Collision-induced scattering in CO2 gas

Collisioninduced scattering in CO2 gas

Victor Teboul, Yves Le Duff, and Tadeusz Bancewicz

Citation: J. Chem. Phys. 103, 1384 (1995); doi: 10.1063/1.469761 View online: http://dx.doi.org/10.1063/1.469761

View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v103/i4 Published by the American Institute of Physics.

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2 Victor Teboul and Yves Le Duff

Laboratoire des Proprie´te´s Optiques des Mate´riaux et Applications, Universite´ d’Angers, Faculte´ des Sciences, 2 Boulevard Lavoisier, 49045 Angers, France

Tadeusz Bancewicz

Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University, Grunwaldzka 6, 60-780 Poznan´, Poland

~Received 6 March 1995; accepted 18 April 1995!

Carbon-dioxide gas rototranslational scattering has been measured at 294.5 K in the frequency range 10–1000 cm²¹at 23 amagat. The depolarization ratio of scattered intensities in the frequency range 10–1000 cm²¹is recorded. The theoretical and experimental spectra in the frequency range 10– 470 cm²¹are compared. The anisotropic double differential cross section for scattered light is calculated theoretically considering first- and second-order dipole-induced dipole, first-order dipole–induced octopole, and first-order dipole–dipole–quadrupole light scattering mechanisms as well as their cross contributions. © 1995 American Institute of Physics.

I. INTRODUCTION

Light scattering from low density linear molecule fluids emerges due to permanent monomolecular polarizabilities as well as excess polarizability of molecular pairs caused by intermolecular interactions ~collisions!.^{1– 4} Both above dis- cussed sources of scattered radiation create however quite different spectral shapes of the scattered light. Permanent polarizabilities are mainly responsible for spectra composed of individual rotational lines. Due to Boltzmann distribution of the rotational energy levels these spectra decay relatively fast. On the other hand, the excess short-range interaction- induced part of the pair polarizability creates a continuous rototranslational relatively weak but very broad spectral dis- tribution of scattered light. Finally, in experiment, both these spectral shapes overlap. At low frequencies rotational spectra due to permanent molecular polarizabilities dominate sub- stantially but when we shift to high frequencies interaction- induced contributions become predominant. It has been found that the interaction-induced first-order dipole–induced dipole ~DID! light scattering mechanism can explain the greatest part of the integrated interaction-induced intensity as well as the low frequency interaction-induced spectra. How- ever, when high frequency scattering is involved, DID inter- actions cannot explain the observed intensities. This was shown for atoms,^5,6 for several isotropic molecules @CH₄, CF₄, C~CH₃!4and SF₆#,^7–9and for anisotropic molecules be- longing to the symmetry group C_2V¹⁰ or D_`h ~N₂!.^11,12The special case of very light molecules H₂ and D₂ was also investigated.^13–17

In this paper experimental depolarized carbon dioxide intensities are measured in the frequency range up to 470 cm²¹, not previously studied for CO₂ gas. To explain high frequency spectra, shorter than first-order DID range interaction-induced light scattering mechanisms have to be considered. Here for comparison of our experimental spec- trum with the theoretical one we calculate the latter taking into consideration the first- and second-order ~back- induction! dipole–induced dipole light scattering mecha- nisms as well as the dipole–induced octopole effect related with the nonuniformity of the local field acting on the mol-

ecule and the octopole moment induced in the CO₂molecule by the field of surrounding molecules and moreover the non- linear response of the CO₂ molecule to the incident laser field and to the field due to the permanent quadrupole mo- ments of other CO₂ molecules. The irreducible spherical components of the second-order DID, dipole–induced octo- pole and hyperpolarizability–permanent quadrupole (BQ) light scattering mechanisms are considered for the first time.

Our paper is organized as follows. Section II describes experimental details and the procedure used for measuring the spectrum in absolute units. The theoretical model of the rototranslational interaction-induced CO₂ spectrum and de- tails of our computations are given in Sec. III. Experimental spectra and their comparison with the theoretical spectrum are discussed in Sec. IV. Analytical formulas for the nonzero spherical components of the model pair polarizability used in this work are assembled in the Appendix.

