JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 22 8 DECEMBER 1999
The effect of intermolecular interactions on the electric propertiesof helium and argon. II. The dielectric, refractivity, Kerr,and hyperpolarizability second virial coefficients
Henrik KochDepartment of Chemistry, University of Southern Denmark, Odense, DK-5230 Odense M, Denmark
Christof Hattig,a) Helena Larsen, Jeppe Olsen, and Poul Jo”rgensenDepartment of Chemistry, University of Aarhus, DK-8000 Aarhus C, Denmark
Berta FernandezDepartment of Physical Chemistry, Faculty of Chemistry, University of Santiago de Compostela,E-15706 Santiago de Compostela, Spain
Antonio RizzoIstituto di Chimica Quantistica ed Energetica Molecolare del Consiglio Nazionale delle Ricerche,Via Risorgimento 35, I-56126 Pisa, Italy
~Received 21 July 1999; accepted 16 September 1999!
In a recent study,1 which in the following is referred toas ‘‘Part I,’’ we calculated the frequency-dependent intertion induced polarizabilities and second hyperpolarizabilitfor the He2 and Ar2 van der Waals complexes using thcoupled cluster singles and doubles~CCSD! and, for He2,also the full configuration interaction~FCI! method. A FCIpotential energy curve for He2 was obtained. Ferna´ndez andKoch2 determined a CCSD including connected triples crections~CCSD~T!! potential energy curve for the Ar2 dimer.All the calculations were carried out with extended basets. We exploit here these accurateab initio data to evaluatethe dielectric, the refractivity, the Kerr, and the hyperpolizability second virial coefficients of the helium and arggases in a wide range of temperatures.
Much work has been carried out on the determinationthese virial coefficients, both from an experimental andtheoretical point of view. However, the accuracy of the cculations for the virial coefficients has been hamperedparticular for argon, by the limited accuracy with which iteraction induced changes of polarizabilities and hyperpoizabilities could be computed and by the fact that the disp
a!Present address: Forschungszentrum Karlsruhe, Institute of Nanotecogy, P.O. Box 3640, D-76021 Karlsruhe, Germany.
10100021-9606/99/111(22)/10108/11/$15.00
-s
-
s
-
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r-r-
sion contribution to these frequency-dependent properhas not been considered.
A detailed review of previous investigations on the agon virial coefficients was given in Ref. 3, where the dieletric, refractivity, Kerr, and hyperpolarizability second viriacoefficients were evaluated and reasonable agreementexperiment was obtained. A rather extensive analysis ofeffects of basis set size, electron correlation, and frequedependence of the properties suggested that the observwere especially sensitive to the basis set and that new calations with more extended and diffuse basis sets wouldneeded to improve the agreement with experiment. Thesults of such calculations are reported in Part I, andevaluation of the virial coefficients is discussed here.
The dielectric virial coefficient of helium has been dtermined both by direct4–7 and indirect8–11 methods at tem-peratures ranging from as low as 4.22 K up to 323.15Data are also available at 3.799 K.12 The low temperaturedata are highly scattered. The refractivity virial coefficienwere measured at 303 and 323 K by Achtermannet al.usinga differential-interferometric technique.9,11 The comparisonwith theoretical values—based on a classical dipole-inducdipole ~DID! model—gave rather poor results, whereasmore refined semiclassical approach, using both a Dmodel and estimates taken from theab initio results ofDacre13 for the trace of the polarizability tensor, was showto give much better agreement. Orcutt and Cole5 also mea-ol-
10109J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Properties of He2 and Ar2 . II
sured the refractive coefficient of helium at 323 K.Buckingham and Dunmur measured the Kerr elect
optic effect of rare gases atl5632.8 nm in 1967.14 Theydetermined the dependence of the Kerr constant on thesity and found no deviations from a linear law for both hlium and neon, whereas a quadratic behavior was evidenargon. Tammeret al.15 studied the dc-Kerr effect of heliumat 632.8 nm again some 25 years later. Their experimmade at a temperature of 303.7 K and in a range of pressbetween 76 and 97 bar, confirmed that virial coefficientsorder higher than first~thus includingBK ,) are so small thatthey could not be determined at ‘‘~...! the present stage oexperimental accuracy.’’
Previous theoretical studies have shown that the demination of the virial coefficients for linear and nonlineoptical properties is a challenging task. Coupled HartreFock results for the second dielectric virial coefficienthelium were obtained by Fortuneet al.16,17Dacre has studiedthe second dielectric virial coefficients of both helium@at theSCF and correlated—configuration interaction singlesdoubles~CISD!—levels#13 and argon~at the SCF level!.18
Hohm determined the frequency- and temperatudependencies of the second refractivity virial coefficientsseveral atoms and small molecules.19 He used a DID basedtheoretical model and observed significant deviations wrespect to the available experimental results. Hohmet al.20
studied also in detail the temperature dependence of theond dielectric virial coefficients of rare gases, and found tit could not be reproduced with then existing theories.
