sec-ond virial coefficient, B2(T), hereinafter referred to as the‘pressure’ virial coefficient, are often available over a fairlyextensive temperature range. Values of the pressure virialcoefficient are obtained from fits of pressure-volume datato the virial equation

PVRT

= 1 + B2(T ) ρ + B3(T ) ρ2 + · · · , (2)

in which P is pressure,V is molar volume and ρ its recip-rocal,R is the universal gas constant,T is the temperature,and B2(T), B3(T),… , are the pressure virial coefficients.We shall be concerned here solely with the second virialcoefficient, B2(T), as it depends only upon the two-bodyinteraction.

At sufficiently high temperatures, classical mechani-cal values for second virial coefficients may be computedwith an accuracy that is better than the experimentaluncertainty. At lower temperatures, however, quantummechanical corrections may become important enoughto require their inclusion. Except for gaseous He at alltemperatures, and for Ne at very low temperatures, it suf-fices to add the semi-classical Wigner-Kirkwood correc-tions, which leads to the expansion [66]

B2(T ) = B2,cl(T ) +(

�2

2μ

)B(1)2,qm(T )

+(

�2

2μ

)2

B(2)2,qm(T ) + . . . , (3)

with � the Planck constant divided by 2π , and μ thereduced mass of the interacting pair. The term B2,cl(T ) isthe classical contribution, while B(1)

2,qm(T ) and B(2)2,qm(T )

are the first and second quantum corrections, respec-tively. These quantities are given by

B2,cl(T ) = −2πN0

∫ ∞

0

[e−βV (r) − 1

]r2 dr, (4)

B(1)2,qm(T ) = 2πN0β

∫ ∞

0[βV ′(r)]2e−βV (r)r2 dr, (5)

and

B(2)2,qm(T ) = −πN0β

2

6

∫ ∞

0

{110

[βV ′′(r)]2+ 15r2

[βV ′(r)]2

+ 19r[βV ′(r)]3 − 1

72[βV ′(r)]4

}e−βV (r)r2 dr,

(6)

1602 P. T. MYATT ET AL.

in which V′(r) and V′′(r), respectively, are the first andsecond radial derivatives of the PEF. Straightforwardapplications of calculus readily yield the analogous inte-gral expressions for the partial derivatives with respectto potential function parameters, �B2(T; {pi})/�pj , thatare required for incorporating this property into a least-squares-fit procedure. Note that although third quantumcorrections to the pressure virial coefficients are not dif-ficult to compute, their contributions are typically muchsmaller than the experimental uncertainties at all temper-atures for which argon pressure virial coefficient data areavailable: for this reason, they have been neglected.

While spectroscopic data relate directly to the inter-action energy (pair-potential) in that each spectroscopictransition energy (1) is determined by the differencebetween the eigenenergies of the initial and final molecu-lar states, virial coefficient data are encumbered by a layerof averaging that involves a thermally weighted integra-tion over all possible interatomic separations. This aver-aging makes the virial data inherently less sensitive to thedetailed shape of the PEF. Moreover, pressure virial coef-ficient measurements typically have experimental uncer-tainties of the order of 1%, and are thus relatively lessprecisely determined than spectroscopic transition fre-quencies. These disadvantages are nonetheless partiallycountered by the ability to obtain virial coefficient dataover extensive temperature ranges (of order 100–3000K),thereby enhancing their overall sensitivity to the nature ofthe short-range repulsive wall.

Due to relatively recent improvements in acousticresonators, adiabatic speed-of-sound measurements cannow be carried out much more accurately than equation-of-state pressure-volume measurements, with experi-mental uncertainties typically of order 0.1%. Speed-of-sound datamay be represented by a virial equation havinga form [67,68] similar to that of Equation (2), namely,

u2ad(T, ρ) = RTγ ◦

M[1 + β2a(T ) ρ + β3a(T ) ρ2 + · · · ]

,

(7)

in which uad is the adiabatic speed of sound, γ ° is theideal gas value of the heat capacity ratio γ � CP/CV =5/3 ,M is the molecular weight, (RTγ ◦/M)

12 = u◦

ad is theadiabatic speed of sound for an ideal gas, while β2a(T)and β3a(T) are the second and third acoustic virial coef-ficients, respectively. We shall be concerned here onlywith β2a(T), which is related to the second pressure virialcoefficient B2(T) and its first and second derivatives withrespect to temperature by [67–69]

β2a(T ) = 2B2(T ) + 2(γ ◦ − 1)TdB2

dT

+ (γ ◦ − 1)2

γ ◦ T 2 d2B2

dT 2 . (8)

Integral expressions for classical and quantum correctioncontributions to β2a(T) and their partial derivatives withrespect to potential function parameters may be obtainedreadily from Equations (3)–(6) and (8).

In the 1960s–1970s great hopes were placed on thepossible use of atomic/molecular beam scattering mea-surements as a source of information on interatomic andintermolecular potentials. Unfortunately, almost all scat-tering measurements for rare gas pairs were carried outfor collision energies in the range 50 cm−1 �Escatt �800 cm−1 that is already adequately covered by avail-able pressure and acoustic second virial coefficient data.Moreover, for most differential scattering cross-sectionalmeasurements, comparisons between experimental andcalculated results are hampered by a requirement for aver-aging over a set of experimental conditions that oftenwere not specified explicitly in the literature. In suchcases, comparisons could only be carried out by util-ising centre-of-mass scattering pseudo-data generatedfrom the (relatively crude) PEF that had been reported inthe original experimental work, a distinctly unsatisfyingprocedure.

The only reported experimental data that dependalmost exclusively on short-range repulsive rare gas atominteractions are a set of very high-energy atomic beamtotal integral scattering cross section measurements car-ried out by Rol and coworkers during the early 1970s.They obtained results for all possible noble gas pairsand fitted them to the short-range analytic form V(r) =A e−αr for particular ranges r � [rmin, rmax]. Values of A,α, rmin and rmax for the He–He system were reported inRef. [70], while analogous parameter sets for the otherfourteen high energy atom–atom interactions were com-municated to Aziz, and cited by him in a major reviewpaper in 1984 [71]. A set of 10 short-range interactionenergy pseudo-data generated from those Ar–Ar param-eters were also employed by Aziz [57] as part of the data-set utilised to determine his HFD-ID PEF for Ar2.

In the present work, spectroscopic data, equation-of-state pressure virial coefficients and acoustic second virialcoefficient data are treated as ‘primary’ data for optimis-ing the parameters that define the PEF for the groundstate of Ar2. Both B2(T) and β2a(T) probe the interac-tion potential energy inmuch the same fashion.However,as speed-of-sound measurements normally can be mademuch more accurately than equation-of-state pressure-volume measurements, acoustic virial coefficient valuesextracted from them are also considerably more accuratethanpressure-volume virial coefficient values (typically ofthe order of ±0.1% vs. ±1%).

MOLECULAR PHYSICS 1603

Transport properties such as viscosity, thermal con-ductivity and diffusion coefficients also depend on ther-mal averages of the PEF over all distances and hencemay be expected to provide insight regarding the natureof the short-range potential wall. However, their simu-lation is encumbered by three layers of integration, overatomic pair-separation, impact parameter and the relativekinetic energy of colliding atom pairs. They are also sub-ject to experimental accuracy/precision problems. Con-sequently, they are treated as ‘secondary’ data in thepresent analysis, in that they are utilised primarily to testthe predictive power of the PEF determined from a fit tothe primary data and in a manual (as opposed to auto-mated) fit to refine the final recommended PEF.

The empirical analysis reported in the present workwas carried out using programdPotFit of Ref. [72], whichperforms non-linear least-squares fits that employ theexact partial derivatives of the observables with respectto the PEF parameters, pj, to predict parameter changesfrom one fit-cycle to the next, until a specified degree ofconvergence has been attained. The quality of a multi-parameter fit of a PEF to a set (or sets) of data is typ-ically characterised by the value of the dimensionlessroot-mean-square deviation (DRMSD, or dd) betweencalculated, ycalc(i), and experimental, yobs(i), values forthe input data. This quantity, denoted herein by dd, isdefined as

dd ≡√√√√ 1

Ndat

Ndat∑i=1

(ycalc(i) − yobs(i)

ui

)2

, (9)

in which Ndat is the number of data fitted and ui is theexperimental uncertainty associated with datum i. Thequality of agreement for any particular subset of datamaybe computed in the same manner. As a result, if data sub-sets are labelled by ‘α’, then the overall dd value is relatedto the subset values ddα by

ddtot ≡ dd =[

1Ndat

∑α

Ndat(α) (ddα)2

] 12

,

with Ndat(α) the number of data of type α, and Ndat =�α Ndat(α) .

3. Models for the potential energy function

The traditional three-parameter Morse [6] PEF, VM(r),for the interaction energy between two chemical speciesseparated by a distance rmay be written as

VM(r) = De[1 − e−β(r−re)

]2, (10)

in which parameters re and De specify the location andstrength of the pair interaction energy at the minimumthat characterises equilibrium, while the third parameter,β , together with De, determines the curvature of VM(r)at re. The Morse PEF was employed by spectroscopiststo represent interaction potential energies of diatomicmolecular species formore than 50 years. Duringmuch ofthis same period, interaction PEFs extracted from analy-ses of the temperature dependence of macroscopic prop-erties, such as the second virial coefficient and transportcoefficients, were represented mainly by LJ PEFs. Thisdichotomy led to the practise of referring to diatomicPEFs extracted from molecular spectroscopic data as‘intramolecular’ and to those extracted frommacroscopic(bulk) property data as ‘intermolecular’.

Extensions of the Morse diatomic PEF were motivatedby the realisation that atomic interaction energies couldbe determined to spectroscopic accuracy by fully quan-tum mechanical direct non-linear least-squares fits of allavailable spectroscopic data to multiparameter algebraicfunctions [73–75]. This contrasted strongly with the tra-ditional route of defining the PEF by sets of turning pointpairs obtained by applying the semi-classical RKR pro-cedure to empirical analytic expressions for the vibra-tional energies and inertial rotational constants. An ini-tial effort in this direction, but still utilising semiclassicalRKR turning points, was the introduction of the ‘general-ized Morse oscillator’ (GMO) PEF by Coxon and Hajige-orgiou [76]. However, as the GMO functional form oftenextrapolates poorly [77], Lee et al. [78] introduced the‘expanded Morse oscillator’ (EMO) PEF, an analogousform that provides stable extrapolation to large and smalldistances, and hence is capable of yielding realistic predic-tions for the energies of high vibrational and/or rotationallevels. The EMO PEF (see also Ref. [26]) has the form

VEMO(r) ≡ De[1 − e−βEMO(r)(r−re)

]2, (11)

with an exponent-coefficient ‘shape’ function βEMO(r)represented by a polynomial expansion

βEMO(r) =Nβ∑i=0

βi [yrefq (r)]i (12)

in powers of a generalised Šurkus variable [21] yrefq (r)defined as

yrefq (r) ≡ rq − rref q

rq + rref q, (13)

with the values of q and of the location of the expansioncentre rref being selected manually.

1604 P. T. MYATT ET AL.

Somewhat later, the ‘Morse/long-range’ (MLR) func-tion, a fully analytic PEF that explicitly incorporates real-istic long-range behaviour, was introduced by Le Roy andHenderson [23]. In its more general current form it maybe written as [25,26]

VMLR(r) ≡ De

[1 − uLR(r)

uLR(re)e−βMLR(r) yeqp (r)

]2. (14)

This function may be thought of as a generalisation ofthe original three-parameter Morse [6] PEF of Equation(10) in which the introduction of the exponential prefac-tor, the use of a dimensionless radial variable yeqp (r), andthe imposition of a fixed limiting-long-range value forthe exponent function βMLR(r) yeqp (r) combine to con-strain its limiting long-range behaviour to have a specifiedform, uLR(r). MLR functions are everywhere continuousand differentiable to all orders, and can incorporate anyplausible form for uLR(r) [24,31].

The dimensionless radial variable yeqp (r) appearing inEquation (14) has the same form as the expansion vari-able of Equation (13), but with the integer q replaced by p,and with the reference distance rref replaced by the equi-librium distance re. It plays the same role in VMLR(r) thatthe difference (r− re) plays in theMorse PEF, except thatyeqp (r) approaches 1 (rather than �) for very large sepa-rations. Similarly, the slowly varying exponent-coefficientshape function

βMLR(r) ≡ β∞ yrefp (r) + [1 − yrefp (r)]Nβ∑i=0

βi[yrefq (r)]i,

(15)

is a generalisation of the shape parameter βEMO(r) ofEquation (12) that is constrained to approach a lim-iting value, β�, at very large r. The dimensionlesspolynomial expansion variable again has the form ofEquation (13), while the ‘switching function’ variableyrefp (r) has that same form, but with the integer q replacedby the same integer p used to define the dimensionlessdistance variable yeqp (r).