II. EXPERIMENT

The scattering intensities of gaseous CO₂have been re- corded at 294.5 K using a typical Raman scattering experiment.⁶The beam of an argon laser ~2 W at 5145 Å! was focused into a four windows high pressure cell contain- ing CO₂gas. The light scattered at 90° by the gas was ana- lyzed with a double monochromator and detected by a pho- tomultiplier connected to a photon counter. A scrambler was inserted in front of the monochromator to make the response of the apparatus independent of the polarization of the light scattered by the CO₂gas. In this way we were able to record two different types of scattered intensity for each frequency shift n. A first type I_H~n! called parallel intensity was ob- tained with a laser beam polarized parallel to the scattering plane. A second type, I_V~n!, called perpendicular intensity, was obtained when the laser beam was polarized perpendicu- lar to the scattering plane. These intensities I_Hand I_Vdepend generally on the solid angle of the incident beam and the scattered beam.⁵We shall refer to the depolarized intensity I_d and the polarized intensity I_p as the limits of I_H and I_V, respectively obtained with convergence angles close to zero for the incident and scattered beams. In our experiment the

scattered beam was collected in a cone with a half-angle about 6.2° and the effect of the convergence of the laser beam into the cell was negligible. Under these conditions we have

I_d51.006I_H20.006I_V, ~1!

I_p51.003I_V20.003I_H. ~2!

The absolute values of the spectral scattering intensities were obtained from a comparison with a reference line ac- cording to a procedure described previously.⁶ We compared the integrated intensity of the S₀~1!rotational line of gaseous D₂at 6 amagat with the scattering intensity of gaseous CO₂ at 6 amagat and n5210 cm²¹. Using the polarizability anisotropy^{18 –20} g50.306 ų for the S₀~1! line of D₂, we calculated the scattering intensities I_d~n!for CO₂in absolute units~cm⁶!forn5210 cm²¹and then for the other frequency shifts considered.²¹

The CO₂gas was purchased from Union Carbide with a purity of 99.99%. The densities were calculated from PVT data compiled in Ref. 22.

III. THEORY

Consider scattering of light linearly polarized in the scat- tering plane and detected with no analyzer. Then the double differential cross section of the scattered spectrum reads

S

^]]V2]vs

D

H

51 5 k_ik_s³ 1

2p

E

^{exp~2ivt!F220 ~t!dt, ~3!}

where

F₂₂⁰ ~t!5^^A~2!~0!(A^~2!~t!& ^~4!

is the autocorrelation function of the irreducible second rank pair polarizability tensor A^~2! of the scattering molecules.

The pair polarizability tensor A^~2! is composed of two parts

A^~2!5~1!A^~2!1~2!A^~2!1DA^~2! ~5! the permanent pair polarizability ₍₁₎A^~2!1_~2!A^~2! and the ex- cess pair polarizabilityDA^~2!, originating in the intermolecu- lar interactions. Generally, when dealing with molecular ro- tations it is very desirable to expressDA^~2!in the language of spherical tensors; for linear moleculesDA^~2!has the form of an expansion in spherical harmonics Y_m^l ~V!:

DA_m^~r!5 ~4p!^3/2

~2r11!^1/2 _l

(

1,l2,L,L

B_l

1,l₂,L,L

~r! ~R₁₂!$^Y~L!~Rˆ

12!

^@Y^~l1!~V1!^Y^~l2!~V2!#^~L!%~mr!, ~6! whereV1,V2, and R₁₂denote, respectively, the orientations of molecules 1 and 2 and the relative molecular separations.

On the other hand, however, Cartesian tensor notation is much more pedagogical, so to start with we briefly discuss the inductional variations DA in the pair polarizability in Cartesian tensor notation and then rewrite the successive light scattering mechanisms into spherical tensor language of Eq.~6!. In the range of intermolecular separations R₁₂where the molecular charge distributions do not overlapDA, up to order R₁₂²⁶, reads^23–25

DA_ab5~11`¹²!$^Aag~1!T_gd~R₁₂!A^~_db^2! 1A^~_ag^1!T_gd~R₁₂!A_de^~2!T_ef~R₂₁!A^~_fb^1!

115¹A_ag^~1!T_gdef~R₁₂!E_b,^~2_def^! 115¹E_a^~2_,gde^! T_gdef~R₂₁!A_fb^~1!