In their experimental study of the hyperpolarizabilitiof interacting molecular pairs, Donley and Shelton estimathe second virial coefficients of the ESHG hyperpolarizabity ratios using a Lennard-Jones 6-12 potential and the Dmodel. The procedure was not sufficiently adequate to pvide good accuracy of the properties.21,22
Using quantum statistical mechanics, Bruchet al. de-rived the second dielectric and upper and lower boundsthe second Kerr virial coefficients for helium.23 For theformer, at 4 K, they obtained an agreement with respecexperiment only on the order of magnitude. This reflectlack of precision in the estimated interaction polarizability
The second dielectric and the second hyperpolarizabvirial coefficients of helium were evaluated by Bishop aDupuis from fourth-order Mo” ller–Plesset perturbation theor~MP4!.24 For the former, good agreement was obtained wthe experimental data and with a previous theoretical CIresult. No experimental data are available for the latter.
Moszynski et al. used symmetry-adapted perturbatitheory ~SAPT! ~Refs. 25, 26! results for the potential curveand for the interaction property to compute the dielecsecond virial coefficient of helium.27–29 First and secondquantum corrections to the semiclassical results wcalculated.28 A considerable discrepancy was then fouwith respect to experiment at low temperatures, wheresemiclassical expansion is divergent. Full quantum-statistcalculations27 allow to correct these discrepancies, anddivergence vanishes. The quality of the results allowedauthors to get some understanding of the scattering of exp
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–
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toa
y
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c
re
ealeeri-
mental data at low temperatures, and to suggest new exmental measurements.
In the next section, we summarize the formulas usedevaluate the virial coefficients semiclassically. In Sec. III wbriefly describe the computational approach. The resultsgiven and discussed in Sec. IV. The last section contasome concluding remarks.
II. THEORY
In Part I we have discussed the evaluation of tfrequency-dependent interaction induced polarizabilitieshyperpolarizabilities.3 Here we only summarize the relationbetween these properties and the different virial coefficieAs in Part I, we denote the interaction induced electric dipaverage polarizability, polarizability anisotropy, second dKerr hyperpolarizability, and isotropic parallel second hyppolarizability with Daave(v,R), Daani(v,R), DgK(v,R),andDg i(v0 ,v1 ,v2 ,v3 ,R), respectively. Allab initio datafor the interaction induced properties are counterpoise crected results and their frequency dependence is descrthrough dispersion relationships involving the static valuand the Cauchy momentsDS(24) or the hyperpolarizabilitydispersion coefficientsA, see Part I.
The second dielectric and the second refractivity vircoefficients are obtained from the expansion of the genezation of the Clausius–Mossotti function in orders of tmolar densityr ~the inverse molar volume! of the gas.30 FortemperaturesT for which the thermal energykT—with k in-dicating the Boltzmann constant—is large compared todissociation energy of the van der Waals dimer the secrefractivity virial coefficient can be calculated from the clasical statistical mechanics expression,31
BR~v,T!5NA
2
6e0Xaave
~v,T!
5NA
2
6e04pE
0
`
Daave~v,R!
3exp~2V~R!/kT!R2dR, ~1!
whereV(R) is the interatomic potential.NA is Avogadro’snumber, ande0 the vacuum permittivity. For very low temperatures, quantum corrections, which account for thecrete rovibrational levels of the van der Waals molecumust be included, see, for instance, Refs. 27–29. Hereassume that the classical approximation can be used. Twe restrict our study to the determination of the zeroth-orexpansion in powers of\23,28,32as given by the semiclassicaexpression in Eq.~1!. We approximate the frequency depedence of the refractivity virial coefficient by expanding ita power series inv up to second order,
BR~v,T!5BR~0!~T!1v2BR
~2!~T!1¯ , ~2!
where the static limitBR(0)(T) is identical to the second di
electric virial coefficientBe(T).33 The coefficientsBR(0)(T)
and BR(2)(T) are calculated analogous toBR(v,T), Eq. ~1!,
but with Daave(v,R) replaced by the interaction-inducestatic polarizability and the Cauchy moment,Daave(0,R) andDSave(24,R), respectively.
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10110 J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Koch et al.
The interaction-induced polarizability anisotropDaani(v,R) is related to the pressure-dependence of theKerr or electro-optical effect.34 The latter is convenientlyexpressed by an expansion of the molar Kerr constantmK asa power series in the density.14 The second Kerr virial coef-ficient BK(v,T) can be calculated as33,14
BK~v,T!5NA
2
162e0S 1
5kTXK
a~v,T!1XKg ~v,T! D
5BKa~v,T!1BK
g ~v,T!, ~3!
with the integralsXKa(v,T) andXK
g (v,T) given by
XKa~v,T!54pE
0
`
Daani~v,R!Daani~0,R!
3exp~2V~R!/kT!R2dR, ~4!
XKg ~v,T!54pE
0
`
DgK~v,R!exp~2V~R!/kT!R2dR.
~5!
Bruch et al.23 have determined an expression for the upand lower bounds to the classical expression, Eq.~4!, withina quantum statistical approach, and they have discusseimplications in the calculation of the second Kerr virial cefficient of helium at very low temperatures.
