Isotropic collision-induced scattering by CF4 in a Raman vibrational band

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Isotropic collision-induced scattering by CF 4 in a Raman vibrational band

J.-L. Godet, A. Elliasmine, Y. Le Duff, and T. Bancewicz

Citation: The Journal of Chemical Physics 110, 11303 (1999); doi: 10.1063/1.478004 View online: http://dx.doi.org/10.1063/1.478004

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/110/23?ver=pdfcov Published by the AIP Publishing

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Isotropic collision-induced scattering by CF

₄

in a Raman vibrational band

J.-L. Godet,^a) A. Elliasmine, and Y. Le Duff

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

T. Bancewicz

Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan´, Poland

~Received 10 November 1998; accepted 18 March 1999!

Light-scattering intensities and depolarization ratio data have been recorded at room temperature for gaseous tetrafluoromethane in the collision-induced band of the Raman vibrational line n1

5908 cm²¹. For the first time, binary isotropic intensities of the n1-Raman band are reported in absolute units for Stokes frequency shifts up to 110 cm²¹. From comparison with theoretical semiclassical computations of the dipole-multipole spectrum contributions, evaluations of the first derivatives of the successive dipole-multipole polarizability tensors are provided. These results are coherent with those obtained from corresponding depolarized intensities. Moreover, the agreement with ab initio quantum-chemistry calculations of the CF₄multipolar polarizabilities is satisfactory.

I. INTRODUCTION

Since a molecule in a gas interacts with its neighbors, collision processes induce polarizability modifications and collision-induced scattering ~CIS! may be observed in the vicinity of spectral lines. For the optically symmetric molecule CF₄, depolarized Rayleigh^1,2 and n1-Raman³ intensities due to CI binary interactions have been reported in absolute units. In these cases, monomolecular depolarized scattering is forbidden and the CI polarizability anisotropy may be regarded as responsible for all observed scattered light. On the contrary, for polarized scattering, Rayleigh and Raman Q lines due to monomer polarizability contribute to corresponding spectra. Moreover, Rayleigh and Raman polarized bands are due as well to the trace and to the anisotropy of Rayleigh and Raman CI-polarizability tensors. Re- cently, the isotropic contribution to the CF₄ Rayleigh scattering has been obtained from the measurement of the depolarized and polarized scattering intensities around the laser frequency.⁴ This procedure needs good accuracy for intensity data, especially when the depolarization factor value is high and the isotropic contribution relatively weak.

In the present work, for the first time, we present our results concerning the CI isotropic intensities in then1-Raman band.

We compare these experimental data with theoretical semiclassical computations. In our analysis, we consider the influence of successive multipolar mechanisms from comparison between the theoretical and the experimental spectra.

Taking into account previous data extrapolated from Rayleigh² andn1-Raman³ depolarized spectra and from the isotropic Rayleigh spectrum,⁴we deduce the values (A8^,E8⁾ of the derivatives of the dipole-multipole polarizability tensors. Then these values are compared with quantum- mechanical values recently calculated by Maroulis.^5,6

II. THEORETICAL SPECTRUM

We consider a macroscopically isotropic system com- posed of N-like globular molecules in an active scattering volume V illuminated by laser radiation of angular frequency vi52pni, linearly polarized in the direction e. We analyze the secondary electromagnetic radiation emitted by the system in response to that perturbation. The radiation scattered with an angular frequencyvs52pns is measured at a point R from the center of the sample located behind an analyzer of polarization n. Then, the quantum-mechanical expression for the differential scattering cross section for scattering into an angular-frequency interval dv and a solid angle dV has the form⁷

]²s

]v]V5vivs 3

c⁴

(

_{i, f} ^rⁱû^{^}ⁱûê^–P–ⁿû^f^&û²^d^~^v²^v^{f i}^!^, ^~¹^!

where ri denotes the density matrix element of the initial state i and P is the polarizability tensor. Using the Fourier transform ofd⁽v2vf i) we rewrite Eq.~1!as

~2! where^ &denotes a canonical average. After Ref. 8, the cor- relation function F(t) for macroscopically isotropic systems may be expressed as the sum of components each of which is the product of two zero rank tensors, one dependent of the scattering geometry and the other on the internal properties of the scattering system. The polarizability tensor P_ab, when completely symmetric to its indexes, decomposes into an isotropic part and an anisotropic part. Expressing the same in the language of irreducible spherical tensors, the polarizability tensor P splits into two irreducible parts of ranks zero and two:

a!Electronic mail: [email protected]

11303

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P⇒P^~⁰^!%P^~^2!.