2¹9B_ab^~1!_,gdT_gd,ef~R₁₂!Q_ef^~2!1•••%^{, ~7!} where A is the dipole–dipole polarizability tensor of the un- perturbated molecule, E its dipole–octopole polarizability tensor, B the dipole–dipole–quadrupole hyperpolarizability tensor, Q the permanent quadrupole moment of the mol- ecule, moreover T_a

1,...,a_n(R₁₂) 5 ¹_a1,...,¹_an(R₁₂²¹) whereas`¹² permutes the indices 1 and 2. The first and sec- ond term of inductional polarizability~7!originates, respec- tively, in the first- and second-order dipole–induced dipole

~DID!light scattering mechanism. The third and fourth terms are due to the dipole–induced octopole ~DIO! mechanism whereas the last term comes from the permanent quadrupole moment of the molecule and its dipole–dipole–quadrupole hyperpolarizability tensor B.

The irreducible spherical tensors version of Eq. ~7! has been elaborated elsewhere.^26,11,27The induced polarizability DA^~2!is fully defined in spherical tensor language if the ex- plicit form of all nonzero B_l

1,l₂,L,L

(r) (R₁₂) coefficients of Eq.

~6! is available. Having the spherical tensor version of Eq.

~7!we wrote a symbolic program in Mathematica²⁸calculat- ing the explicit form of all nonzero B’s of Eq. ~7! and as- sembled them in the Appendix.

We calculate the pure rotational and translational parts of our spectrum assuming negligible translational–rotational coupling between CO₂ molecules. Then the resulting spec- trum takes the form of the convolution of the rotational and translational parts:

]²s

]V]v^5kiks3

E S

^]V]2]vs8

D

^rot

S

^]V]~]v2s2v8!

D

^{trdv8. ~8!}

We easily note that all B_l

1,l₂,L,L

(r) (R₁₂) coefficients of the Appendix are of the form:

B_l

1l₂LL

~2! ~R₁₂!5

(

n

~n!B˜

l₁l₂LL

~2! R₁₂²ⁿ. ~9!

We now calculate the rotational part of our spectrum by means of the slightly modified Eq. ~38!of Ref. 27. For the anisotropic I_Hspectrum calculated here we have to change in Eq. ~38! of Ref. 27 the numerical ~‘‘geometrical’’! factor from ‘‘isotropic scattering’’ ¹₃at Eq.~38!of~Ref. 27!to ¹₅in agreement with Eq. ~3!. Moreover, obviously, we now have to use second-order B_l

1,l₂,L,L

(2) (R₁₂) spherical components of Eq.~6!listed in the Appendix instead of the zeroth-order components B_l

1,l₂,L,L

(0) (R₁₂) discussed in Ref. 27. From the above procedure we note that the following molecular pa- rameters

C_l

1l₂L

~r! ~R₁₂!5

(

_L ^{~B~lr1!l2LL~R12!B~lr1!l2LL~R12!! ~10!}

J. Chem. Phys., Vol. 103, No. 4, 22 July 1995

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govern the interaction-induced rotational branches of our stick spectrum. We calculate these parameters using spheri- cal components of Eq. ~6! listed in the Appendix. Conse- quently the translational part of our spectrum is described by Eq.~37!of Ref. 27. The translational functions depend on L, n, and k. However, since

~L,n,k!S^tr~t!5^~L,k,n!S^tr~t! ~11!

different translational functions emerge only for different pairs of indices L and n1k. Finally, 31 rotational stick spec- tra are computed resulting from different rotational branches:

DJ₁50,62,6l1; DJ₂50,62,6l2 and within each branch from different translational functions. To deal with transla- tional motion we calculate the Birnbaum–Cohen²⁹ model spectrum of^(L,n,k)S^tr(t) which, by way of the zeroth, second, and fourth translational moments is given by Eqs.~50!–~54! of Ref. 27. We calculate the successive spectral moments for successive L and n1k by means of the formulas ~17! and