Since in particular the dispersion contributionBK
g (v,T) makes only a small correction to the dominant cotribution BK
a(v,T), we have assumed here—as in R3—that Kleinman symmetry35 is valid in the calculation ofXK
g (v,T) ~see Sec. III!. Thus 5g i5gzzzz14gxxzz18gxxyy
andDgK(v,R)'Dg i(2v,0,0,v,R). Kleinman symmetry isexact in the static limit, and a~often very good! approxima-tion for the frequency-dependent part. Again we expaXK
a(v,T) and XKg (v,T) as power series inv up to second
order using the corresponding expansion of the properDaani(v,R) andDgK(v,R).
Following the general treatment by Buckingham aPople,33 in Ref. 3 a virial expansion for the second hyperplarizability was introduced,
g~V,r,T!5g~V!1Bg~V,T!r1¯ , ~6!
whereV is a shorthand notation for the usual sequencefrequency argumentsv0 , v1 , v2 , andv3 . Within the semi-classical approximation the coefficientBg(V,T) is given by
Bg~V,T!5NA
8pe0Xg~V,T!, ~7!
with
Xg~V,T!54pE0
`
Dg i~V,R!exp~2V~R!/kT!R2dR.
~8!
In ESHG experiments the hyperpolarizability of a gas, athus its pressure-dependence, can only be measured reto that of a second reference gas. Thus, for comparisonexperiment we report in the tables and figures of Sec. IVthe quantitybA(V,T)5Bg,A(V,T)/gA(V), where the sub-script ‘‘A’’ identifies the system. As for the interaction
c-
r
the
-.
d
s
-
f
dtivethC
induced polarizabilities, we treat the frequency dependeof g i and Dg i approximately by using a power seriethrough second-order in the frequencies,
Bg~V,T!5Bg~0!~T!1vL
2Bg~2!~T!1¯ , ~9!
b~V,T!5Bg
~0!~T!1vL2Bg
~2!~T!1¯
g i~0!~11vL2A1¯ !
, ~10!
and
vL25v0
25v121v2
21v32 .
III. COMPUTATIONAL DETAILS
For the evaluation of the helium and argon dielectrrefractivity, Kerr, and second hyperpolarizability virial coeficients we employ the frequency-dependent interactionduced polarizabilities and second hyperpolarizabilitiesported in Part I and in Ref. 3 for the dimers He2 and Ar2.
For He2, this includes FCI results for Dunning’d-aug-cc-pVTZ-3321 basis set at 25 internuclear distancethe range of 3.00–13.00 a.u. As discussed in Part I, we adthe difference between the CCSD results evaluated withd-aug-cc-pVTZ-3321 and thed-aug-cc-pV5Z-3321 basissets to the FCI results for the interaction induced properto account for remaining basis set errors. In the followingrefer to results obtained using these ‘‘adjusted’’ properties‘‘extrapolated FCI/d-aug-cc-pV5Z’’ data.
In Part I we determined for the helium dimer boCCSD~T! and FCI potential energy curves which weshown to be very close to the reference curve determinedAziz et al.36 in the 3.00–13.00 a.u. internuclear distanrange. Nevertheless, in order to reduce the possible souof inaccuracy, we employed Aziz’s potential36 as V(R) toobtain the results for the virial coefficients of He2 given anddiscussed in the next section. The dependence of theresults on the choice of the potential energy curve is qunegligible except for the integral of Eq.~1! at low tempera-tures. As an example, the second dielectric virial coefficieof helium as obtained semiclassically employing our Fpotential energy curve are on the average 1%–2% smalle~inabsolute value! at temperatures below 100 K than those otained employing the Azizet al. potential.36 The differencesare on the order of fractions of a percent at higher tempetures, or for other integrals.
For the Ar2 complex, ad-aug-cc-pV5Z-33211 CCSD~T!potential was available2 and the interaction induced polarizabilities and second hyperpolarizabilities have been coputed at the CCSD level with thed-aug-cc-pVQZ-33211~static contributions! and thed-aug-cc-pVTZ-33211~disper-sion coefficients! basis sets.3 The basis set errors in the dispersion coefficients are larger than in the static part. Hoever, this has a negligible influence on the final results, sithe dispersion contributions were found in Ref. 3 to be aban order of magnitude smaller than the static terms.3 TheCCSD~T! potential for Ar2, taken from Ref. 2, is availablefor a total of 59 internuclear distances in the range betw3.8 and 20.0 a.u. The static interaction properties were cputed for 12 distances between 5.5 and 30.00 a.u.
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10111J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Properties of He2 and Ar2 . II
In Ref. 3 we proceeded by a determination of the valuof the integrands in Eqs.~1!, ~4!, ~5!, and~8! for each tem-perature at the internuclear distances of theab initio calcu-lated interaction properties, followed by a cubic spline intpolation and a numerical integration. Whereas interchangthe first ~calculation of the integrand function! and the sec-ond ~spline interpolation! steps did not lead to appreciabchanges in the final results in the case of argon, for thelium dimer the integrand calculation-spline interpolation squence led to an inaccurate interpolation of the integrfunction at the lower end of the integration range. We fouthat reversing the sequence, i.e., first obtaining cubic spfits of the interaction induced properties~separately for thestatic and the frequency-dependent contributions! and thendetermining the integrand functions in a grid of 500 pointsthe integration range led to a much more stable numerintegration.