The constraint that causes VMLR(r) to assume thedesired long-range form

VMLR(r) ∼ De − uLR(r) + 1De

[uLR(r)]2 (16)

is the requirement that the limiting long-range value ofβMLR(r), and hence also of the overall exponent functionβMLR(r) yeqp (r) , be defined as

β∞ = ln[

2De

uLR(re)

]. (17)

The shape function βMLR(r) is thus defined in terms of aset ofNβ + 1 empirical shape parameters β i (i = 0, 1, … ,Nβ), the reference separation rref, and the (small, positive)integers p and q.

The long-range function uLR(r) that defines the expo-nential prefactor in the MLR function of Equation (14)is normally written as a sum of individually-dampedinverse-power terms, namely,

uLR(r) ≡mmax∑

m=mmin

D(s)m (r)

Cm

rm. (18)

The damping functions D(s)m (r) used herein are gener-

alised [25] Douketis–Scoles-type [8] damping functionswhose form causes VMLR(r) to behave as r2s for verysmall r. Rather than setting s = −1, as was suggested inRef. [25], the value s = −1/2 is employed in the presentwork, as it gives a Coulombic r−1 PEF behaviour at ultra-short separations. Values formmin , for the inverse powersof the leading long-range interaction terms, and for theassociated dispersion and/or induction coefficients them-selves, depend upon the specific natures of the interactingatoms. Choice of the value for mmax is, however, some-what more subjective.

The MLR functional form thus involves a total ofNβ + NLR + 7 free parameters: in addition to the Nβ +4 parameters (rref, q, {β i|i = 0, … , Nβ}, and Nβ) thatappear in the shape function β(r), there are NLR long-range dispersion/induction coefficients, the equilibriumseparation, re , andwell-depth,De, defining theminimumin the PEF, plus the power p determining the specific formfor the dimensionless radial switching function ypref(r).However, only re , De and the Nβ + 1 exponent expan-sion parameters {β i|i= 0, … ,Nβ} are varied freely in thepresent least-squares fitting procedure, while the power qand expansion centre rref were optimised manually, andthe power p is constrained to be greater than mmax −mmin so that the leading long-range contribution fromthe exponential factor in VMLR(r), which is proportionalto 1/rmmin+p, will not modify the long-range interactionenergy from the specified form, uLR(r) [23]. Because reli-able theoretical values for the Cm coefficients are oftenavailable from the literature most, if not all, of the rele-vant Cm values may normally be fixed at predeterminedvalues in the MLR model.

4. Determination of the Ar2(X 1�+g ) potential

energy function

4.1. Experimental data for gaseous argon

The first discrete spectra obtained for a ground-state rare-gas pair were the band heads for nine electronic band

MOLECULAR PHYSICS 1605

Table . Primary experimental data.

Literature source

Data type #Data Data range Unc.a Lead author (year) Ref.

UV spec. A 1u → X 0+g v ′ = – , v ′′ = – .–. Herman () []

UV spec. B 0+u → X 0+

g v ′ = – , v ′′ = – .–. Herman () []UV spec.C 0+

u → X 0+g v ′ = – , v ′′ = – .–. Herman () []

Pressure virial – K .–. Michelsb () []Coefficients – K .–. Michelsb () []

– K .–. Byrne () [] – K . Kestin () [] – K .–. Sevast’yanov () [] – K .–. Ewing () [] – K .–. Gilgenb () [] - K .–. Estrada-Alexandersb () [] – K .–. Klimeckb () [] – K .–. Cencek () []

Acoustic virial – K .–. Moldover () []Coefficients – K .–. Ewing () []

– K .–. Ewing () [] – K .–. Ewing () [] – K .–. Estrada-Alexanders () [] – K .–. Moldover () [] – K .–. Benedetto () []

aUncertainties in units [cm−] for VUV transitions (rows –), and [cm/mole] (rows –end) for second virialcoefficients.

bEquation-of-state pressure virial coefficients determined as explained in the text.

systems of Ar2 reported by Tanaka and Yoshino [34]in 1970. These results yielded estimates of the spacings�Gv + 1/2 for the five lowest vibrational levels, and analy-sis of the associated four vibrational level spacings yieldedestimates of the ground-state dissociation energy,D0 andwell depth De , and the leading vibrational constants ωeand ωexe [34,35]. Six years later, a re-examination of oneof the band systems at higher resolution enabled Col-bourn and Douglas [36] to carry out an analysis thatyielded rotational constants and improved values for thespacings amongst the first six vibrational levels, which inturn led to an RKR-based pointwise potential function.Finally, a decade later Herman et al. [37] used tunablecoherent vacuum ultraviolet (VUV) radiation producedwith four-wave-sum mixing in a nonlinear medium toexcite fluorescence fromcold (∼ 40K)Ar2 dimers formedin a supersonic beam expansion of argon gas. Theirresults had slightly higher precision than those of Col-bourn and Douglas [36], and their 1397 reported transi-tion energies comprise the spectroscopic data used in thepresent work (see the first segment of Table 1).

The earliest determinations of virial coefficients fromequation-of-state isotherm data for argon were reportedover a century ago. Since that time, equation-of-state datahave been collected in numerous laboratories and anal-ysed to extract second,B2(T), and third,B3(T), virial coef-ficients from them. Such virial data obtained for argonprior to 1998 have been summarised in the critical com-pilation of Dymond et al. [91]. For convenience, we shall

henceforward refer to virial coefficients determined fromequation-of-state data as ‘pressure’ virial coefficients.

We have chosen to include only pressure second virialcoefficient values that have been determined after 1940and with precisions better than 1%. The sources andtemperature ranges for the 178 data thereby selected forthe determination of a new empirical argon (MLR) PEFhave been summarised in the second segment of Table 1.The equation-of-state data of Michels et al. [79,80] andof Gilgen et al. [84] have been re-fitted using a non-linear least-squares procedure to obtain values of B2(T)with uncertainties that take into account all uncertain-ties in the experimental data and statistical uncertain-ties introduced through model dependence. Similarly,the equation-of-state data of Klimeck et al. [86] and thecompressibility values generated by Estrada-Alexandersand Trusler [85] from speed-of-sound data have beenanalysed using the same procedure to generate values ofthe second pressure virial coefficient plus correspondinguncertainties.

Reliable and accurate measurements of the speed ofsound in argon as a function of temperature and densityhave become available during the past 30 years. Nonlinearleast-squares fits to a virial expansion of such speed-of-sound data have enabled the extraction of second acousticvirial coefficients,β2a(T), with small statistical uncertain-ties. The sources and temperature ranges for speed-of-sound data and the 59 values for the second acoustic virialcoefficient obtained from these data are summarised in

1606 P. T. MYATT ET AL.

the third segment of Table 1. Indeed, the uncertaintyranges presented in Table 1 for the two types of virialcoefficient clearly demonstrate typical differences in pre-cision.

The data described in Table 1 are the ‘primary data’that may be employed for DPF analyses using the cur-rent version of program dPotFit. In addition, however,transport property measurements, especially those madeat elevated temperatures, are also sensitive to the natureof the repulsive wall at distances r shorter than the sep-aration σ at which V(r) changes sign. Although trans-port property data cannot presently be utilised directlyin dPotFit, they have been employed in ‘forward’ calcu-lations to test the various PEFs and to ‘tune’ and improvethe PEF being determined in the final stage of the presentanalysis. Three classes of experimental results are con-tained in the transport property data-set. The first dataclass consists of sets of independent measurements, andcomprises 252 viscosity, 99 thermal conductivity, 26 self-diffusion and 45 thermal diffusion data. The second dataclass consists of a set of 22 relative viscosity and 11 rel-ative self-diffusion values, defined by the measurementof the ratio of a value of this property relative to a speci-fied reference value. The third data class consists of valuesobtained via smoothed fits to previously obtained exper-imental data: specifically, the data analysed in Ref. [82](values of η, λ, DA′A, α0 for 29 temperatures, 80K � T� 3273K), Ref. [92] (values of λ for 54 temperatures,80K�T� 2000K), and Ref. [93] (values of λ for 13 tem-peratures, 200K � T � 500K) make up this class. Theoverall set of 609 transport coefficient data employed inthis way, and consisting of 274 values for the (shear) vis-cosity coefficient, η, 195 values for the thermal conductiv-ity coefficient, λ, 66 values of the self-diffusion coefficient,DA′A, and 74 values for the reduced thermal diffusioncoefficient, α0, is summarised and sourced in Table 2.

4.2. Parameters to be fixed or varied in theMLRmodel

Of the Nβ + NLR + 7 fit-parameters that define the MLRfunctional form, the NLR + 1 parameters correspondingto the values of the long-range Cm coefficients and of theinteger power p are predetermined by the form chosenfor the long-range interaction energy uLR(r). In particu-lar, mmin = 6 because the long-range interaction of a pairof neutral closed-shell atoms is governed entirely by dis-persion interactions. As reliable values for the three lead-ing dispersion coefficients for Ar are available from theliterature [135,136] (see Table 5), we chose to setNLR = 3and fix the values for C6, C8 and C10 at literature values.Moreover, as the (integer) power p appearing in yeqp (r)

must exceed mmax − mmin = 4, for this choice of long-range coefficients, it is reasonable to set p = 6 (ratherthan to theminimal value 5) in order that the power of theleading long-range contribution arising from the expo-nential term in theMLR formmatch that of the r−12 termexpected to arise next in the dispersion energy series. Fol-lowing the ‘recipe’ of Douketis et al. [8], the dampingfunction range scaling parameter was set at ρ = 1.10 and,as mentioned near the end of Section 3, the very-short-range power parameter was fixed at s = −1/2.

Of the remaining Nβ + 6 parameters, De, re and theNβ + 1 exponent coefficient PEF ‘shape’ parameters of theMLR form are optimised automatically, while the expo-nent polynomial order Nβ , the integer parameter q andthe reference centre rref for the β(r) expansion are deter-mined manually. Initial values were Nβ = 0 and q = 3 ,and rref equal to the geometricmean of the turning-pointsfor the highest observed vibrational level (v = 5) of Ref.[37], a distance of 4.6 A. Manual refinement of rref led torref = 5.3 A as an optimal choice for the X 1�+

g groundelectronic state of Ar2. As changes in q had no significanteffect on the quality-of-fit, q = 3 has been employed inthe remainder of this work.

4.3. Fitting to spectroscopic data alone, andtreatment of upper state levels

The fluorescence spectra observed and assigned by Her-man et al. [37] consisted of transitions involving threeexcited electronic states. Having settled on the type ofPEF to be used to represent ground-state Ar2, the nextquestion to be addressed is how to treat the upper elec-tronic states. The simplest approach would be to treat allof the upper-state levels as independent term values. Forthis case, the uppermost segment of Table 3 shows howthe quality-of-fit to the data improves as the order of theMLR exponent polynomial increases from Nβ = 0 to 3.As 165 of the 1397 transitions are connected to only asingle X-state level, they carry no information about theX-state potential, and hence were omitted from this anal-ysis. The total number of free parameters in these fits thenconsisted of 302 upper-state term values plus the X-stateMLR parameters.

Since all of the 1397 data were assigned as mem-bers of rotational branches in vibrational bands, follow-ing Herman et al. [37], it would clearly be appropri-ate to represent those upper state level energies usinga set of band constants for each observed vibrationallevel of each upper state. When the energies of the rota-tional sublevels for each vibrational level of each state arerepresented by the three leading ‘mechanical’ bandconstants {Gv ′, Bv ′, Dv ′ }, with the constants for the A 1u

MOLECULAR PHYSICS 1607

Table . Secondary experimental (transport) data.

Temperature Literature source

Data type #Data Range /K % Unc. Lead author (year) Ref.

Viscosity – . Johnston () [] – . Flynn () [] , . Kestin () [] – . Clarke () [] – . Gracki () [] – . Guevera () [] – . Dawe () [] – . May () [] – . Kestin () [] – . Hellemans () [] – . Clifford () [] – . Vargaftik () [] – . Kestin () [] – . Vogel () [] – . Wilhelm () [] – . Evers () [] . . Berg () [] . . Berg () [] – . Lin () []

Thermal conductivity – . Faubert () [] – . Haarman () [] – . Springer () [] – . Chen () [] . . Fleeter () [] – . Clifford () [] – . Ziebland () [] . . Kestin () [] – . Haran () [] – . Kestin () [] – . Johns () [] – . Mardolcar () [] – . Younglove () [] – . Millat () [] – . Hemminger () [] – . Perkins () [] – . Roder () [] – . Sun () [] – . May () []

Thermal diffusion – . Stier () [] – . Moran () [] – . Paul () [] – . Stevens () [] – . Taylor () [] – . Rutherford () [] . Taylor () []

– . Kestin () []Isotopic diffusion – . Winn () []

– . De Paz () [] – . Vugts () [] – . Kestin () []

state extended to include the leading �-doubling con-stant qv ′ , the results appearing in the second segment ofTable 3 are obtained. While the number of fitted parame-ters is reduced bymore than a factor of four, this approachis accompanied by only about 10% increases in the valuesof dd, the ‘cost’ of imposing realistic physical models onthe energy levels of the three upper electronic states.