~18! of Ref. 30 and Eqs. ~42! and ~43! of Ref. 27. In our spectrum we have seven different translational ‘‘subspectra’’

with different pairs of indices L and n1k. Our formulas for the translational moments³⁰ require the isotropic part of the potential between the CO₂ molecules. We selected for this purpose an isotropic Lennard-Jones ~12-6! type potential withe^/kB5245.32 K ands53.769 Å taken from the corre- sponding state principle³¹ ~LJ CSP!. We calculated as well the isotropic part of the 3LJQ potential of³²using³³

U₀~R₁₂!54

(

_ab ^eab

FS

^sRab12

D

^12F

S

^{6, 112 , 32,32;}

S

^Rra12

D

^2,

S

^Rr12b

D

²

D

²

S

^sRab12

D

⁶

3F

S

^{3, 52, 32,32;}

S

^Rr12a

D

^2,

S

^Rr12b

D

²

DG

^{, ~12!}

where F is the Appell function. However, the isotropic part of 3LJQ potential U₀(R₁₂) calculated using Eq. ~12! was much shallower in comparison with the LJ CSP potential.

Such a potential predicts shapes for all ^(L,n,k)S^tr~n! stepper than those of LJ CSP origin and consequently gives worse comparison of the theoretical spectrum and experimental data. All spectral moments for light scattering mechanisms

used in our computations, together with the characteristic times t1

(L,n,k)

andt2 (L,n,k)

of the respective Birnbaum–Cohen profiles are listed in Table I. Moreover Fig. 1 shows transla- tional spectra for some selected light scattering mechanisms.

We note that when dealing with shorter and shorter range light scattering mechanisms the calculated translational spec- tra are more and more diffuse analogically to the results of quantum-mechanically computed translational profiles of hy- drogen spectra.¹⁷ In Table II we list the values of CO₂mul- tipole polarizabilities and hyperpolarizability tensor compo- nents used in our computations.

Finally having calculated the rotational stick spectra and appropriate translational profiles, we compute with Eq. ~8! our resulting spectrum as the convolution of the rotational and translational spectra. From our calculations it results that for CO₂ the second-order DID light scattering mechanism contributes 9% to the total integral intensity of the I_Hspec- trum. This mechanism, however, exerts a diversified influ- ence on the average values of the individual spherical com- ponents ^^C_l1l2L

(2) (R₁₂)&^{, Eq.} ~10!. It contributes 20% to the

FIG. 1. Translational spectra ^(L,n,k)SˆBC(n⁾5[^(L,n,k)SBC(n^{)]/[ M}0(L,n,k)]

for some selected light scattering mechanisms. From top to bottom we plot 1:^~4,6,6!SˆBC~n!, 2:^~4,5,5!SˆBC~n!, 3:^~2,6,3!SˆBC~n!, 4:^~2,3,3!SˆBC~n!.

TABLE I. The zeroth M 0(L,n,k), and normalized second Mˆ 2(L,n,k), and fourth Mˆ4(L,n,k) translational moments for the successive light scattering mechanisms discussed in this work. The last two entries give the respective Birnbaum–Cohen profile characteristic timest1

(L,n,k)andt2 (L,n,k).

M 0~233!

^R1226& M 0~236!

^R1229& M 0~455!

^R12210& M 0~456!

^R12211& M 0~066!

^R12212& M 0~266!

^R12212& M 0~466!

^R12212&

0.135 Ų³ 1.47310²³Å²⁶ 3.45310²⁴Ų⁷ 8.21310²⁵Ų⁸ 1.98310²⁵Ų⁹ 1.98310²⁵Ų⁹ 1.98310²⁵Ų⁹

Mˆ 2~233! Mˆ 2~236! Mˆ 2~455! Mˆ 2~456! Mˆ 2~066! Mˆ 2~266! Mˆ 2~466! 7.93310²⁴s²² 1.49310²⁵s²² 2.88310²⁵s²² 3.27310²⁵s²² 2.40310²⁵s²² 2.81310²⁵s²² 3.74310²⁵s²²

Mˆ 4~233! Mˆ 4~236! Mˆ 4~455! Mˆ 4~456! Mˆ 4~066! Mˆ 4~266! Mˆ 4~466! 8.54310⁵⁰s²⁴ 2.72310⁵¹s²⁴ 6.18310⁵¹s²⁴ 8.00310⁵¹s²⁴ 6.91310⁵¹s²⁴ 7.83310⁵¹s²⁴ 1.03310⁵²s²⁴

t1~233! t1~236! t1~455! t1~456! t1~066! t1~266! t1~466!