The behavior of the integrands involved in the semiclsical determination of the virial coefficients of argon wdiscussed in some detail in Ref. 3. The changes inducethe use of a larger basis set are not such as to deserve fudiscussion. The functions were integrated between 5.0030.00 a.u., an interval which was found to be suitable aguaranteed good convergence of the results already in Re
The functions which appear as integrands in the evation of the second refractivity@Eq. ~1!# and the hyperpolar-izability @Eqs. ~4! and ~8!# virial coefficients of helium aredisplayed in Figs. 1 and 2, respectively.
Figure 1 shows the integrand in Eq.~1! in the 3.00–13.00 a.u. internuclear distance range for both the temptures of 7.199 K and 323.15 K and forl5632.8 nm. Thebalance between the positive and negative regions of thetegrands is not as critical as it was found to be for the2dimer.3 The region below'6 a.u. largely dominates, and it iapparent that the behavior of the function at short range~be-low 3 a.u.! has no influence on the calculation of the secodielectric and refractivity virial coefficients of helium. As thtemperature increases, the integrand is displaced towardshort internuclear distance region, which makes this reg
FIG. 1. He2. The functionDaave(v,R)e2@V(R)/kT#R2 ~a.u.! against the inter-atomic distanceR ~a.u.! for T5323.15 K andT57.199 K, and forl5632.8 nm.
s
-g
e--dde
al
-
byherndd3.
a-
a-
in-
d
then
more and more important for the description of the propeasT increases.
The ‘‘effective’’ region of integration becomes mornarrow at low temperatures, and the minimum moves towthe equilibrium distance of the dimer~5.61 a.u.!. This, to-gether with the previously mentioned fact that the propeseems to be more sensitive to the choice of the internucpotential asT decreases, is an indication that even if minmal, the differences between our FCI potential and thedetermined by Aziz~and effectively employed in the calculation of the virial coefficient! are important in the regionaround equilibrium.
The integrand in the evaluation of the polarizability cotribution to the second Kerr@Eqs.~4!# closely resembles thecorresponding function seen for argon in Ref. 3. The valuethe integrand is far smaller; the maximum, which in Ar2 waslarger than 300 a.u., reduces to about 0.15 a.u. for He2. Fre-quency dispersion is very small, practically undetectablelow 4 a.u. or beyond 8 a.u. An effect of frequency depedence is instead observable in Fig. 2, where the integracorresponding to the polarizability contribution to the secoKerr virial coefficient and to the ESHG second virial coefcient are displayed. Note that—as for Fig. 1—the negatarea largely overcomes the positive area lying beyond'7.5a.u., and that there is no sign of a quasicancellation effecseen for the argon dimer.3
In order to ascertain the importance of long range effeon the virial coefficients of helium, the 25 extrapolateFCI/d-aug-cc-pV5Z frequency-dependent interactionduced polarizabilities and second hyperpolarizabilities wsupplemented with a set of points computed using the alytic fits determined in Part I~see Tables VII, VIII and IX inRef. 1!. Twenty three extra points, equally spaced in t13.0–36.0 a.u. interval were added, and the integrationthen extended first from 3.0 a.u. to 27.0 a.u. and then u36.0 a.u.
The differences between results obtained with an ingration range extended to 27.0 a.u. and 36.0 a.u. are tonegligible. Extending the upper limit of to 36.0 a.u. modifi
FIG. 2. He2. The functionDg i(v,R)e2@V(R)/kT#R2 ~a.u.! against the inter-atomic distanceR ~a.u.! for T5298.15 K andl5632.8 nm.
10112 J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Koch et al.
TABLE I. Helium. Extrapolated FCI/d-aug-cc-pV5Z results for the second dielectric virial coefficientBe(T),the dispersion contributionBR
(2)(T) and the second refractivity virial coefficientBR(v,T) at l5632.8 nm. Thepotential used is that of Azizet al. ~Ref. 36!. Temperatures are in K, virial coefficients in cm6 mol22 withfrequencies assumed to be given in a.u. We also list theab initio results of Moszynskiet al. ~Ref. 27! for Be andexperimental data, where available.
T Be(T) Be(T)a Be(T) ~expt! Ref. BR(2)(T) BR(632.8 nm,T)
aReference 27.bAlso 20.03060.004, Ref. 12 for3He at 4 K.cCf. 20.0614, calculated at the MP4 level by Bishop and Dupuis~Ref. 24!.dDetermined as20.06860.010 by Achtermannet al. ~Ref. 11!.eDetermined as20.06860.010 by Achtermannet al. ~Ref. 9!.
iaob
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the results for the second dielectric and refractivity vircoefficients discussed in Sec. IV A with respect to thosetained integrating to 13.0 a.u. by an average of'2.5% attemperatures greater than 100 K. At low temperatures, whthe virial coefficients become small in absolute value,percentage changes are more substantial~'20% at 27.098K!. The variation is of about 4%–4.5% for the polarizabilicontribution and of 1%–1.5% for the hyperpolarizabilicontribution to the second Kerr virial coefficients of SeIV B ~both static and dispersion!. Due to the near cancellation of the two contributions, the percentage effects becomlarge in the sumBK(v,T) where this coefficient is smallFinally, for the second hyperpolarizability coefficient, SeIV C, the average change is of about 1%–1.5% both forstatic and the dispersion contributions. The helium virial cefficients in the tables and figures in Sec. IV are thosetained extending the upper limit of integration in all integraover the internuclear coordinates up to 36.0 a.u., with a cuspline grid of 1000 points.