Then the correlation function F(t) reads

F~t!51

3~e–n!²^P^~0^!~0!(P^~0!~t!&

131~e–n!²

30 ^P^~²^!~0!(P^~²^!~t!&^. ~3!

In our experiment, we study the 90° scattering intensities and we observe scattered radiation without any analyzer.

However, the polarization of the incident laser beam is either perpendicular or parallel to the scattering plane. In the perpendicular case~'!, the correlation function becomes

F^'~t!5F^iso~t!1⁷6F^d~t!, ~4! where

F^iso~t!5¹3^^P^~⁰^!~0!(P^~⁰^!~t!&^, ^~⁵^!

F^d~t!5¹5^P^~2^!~0!(P^~2^!~t!&^, ~6!

are responsible for the isotropic and the depolarized spectral intensities, respectively. From Eq. ~2!, we note that the double differential cross section (]²s^/]v]V)_jof the respec- tive kind of scattered radiation of interest and the appropriate time correlation function F^j(t) are connected by a Fourier transform

S

^]v]^]²^s^V

D

_j⁵^v^cⁱ^v⁴^s³ ²¹^p

^E

2`

`

dt e²ⁱ^v^tF^j~t!, ~7!

withjequal to d,'or iso. From Eqs.~4!and~6!we deduce the frequency-dependent depolarization ratio for a 90° scattering experiment:

h~n!5I^d~n!

I^'~n!5FtF^d~t!

FtF^'~t!5 I^d~n!

I^iso~n!1⁷6I^d~n!, ~8! wheren is the frequency shift that is defined relative to the incident-beam frequency ni and Ft symbolizes the Fourier transform. Therefore, the isotropic intensities I^isomay be deduced from depolarization ratios and depolarized intensities as

I^iso~n!5

S

^h^~¹ⁿ^!²⁷⁶

D

^I^d^~ⁿ^!^. ^~⁹^!

Within the multipolar polarizability model and for the particular case of the CF₄ n1-Raman band, ~multi!polarizabilities ~dipolara, dipole-quadrupole A, dipole-octopole E, etc.! as well as their normal coordinate derivatives are responsible for the incremental CI pair polarizability tensor as is explained in detail in our previous paper on a depolarized spectrum.³ In Table I, we provide the successive multipolar contributions to the correlation function F^iso(t) involving the dipolar polarizabilitya and independent components (A,E) of the tensors ~A,E! together with their normal-coordinate derivativesa8^{, A}8^{, and E}8 ~it is worth noting that, contrary to the depolarized spectrum,³terms due to hyperpolarizabili- ties do not contribute to the isotropic spectrum in the first- order approximation considered here!.

In this work, the Fourier transforms of the successive multipolar correlation functions provided in Table I ~and therefore, the theoretical isotropic spectrum!are numerically computed by the method, the approximations, and the intermolecular potentials^9–11already described, used and justified in Refs. 2–4.

III. EXPERIMENTAL PROCEDURE

The experiment and the experimental setup used are very similar to the one described in a recent paper.³Provided by L’Air Liquide with an initial purity of 99.995%, the CF₄gas was pumped at low temperature in order to improve its purity. Then it was compressed inside a four-window high- pressure cell at 294.5 K and illuminated by the lL5514.5 nm line of an argon-ion laser operating at 2 W. The polarization of the laser was adjusted relative to the horizontal scattering plane defined by the laser beam and the axis of the 90° scattered beam. According to the laser polarization perpendicular or parallel to the scattering plane, we obtained an experimental polarized spectrum I^V or an experimental de- polarized spectrum I^H, respectively. At each frequency, intensities have been measured at various densities up to 250 amagat and the pair intensities I₂ have been deduced as co- efficients proportional to the square density in the virial ex- pansion given by