6.67310²¹³s 4.54310²¹³s 2.27310²¹³s 2.13310²¹³s 3.53310²¹³s 2.86310²¹³s 1.37310²¹³s t2~233! t2~236! t2~455! t2~456! t2~066! t2~266! t2~466!

1.89310²¹³s 1.47310²¹³s 1.53310²¹³s 1.43310²¹³s 1.18310²¹³s 1.24310²¹³s 1.36310²¹³s

most anisotropic component ^^C244

(2)(R₁₂)&, 14% to the

^^C204

(2)(R₁₂)&component, 7% to the^^C202

(2)(R₁₂)&^component,

5% to the ^^C222

(2)(R₁₂)&, and 9% to the isotropic component

^^C002

(2)(R₁₂)&. Overall, the hyperpolarizability (BQ) light

scattering mechanism gives a negative contribution to the integrated I_Hintensity lowering its value by 3%. It has, how- ever, a quite substantial and varied influence on the indi- vidual spherical components ^^C_l1l2L

(2) (R₁₂)&. It lowers the

value of the component ^^C224

(2)(R₁₂)& down to 40% of its

value calculated when the (BQ) mechanism is neglected.

The same concerns the^^C024

(2)(R₁₂)& component but here the

reduction is less pronounced and amounts to 68% of that calculated if the (BQ) mechanism is neglected. Quite differ- ent is the influence of the (BQ) mechanism on the most anisotropic component ^^C244(2)(R₁₂)&: its value increases by 40%.

IV. RESULTS AND DISCUSSION

We show in Fig. 2 the depolarized Stokes scattering spectrum of CO₂at 23 amagat and 294.5 K from 10 to 1000 cm²¹. Up to about 450 cm²¹ the scattering intensities de- crease when frequency shifts increase. Then a band is ob- served centered about 670 cm²¹due to the Raman forbidden line n2.³⁴ For the low frequency part, up to 100 cm²¹, the observed intensities come mostly from contribution of the pure rotational Raman lines of the linear molecule CO₂.³⁵In this region the intensities change roughly proportionally to the gas density. This density behavior changes at high fre- quencies. We show in Fig. 3 that the ratio of the experimental intensities measured at 160 cm²¹divided by the gas density increase linearly with density. The same behavior may be observed at every frequency of this high frequency part up to 450 cm²¹ and for the induced Raman bandn2.³⁴ These re- sults show for the first time in the region of 150– 450 cm²¹ that beyond the low frequency scattering corresponding to the individual rotational motion of each CO₂ molecule we observe scattering intensities completely induced by binary interactions.

In Fig. 4 we show the spectral depolarization ratiohn~n! up to 1000 cm²¹calculated from the depolarized and polar- ized intensities. We have

hn~n!5I_d~n!

I_p~n!. ~13!

For completely depolarized scatteringhn5⁶7. In our case we observed a slow decrease of hn~n! up to 400 cm²¹ and then a sudden change in depolarization ratio when the con- tribution of then2Raman band becomes significant. Finally,

TABLE II. Numerical values of the carbon-dioxide multipole polarizabil- ities and the CO2permanent quadrupole moment used in our calculations

~all values are in atomic units and from Ref. 37!.

E_x,xxx 264.10

E_z,zzz 176.56

B_xx,xx 2140

B_xx,zz 79

B_zz,zz 2305

Bxz,xz 2205

u 23.239

a ^17.626

g ^14.271

FIG. 2. Experimental depolarized intensities Idin absolute units~cm⁶!for the stokes scattering of CO₂gas vs frequency shiftn^{in cm21}at 23 amagat and 294.5 K.

FIG. 3. Depolarized intensity Id ~arbitrary units! divided by density for several densities for CO2gas at 294.5 K and two frequency shifts~n: 160 cm²¹;3: 660 cm²¹!.

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around 670 cm²¹, hn~n! fluctuates around an average value of 0.56 in agreement with previous studies of the n2band.³⁴ From the behavior ofhn~n!in the frequency region 100– 450 cm²¹ we estimate that the isotropic intensity at each fre- quency of this region is at the most about 6% of the depo- larized intensity at the same frequency.³⁶ However the ex- perimental incertitudes are too large to propose an experimental spectral shape for this isotropic spectrum.