All numerical integrations were carried out wit
l-
ree
.
es
.e--
ic
Mathematica.37 The conversion factors between a.u., SI aesu units are given in the Appendix.
IV. RESULTS AND DISCUSSION
The results are presented in Tables I–VI and in Fi3–6. In the following, we discuss each property separafor both rare gases.
A. The second dielectric and refractivity virialcoefficients
Table I displays the temperature dependence of ourtrapolated FCI/d-aug-cc-pV5Z results for the second dieletric virial coefficient, Be , the dispersion contributionBR
(2)
and the second refractivity virial coefficientBR in helium.BR is evaluated at the wavelength of 632.8 nm. The availaexperimental data are included in Table I. ForBe comparisonis made with the fully quantum statistical results of Moszyski et al.,27 see also Fig. 1. The range of temperaturestends from 3.799 K up to 407.60 K. In the low temperatu
nioe
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em
esiea
a.
y
eetho
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n
inht
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10113J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Properties of He2 and Ar2 . II
region the semiclassical result, obtained from Eq.~1!, corre-sponds to the zero-order term in a perturbative expansioorders of\.23,28,32It has been shown that such an expansis divergent for very low temperatures, whereas quantumfects altogether are negligible for temperatures greater70 K, as Fig. 3 clearly confirms.28 Comparison with theBe
(0)
results of Moszynski and co-workers is reasonable at tperatures beyond 100 K.28 At 3.799 K theirBe
(0) is 20.0961cm6 mol22, compared to our20.0731 cm6 mol22 ~cf. quan-tum statistical result in Table I,20.0081 cm6 mol22!. Takinginto account the level of sophistication of theab initio analy-sis, the conclusion of Ref. 27 that the measures at somthe low temperature data should be repeated seems plauThis recommendation may be extended also to the expmental points at higher temperature, which carry error bthat do not enclose theab initio results.
Our result at 296 K is more than 10% less negative ththat obtained at the MP4 level by Bishop and Dupuis24
Around that temperature, our result is within'8% of that ofMoszynski et al., which, incidentally, are instead in vergood agreement with results~not shown here! obtained withFCI/d-aug-cc-pVTZ-33221 data. These differences betwthe ab initio results are a measure of the dependence ofsecond dielectric virial coefficients on different treatmentselectron correlation and on the basis set.
For the second refractivity virial coefficient an expemental value of20.06860.010 cm6 mol22 has been ob-tained by Achtermannet al.9,11 at the wavelength of 632.99nm and for temperatures of 303 and 323 K. Our correspoing results are20.0559 and20.0582 cm6 mol22, respec-tively, and compare reasonably well with experiment. Usa more refined semiclassical approach, see above, Acmannet al. obtained a coefficient of20.059 cm6 mol22 at303 K.
The results for the second dielectric and second reftivity virial coefficients of argon are displayed in Table II. Aclearly stated above, in this paper we are not concernedthe frequency dependence of the properties for argon, anassume the same dispersion contribution as computed in
FIG. 3. Helium. Temperature dependence of the second dielectric vcoefficientBe(T). Comparison with experiment and with the fully quantustatistical results of Moszynskiet al. ~Ref. 27!.
innf-an
-
ofble.ri-rs
n
nef
d-
ger-
c-
ithweef.
3. Thed-aug-cc-pVQZ-33211 (‘‘dQZ’’ in Table II and Fig.5! basis set correction increases the second dielectric cocients ~on the average by 2%–3%!. The comparison withexperiment is difficult due to the large uncertainty of somethe experimental data. In Ref. 3 we discussed the particsensitivity of the classical integrals to the balance betwpositive and negative regions of the integrands in argonfeature that depends on the temperature and which is prcally absent in the case of helium. The changes introduby the change of basis set are not such as to alter sigcantly this ratio. This, on the other hand, was shown toless critical for these properties than for the polarizabilcontribution to the Kerr constant and for the hyperpolarability coefficients.3 As an example, atT5298.15 K the ratiobetween the positive and the negative regions in the intefor Be of argon is of about 4:1.
al
TABLE II. Argon. The second dielectric viral coefficientBe(T), the disper-sion contribution BR
(2)(T) and the second refractivity viral coefficienBR(v,T) at l5632.8 nm. Results with two different basis sets~d-aug-cc-pVTZ-33221—dTZ below—vsd-aug-cc-pVQZ-33221—dQZ in the Tablesheading! are shown forBe(T). The dispersion@BR
(2)(T)# it obtained with thedTZ set. The viral refractivity coefficients at the wavelength of 632.8 nmcomputed fromdQZ dielectric contributions plus thedTZ dispersion.T isgiven in K, and the virial coefficients in cm6 mol22 with frequencies as-sumed to be given in a.u.