The two upper segments of Table 3 show that the thequality-of-fit parameter dd does not improve significantlyupon increasing the X-state exponent polynomial orderbeyond Nβ = 2. Moreover, most of the β i coefficientsyielded by the Nβ = 3 fits had uncertainties greater than

100%.Thus,we conclude that the highest-orderMLRPEFfor ground-state Ar2 that can be determined from thespectroscopic data alone is one for which Nβ = 2.

A final level of sophistication for treating the excimerenergy levels would be to represent them as levels of ana-lytic model PEFs. The fact that Herman et al. [37] wereable to provide absolute vibrational assignments for thoseupper-state vibrational levels, v ′ = 24 − 30 for the A 1ustate, v ′ = 20 − 27 for the B 0+

u state, and v ′ = 0 −4 for the C 0+

u state, made this a tantalising possibility.However, it was only partially realised, in that we wereable to determine a PEF model for theC 0+

u state, but not

1608 P. T. MYATT ET AL.

Table . Summary of fits to spectroscopic data alone, performedwith X 1�+

g state represented by an MLR PEF. Bold font indicatethe selected optimum result for each case.

dd for Total #

Stage #Data A u B 0+u C 0+

u Total param.

NXβ

Term values for all excimer energy levels . . . . . . . . 2 0.659 0.914 0.833 0.841 307 . . . .

NXβ

Band constants for all excimer energy levels

. . . . . . . . 2 0.844 0.973 0.850 0.919 72 . . . .

NCβ

EMO PEF forC 0+u excimer state, with NX

β= 2

. . . . . . . . . . . . 4 0.844 0.972 0.889 0.923 64 . . . .

for the other two. In particular, with the levels of theX 0+u

state defined by anNXβ = 2MLR function with q= 3, p=

6 and rref = 5.3 A, but its other parameters allowed to vary,the fits were repeated using band constants for the A 1uand B 0+

u , states, but with an EMO PEF used to representthe levels of theC 0+

u state. For various values of the expo-nent polynomial order, the results of fits performed withq = 6 and rref = re are presented in the bottom segmentof Table 3. It is clear that those fits converge for NC

β = 4 ,and that the quality of fit there was essentially identical tothat obtained using band constants for theC 0+

u state. Theparameters defining the resulting C-state EMO potentialare reported in Table 5.

Table . Fitted parameters defining the recomen-ded for MLR potential a for ground-state (X 1�+

g )Ar and the EMO PEF b for the C 0+

u state discussedin Section ..

MLR PEF a for X 1�+g EMO PEF b forC 0+

uNX

β= NC

β=

Parameter (spec/vir/PP) (spec)

rref /A . re(C 0+u )

De /cm− .() .()

re /A .() .()β . .β . .β . −.β . −.β . .

aPredetermined MLR parameters: p= , q= , C =.× cm−A [], C = .× cm−A, C =.× cm−A [], and rref = . A.

bPredetermined q= , and an asymptote energy,. cm− above that for the ground state [].

4.4. Fitting to the full primary data set

A basic starting point for the inclusion of experimentalpressure and acoustic virial coefficients and other data inthe fitting procedure was a forward calculation of theirquality of agreement with predictions generated from theoptimum PEF determined from fits to the spectroscopicdata alone, namely, that associated with the bold-fontresults in the middle segment of Table 3. The dd valuesobtained in this way are shown in square brackets in thefirst column of Table 4. Since an ideal dd value for anydata-set or subset would be a number close to 1, it is clearthat this ‘spectroscopy-alone’ PEF yields poor predictionsfor the virial coefficients, especially for the higher qual-ity acoustic virial data. As it was partially based on valuesfor data not included in the analysis, the first entry in thefourth row of this table has also been presented in squarebrackets.

Table . Summary of overall fit results. a Numbers without bracketsare dd values for sets of data included in the fits; those in squarebrackets were obtained as predictions from a fitted potential.

dd values

spec spec / vir spec / vir / PP b

# Data Property NXβ

= 2 NXβ= NX

β= NX

β=

spectroscopy . . . . pressure virials [.] . . . acoustic virials [.] . . . all primary data [.] . . . transport data [.] [.] [.] . All Data [.] [.] [.] .

aAll MLR PEFs have rref = . A, p= , q= , and employ s= −½ dampingfunctions.

bAmanually tuned ‘pull-point’ (PP) of V(r= . A)= cm− was treated asan added datum in the fit.

MOLECULAR PHYSICS 1609

The next step was to re-optimise the NXβ = 2 X-state

MLR potential parameters while incorporating the virialcoefficients in the set of fitted data . This yielded the sec-ond columnof dd values shown inTable 4. As inclusion ofthe virial data caused a significant increase in the dd valuefor the spectroscopic data, the fit was repeated with theexponent polynomial order increased to NX

β = 3, whichyielded the third column of dd values in Table 4. Increas-ingNX

β further yielded no additional improvements in thedd values for the full primary data-set. The fact that theirdd values converge to numbers somewhat larger than 1implies that the uncertainties used for the two types ofvirial coefficient data may be somewhat too small.

4.5. Incorporating transport property data into theanalysis

While they cannot be treated as experimental data inDPFprogram dPotFit, transport properties have long beenconsidered to provide a useful test of the accuracy of rare-gas-pair PEFs, particularly in the short-range repulsivewall region. The first three entries on the fifth row ofTable 4 therefore list the dd values calculated for the 609transport property data of Table 2 using the three fittedpotentials described above. These ‘forward’ calculationswere carried out using a program based upon code listedin Appendix 12 of Ref. [137]. The trend seen there showsthat extending the MLR exponent polynomial order toNX

β = 3 in order to obtain improved agreement withthe virial data yielded a PEF whose predictions for thetransport properties were substantially worse (by 20% !).This indicated that a higher-order MLR function wouldbe needed in order to attain good fits to both the pri-mary (spectroscopy and virial coefficients) and secondary(transport property) data. Moreover, as the shape of thePEF in the well region is sharply defined by the spectro-scopic data and the theoretically known long-range coef-ficients, this (mis)behaviour was attributed to a problemwith the short-range repulsive wall.

As the final stage of the analysis, a series of fits to all ofthe primary data were then performed with the repulsivewall constrained to pass through amanually selected ‘pullpoint’ PEF value set at a distance of 2.8 A, at which therepulsivewalls of the first three potentials of Table 4 beganto diverge. In particular, a value of dd for the transportdata was calculated from each of the fitted PEFs obtainedas the energy at that ‘pull point’ was varied, in order tooptimise the dd value calculated for ‘All Data’ (row 6 inTable 4). As indicated by the header of the final columnin Table 4, retaining good agreement with the primarydata while applying this additional constraint requireda further increase in the order of the MLR exponent

polynomial to NXβ = 4. No further improvement in the

quality of fit was obtained for NXβ = 5.

In a comparison between the dd values seen in the lasttwo columns of Table 4, the 2.3% net improvement in ddfor ‘All Data’ (row 6) is mainly due to the 17.5% improve-ment in dd for the transport property data (row 5), coun-tered by a very small (0.18%) degradation in the dd valuefor the primary data (row 4). This NX

β =4 MLR PEFthus provides a statistically significant improvement inthe representation of the 2243 experimental data utilised(1634 directly, 609 indirectly) in its determination, andtherefore we deem it the new best empirical PEF forground-state Ar2. The parameter values defining thisrecomended MLR PEF for ground-state Ar2, togetherwith those for the EMO PEF for the C 0+

u state associ-ated with the second-last row of Table 3, are presentedin Table 5. The numbers in parentheses there are the 95%confidence limit uncertainties in the last digits shown forparameters with explicit physical significance. The com-pact final form of these parameter values was obtained byapplication of the sequential rounding and refitting pro-cedure of Ref. [139], whichminimises the numbers of sig-nificant digits required to describe a fitted data set with nosignificant loss of precision.

5. Comparisons with previous PEFs forAr2(X 1�+

g )

5.1. Quality-of-fit comparisons

Because of its importance (together withmolecular nitro-gen) as a primary calibration substance for equation-of-state and transport property measurements, argon hasalso served as a primary testing ground for the deter-mination of reliable PEFs. It is therefore appropriate tocompare the recommended empirical PEF of Table 5 withsome of the best previously reported functions. Table 6compares predictions of the properties of Tables 1 and 2taken from the last column of Table 4 with predictionsgenerated from two previous empirical PEFs, from tworecent high quality ab initio PEFs, and from a commonlyused semi-empirical analytic function.

The oldest empirical PEF considered here (column 4of Table 6) was an HFD-type function with individu-ally damped long-range dispersion contributions that wasreported by Aziz [57] in 1993. It was determined pri-marily via manual fitting to a combination of molecularbeam scattering data and second pressure virial coeffi-cient data, together with transport property data and keyspectroscopic properties (the inertial Bv constants andvibrational spacings, �Gv+ 1

2). Around that time, how-

ever, it was realised (see, e.g. Equation (8) ) that acoustic

1610 P. T. MYATT ET AL.

Table . Dimensionless root-mean-square deviations dd between calculated andexperimental values of four types of argondata, for six PEFmodels.dd values in squarebrackets in the first three columns, and all dd values in last three columns, were gen-erated by forward calculations using fixed potential functions.

dd

Empirical PEFs Ab initio PEFs TT model

PEF type: MLR HFD-ID TTmod TTmod TT

data type # Data NXβ= Aziz [] Boyes [] [] [] []

Spectroscopy . . . . . .Pressure virials . . [.] . . .Acoustic virials . [.] . . . .Primary data . . . . . .Transport data . . . . . .

All data . . . . . .

De /cm−1 .() . . . . .re / A .() . . . . .

virial coefficients probe essentially the same regions of thePEF as do pressure virial coefficients, and that speed-of-sound measurements can be made much more preciselythan equation-of-statemeasurements. As second acousticvirial coefficients had not been used in its determination,calculations of their values and their temperature depen-dence would offer a stringent test of the Aziz PEF. Such acomputation was carried out by Boyes [140], who foundthat the Aziz PEF failed badly in predicting β2a(T) val-ues (see row 3 of column 5 of Table 6). Boyes also carriedout a redetermination of that HFD-type PEF based uponthe input data employed by Aziz, but with the pressurevirial coefficient data replaced by the new acoustic virialdata, and a more quantitative treatment of the spectro-scopic data. This yielded a new HFD-type PEF that wasconsistentwith acoustic virial and spectroscopic data and,at the same time, gave a much improved level of agree-ment with the pressure virial and transport property data(see column 3 of Table 6). However, the present recom-mended MLR potential (see column 3 of Table 6) is indistinctly better overall agreement with the data than areeither of these older empirical PEFs.

The three final columns of Table 6 then present theresults of forward calculations of the properties of Tables1 and 2 using the recommended analytic representa-tions of the ab initio PEFs of Jäger et al. [15] and ofPatkowski and Szalewicz [16], and the widely cited HFD-type potential of Tang and Toennies [141]. While the‘TT’ model of the very last column had the very basicTang–Toennies-type PEF form, in which the exponentcoefficient of the repulsive Born–Mayer term was thesame as the scaling parameter in the dispersion damp-ing functions, the modified TT-type functions deter-mined fromfits to the ab initio points hadmultiparameterpre-exponential [16] or exponent coefficient [15] factors.

The predictions generated from these recent ab initio-based PEFs are clearly quite good, the one by Jäger et al.yielding a dd value for ‘All Data’ only 10% larger that thatfor our best empirical potential. This result attests to theincreasing reliability of high quality ab initio calculationsfor such a large many-electron system.

Results obtained using the two-parameter (De, re)TT03 model PEF [141] have been included in Table 6(last column) primarily to emphasise the fact that no two-parameter PEF, not even one with accurate long-rangecoefficients and parameter values (De and re) taken fromprevious empirical studies, is able to represent the Ar2interaction energy well enough to predict actual spec-troscopic transition energies, acoustic and pressure sec-ond virial coefficients, or transport coefficients close tobeing within the uncertainties of the experimental val-ues. Of course, this has been well known for simpletwo-parameter LJ and exp-6 model PEFs for many years[41,42], but it may not have been fully appreciated thatthis shortcoming also holds for more sophisticated two-parameter model PEFs, even those that include, as doesthe TT03 model, a sensible representation of the long-range part of the interaction.