Figure 5 shows the comparison of the experimental and theoretical depolarized spectra. We note that the experimen- tal and theoretical spectra have almost the same shape

~slope!but the intensities of the theoretical spectrum at high frequencies are roughly of one order of magnitude smaller than the experimental ones. Similar slopes of the experimen- tal and theoretical spectra would tempt us to the conclusion that a CO₂ molecule with higher values of multipole polar- izabilities components than computed³⁷ and used in this pa- per could fit the experimental data. However, we prefer to attribute the discrepancy to overlap effects during CO₂col- lisions but, to our knowledge, numerical details of this part of the excess pair polarizability are not yet available for CO₂. Obviously the consideration of the next multipole con- tributions ~shorter range than R₁₂²⁶!could lower the discrep- ancy as well. Moreover our conclusion relating slope and intensity could suggest that the angular dependence of excess overlap polarizability should resemble the angular depen- dence of the relatively long range electrostatic multipole–

induced multipole light scattering mechanisms considered here by us. However, in the case of CO₂ the discrepancy between the multipolar high frequency part of the theoretical spectrum and the experimental one is substantially less pro-

nounced than in the nitrogen case.¹²This suggests that over- lap effects are less important in the case of CO₂molecules in comparison with N₂.

Figure 5 shows that second-order DID CO₂contributions though not great in value visibly improve the comparison between the theoretical and experimental spectra. Moreover, in the course of our computations we note that the high fre- quency CO₂ rototranslational spectrum, in comparison with the nitrogen one,^11,12 is created much more strongly by its translational part whereas the rotational spectrum located at lower frequencies due to the relatively small value of the CO₂rotational constant plays a less significant role in creat- ing the resulting high frequency spectrum. However this last contribution cannot be excluded as shown by recent studies of diatomic molecules scattering.³⁸

APPENDIX

B₀₀₀₂^~2! ~R₁₂!52

A

⁶a²

R₁₂³ 12

A

⁶a³~51k²!

5R₁₂⁶ , ~A1!

B₂₀₂₀^~2! ~R₁₂!5B₀₂₂₀^~2! ~R₁₂!5

A

⁶a³k~105150k1k²!

25R₁₂⁶ ,

~A2!

FIG. 4. Depolarization ratiohnvs frequency shiftn^{for CO}2gas at 294.5 K and 23 amagat. The dashed line~---!corresponds to a depolarized scattering withhn56/7.

FIG. 5. Theoretical and experimental CO2spectra in the frequency range 10– 470 cm²¹at 23 amagat and 294.5 K. Experimental results~111!for Id~n!are similar to Fig. 2 except for the points fromn5440 cm²¹ton5470 cm²¹for which an estimation of then2Raman band contributions have been subtracted. Theoretical double differential cross section~---!due to the per- manent polarizabilities only with permanent anisotropy rotational lines broadened as in Ref. 11; theoretical double differential cross section~– – –! calculated using the first-order DID1the first-order DIO1the first-order BQ light scattering mechanisms;~—!theoretical double differential cross sec- tion calculated using the full expression of Eq.~7! ~including second-order DID!for the light scattering mechanism in the form listed in the Appendix.

B₂₀₂₂^~2! ~R₁₂!5B₀₂₂₂^~2! ~R₁₂! 522

A

¹⁰⁵a²k

5R₁₂³

12

A

¹⁰⁵a³k~23015k1k²!

175R₁₂⁶ , ~A3!

B₂₀₂₄^~2! 5B₀₂₂₄^~2! 522

A

²¹~²7a^E21¹5B20Q!

3R₁₂⁵

136

A

²¹a³k~51k²!

175R₁₂⁶ , ~A4!

B₂₂₂₀^~2! ~R₁₂!54

A

¹⁰⁵a³~271k!k²

175R₁₂⁶ , ~A5!

B₂₂₀₂^~2! ~R₁₂!52

A

³⁰a²k²

25R₁₂³ 14

A

³⁰a³k²~718k!