10114 J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Koch et al.
B. The second Kerr virial coefficients
As discussed in Sec. I, in experimental studies evenrecent as 1992 no appreciable deviations from the linearpersion law with the density has been detected for the Kconstant of helium.14,15 The contributions to the second Kevirial coefficient of helium are reported in Table III anshown in Fig. 4. The coefficientsBK are evaluated for thewavelength of 632.8 nm. The range of temperatures extefrom 77.40 K to 407.60 K. We computed the coefficien
FIG. 4. Helium. Temperature dependence of the second Kerr virial cocient BK(v,T)5BK
a(v,T)1BKg (v,T). Also, BK
a(v,T)5BK(0),a(T)
1v2BK(2),a(T) and BK
g (v,T)5BK(0),g(T)12v2BK
(2),g(T). The second Kerrvirial coefficient is shown for a wavelengthl5632.8 nm.
TABLE III. Helium. The second Kerr viral coefficientBK(v,T)5BK
a(v,T)1BKg (v,T). The two contributions are further expanded
BKa(v,T)5BK
(0),a(T)1v2BK(2),a(T) and BK
g (v,T)5BK(0),g(T)
12v2BK(2),g(T). BK(v,T) is also given for the wavelength of 632.8 nm.T
is given in Kelvin, and virial coefficients in V22 m8 mol22310236 with fre-quencies assumed to be given in a.u.
also at low temperature, and observed that—as it couldexpected—the polarizability contributionBK
a becomes rap-idly dominant, being at 3.799 K two order of magnitudlarger than the one computed at 77.4 K, and about 500 tilarger than the contribution at 303 K. The increase is oorder of magnitude ‘‘slower’’ for the hyperpolarizabilitycontribution. Due to near cancellation of the two contribtions (BK
a andBKg ), the ratio for the actual Kerr virial coef
ficient BK at 3.799 K and 303 K is if'10 000. Bruchet al.23
estimated from their quantum statistical analysis that thetio of BK
a(4.2 K) andBKa(300 K) should be of'39. At 300 K
the authors obtain, through a coupled perturbed HartrFock approach, a value of 2.56310236V22 m8 mol22 fortheir BK , which includes however only the polarizabilitterm. Our result at 300 K lies around 2.310236V22 m8 mol22, in reasonable agreement with the etimate by Bruchet al.23 In conclusion, the frequency dependence ofBK for helium is quite negligible in the whole rangof temperatures, and the near cancellation of the polarizaity contribution with the hyperpolarizability contributionleads to predict that there is a temperature~our estimate isaround 295 K! where the second Kerr virial coefficient ohelium goes to zero. There is thus a striking difference wrespect to what is seen for argon, whereXK
a is two orders ofmagnitude larger thanXK
g and clearly dominatesBK up tovery high temperatures.
In Table IV we list the results, both the different contrbutions and the overall coefficient, for the second Kerr vircoefficient of argon evaluated at a wavelength of 458.0 nThe comparison with experiment38 is central to the scope oFig. 5. Improving the basis set has minor effects on thelarizability anisotropy contribution, whereas the hyperpolizability term changes substantially. Since the total coecient is dominated by the polarizability contribution, thoverall variation is in the order of about a 3%. The expemental second Kerr virial coefficients by Dunmuret al.38 at458 nm for several temperatures carry large error bars, whnevertheless do not include either of our results~cf. Fig. 5!.The currentdQZ results are only slightly closer to thesexperimental values than the previousdTZ estimates.3 Onthe other hand, if we consider the much more recent Kstudy of Shelton and Palubinskas,39 and as in Ref. 3 computethe ratiobfit /afit of the virial expansiongeff
K 5afit1bfitr fromthe dc-Kerr hyperpolarizability atl5632.8 nm which istaken from Ref. 39 (77.8310263C4 m4 J23'1248 a.u.) andfrom our new value of BK at 632.8 nm (5.57310233V22 m8 mol22), we obtain a value of 8.7031025 m3 mol21 at 296 K. This compares very favorablwith the experimental measurement~bfit /afit58.960.331025 m3 mol21 at l5632.8 nm andT5296.15 K).
The neat dominance of the contribution arising from tintegration in Eq.~4! with respect to that arising from thKerr interaction hyperpolarizability and which is relatedthe integral in Eq.~5!, rules out possible causes of inaccurcies in our numbers due to the critical balance of positive anegative contributions to the hyperpolarizability classicaltegrals mentioned above. Also, the neglect of long raneffects @i.e., having limited the integration in Eq.~4! to 30a.u.# is of no consequence for our conclusions; at 298.15
-
edo
10115J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Properties of He2 and Ar2 . II
TABLE IV. Argon. The second Kerr viral coefficientBK(v,T)5BKa(v,T)1BK
g (v,T). The two contributionsare further expanded asBK
a(v,T)5BK(0),a(T)1v2BK
(2),a(T) and BKg (v,T)5BK
(0),g(T)12v2BK(2),g(T). The
static contributions are computed with thed-aug-cc-pVQZ-33221~dQZ! set, whereas the dispersion is obtainwith the smallerd-aug-cc-pVTZ-33221~dTZ! basis set.BK(v,T) as obtained from the results for the twcontributions is given forl5458 nm. T is given in Kelvin, and the virial coefficients in V22 m8 mol22 withfrequencies assumed to be given in a.u.
when we carry out the integration forXKa in a interval re-
duced by 10%~i.e., up to 28.8 a.u. instead of 30 a.u.!, theintegral is reduced by'0.08%.