5.2. Direct comparisons between PEFs

Visual comparisons of the present recommended NXβ =4

MLR PEF of Table 5 with two precursor best-fit PEFsfrom the earlier columns of Table 4, namely, the NX

β =2MLR PEF fitted to spectroscopic data alone and theNX

β =3 MLR PEF fitted to spectroscopic and virial data,as well as with the other PEFs of Table 6, are presented inFigures 1 and 2. While their recommended analytic rep-resentations were used to generate the results in the ‘AbInitio PEF’ columns of Table 6, only the ab initio points

MOLECULAR PHYSICS 1611

4 5 6 7 8-100.0

-80.0

-60.0

-40.0

-20.0

0.0

ener

gy/c

m−1

r / Å

1.8 2 2.2 2.4 2.6 2.8

104

105

ener

gy/c

m−1

r / Å

Nβ = 2{spec alone}

Nβ = 4{transport tuned}

TT03

Nβ = 3{spec & virial data}

Boyes94

Rol pointsfrom Aziz93

Aziz93

Figure . Potential energy functions for the interaction betweentwo ground-state argon atoms. Upper panel: short-range (. –. A) behaviour of V(r). Short-dashNβ= curve:MLR PEF (fitted tospectroscopic data alone); long-dash Nβ= curve: MLR PEF (fittedto spectroscopic and virial data); solid Nβ= curve: MLR PEF (fit-ted to spectroscopic and virial data, and tuned to transport data);dash-dot curve: Aziz [] PEF; dash-double-dot curve: Boyes []PEF; upper short-dash curve: TT model;� (hollow square) sym-bols: ab initio values from Ref. []; (solid, triangular) symbols:ab initio values from Ref. []; (black, diamond) symbols: Rolpseudo-data fromRef. []. Lower panel: intermediate-range (. –. A) behaviour of V(r).

themselves are shown in Figure 1. It is immediately evi-dent in the lower panel of Figure 1 that on an energyscale which spans the full well depth, all of these functionsbecome indistinguishable. However, on the logarithmicscale used for the short-range region in the upper panelof Figure 1 and the expanded-scale view of the potentialminimum in Figure 2, some distinctions become clear.

In particular, most PEFs are distinguishable from oneanother in the short-range repulsive (r < σ ) region con-sidered in the upper panel of Figure 1. The three fittedMLR PEFs shown here are: the NX

β =2 function that wasdetermined from a fit to the spectroscopic data alone(from column 1 of Table 4), the NX

β =3 MLR functiondetermined from a fit to the full primary data set (col-umn 3 of Table 4), and the recommendedNX

β =4 functionobtained by tuning PEFs fitted to the primary data andtuned to yield improved predictions of the transport data

ΙΙ

Nβ = 2{spec alone}

Nβ = 4{transport tuned}

TT03

Nβ = 3{spec & virial data}

Aziz 93

Boyes94

3.70 3.75 3.80

-99.5

-99.0

-98.5

ener

gy/c

m−1

r / Å

Figure . Potential energy functions in the vicinity of the min-imum for the Ar2(X

1�+g ) ground electronic state interaction

energy. Except for the fitted ab initio PEFs from Ref. [] (dottedcurve and solid trianglular points) and Ref. [] (dotted curve andopen square points with error bars), the curves and symbols areas identified in the caption for Figure . The minimum, (De =99.50 ± 0.04 cm−1, re = . ± . A), for the recommendedbest-fit Nβ= MLR PEF is indicated by the vertical and horizontalerror bars.

(last column of Table 4). As a historical comparison, a setof short-range Ar–Ar interaction energies extracted byRol and co-workers frommolecular beammeasurementsof high-energy total integral scattering cross sections (asgiven in Table 2 of Ref. [57]) are shown as diamond-shaped points with error bars. However, they systemati-cally lie below both the recent ab initio results (solid trian-gles and open square points), and from the recommendedbest empirical PEF (solid curve), and are not consideredfurther here.

The five empirical PEFs and the two sets of ab initiopoints are also clearly distinguishable on the expandedscale of Figure 2, in which the distinct natures of thevarious PEFs in the vicinity of the the minima of theirinteraction energies can be seen clearly. Both the ab ini-tio points (solid triangles [15] and open square pointswith error bars [16]) and their recommended analyticrepresentations (dotted curves) are shown in this figure.Except for the NX

β = 2 MLR function determined fromspectroscopy alone (short-dash curve), the simple Tang–Toennies function [141] (dashed curve), and the venera-ble Aziz potential (dash-dot curve), all of these functionshaveminima lying within the uncertainties of that for ourrecommended NX

β = 4 MLR function (solid curve, withhorizontal and vertical error bars designating its mini-mum).

1612 P. T. MYATT ET AL.

0.5 1 1.5 2 2.5103

104

105

106

107

108

109

ener

gy/c

m−1

r / Å

Nβ = 2{spec alone}

Nβ = 4{transport tuned}

TT 03

Nβ = 3{spec & virial data}

Aziz 93 &Boyes 94

Coulomb q1 q2/r limit

0.1

Figure . Comparisons between the ultra-short distance (. –. A) behaviours of the empirical Nβ=, and fitted MLR PEFs,the Aziz [], Boyes [] HFD-ID empirical fit-PEFs, and the Tang–Toennies two-parameter model for the Ar2(X

1�+g ) interaction, as

identified in Figure . The Coulomb-limit for the Ar–Ar interaction(q = q = ) is represented by the dotted line. Symbols:� (smallhollow squares) from Ref. []; (large hollow squares) from Ref.[]; (solid triangles) from Ref. [].

Finally, the ultra-short-range (r < 1.5 A) behavioursof the various PEFs are shown in Figure 3, together witha number of additional recent ab initio points for thisregion (open square points [65]). The dotted line on thislog-log plot shows the limiting very short-rangeCoulombrepulsion behaviour of two Ar nuclei with assumed effec-tive charges of +18. All threeMLR functions become par-allel to this limiting line, by construction, because of thechoice of s= −½ in the definition of their damping func-tions, while the ab initio points appear to be approachingthat limiting Coulomb behaviour.

Figure 2, taken in conjunction with the lower panel ofFigure 1 and the DRMSD values given in Table 6, makesit clear that the experimental data employed to determinethe PEF for the electronic ground-state Ar–Ar interactionpotential are sensitive to relatively small differences in thewell and the repulsive wall for interaction energies lessthan approximately 1000 cm−1.

6. Discussion and conclusions

The relative performances of the two recent ab initio PEFs[15,16] in the comparisons of Table 6 is a testament to theexcellent quality of current coupled-cluster ab initio com-putational methodology. Note that while Table 6 showsthat the ab initio PEF of Jäger et al. [15] describes nei-ther the primary experimental data set nor the transportdata as well as does the empirical Nβ=4 MLR PEF, itnonetheless does impressively well, with a dd value for‘All Data’ only 10% larger than that for our best empirical

PEF. Moreover, it gives better agreement with all cate-gories of data than does either of the HFD-ID empiricalPEFs that had actually been fitted to similar sets of exper-imental data. It is, however, puzzling that the ab initioPEF of Ref. [16] fares much worse in these comparisonsthan does that of Ref. [15], particularly for the acousticvirial data (dd = 4.879 vs. 2.479), given that the ab initiointeraction energies computed in Ref. [15] typically fallwithin the fairly tight uncertainties assigned to the ab ini-tio results reported in Ref. [16].

From qualitative comparisons of the curves shown inFigure 1, it appears that the largest differences betweenthe ab initio results and the various empirical potentialsare associated with the relatively few (10 for Ref. [15], 7for Ref. [16]) points obtained at Ar–Ar separations onthe repulsive wall (i.e. r < σ ). More quantitative com-parisons of the degree of agreement between these tworecent high-level ab initio calculations [15,16] and vari-ous empirical PEFs are provided by Table 7, for which thedd values were obtained by treating the recent higher-level ab initio values for the Ar2(X 1�+

g ) interaction ener-gies [15,16] as external data sets, while assuming that theuncertainties reported by Patkowski and Szalewicz [16]could also be used for the results of Jäger et al. [15,142].

The first pair of columns in Table 7 presents the RMSdifferences across the whole domain, while the secondpair of columns give the corresponding dd-values for theab initio results in the region of the attractive potentialenergy well, and the third pair of columns presents theanalogous results for the repulsive wall region, r < σ .It is important to remember that the dd values given inTable 7 do not represent tests of the quality of the var-ious potentials, but are simply measures of the level ofagreement between the five empirical PEFs and the twosets of ab initio points. As might be expected (see lasttwo rows of Table 7), the modified-TT-type PEFs actu-ally fitted to these sets of points by the authors of Refs.[15] and [16] are in much closer agreement with their abinitio points in all regions. Note, however, that this is notdue to the choice of potential form, asNβ = 3 MLR PEFswith equivalent numbers of free parameters (i.e. also withC6 and C8 free) fitted to those ab initio data give overalldd values of 0.43 and 0.34, respectively. Finally, the smallsize of the dd value obtained on applying the fitted PEFof Ref. [16] to the ab initio points of Ref. [15] and viceversa confirms the statement by Patkowski and Szalewiczthat the two sets of ab initio points agree to within theirestimated uncertainties.

In summary, the present work has determined animproved empirical PEF model for the Ar–Ar interac-tion energy from a combination of a conventional DPFtreatment of spectroscopic and second virial coefficient

MOLECULAR PHYSICS 1613

Table . Comparisons of differences between two sets of ab initio points and predic-tions of various empirical PEFs.

dd {all points} dd {well: r> σ } dd {wall: r< σ }

PEF Jäger [] Patk. [] Jäger [] Patk. [] Jäger [] Patk. []

Nβ= MLR . . . . . .

Nβ= MLR . . . . . .

Nβ= MLR . . . . . .

Aziz HFD-ID [] . . . . . .Boyes HFD-ID [] . . . . . .TT (Jäger []) . . . . . .TT (Patkowski []) . . . . . .

data with a manual ‘tuning’ of the repulsive wall to incor-porate optimisation of agreement with transport proper-ties. The resulting function is in distinctly better over-all agreement with this body of 2243 data than were themost recent previous empirical potentials (by 7.8% [140]and 182% [57]), or two recent very high quality ab initioPEFs (by 1.8% [15] and 28% [16], respectively). We alsofind that a two-parameter PEF with a widely used formtaken from a highly quoted source [141] (� 250 citationsto date) has a dd value for this data set some 17 timeslarger than that for the present MLR(Nβ=4) empiricalPEF. This reaffirms our reservations about the utility ofany two-parameter model potential form. A comprehen-sive review of the degree of agreement between predic-tions generated from the various empirical potentials andthe experimental data is presented in the Appendix.

The present empirical MLR PEF also enables the gen-eration of smooth sets of predicted values of the twotypes of virial coefficients and four types of transportproperty coefficients considered herein over an exten-sive temperature range. A compact illustrative version ofthese results is presented in Table 8: a more extensive

tabulation has been included in the Supplemental Data.Such a tabulation should likely supersede the use of frag-mentary experimental data-sets for the archival represen-tation of our empirical knowledge of these properties.

In contrast with the early near-dissociation theoryanalysis prediction of Ref. [35], the present recommendedMLR potential supports only eight, rather than ninevibrational levels. Note, however, that the prediction ofRef. [35] was based solely upon the low-resolution ear-liest observations of the vibrational spacings of levelsv = 0 − 4 [34], and its predicted binding energy forv = 8 was, within its uncertainties, 0 cm−1. Finally, wehave employed the algorithm of Ref. [143] to compute avalue of −714 A for the 40Ar s-wave scattering length forour recommended PEF. Listings of the binding energiesand calculated rotational constants for the eight boundvibrational levels, and of the energies of all 173 trulybound levels and 61 quasibound rotational levels sup-ported by this PEF (together with the tunneling lifetimesof the latter), have been included as Supplementary Data.Also included with the Supplementary Data is a listingof a stand-alone Fortran subroutine for generating our

Table . Values of virial and transport property coefficients for Ar calculated from theempirical MLR{Nβ = } PEF.

Temp./K Press.vir Acou.vir. Viscosity Therm.cond. Self diff. Therm.diff.T B βa η λ DAA′ α[K] [cm mol−] [cm mol−] [μPa · s] [mWm− K−] [−m s−]

. − . − . . . . .. − . − . . . . .. − . − . . . . .. − . − . . . . .. − . − . . . . .. − . . . . . .. − . . . . . .. − . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .

1614 P. T. MYATT ET AL.

recommended MLR PEF and tabulations of the primaryand secondary data that have been employed in the deter-mination of the MLR PEF for the Ar2 dimer.

Acknowledgments

This research has been supported by the Natural Sciences andEngineering Research Council of Canada by way of a ‘Discov-ery’ grant awarded to Robert J. Le Roy.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

Natural Sciences and Engineering Research Council of Canada[grant number RPGIN-3929-2014].