175R₁₂⁶ , ~A6! B₂₂₂₂^~2! ~R₁₂!522

A

⁶a²k²

5R₁₂³ 12

A

⁶a³k²~217195k! 245R₁₂⁶ ,

~A7! B₂₂₄₂^~2! ~R₁₂!536

A

³⁰a²k²

25R₁₂³ 136

A

³⁰a³k²~28117k! 1225R₁₂⁶ ,

~A8! B₂₂₂₄^~2! ~R₁₂!54

A

³⁰~2ak^E21B22Q!

105R₁₂⁵

1144

A

³⁰a³~271k!k²

1225R₁₂⁶ , ~A9!

B₂₂₄₄^~2! ~R₁₂!52

A

³³~2ak^E21B22Q!

21R₁₂⁵

172

A

³³a³~271k!k²

245R₁₂⁶ , ~A10!

B₂₄₂₀^~2! ~R₁₂!5B₄₂₂₀^~2! ~R₁₂!536

A

²¹a³k³

175R₁₂⁶ , ~A11! B₂₄₂₂^~2! ~R₁₂!5B₄₂₂₂^~2! ~R₁₂!572

A

³⁰a³k³

1225R₁₂⁶ , ~A12! B₂₄₃₂^~2! ~R₁₂!52B₄₂₃₂^~2! ~R₁₂!536

A

²¹a³k³

245R₁₂⁶ , ~A13! B₂₄₄₂^~2! ~R₁₂!5B₄₂₄₂~R₁₂!536

A

³³a³k³

245R₁₂⁶ , ~A14! B₂₄₂₄^~2! ~R₁₂!5B₄₂₂₄^~2! ~R₁₂!5

A

⁶~6ak^E42¹5 B24Q!

189R₁₂⁵

136

A

⁶a³k³

1225R₁₂⁶ , ~A15!

B₂₄₃₄^~2! ~R₁₂!52B₄₂₃₄^~2! ~R₁₂! 5236

A

²¹⁰a³k³

1225R₁₂⁶

1

A

²¹⁰~212ak^E41B24Q!

945R₁₂⁵ , ~A16! B₂₄₄₄^~2! ~R₁₂!5B₄₂₄₄^~2! ~R₁₂!

52

A

³⁰~24ak^E419B24Q!

945R₁₂⁵

1324

A

³⁰a³k³

1225R₁₂⁶ , ~A17!

B₂₄₅₄^~2! ~R₁₂!52B₄₂₅₄^~2! ~R₁₂!5

A

³³⁰~6ak^E41B24Q!

135R₁₂⁵

236

A

³³⁰a³k³

175R₁₂⁶ , ~A18! B₂₄₆₄^~2! ~R₁₂!5B₄₂₆₄^~2! ~R₁₂!52

A

³⁹⁰~12ak^E42B24Q!

135R₁₂⁵

172

A

³⁹⁰a³k³

175R₁₂⁶ , ~A19! B₄₀₄₂^~2! ~R₁₂!5B₀₄₄₂^~2! ~R₁₂!536

A

²¹a³k³

35R₁₂⁶ , ~A20! B₄₀₄₄^~2! ~R₁₂!5B₀₄₄₄^~2! ~R₁₂!522

A

²³¹⁰a^E4

63R₁₂⁵ . ~A21! We use the following notation:

a5azz12axx

3 , ~A22!

k5azz2axx

3a ^{, ~A23!}

E258E_x,xxx23E_z,zzz, ~A24!

E45E_z,zzz12E_x,xxx, ~A25!

Q5Qzz, ~A26!

B205B_zz,zz14B_xz,xz1B_xx,zz14B_xx,xx,

B2252B_zz,zz26B_xx,zz14B_xz,xz24B_xx,xx, ~A27! B2453B_zz,zz28B_xz,xz22B_xx,zz12B_xx,xx.

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J. Chem. Phys., Vol. 103, No. 4, 22 July 1995

Downloaded 14 Dec 2011 to 193.52.40.1. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

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~1988!.

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~1966!.

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21Collision-induced pair contributions to scattering intensities Id~n!, mea- sured in cm⁶, are independent of the CO2density.

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23A. D. Buckingham, Adv. Chem. Phys. 12, 107~1967!.

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~1988!.

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~Addison-Wesley, Redwood City, CA, 1991!.

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