In conclusion the experimental Kerr data of Dunmet al.38 are most likely inaccurate, and should thereforere-examined.
C. The second hyperpolarizability virial coefficients
The results for the helium hyperpolarizability virial coefficients are given in Table V and Fig. 6. Once again,range of temperatures goes from 77.4 K to 407.60 K, buthave results also for lower temperatures. Due to the absof a proper quantum statistical treatment, we refrain fr
FIG. 5. Argon. Temperature dependence of the second Kerr virial cocient BK(v,T) for l5458 nm. Comparison of results of Ref. 3—obtainewith thed-aug-cc-pVTZ (dTZ) basis set—and of experiment~Ref. 38! withthe currentd-aug-cc-pVQZ~static!1d-aug-cc-pVTZ~dispersion! results.
e
eece
discussing them at this stage. The dispersion contributiofar more important here than for the former coefficients.296 K Bishop and Dupuis24 determined the staticb @Eq.~10!# at the MP4 level as21.41 cm3 mol21. This comparesreasonably well with our corresponding static result
-
TABLE V. Helium. The second hyperpolarizability virial coefficienBg
(0)(T), Bg(2)(T), and b(V,T) for an ESHG process (v15v25v,v3
50,vL256v2) at 632.8 nm as a function of temperature~Kelvin!. To com-
pute b, we assumedg i(22v;v,v,0)543.104 a.u., as obtained by Bishoand Pipin ~Ref. 50!. Bg is given in C4 m7 J23 mol21310268 and b incm3 mol21. Frequencies are assumed to be given in a.u.
aBishop and Dupuis~Ref. 24! determined MP4 staticb521.41 cm3 mol21.
ts
aio-
ndn-
a
a-
he
toingt onn
ge
ins
di-f-theidere-
lar-
re
ta.ef-ent
offrac-t
riz-e-
histhe
euesrlar-he, ittheblybut
ticundsees.n-
bil
t.
e
ue
10116 J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Koch et al.
21.3349 cm3 mol21. The lack of experimental data prevena further discussion of our results forb of the individualgases. The behavior of the integrand in Fig. 2 is—attemperatures—very smooth, with the negative contributto the integral in Eq.~8! largely dominating the positive contribution.
In Table VI we summarize the results for the secohyperpolarizability virial coefficient of argon. The static cotribution changes considerably going from thed-aug-cc-pVTZ-33211 to thed-aug-cc-pVQZ-33211 basis set. Asconsequence a significant change is seen in theb(v,T) re-sults. No experimental data are available forBg , but usingthe same procedure as in our previous paper,3 we can esti-
FIG. 6. Helium. Temperature dependence of the second hyperpolarizavirial coefficients Bg
(0)(T), Bg(2)(T), and b(v,T) for an ESHG process
(v15v25v; v350, vL256v2) at l5632.8 nm.
TABLE VI. Argon. The second hyperpolarizability virial coefficienBg
(0)(T), Bg(2)(T), Bg(T), and b(v,T) for the ESHG process (v15v2
5v,v350,vL256v2) at l5514.5 nm as a function of temperatur
~Kelvin!. To computeb, we usedg i(22v;v,v,0)51577 a.u.~Ref. 51!.Static results are obtained with thed-aug-cc-pVQZ-33221 (dQZ) basis set,whereas the frequency dispersion was that computed using ad-aug-cc-pVTZ-33221 (dTZ) basis set.Bg is given in C4 m7 J23 mol21310268 andbin cm3 mol21. Frequencies are assumed to be given in a.u.
aThe static value is 0.8074, which can be compared to the estimated val229.8610.8, Ref. 24, obtained from theab initio result of He and theexperimental density ratios of Ref. 21.
lln
mate the quantityb(ArH2)1b(H2He). At l5514.5 nm andat T5298 K we obtainb50.188 cm3 mol21 for argon andb521.58 cm3 mol21 for helium. The ratiosg(Ar)/g(H2)andg(H2)/g(He) as a function of the gas density were mesured by Donley and Shelton.21 Combining these data, wecomputeb(ArH2)1b(H2He)523.20 cm3 mol21, in contrastto the experimental value21.468.0 cm3 mol21.21
In Ref. 3 we made a conservative estimate of tchanges expected in the integralXg for argon at 298.15 Kbased upon improving the basis set from triple-zetaquadruple-zeta quality. We made this prediction considerthe balance of negative and positive regions and its effecthe result in Eq.~8!. We predicted a possible change of sigand a decrease of about one order of magnitude inBg(v,T)and b. The integral ~which at T5298.15 K and l5514.5 nm was'24 000 a.u., and was predicted to chanto ' 22000 a.u.! becomes instead'11000 a.u. Thus, ourprediction was certainly conservative, but the effect remaimpressive.