References

[1] J.C. Maxwell, Phil. Trans. R. Soc. 157, 49 (1867).doi:10.1098/rstl.1867.0004

[2] J.E. (Lennard-)Jones, Proc. R. Soc. (London) A 206, 441(1924).

[3] J.E. (Lennard-)Jones, Proc. R. Soc. (London) A 206, 463(1924).

[4] R.A. Buckingham, Proc. R. Soc. (London) A 168, 264(1938). doi:10.1098/rspa.1938.0173

[5] E.A. Mason, J. Chem. Phys. 22, 169 (1954).doi:10.1063/1.1740026

[6] P.M. Morse, Phys. Rev. 34, 57 (1929).doi:10.1103/PhysRev.34.57

[7] G.C.Maitland and E.B. Smith, Chem. Phys. Lett. 22, 443(1973). doi:10.1016/0009-2614(73)87003-4

[8] C. Douketis, G. Scoles, S. Marchetti, M. Zen, andA.J. Thakkar, J. Chem. Phys. 76, 3057 (1982).doi:10.1063/1.443345

[9] K.T. Tang and J.P. Toennies, J. Chem. Phys. 80, 3726(1984). doi:10.1063/1.447150

[10] K.-C. Ng, W.J. Meath, and A.R. Allnatt, Chem. Phys. 32,175 (1978). doi:10.1016/0301-0104(78)87049-9

[11] K.-C. Ng, W.J. Meath, and A.R. Allnatt, Mol. Phys. 38,375 (1979).

[12] R. Hellmann, E. Bich, and E. Vogel,Mol. Phys. 105, 3013(2007). doi:10.1080/00268970701730096

[13] M. Jeziorska, W. Cencek, K. Patkowski, B. Jeziorski,and K. Szalewicz, J. Chem. Phys. 127, 124303 (2007).doi:10.1063/1.2770721

[14] R. Hellmann, E. Bich, and E. Vogel, Mol. Phys. 106, 133(2008). doi:10.1080/00268970701843147

[15] B. Jäger, R. Hellmann, E. Bich, and E. Vogel, Mol. Phys.107, 2181 (2009); erratum, ibid, 108, 105 (2010).

[16] K. Patkowski and K. Szalewicz, J. Chem. Phys. 133,094304 (2010). doi:10.1063/1.3478513

[17] J.L. Dunham, Phys. Rev. 41, 713, 721 (1932).doi:10.1103/PhysRev.41.713

[18] R. Rydberg andZ. Physik, 73, 376 (1931),O.Klein andZ.Physik, 76, 226 (1932); R. Rydberg, ibid., 80, 514 (1933);A.L.G. Rees, Proc. Phys. Soc.59, 998 (1947).

[19] J.A. Coxon andP.G.Hajigeorgiou, J.Mol. Spectrosc. 150,1 (1991); Can. J. Phys. 70, 40 (1992).

[20] J.F. Ogilvie, Proc. R. Soc. (London) A 378, 287 (1981).doi:10.1098/rspa.1981.0152

[21] A.A. Šurkus, R.J. Rakauskas, and A.B. Bolotin, Chem.Phys. Lett. 105, 291 (1984).

[22] P.G. Hajigeorgiou and R.J. Le Roy, in 49 th Ohio StateUniversity International Symposium on Molecular Spec-troscopy (Ohio State University, Columbus, OH, 1994),paper WE04; J. Chem. Phys. 112, 3949 (2000).

[23] R.J. Le Roy and R.D.E. Henderson, Mol. Phys. 105, 663(2007). doi:10.1080/00268970701241656

[24] R.J. Le Roy, N. Dattani, J.A. Coxon, A.J. Ross, P. Crozet,and C. Linton, J. Chem. Phys. 131, 204309 (2009).doi:10.1063/1.3264688

[25] R.J. Le Roy, C.C. Haugen, J. Tao, and H. Li, Mol. Phys.109, 435 (2011). doi:10.1080/00268976.2010.527304

[26] R.J. Le Roy and A. Pashov, J. Quant. Spectrosc. Radiat.Transfer 186, 210 (2016).

[27] R.J. Le Roy, Y. Huang, and C. Jary, J. Chem. Phys. 125,164310 (2006). doi:10.1063/1.2354502

[28] R.D.E. Henderson, A. Shayesteh, J. Tao, C.C. Haugen,P.F. Bernath, and R.J. Le Roy, J. Phys. Chem. A 117,13373 (2013). doi:10.1021/jp406680r

[29] J.A. Coxon and P.G. Hajigeorgiou, J. Quant.Spectrosc. Radiat. Transfer 151, 133 (2015).doi:10.1016/j.jqsrt.2014.08.028

[30] Y.-S. Cho and R.J. Le Roy, J. Chem. Phys. 144, 024311(2016). doi:10.1063/1.4939274

[31] N. Dattani and R.J. Le Roy, J. Mol. Spectrosc. 268, 199(2011). doi:10.1016/j.jms.2011.03.030

[32] V.V. Meshkov, A.V. Stolyarov, M.C. Heaven, C. Haugen,and R.J. Le Roy, J. Chem. Phys. 140, 064315 (2014).doi:10.1063/1.4864355

[33] H. Knöckel, S. Rühmann, and E. Tiemann, J. Chem.Phys. 138, 094303 (2013); 138, 189901 (2013).

[34] Y. Tanaka and K. Yoshino, J. Chem. Phys. 53, 2012(1970). doi:10.1063/1.1674282

[35] R.J. Le Roy, J. Chem. Phys. 57, 573 (1972).doi:10.1063/1.1678005

[36] E.A. Colbourn and A.E. Douglas, J. Chem. Phys. 65,1741 (1976). doi:10.1063/1.433319

[37] P.R. Herman, P.E. LaRocque, and B.P. Stoicheff, J. Chem.Phys. 89, 4535 (1988). doi:10.1063/1.454794

[38] M.V. Bobetic and J.A. Barker, Phys. Rev. B 2, 4169(1970). doi:10.1103/PhysRevB.2.4169

[39] J.A. Barker, R.A. Fisher, and R.O. Watts, Mol. Phys. 21,657 (1971). doi:10.1080/00268977100101821

[40] G.C.Maitland andE.B. Smith,Mol. Phys. 22, 861 (1971).doi:10.1080/00268977100103181

[41] J.A. Barker, W. Fock, and F. Smith, Phys. Fluids 7, 897(1964). doi:10.1063/1.1711301

[42] J.A. Barker and M.L. Klein, Chem. Phys. Lett. 11, 501(1971). doi:10.1016/0009-2614(71)80394-9

[43] J.M. Parson, P.E. Siska, and Y.T. Lee, J. Chem. Phys. 56,1511 (1972). doi:10.1063/1.1677399

[44] J.J.H. van den Biesen, R.M. Hermans, and C.J.N.van den Meijdenberg, Physica 115, 396 (1982).doi:10.1016/0378-4371(82)90031-0

[45] M.R. Moldover and J.P.M. Trusler, Metrologia 25, 165(1988). doi:10.1088/0026-1394/25/3/006

MOLECULAR PHYSICS 1615

[46] M.B. Ewing, A.A. Owusu, and J.P.M. Trusler, Physica A156, 899 (1989). doi:10.1016/0378-4371(89)90026-5

[47] M.B. Ewing and J.P.M. Trusler, Physica A 184, 415(1992). doi:10.1016/0378-4371(92)90314-G

[48] J. Kestin, S.T. Ro, andW.A.Wakeham, J. Chem. Phys. 56,4119 (1972). doi:10.1063/1.1677824

[49] D.W. Gough, G.P. Matthews, and E.B. Smith,J. Chem. Soc. Faraday Trans. 1 72, 645 (1976).doi:10.1039/f19767200645

[50] E. Vogel, Ber. Bunsenges. Phys. Chem. 88, 997 (1984).[51] E. Bich and E. Vogel, Int. J. Thermophys. 12, 27 (1991).

doi:10.1007/BF00506120[52] H.M. Roder, R.A. Perkins, andC.A.Nieto deCastro, Int.

J. Thermophys. 10, 1411 (1989).[53] R.A. Perkins, D.G. Friend, H.M. Roder, and C.A.

Nieto de Castro, Int. J. Thermophys. 12, 965 (1991).doi:10.1007/BF00503513

[54] M.R. Moldover, S.J. Boyes, C.W. Meyer, and A.R.H.Goodwin, J. Res. Natl. Inst. Std. Tech. 104, 11 (1999).doi:10.6028/jres.104.002

[55] R.A. Aziz and M.J. Slaman, Mol. Phys. 58, 679 (1986).doi:10.1080/00268978600101501

[56] R.A. Aziz and M.J. Slaman, J. Chem. Phys. 92, 1030(1990). doi:10.1063/1.458165

[57] R.A. Aziz, J. Chem. Phys. 99, 4518 (1993).doi:10.1063/1.466051

[58] G. Chałasinski, D.J. Funk, J. Simons, and W.H.Breckenridge, J. Chem. Phys. 87, 3569 (1987).

[59] A.D. McLean, B. Liu, and J.A. Barker, J. Chem. Phys. 89,6339 (1988). doi:10.1063/1.455400

[60] D.E. Woon, J. Chem. Phys. 100, 2838 (1994).doi:10.1063/1.466478

[61] T. VanMourik, A.K.Wilson, and T.H. Dunning, Jr, Mol.Phys. 96, 529 (1999). doi:10.1080/00268979909482990

[62] B. Fernández and H. Koch, J. Chem. Phys. 109, 10255(1998).

[63] S.M. Cybulski andR.R. Toczylowski, J. Chem. Phys. 111,10520 (1999). doi:10.1063/1.480430

[64] P. Slavícek, R. Kalus, P. Paška, I. Odvásková, P. Hobza,and A. Malijevský, J. Chem. Phys. 119, 2102 (2003).

[65] K. Patkowski, G. Murdachaew, C.-M. Fou, andK. Szalewicz, Mol. Phys. 103, 2031 (2005).doi:10.1080/00268970500130241

[66] E. Mason and T. Spurling, in International Encyclope-dia of Physical Chemistry and Chemical Physics, Topic10, “The Fluid State”, edited by J. Rowlinson (PergamonPress, Oxford, 1969), Vol. 2, Chap. 2, pp. 17–66.

[67] J.P.M. Trusler, Physical Acoustics andMetrology of Fluids(Hilger, Bristol, 1991).

[68] M.R. Moldover, K.A. Gills, J.J. Hurly, J.B. Hehl, andJ. Wilhelm, in Measurements in Gases. Chapter 10 ofModern Acoustical Techniques for the Measurement ofMechanical Properties, edited by M. Levy, H.E. Bass, R.Stern, Vol. 39 Experimental Methods in the Physical Sci-ences (Academic Press, New York, 2001).

[69] F.R.W. McCourt, in Virial Coefficients, Chapter 27,pp. 673–711 of Handbook of Molecular Physics andQuantum Chemistry, edited by S. Wilson, Volume P.F.Bernath, Volume 3: Molecules in the PhysicochemicalEnvironment: Spectroscopy, Dynamics and Bulk Proper-ties, Volume Editor (Wiley, 2003).

[70] B. Foreman, P.K. Rol, and K.P. Coffin, J. Chem. Phys. 61,1658 (1974). doi:10.1063/1.1682159

[71] R.A. Aziz. Interatomic Potentials for Rare-Gases: Pureand Mixed Interactions, Chapter 2, pp. 5–86 in ‘InertGases: Potentials, Dynamics, and Energy Transfer inDoped Crystals’, Springer Series in Chemical Physics,Vol. 34, edited by M.L. Klein (Spinger-Verlag, Berlin,1984).

[72] R.J. Le Roy, J. Quant. Spectrosc. Radiat. Transfer 186,179 (2016).

[73] J.A. Coxon andP.G.Hajigeorgiou, J.Mol. Spectrosc. 150,1 (1991). doi:10.1016/0022-2852(91)90189-H

[74] H. Richter, H. Knöckel, and E. Tiemann, Chem. Phys.157, 217 (1991). doi:10.1016/0301-0104(91)87146-M

[75] R. Brühl, J. Kapetanakis, and D. Zimmermann, J. Chem.Phys. 94, 5865 (1991).