I5I₀1I₁r1I₂r²1I₃r³^. ~10! From I₂^H(n^{) and I}2

V(n), we deduce the experimental depolarization ratio h2

exp(n⁾5I₂^H(n^)/I2

V(n). However, due to the non-

TABLE I. Isotropic Raman light-scattering correlation functions F_n

1

iso(t) for successive multipolar induction operators. The functions SN(t), Rj(t), and F₁are defined in Refs. 2–4 whereas R12 is the intermolecular distance. ~DQ: dipole-quadrupole; DO: dipole-octopole; QQ: quadrupole-quadrupole; QO: quadrupole- octopole; OO: octopole-octopole.!

Mechanism Raman case F_n

1 iso(t)

First-order DID a8^Ta1aTa8 ⁰

Second-order DID a8^TaTa1... 216a⁴a8²^^R¹²⁽⁰⁾²⁶^R¹²^(t)²⁶&^F1

DQ a8^TA1a^TA8 ¹⁶⁰₇ _@(a8Â)²1(aÂ8⁾²#S3(t)R0(t)R3(t)F₁ DO a8^TE1a^TE8 ²²⁴₉ _@(a8Ê)²1(aÊ8⁾²#S4(t)R0(t)R4(t)F₁ QQ A8^TA1ATA8 ²⁸¹⁶₁₈₉(AA8⁾²^S⁴^(t)R³^(t)R³^(t)F1

QO A8^TE1ATE8 ⁴¹⁶₂₁_@(AE8⁾²1(A8^E)²#S5(t)R3(t)R4(t)F₁

OO E8^TE1ETE8 ¹¹⁰₃ (EE8⁾²^S⁶^(t)R⁴^(t)R⁴^(t)F1

11304 J. Chem. Phys., Vol. 110, No. 23, 15 June 1999 Godetet al.

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zero value of the collection angle Qused for the scattering beam, the depolarized intensity I₂^d(n) is not exactly I₂^H(n^), the polarized intensity I^'₂(n) is not exactly I₂^V(n^{), and the} isotropic intensity I₂^iso(n) is not exactly given by Eq.~9!.^12,13 In our setup, Q56.2°. Consequently, according to the correction formulas calculated in Ref. 13, we get the depolarized pair intensity by using³ I₂^d(n⁾5$^1.006 20.006/h2

exp(n⁾%^I2

H(n). In a similar way, we obtain the isotro- pic pair intensity I₂^isousing an equation derived from Eq.~9! for the particular collection-angle value of our setup:⁴

I₂^iso~n!5

S

^h^1.004²^~ⁿ^!^exp²^1.1695

D

^I²^d^~ⁿ^!^. ^~¹¹^!

IV. RESULTS AND DISCUSSION

We present in Fig. 1 the behavior of the experimental pair depolarization ratio h2

exp(n⁾ ^measured ⁱⁿ ^the 10– 110 cm²¹ frequency-shift range of the n1-Raman band.

Also in Fig. 1, we show the previously published depolarization ratio for the CF₄ Rayleigh band.⁴ The Raman depolarization-ratio data are significantly different from these observed in the Rayleigh band. At low frequencies (10– 20 cm²¹) the upper Raman value measured is about 0.5 when we obtained almost 6/7 for the Rayleigh band. At higher frequencies (n>60 cm²¹), h2

exp(n) is measured at about 0.2. This is approximately three times less than the value obtained for Rayleigh scattering in the 60– 100 cm²¹ frequency region. Such behavior shows first that the diode- induced-dipole ~DID! interactions alone fail to reproduce scattering intensities and second that the multipolar polarizability effects and/or other short-range effects contribute dif- ferently in the Raman and the Rayleigh bands. From the pair depolarization-ratio data and the pair depolarized intensities