V. CONCLUSIONS
We have carried out a detailed study of the secondelectric, refractive, Kerr, and hyperpolarizability virial coeficients of helium and argon. Our study, which assumesvalidity of a semiclassical approach, is extended to a wrange of temperatures and employs the FCI and CCSDsults for the frequency-dependent interaction induced poizabilities and hyperpolarizabilities for He2 and Ar2, respec-tively, which were reported in Refs. 1 and 3. We compaour virial coefficients with those obtained in other recentabinitio calculations and with the available experimental da
The present results for the second dielectric virial coficient of helium are in reasonable agreement with a recquantum perturbation study of Moszynskiet al.,28 and thefully quantum statistical results of Ref. 27 in the rangetemperatures between 70 and 410 K. For the second retivity virial coefficient of helium our study is the first reporof a theoretical determination based onab initio results forthe frequency-dependence of the interaction induced polaability of He2. For temperatures higher than 70 K the agrement with most of the experimental data is satisfactory. Tis also the case for the very few experimental data forrefractivity virial coefficient.
For the second dielectric virial coefficients for argon wobtain results which are about 3% larger than the valobtained in a previous study3 where we employed a smallebasis set in the calculation of the interaction induced poizability of Ar2. Since this is a small change compared to terror bar and the scattering of the experiment resultsmakes no significant effect on the agreement betweentheoretical and experimental results, which is reasonagood for the temperature range between 242 and 303 K,unsatisfactory for higher temperatures.
For the Kerr constant of helium the very weak quadradependence on the gas density is confirmed; in fact, aroroom temperature the second Kerr virial coefficient—whofrequency dependence is negligible—practically vanishThis explains the difficulties the experimentalists still e
ity
of
u
-lityts
en8.digfotltc
iv
tel
om-thex
thla
ricla
tio
.,
-ty
i
rka-ann
.
an,
r-
rd,
s.
rd,
10117J. Chem. Phys., Vol. 111, No. 22, 8 December 1999 Properties of He2 and Ar2 . II
counter in detecting any deviations from the linear pressdependence ofmK.
For the second Kerr virial coefficient of argon, the improved calculations for the interaction induced polarizabiand hyperpolarizability of Ar2 reduced the theoretical resulfor BK by about 3%. Our results of 8.7031025 m3 mol21 forBK /AK at 632.8 nm and 296.15 K is in favorable agreemwith the recent accurate experimental Kerr result of (60.3)31025 m3 mol21 which was obtained by Shelton anPalubinskas.39 The present calculations substantiate the snificant disagreement with earlier experimental results458 nm by Dunmuret al.38 found in Ref. 3. We suggestherefore an experimental reinvestigation of the latter resu
As anticipated in Ref. 3, the reevaluation of the interation induced second hyperpolarizability of Ar2 leads to asignificant change in the theoretical results for the relathyperpolarizability second virial coefficientb(v,T) ofargon—at 298 K and 514.5 nm from 0.6822 cm3 mol21 to0.1878 cm3 mol21.
Combining the present results forb for argon and heliumwith the pressure ratios employed in the experimental demination of b(ArH2) and b(HeH2), we obtain a theoreticaestimate forb(ArH2)1b(HeH2) of 23.2 cm3 mol21, which iseven less in agreement with the experimental value frRef. 21 of21.468.0 cm3 mol21, than the previous theoretical result of Ref. 3. The remaining discrepancy betweentheoretical and the experimental result can likely not beplained by shortcomings in theab initio results forDg i ofHe2 and Ar2, since the present calculations account forfrequency-dependence of the interaction induced hyperpoizability of both He2 and Ar2 and they are—in particular foHe2—essentially converged with respect to the one-partbasis sets and the correlation treatment. More likely theydue to either the neglect of quantum effects in the calculaof the hyperpolarizability virial coefficients23 or to approxi-mations made in the comparison with the experiment, e.gthe ansatz for the~pressure-dependent! local field factors.21
ACKNOWLEDGMENTS
One of the authors~C.H.! thanks the European Commission for financial support through the Training and Mobiliof Researchers~TMR! Program~Grant No. ERBFMBICT96.1066!. B.F. acknowledges a grant from the Spanish Dreccion General de Ensenanza Superior~Ref. No. PB95-0861! and supercomputer time from the CESGA. This wohas been supported by the Danish Natural Science ReseCouncil ~Grant No. 9600856!. H.K. acknowledges the Carlsberg foundation and the European Commission for a grA.R. acknowledges fruitful discussions with Robert Moszyski.
APPENDIX: CONVERSION FACTORS
Conversion factors from atomic units to SI and esu40
See also Ref. 41.
~1! 1 a.u. of r[a023 mol>6.7483331030 m23 mol
56.7483331024 cm23 mol ~esu!;~2! 1 a.u. ofv[\a0
22me21>4.1341431016 s21;
re
t9
-r
s.-
e
r-
e-
er-
eren
in
-
rch
t.-
~3! 1 a.u. of a[e2a04me\
22>1.64878310241 C2 m2 J21
51.48186310225 cm3 ~esu!;~4! 1 a.u. of g[e4a0
10me3\26>6.23538310265 C4 m4 J23
55.03669310240 esu;~5! 1 a.u. of AR(v,T), b[a0
3 mol21>1.48135310231
m3 mol2151.48135310225 cm3 mol21 ~esu!;~6! 1 a.u. of BR(v,T)[a0
6 mol22>2.19587310262
m6 mol22
52.19587310250 cm6 mol22 ~esu!;~7! 1 a.u. of mK(v,T), AK(v,T)[e2a0
9me2 mol21\24
>5.60408310255 V22 m5 mol2155.03672310240 esu;~8! 1 a.u. of BK(v,T)[e2a0
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