[76] J.A. Coxon andP.G.Hajigeorgiou, J.Mol. Spectrosc. 139,84 (1990). doi:10.1016/0022-2852(90)90244-K

[77] P.G. Hajigeorgiou and R.J. Le Roy, J. Chem. Phys. 112,3949 (2000). doi:10.1063/1.480946

[78] E.G. Lee, J.Y. Seto, T. Hirao, P.F. Bernath, andR.J. Le Roy, J. Mol. Spectrosc. 194, 197 (1999).doi:10.1006/jmsp.1998.7789

[79] A.Michels, H.U.B.Wijker, and H.K.Wijker, Physica 15,627 (1949). doi:10.1016/0031-8914(49)90119-6

[80] A. Michels, J.M. Levelt, and W. De Graaff, Physica 24,659 (1958). doi:10.1016/S0031-8914(58)80080-4

[81] M.A. Byrne, M.R. Jones, and L.A.K. Staveley, Trans.Faraday Soc. 64, 1747 (1968). doi:10.1039/tf9686401747

[82] J. Kestin, K. Knierim, E.A.Mason, B. Najafi, S.T. Ro, andM. Waldman, J. Phys. Chem. Ref. Data 13, 229 (1984).doi:10.1063/1.555703

[83] R.M. Sevast’yanov and R.A. Chernyavskaya, Inzh.-Fiz.Zh. 52, 974 (1987).

[84] R. Gilgen, R. Kleinrahm, and W. Wagner, J. Chem.Thermo. 26, 383 (1994). doi:10.1006/jcht.1994.1048

[85] A.F. Estrada-Alexanders and J.P.M. Trusler, Int. J. Ther-mophys. 17, 1325 (1996). doi:10.1007/BF01438673

[86] J. Klimeck, R. Kleinrahm, and W. Wagner, J. Chem.Thermo. 30, 2571 (1998).

[87] W. Cencek, G. G.A.H. Harvey, M.O. McLinden, andK. Szalewicz, J. Phys. Chem. A 117, 7542 (2013).doi:10.1021/jp4018579

[88] M.B. Ewing and A.R.H. Goodwin, J. Chem. Thermo. 24,531 (1992). doi:10.1016/S0021-9614(05)80123-5

[89] A.F. Estrada-Alexanders and J.P.M. Trusler, J. Chem.Thermo. 27, 1075 (1995). doi:10.1006/jcht.1995.0113

[90] G. Benedetto, R.M. Gavioso, R. Spagnolo, P.Marcarino, and A. Merlone, Metrologia 41, 74 (2004).doi:10.1088/0026-1394/41/1/011

[91] J.H. Dymond, K.N. Marsh, R.C. Wilhoit, and K.C.Wong, Landolt-Börnstein, New Series, Group IV: Phys-ical Chemistry, 21A, Virial Coefficients of Pure Gases(Springer, Berlin, 2002).

[92] H. Ziebland, Pure Appl. Chem. 53, 1863 (1981).[93] B.A. Younglove and H.J.M. Hanley, J. Phys. Chem. Ref.

Data 15, 1323 (1986). doi:10.1063/1.555765[94] H.L. Johnston and E.R. Grilly, J. Phys. Chem. 46, 948

(1942). doi:10.1021/j150422a019[95] G.P. Flynn, R.V. Hanks, N.A. Lemaire, and J. Ross, J.

Chem. Phys. 38, 154 (1963). doi:10.1063/1.1733455

1616 P. T. MYATT ET AL.

[96] J. Kestin and A. Nagashima, J. Chem. Phys. 40, 3648(1964). doi:10.1063/1.1725066

[97] A.G. Clarke and E.B. Smith, J. Chem. Phys. 48, 3988(1968). doi:10.1063/1.1669725

[98] J.A. Gracki, G.P. Flynn, and J. Ross, J. Chem. Phys. 51,3856 (1969). doi:10.1063/1.1672602

[99] F.A. Guevara, B.B. McInteer, and W.E. Wageman, Phys.Fluids 12, 2493 (1969). doi:10.1063/1.1692386

[100] R.A. Dawe and E.B. Smith, J. Chem. Phys. 52, 693(1970). doi:10.1063/1.1673042

[101] E.F. May, R.F. Berg, and M.R. Moldover, Int. J. Thermo-phys. 28, 1085 (2007). doi:10.1007/s10765-007-0198-7

[102] J. Hellmans, J. Kestin, and S. Ro, Physica 71, 1 (1974).doi:10.1016/0031-8914(74)90043-3

[103] A.A.Clifford, P.Gray, andA.C. Scott, J. Chem. Soc. Fara-day Trans. 1 71, 875 (1975). doi:10.1039/f19757100875

[104] N.B. Vargaftik and Y.D. Vasilevskaya, J. Eng. Phys. 46, 30(1984).

[105] E. Vogel, Int. J. Thermophys. 31, 447 (2010).doi:10.1007/s10765-010-0760-6

[106] J. Wilhelm and E. Vogel, Int. J. Thermophys. 21, 301(2000). doi:10.1023/A:1006667125801

[107] C. Evers, H.W.Lösch, and W. Wagner, Int. J. Thermo-phys. 23, 1411 (2002). doi:10.1023/A:1020784330515

[108] R.F. Berg and M.R. Moldover, J. Phys. Chem. Ref. Data41, 043104 (2012). doi:10.1063/1.4765368

[109] R.F. Berg and W.C. Burton, Mol. Phys. 111, 195 (2013).doi:10.1080/00268976.2012.713132

[110] H. Lin, X.J. Feng, J.T. Zhang, and C. Liu, J. Chem. Phys.141, 234311 (2014). doi:10.1063/1.4903960

[111] F.M. Faubert and G.S. Springer, J. Chem. Phys. 57, 2333(1972). doi:10.1063/1.1678589

[112] J.W. Haarman, AIP Conf. Proc. 11, 193 (1973).doi:10.1063/1.2948425

[113] G.S. Springer and E.W. Wingeier, J. Chem. Phys. 59,2747 (1973). doi:10.1063/1.1680394

[114] S.H.P. Chen and S.C. Saxena, Mol. Phys. 29, 455 (1975).doi:10.1080/00268977500100391

[115] R. Fleeter, J. Kestin, R. Paul, and W.A. Wakeham,Physica A 108, 371 (1981). doi:10.1016/0378-4371(81)90138-2

[116] A.A. Clifford, P. Gray, A.I. Johns, A.C. Scott, and J.T.R.Watson, J. Chem. Soc. Faraday Trans. 1 77, 2679 (1981).doi:10.1039/f19817702679

[117] J. Kestin, S.T. Ro, and W.A. Wakeham, Ber.Bunsenges. Phys. Chem. 86, 753 (1982).doi:10.1002/bbpc.19820860816

[118] K.N. Haran, G.C. Maitland, M. Mustafa, and W.A.Wakeham, Ber. Bunsenges. Phys. Chem. 87, 657 (1983).doi:10.1002/bbpc.19830870810

[119] A.I. Johns, S. Rashid, and J.T.R. Watson, J.Chem. Soc. Faraday Trans. 1 82, 2235 (1986).doi:10.1039/f19868202235

[120] U.V. Mardolcar, C.A. Nieto de Castro, and W.A.Wakeham, Int. J. Thermophys. 7, 259 (1986).doi:10.1007/BF00500153

[121] J. Millat, M. Mustafa, M. Ross, W.A. Wakeham, andM. Zalaf, Physica A 145, 461 (1987). doi:10.1016/0378-4371(87)90005-7

[122] W. Hemminger, Int. J. Thermophys. 8, 317 (1987).doi:10.1007/BF00503945

[123] H.M. Roder, R.A. Perkins, and A. Laesecke, J. Res. Natl.Inst. Std. Tech. 105, 221 (2000). doi:10.6028/jres.105.028

[124] L. Sun and J.E.S. Venart, Int. J. Thermophys. 26, 325(2005). doi:10.1007/s10765-005-4502-0

[125] L.G. Stier, Phys. Rev. 62, 548 (1942).doi:10.1103/PhysRev.62.548

[126] T.I. Moran and W.W. Watson, Phys. Rev. 109, 1184(1958). doi:10.1103/PhysRev.109.1184

[127] R. Paul, A.J. Howard, and W.W. Watson, J. Chem. Phys.39, 3053 (1963). doi:10.1063/1.1734142

[128] G.A. Stevens and A.E. De Vries, Physica 39, 346 (1968).doi:10.1016/0031-8914(68)90089-X

[129] W.L. Taylor and S. Weissman, J. Chem. Phys. 59, 1190(1973); 62, 3837 (1975).

[130] W.M. Rutherford, J. Chem. Phys. 58, 1613 (1973).doi:10.1063/1.1679402

[131] W.L. Taylor, J. Chem. Phys. 62, 3837 (1975).doi:10.1063/1.431025

[132] E.B. Winn, Phys. Rev. 80, 1024 (1950).doi:10.1103/PhysRev.80.1024

[133] M. De Paz, B. Turi, and M.L. Klein, Physica 36, 127(1967). doi:10.1016/0031-8914(67)90085-7

[134] H.F. Vugts, A.J.H. Boerboom, and J. Los, Physica 44, 219(1969). doi:10.1016/0031-8914(69)90223-7

[135] A. Kumar and A.J. Thakkar, J. Chem. Phys. 132, 074301(2010). doi:10.1063/1.3315418

[136] C.Hättig andB.A.Hess, J. Phys. Chem. 100, 6243 (1996).[137] G.C. Maitland, M. Rigby, E.B. Smith, and W.A. Wake-

ham, Intermolecular Forces – Their Origin and Determi-nation (Oxford University Press, Oxford, 1981).

[138] A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASDTeam (2015). NIST Atomic Spectra Database (ver. 5.3),[Online]. Available: http://physics.nist.gov/asd [2016,September 13]. National Institute of Standards andTechnology, Gaithersburg, MD.

[139] R.J. Le Roy, J. Mol. Spectrosc. 191, 223 (1998).doi:10.1006/jmsp.1998.7646

[140] S.J. Boyes, Chem. Phys. Lett. 221, 467 (1994).doi:10.1016/0009-2614(94)00303-3

[141] K.T. Tang and J.P. Toennies, J. Chem. Phys. 118, 4976(2003). doi:10.1063/1.1543944

[142] Although it is clear from the discussion presented inRef. [16] that uncertainties for the results in Ref. [15]should, in principle, be larger than those determinedin Ref. [16], the assignment of larger uncertainties herewould only serve to introduce an unwarranted bias infavour of the results reported in Ref. [15].

[143] V.V. Meshkov, A.V. Stolyarov, and R.J. Le Roy, J. Chem.Phys. 135, 154108 (2011). doi:10.1063/1.3649946

[144] C. Tegeler, R. Span, and W. Wagner, J. Phys. Chem. Ref.Data 28, 779 (1999). doi:10.1063/1.556037

[145] R.D. Weir, I. Jones, J.S. Rowlinson, and G.Saville, Trans. Faraday Soc. 63, 1320 (1967).doi:10.1039/TF9676301320

[146] S. Chapman and T.G. Cowling, Mathematical Theoryof Nonuniform Gases, 3rd ed. (Cambridge UniversityPress, London, 1970).

[147] F.R.W. McCourt, J.J.M. Beenakker, W.E. Köhler, and I.Kušcer,Nonequilibrium Phenonema in Polyatomic Gases(Oxford University Press, Oxford, 1990), Vol. 1.

[148] F.R.W. McCourt, in Gas Transport Properties, Chap-ter 28, pp. 712–752 of Handbook of Molecular Physics

MOLECULAR PHYSICS 1617

and Quantum Chemistry, edited by S. Wilson, Vol. 3:Molecules in the Physicochemical Environment: Spec-troscopy, Dynamics and Bulk Properties, P.F. Bernath,Volume Editor, (Wiley, 2003).

[149] E. Vogel, Ber. Bunsenges. Phys. Chem. 21(M2), 169(1972).

Appendix. Review of agreement with experimentaldata

A.: Comparisons with spectroscopic dataSpectroscopic data are generally believed to provide themost precise information about the potential well andlow-energy repulsive wall. However, previous work [57]tended to focus attention upon properties deduced fromthose data, such as the vibrational level spacings�Gv + 1/2and rotational constants Bv , rather than directly upon theset of actual transition energies, as considered herein. Webelieve that this is likely the reason that the PEFs of Aziz[57] and Boyes [140] have DRMSD values that are sig-nificantly larger (by 4.6%, 7.0%, respectively) than thosegenerated by our recommended MLR PEF (first row ofTable 6), in spite of the fact that those derived proper-ties were included in their fits. Similarly, the fact that thepredictions generated from the PEF of Jäger et al. [15]yield a spectroscopic data dd value within 2% of that aris-ing from our fitted MLR function provides strong confir-mation of the accuracy of their ab initio PEF in the wellregion.

A.: Comparison with virial dataEquation-of-state (pressure) and acoustic second virialcoefficient measurements, especially if they are availableover an extensive temperature range, also serve as impor-tant sources of data from which potential energy infor-mation may be extracted. However, as virial coefficientsinvolve integrations over the atomic separation, r, theyare typically sensitive to broader features of the interac-tion, such as the ‘volume’ of the potential well.