given previously in Ref. 3, we deduced from Eq. ~11! the experimental absolute-unit isotropic intensities I^iso(n^{) which} are reported together with their error bars in Fig. 2 for the 10– 110 cm²¹frequency-shift range. In Fig. 2, we also show our total theoretical spectrum and its successive multipolar contributions for a Lennard-Jones potential,⁹ a set D 5(a^,a8⁾ ^of ^dipolar polarizabilities, and a set M 5(A,A8^,E,E8) of dipole-multipole polarizabilities. The D values,a52.93 Å³anda854.00 Å², are deduced from CF₄ refractive-index measurements¹⁴ and from Raman studies,^15,16respectively. They are close to the static polariz- abilities calculated ab initio by Maroulis^5,6 ~a52.89 Å³; a853.92 Å²!and may be regarded with confidence. TheM values of A, A8^{, E, and E}8 result from quantum-mechanical calculations of Maroulis.^5,6They are reported in Table II. As can be seen in Fig. 2, the total theoretical spectrum calculated with these values lies below the lower error-bar limits, whatever the frequency is. This discrepancy has also been observed, however to a lesser degree, in the depolarized n1-Raman band, and a fitted value of A8 is proposed in Ref.

3. In the present work, we test the compatibility of our theoretical model with both depolarized and isotropic Raman experiments. We check if there exists a new set M8 ^of dipole-multipole polarizabilities since both the depolarized and the isotropic theoretical intensities lie inside the error bars of the corresponding experimental points. Short- distance range effects are expected to contribute significantly at high frequencies. Unfortunately, the overlap and exchange effects are not yet known in the CF₄ case and, therefore, cannot be taken into account by our model. Besides, at low frequencies, it is not possible to study both depolarized and isotropic spectra in order to measure dipole-multipole polarizabilities. First, a first-order DID contribution exists in the

FIG. 1. Experimental depolarization ratioh2

exp(n⁾5I₂^H(n^)/I2

V(n) of the binary n1-Raman band of gaseous CF4 ~d! at 294.5 K in the 0 – 110 cm²¹ frequency-shift range. The corresponding depolarization ratio for the Ray- leigh band is also reported~1!.

FIG. 2. Two-body isotropic scattering Stokes spectrum of the n1-Raman band of gaseous CF4in absolute units at 294.5 K. Closed circles~d!indicate our experimental data including error bars. Theoretical curves~DID – – –, DQ - - -, DO – - –, QQ1QO1OO – - - –, and total theoretical —! are computed using a Lennard-Jones potential ~Ref. 9!, a52.93 Å³, a8 54.00 Å², and a setMof multipolarizability values calculated by Maroulis

~Refs. 5 and 6!given in Table II.

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depolarized case, which predominates over all dipole- multipole contributions in the 0 – 30 cm²¹frequency range,³ whereas only a weak second-order DID mechanism contrib- utes to the isotropic spectrum. Second, it can be noticed in Fig. 2 that the line shape of the experimental isotropic spectrum deviates from the corresponding theoretical one as n tends towards zero. This may be attributed, at least partly, to the vibrational-rotational coupling present in then1 Q line.¹⁷ Consequently, it is not possible to fit our theoretical model and both experimental spectra at lower frequencies as well as at higher frequencies, and we restrict our study to the 30– 80 cm²¹frequency range. As in Ref. 3, we chose forM8 the values of A51.0 Å⁴ and E51.5 Å⁵ that we previously deduced from the analysis of both depolarized and isotropic Rayleigh spectra.⁴Then we adopt a fitting procedure related to the parameter vector (A8^,E8) only. Despite the fact that we study a restricted frequency range, it is not possible to have theoretical curves through all the error bars, or through all but one or two. Taking into consideration the fact that three theoretical points may be slightly outside the error bars for the two spectra together, the parameter vector ~A8 '5 Å³, E8^'^{18 Å}⁴^!becomes a solution which completes set M8 reported in Table II together with previous evaluations of A, A8^{, E, and E}8^.^2–4In Figs. 3 and 4, we present experimental depolarized and isotropic n1-Raman bands, respectively, and corresponding theoretical spectra computed using our evaluated M8 values and the same Lennard-Jones potential.⁹ In the 30– 80 cm²¹frequency range, the theoretical curves lie either near the upper limits of the error bars in the depolarized case of Fig. 3 or in the vicinity of the error- bar lower limits in the isotropic case of Fig. 4. Moreover~i! above 80 cm²¹, the theoretical model cannot reproduce experimental intensities and ~ii! the fitted value E8'18 Å⁴ is far from the E855.53 Å⁴calculated by Maroulis.^5,6We have checked that, in our experiment, leaking and mixing between polarized and depolarized components due to experimental