A visual comparison between computed and exper-imental values of the pressure second virial coefficient,B2(T), is presented in Figure A.1. Computed curves weregenerated from the potential-fit PEF MLRNβ=4 (whoserepulsive wall was ‘tuned’ to improve agreement withtransport property data), two previous empirical PEFs[57,140] that were determined from fits to derived spec-troscopic properties (such as �Gv+ 1

2and Bv values) plus

virial and transport property data, and the two-parameterTang-Toennies [141] model PEF.

On the scale of the main figure it is not possible todistinguish the levels of agreement between the com-puted and experimental values of B2(T). Moreover, as

200 400 600 800 1000

-300

-250

-200

-150

-100

-50

0

B2(

T)

/cm

3 mol

-1T /K

Pressure Virial coefficients

200 300 400 500 600-50

-40

-30

-20

-10

0

10

TT 03

Aziz 93

{MLR , Boyes 94}

Figure A.. Temperature dependence of the equation-of-state(pressure) second virial coefficient. Computed curves: solid, cur-rent MLRN

β=4 PEF; dash-dot, Aziz [] PEF; dash-dot-dot,

Boyes [] PEF; long-dash, TT [] PEF. Experimental data:

(solid triangle) []; (solid inverted traingle) []; (opendiamond) []; ∇ (open inverted triangle) []; (open triangle)[];♦ (solid diamond) []; (open square) []; (solid square)[].

it is typically difficult to determine B2(T) with a preci-sion better than±1%, it remains rather difficult to distin-guish amongst most of the computed curves even on theexpanded scale of the inset figure showing the 200–600Ktemperature range data. It is clear, however, that the TT03curve lies well outside the experimental uncertainties andthat the Aziz HFD-ID curve (dash-dot-dot curve)may bemarginally distinguished from the Boyes HFD-ID curveand the two MLR curves. These visual distinctions areconsistent with the dd-values listed in the second row ofTable 6.

The fact that the Boyes HFD-ID PEF, which wasnot fitted to pressure virial data, yields a dd-valuefor those data that is actually smaller than the dd-value for the Aziz HFD-ID PEF that was fitted toit, quantitatively supports the statement in Sections5.1 and 5.2 that B2(T) and β2a(T) depend upon thePEF in much the same way. The dd-values of 1.180[15] and 1.689 [16] for the ab initio PEFs indicatethat the computed curve obtained with the former abinitio PEF will give agreement with experimental B2(T)data that is similar to the level of agreement obtained for

1618 P. T. MYATT ET AL.

100 200 300 400

-200

-150

-100

-50

0

T /K

β 2a/c

m3 m

ol-1

Acoustic Virial Coefficients

240 260 280 300 320 340-5

0

5

10

15

20

TT 03

Aziz 93

{MLR , Boyes 94}

Figure A.. Temperature dependence of the acoustic second virialcoefficient. Computed curves: solid, current MLRN

β=4 PEF; dash-

dot, Aziz [] PEF; dash-dot-dot, Boyes [] PEF; long-dash,

TT [] PEF. Experimental data: (open circle), []; � (smallopen square),[]; (open triangle), [];� (open right triangle),

[]; (open inverted triangle), []; (open diamond), [];(open square), [].

the present MLR empirical PEF, while the latter ab initioPEF will give agreement better than that obtained withthe Aziz HFD-ID PEF but not as good as that obtainedwith the Boyes HFD-ID PEF.

Acoustic second virial coefficients, obtained fromspeed-of-sound measurements, can be determined witha precision that is at least 10 times tighter than that asso-ciated with typical pressure virial coefficients. A visualcomparison between the computed and experimentaltemperature dependence of the acoustic second virialcoefficient, β2a(T), may be found in Figure A.2, whilerow 3 of Table 6 shows the results of computations thatwere carried out for the same set of three empirical PEFsemployed for the pressure virial comparisons in FigureA.1. It is already clear on the scale of the main panel ofthis figure that the β2a(T) curve computed from the two-parameter TT-model (with an acoustic virial dd value of114.08) lies well outside the experimental uncertainties.Moreover, from the inset figure for the 240–350K tem-perature range, it is also clear that the curve computedfrom theAzizHFD-IDPEF (which did not include acous-tic virial data in its determination) fails to predict values

of the acoustic second virial coefficient within the exper-imental uncertainties (dd value 15.290), while the curvescomputed from the BoyesHFD-IDPEF and theMLRNβ=4PEF (both of which included the acoustic virial data intheir determinations) remain visually indistinguishableon the expanded scale of the inset figure. The dd val-ues for entries 6 and 7 of row 3 of Table 6 then providequantitative measures of the relative levels of agreementbetween the experimental data and the ab initio PEFs.

A.: Comparison with transport dataTransport properties, whose calculation involves averagesover the ‘collision energy’ (i.e. the centre-of-mass relativetranslational energy), are more strongly influenced by thenature of the repulsive wall of the PEF than are the spec-troscopic or virial coefficient data. Indeed, a rough ‘rule-of-thumb’ estimate for the calculation of the appropri-ate thermally averaged collision cross sections is that thethermal averaging over energy-dependent collision crosssections should be carried out for collision energies rang-ing from zero up to approximately 10 times the thermalenergy (kBT) associated with the highest temperature forwhich transport propertymeasurements have beenmade.Thus, for example, for transport property measurementsup to temperature T= 1000 K, appropriate collision crosssection calculations in the range 0 � Ecoll 7000 cm−1

will typically be required to determine the (converged)temperature dependence of transport coefficients.

The dd-values listed in row 5 of Tables 4 and 6 have allbeen obtained from forward calculations of the transportcoefficients using the various PEFs discussed above. Asthe overall transport data dd-value of 1.061 obtained forthe MLRNβ=4 PEF is close to unity (which means that theset of 609 computed values will, on average, lie just out-side the corresponding assigned experimental uncertain-ties), there is thus little incentive for the incorporation oftransport property data into the iterative nonlinear least-squares DPF package dPotFit, given that it would neces-sitate substantial revisions of the code package.

Expressions for the temperature dependence of thetransport properties of a non-equilibriummonatomic gasmay be found in a number of standard monographs (see,for example, [137,146–148]): the simplest such expres-sion may be referred to as the first Chapman–Cowlingapproximation. This first approximation is frequently arather good one, especially for monatomic species, givingtypically 98% or more of the converged value. Althoughsecond-approximation corrections generally suffice fortypical experimental accuracies, so that higher-order cal-culations are seldom necessary, we have nonetheless car-ried out third-approximation computations of viscosityand thermal conductivity coefficients at all temperatures.

MOLECULAR PHYSICS 1619

The first Chapman–Cowling approximation, [η]1, forthe viscosity of a monatomic gas is written simply as[147]

[η(T )]1 kBTcrS(20)

(A1)

in terms of an effective cross section S(20), whichdepends upon the interaction PEF, and the average speedcr ≡ (8kBT/πmr)

12 , with kB the Boltzmann constant and

mr the reduced mass of the colliding pair of atoms. Thisfirst-order expression (A1) may also be written in termsof the more traditional omega integral �(2, 2)(T) usingS(20) = 8�(2,2)/(5cr) as [146]

[η(T )]1 = 5kBT8�(2,2) . (A2)

Higher-approximation expressions for the viscosity arethen traditionally written as

[η(T )]m = [η(T )]1(1 + f [m]η ) , m ≥ 2. (A3)

The most accurate experimental values for ηArhave been obtained using dual-capillary instruments[101,108–110] of the type developed by researchers atNIST [101]. Two of these measurements [108,109], attemperature T = 298.15K, have been assigned uncer-tainties of±0.0060μPa s, corresponding to a precision of0.022%. However, although both measurements, namely22.5666± 0.0060μPa s [108] and 22.5589± 0.0060μPa s[109], fall within the uncertainty intervals assigned to theT = 298.15K values 22.570± 0.019μPa s given in Ref.[101] and 22.562± 0.014μPa s given in Ref. [110], theiruncertainty intervals do not overlap. The current recom-mended empirical MLR PEF and the ab initio PEFs ofRefs. [15,16] all generate computed values of ηAr that lieslightly below the confidence intervals of Refs. [101] and[108]. However, all three PEFs give ηAr values that fallwithin the uncertainty interval of ±0.014μPa s given inRef. [110], while the ηAr values generated by theMLRPEFand the ab initio PEF of Ref. [16] also lie within the tighter(±0.0060μPa s) uncertainty interval of Ref. [109]. Themeasurements of May et al. [101] and Lin et al. [110] arefor 18, respectively, 14 temperatures spanning tempera-ture intervals 202K< T< 395K, respectively, 298K< T< 654K: the correspondingDRMSD (dd) values for thesetwo experimental data-sets are (ddMay = 0.950, ddLin =0.840), (ddMay = 1.091, ddLin = 0.990), and (ddMay =0.835, ddLin = 0.821), respectively, for the MLR, ab ini-tio [15], and ab initio [16] PEFs.

Figure A.3 compares the temperature dependence ofthe viscosity coefficient of argon as computed (predicted)via its second Chapman-Cowling approximation using

+

+

+

+

+

+

++

+

+

+

+

+

+

100 200 300 400 500

5

10

15

20

25

30

35

T /K

η/μ

Pa·

s

ViscosityCoefficients

TT 03

+

+

+

+

+

+

+

+

800 1100 1400

45

50

55

60

65

70

TT 03

+

+

+

+

+

500 600 700

34

36

38

40

42

44

46

TT 03

Aziz 93

MLR

Figure A.. Temperature dependence of the shear viscosity coef-ficient. Computed curves: solid, current MLRN

β=4 PEF; dash-dot,

Aziz [] PEF; dash-dot-dot, Boyes [] PEF; long-dash, TT[]model. Experimental data: (solid triangle) [];� (open tri-angle ) [];� (small open square) []; (solid inverted triangle),

[]; ♦ (open diamond) []; (solid circle) []; © (opencircle) []; � (open right triangle []; (open square) [];� (open left triangle) []; + [],▓ (solid square) [].

the four empirical PEFs of Table 6 with 226 experimen-tal data spanning the temperature range 110K � T �1500K. Themain panel shows the data from 110 to 500Kwhile the two inset panels give comparisons for temper-atures above 500K. As was the case for the pressure andacoustic virial coefficients in FiguresA.1 andA.2, the onlydistinctly different curve on the scale of the main panel isthat generated from the two-parameter TT03 model PEF[141].

It is clear from the inset panels that, particularlyfor higher temperatures, the two-parameter model ofRef. [141] does not successfully predict the viscosity ofgaseous argon for temperatures much above 300K, gen-erating values of η(T) that are significantly too small. Thecurve for the Aziz HFD-ID PEF can also be seen to sep-arate from the curves generated from the Boyes HFD-ID and the present MLRNβ=4 PEF, which remain indis-tinguishable even on the scale of the figure inserts. TheBoyes HFD-ID and the present MLRNβ=4 PEF may, how-ever, be distinguished from one another by the dd-valuesin the first row of Table 8. These dd-values indicate that

1620 P. T. MYATT ET AL.

++

+@

@

@

@

@

x

x

#

###

##

##

#

#

#

#

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#

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#

#

#

#

#

400 600 800 1000 120015

20

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T /K

λ/m

Wm

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-1

ThermalConductivity

TT 03

Aziz 93

{MLR,Boyes 94}

x

x

x

x

x

#

#

#

#

1000 1400 1800

45

50

55

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65

+

+

+

@

@

≤

≤

≤

≤

≤≤≤

≤≤

100 200 3005

10

15

20

Figure A.. Temperature dependence of the thermal conductivitycoefficient. Computed curves: solid, current MLRN

β=4 PEF; dash-

dot, Aziz [] PEF; dash-dot-dot, Boyes [] PEF; long-dash,TT [] model. Experimental data: (solid square) []; + (plussymbol) []; x []; (open square) [];� (small open square)[];� (open left triangle) []; (solid inverted triangle) [];�(solid right triangle) []; # []; (solid triangle) []; ♦ (soliddiamond) []; © (open circle) []; (solid circle) []; @ [];

(solid left triangle) []; ∇ (open inverted triangle) []; �(open right triangle) []; (open triangle) []; (open dia-mond) [].

the MLRNβ=5 PEF represents the viscosity data slightlybetter than does the Boyes HFD-ID PEF, while the nexttwo entries show that the Jäger et al. ab initio PEF [15]represents the viscosity data about as well as does theBoyes PEF, and that the viscosity values computed fromthe Patkowski-Szalewicz ab initio PEF [16] are compara-ble to those obtained using the MLRNβ=4 PEF. The finalentry then shows that the two-parameter TT-model PEFrepresents these data quite poorly. Curves calculated fromthe two ab initio PEFs have not been included in any ofFigures A.3–A.5, as on the scales of those figures theyare indistinguishable from the curves for the Boyes andMLRNβ=4 PEFs.

An examination of the first Chapman–Cowlingapproximation for the shear viscosity of a monatomicgas, coupled with the natures of the set of Ar–Ar PEFsillustrated in Figure 1, enables a correlation to be drawnbetween the temperature dependence of the viscositycoefficient calculated using these PEFs and the nature

500 10000

50

100

150

200

DA

′A/1

0-4m

2s−1

T /K

Isotopic Diffusion

TT 03

Aziz 93

{MLR,Boyes 94}

100 200 300 4000

10

20

30

40

Figure A.. Temperature dependence of the isotopic-diffusioncoefficient in Ar at pressure bar. Computed curves: solid, currentMLRN

β=4 PEF; dash-dot, Aziz [] PEF; dash-dot-dot, Boyes

[] PEF; long-dash, TT []model. Experimental data: (opendiamond) []; (solid square) []; (solid, triangle) [];�(open square) [].

of the repulsive wall of the PEF. Such a comparison willsuffice for this purpose, as higher approximations makeonly minor corrections to the values obtained using thefirst approximation expression: note that f [3]η rangesfrom 0.0005 at 200K to 0.0050 at 600K to 0.0070 at6000K. It will thus be clear from Equation (A1) that for agiven temperature T, a smaller value of η implies a largervalue of S(20): thus, for example, for T = 700K (inthe lower inset panel of Figure A.3) the relative order ofthe calculated values of η(700K) is TT03 < MLRNβ=4 Boyes < Aziz. From the lower panel of Figure 1, it isclear that all five PEFs and the TT03 model of Table 8are roughly equivalent in the well region, while from theupper panel of Figure 1 it is clear that the TT03 model isthe most repulsive and the Aziz PEF is the least repulsive(although not by much more than the Boyes PEF), whilethe MLRNβ=4 PEF lies between the Boyes PEF and TT03model.

Measurement of transport phenomena in amonatomic gas necessarily involves observation ofdifferent isotopes that are assumed to have identicalpair potentials, but must formally be treated as differentspecies. As is well known (see, e.g. Ref. [148] and refer-ences therein), transport phenomena are more complex

MOLECULAR PHYSICS 1621

Table A.. dd Values for individual transport properties.

Potential energy function

Property #Data MLRNβ=4 Aziz Boyes Jäger Patk. TT’

Viscosity . . . . . .Relative viscosity . . . . . .Thermal conductivity . . . . . .Isotopic diffusion . . . . . .Rel. isotopic diffusion . . . . . .Thermal diffusion . . . . . .Overall transport . . . . . .

for gas mixtures than for pure gases. In particular, fora binary mixture of gases A and B the energy and massflux vectors q (heat flux) and Jd (diffusive flux) may bewritten as [147,148]

q = −λT∇ lnT − pxAxB

DT∇xA (A4)

and

Jd ≡ vA − vB = − 1xAxB

[DT∇ lnT + DBA∇xA] (A5)

in terms of the non-equilibrium thermodynamic driv-ing forces −lnT and −xA associated with tempera-ture and mole-fraction (concentration) gradients main-tained across the container. Four transport coefficientsare required to describe these two fluxes: two diagonallinear response coefficients, specifically the thermal con-ductivity coefficient λ associated with the energy fluxcaused by the temperature gradient and the binary diffu-sion coefficient DBA associated with the mass flux asso-ciated with the mole-fraction (concentration) gradient,plus two off-diagonal linear response coefficients, specif-ically the better-known thermal diffusion coefficient DTassociated with a mass flux caused by the presence ofthe temperature gradient in the container and the lesser-known diffusion-thermal (or Soret) coefficient DT asso-ciated with the reciprocal effect, that is, an energy (heat)flux caused by the presence of a mole-fraction gradient inthe container.

Figure A.4 illustrates the level of agreement betweenthe temperature dependence of the thermal conductiv-ity coefficient as calculated from the second Chapman-Cowling approximation and its experimental tempera-ture dependence. As for the viscosity coefficient, thirdChapman-Cowling approximation corrections are rela-tively small, ranging from f [3]λ � 0.0008 for T � 210Kto 0.0075 at 600K to � 0.01 at 2000K. The mainpanel shows the comparison for the temperature range

250K to 1250K, while the two inset panels give com-parisons for lower (100K to 240K) and higher (1000Kto 1950K) temperatures. The λ(T) curve computedusing the two-parameter TT03 model falls significantlybelow the experimental values, especially for tempera-tures of 500K or higher. Indeed, as can be seen fromthe third row of Table 8, the TT03 model has a DRMSDvalue of 3.686, while the three empirical PEFs and bothab initio PEFs generate DRMSD values near unity, rang-ing from 1.455 for the Aziz HFD-ID PEF to 0.686 for theMLRNβ=4 PEF.

Note that the relative ordering of the computed val-ues of λ(T) is the same for a given temperature as thatfound for η(T). That this should be so may also be seenfrom the first Chapman-Cowling approximation expres-sion according to which λ(T) may be written in terms ofthe effective cross sections S(11) or S(20), or in termsof the ‘Omega integral’ �(2, 2) [147]

[λ(T )]1 = 5k2BT2mcr S(11)

= 15k2BT4mcr S(20)

= 75k2BT32m�(2,2)

(A6)

withm the mass of an (argon) atom, because of the exactrelation S(11) = 2

3 S(20) [147]. From Equation (A6)it will be clear that, apart from rather small differencesassociated with higher-approximation corrections, therelative abilities of the PEFs under consideration to repre-sent the temperature dependence of the thermal conduc-tivity coefficient of gaseous argon should be the same asthat obtained for the viscosity coefficient. As may thus beanticipated, this conclusion is confirmed by an examina-tion of the first and third rows of Table 8.

Figure A.5 compares experimental values for the iso-topic diffusion coefficient DA′A(T ) for 36Ar (A′) diffus-ing through a dilute gas of 40Ar (A) with values com-puted from the same set of three empirical PEFs andTT03model that were shown in Figures A.3 and A.4 for thetemperature dependences of the viscosity and thermalconductivity coefficients. Values for DA′A(T ) have been

1622 P. T. MYATT ET AL.

computed using the second Kihara approximation forf [2]D [137], as the Chapman–Cowling second approxima-tion for this correction is considerably more complex andgives values that do not differ by much numerically fromthose obtained using themuch simpler Kihara expression[137]. Values computed using the Kihara expression forf [2]D range from 0.0001 at T = 77K to 0.0046 at 600K to0.0062 at 1600K.

Note that because isotopic diffusion rates are ratherdifficult tomeasure accurately, and as naturally-occurringargon is almost isotopically pure 40Ar (containing only0.3365% 36Ar and 0.0632% 38Ar), isotopic diffusion inargon presents a particularly challenging case for theexperimentalist. Only two direct experimental measure-ments have been carried out, one by Winn [132] in 1950and one by De Paz et al. [133] in 1967. The experimen-tal uncertainties are quite large, typically of the orderof ±3%. Also shown in Figure A.5 are smoothed val-ues of DA′A(T ) determined by Kestin et al. [82] froman experimental correlation. As may be seen from thisfigure, the temperature dependence predicted by theTT03 two-parameter model lies significantly below theexperimental values, especially for temperatures above500K.

In 1969, Vugts, Boerboom, and Los [134] carriedout relative measurements of the isotopic diffusioncoefficient DA′A(T ) at 10 temperatures in the range234K < T < 418K, with a reported uncertainty of±0.1%. Their relative data have been converted into abso-lute data by employing their reference value DA′A =18.25 ± 0.090 cm2 s−1 at T = 298.15K, and are rep-resented in Figure A.5 by solid squares. Although it isdifficult to discern on the scale of the figure, but it isclear from Table 8, values of DA′A(T ) computed usingthe five PEFs and the TT03 model lie well outside theexperimental uncertainties assigned to themeasurementsby Vugts et al. [134]: this suggests that the assignmentof ±0.1% uncertainties to their data was likely overlyoptimistic.

As for the viscosity and thermal conductivity coef-ficients, the temperature dependence of DA′A(T ) pre-dicted by computations based upon the Boyes HFD-IDand MLRNβ=4 PEFs are essentially indistinguishable onthe scales of Figure A.5, while the DA′A(T ) curve com-puted using the HFD-ID PEF of Aziz [57] is marginallydistinguishable from those computed using the other twoempirical PEFs. Here, as has been seen for η(T) and λ(T)in Figures A.3 and A.4, respectively, the results in row 4of Table 8 show that the dd values for the isotopic diffu-sion experimental data set allow a ranking to be made ofthe predictions: in particular, the dd values for the twoab initio PEFs are only 8 − 10% larger than that for theMLRNβ=4 PEF, and those for the Boyes and Aziz PEFs are

25 and 45% larger, respectively, and the TT03 model is69% larger,

Because the binary diffusion coefficient DAB dependsonly upon the unlike-atom interaction, no simplificationof its expression in terms of effective cross sections(or equivalently, of omega integrals) is achieved forisotopic diffusion, for which the interaction energiesbetween all atomic isotopes are identicalwithin theBorn–Oppenheimer approximation. The relative ordering ofthe curves representing the temperature dependence ofthe isotopic diffusion coefficient can thus be understoodfrom the first Chapman-Cowling approximation for thebinary diffusion coefficient which, for isotopic diffusionof A′ in A is given [147] by

DA′A(T ) = kBTmA′

1nA cA′A S(10|A′)A′A

, (A7)

with mA′ the mass of A′, nA the number density of A,cA′A the average relative speed for A′ A collisions, and theeffective cross section S(10|A′)A′A for mass diffusion isrelated to the more traditional omega-integral �

(1,1)A′A by

[147]

S(10|A′)A′A = 163 cA′A

y2A �(1,1)A′A , (A8)

with y2A ≡ mA/(mA + mA′ ) a mass fraction. The relativeordering of the computed values of DA′A(T ) from small-est to largest at a given temperature T will be dictated bythe values obtained for S(10|A′)A′A or, equivalently, of�

(1,1)A′A , from largest to smallest, precisely as for η(T) and

λ(T).The lowest non-zero approximation for the thermal

diffusion coefficient, or equivalently, the thermal diffu-sion factor αT related to DT by DT = xAxBαT, involvesa rather complicated combination of effective cross sec-tions or of omega integrals, mole fractions, and massratios (see, for example, Equation (2.3–58) of Ref. [147]or Equation (A5.2.36) of Ref. [137]). Some simplificationdoes occur for isotopic atomic mixtures, as all interac-tion energies are the same (i.e. independent of the atomicmasses), but the expression is still relatively complex.However, if the isotopes are sufficiently heavy that themass ratio RM ≡ (mA − mA′ )/(mA + mA′ ) is small, thento a good approximation (upon neglecting contributionsof order R2

M or higher) the thermal diffusion factor can bewritten as [137,146]

αT α0RM, (A9)

MOLECULAR PHYSICS 1623

500 10000.00

0.10

0.20

0.30

0.40

0.50

T /K

α 0

Reduced Thermal Diffusion Factor

TT 03

{Aziz 93, Boyes 94}

MLR

Jäger 09

Figure A.. Temperature dependence of the reduced thermal dif-fusion coefficient α. Computed curves: solid, MLRN

β=4 PEF; dash-

dot, Aziz [] PEF; dash-dot-dot, Boyes [] PEF; long-dash, TT’[] model, dotted, ab initio PEF of Jäger et al.. Experimental data:

(solid triangle) []; (solid inverted triangle) []; (solidright triangle) []; (open triangle) []; (solid square) [];� (solid left triangle) []; (solid circle) []; (diamond)[].

in which α0, which is independent of isotopic mixturecomposition, is referred to as the reduced thermal diffu-sion factor.

A comparison between 45 experimental values of thereduced thermal diffusion factor α0 and the temperaturedependences generated from the Chapman-Cowling firstapproximation expression for [α0]1 and a second approx-imation correction using the Kihara expression for f [2]α

for the empirical-fit PEFs is presented in Figure A.6. Thesecond-approximation corrections are significantly largerfor this cross-coefficient, ranging from 0.0034 for T =100K to 0.0225 at 600K to 0.035 at 1600K. An addi-tional 29 values of α0 determined by Kestin et al. [82]based upon statistical least-squares fits to experimentaldata have also been included in this figure, as have predic-tions of the temperature dependence of [α0]2 based uponthe TT03 two-parameter model PEF (dashed curve) andthe Jäger et al. [15] modified TT form PEF that was fittedto their ab initio pair-interaction energies (dotted curve).As the isotopic thermal diffusion factor is extremely dif-ficult to measure with high precision, the experimentaluncertainties assigned to α0 are typically between ±5%and ±10%. However, in order not to clutter Figure A.6unnecessarily, typical error bars have been shown onlyfor the Stevens and De Vries [128], Rutherford [130] andStier [125] data, and for the Taylor [131] datumat temper-ature 215K. Note also that as the value reported by Stierfor αT at T = 154K appears to be erroneous, it has beenomitted from Figure A.6.