polarization errors do not significantly affect our data.¹⁸ Therefore, deviations in our calculations with regard to experiment may be due to short range effects, to the aforementioned vibrorotational coupling, and/or to uncertainties as to the potential. Several CF₄potentials exist. Most of them have been presented in previous papers.^3,4The choice of potential slightly modifies the integrated intensity for each dipole- multipole contribution.^3,4 Moreover, considering the strong first-order DID contribution to the depolarized Rayleigh and

FIG. 3. Two-body depolarized scattering Stokes spectrum of then1-Raman band of gaseous CF4in absolute units at 294.5 K reported in Ref. 3. Closed circles~d!indicate experimental data including error bars. The theoretical curves ~DID – – –, DQ - - -, DO – - –, QQ1QO1OO – - - – and total theoretical —!are computed using a Lennard-Jones potential~Ref. 9!,a 52.93 Å³, a854.00 Å², and a setM8of multipolarizability values given in Table II.

TABLE II. Theoretical and experimental values of the CF₄multipolarizabilities and of their bond-length R derivatives~which are related to normal-coordinate derivatives; see Ref. 3!. Theoretical values~setM!com- puted ab initio by Maroulis~Refs. 5 and 6!and used in Fig. 2 are labeled by an asterisk~*_!. Experimental values

~setM8!deduced from our CIS experiments~Refs. 2–4 and the present work!and used in Figs. 3 and 4 are labeled by a diamond (^L). These labels correspond to a CF₄potential given in Ref. 9. The circles~°!and the stars (^!) refer to two other potentials that are given in Refs. 10 and 11, respectively.

Polarizability Theory Experiment CIS spectrum

uAu 0.972 Å⁴* 1 Å⁴^L isotropic Rayleigh~Ref. 4!

1.2 Å⁴ depolarized Rayleigh~Ref. 2!

uA8u5

U

^]]^A^R

U

^{4.09 Å}³^* ^{5 Å}^3L isotropic Raman~this work! 4.7 Å³° isotropic Raman~this work! 4.7 Å³! isotropic Raman~this work!

5.3 Å³ depolarized Raman~Ref. 3!

uEu 1.15 Å⁵* 1.5 Å⁵^L isotropic Rayleigh~Ref. 4!

3.5 Å⁵ depolarized Rayleigh~Ref. 2!

uE8u5

U

^]]^ER

U

^{5.53 Å}⁴^* ^,^{18 Å}⁴^L! isotropic Raman~this work! ,16 Å⁴° isotropic Raman~this work! ,28 Å⁴ depolarized Raman~Ref. 3!

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Raman spectra, we found that several potentials^19–22provide theoretical integrated intensities which are bigger than the upper limit of the corresponding experimental ones.³ Only three available potentials are compatible with these measurements: the Lennard-Jones potential used in Figs. 2–4, a second Lennard-Jones potential reported in Ref. 10, and the isotropic part of potential calculated ab initio by Palmer and Anchell.¹¹ On the other hand, whatever the potential is, the spectral shapes of dipole-multipole contributions are not significantly modified, as can be seen in Figs. 3 and 4 of Ref. 3.

This is due to the fact that potential-independent rotational stick spectra of the successive light scattering mechanisms are mainly responsible for the dipole-multipole line shapes.^3,23 The choice of potential simply shifts upward or downward slightly each theoretical dipole-multipole contribution like those represented in the logarithmic scale of Figs.

2–4. For the three aforementioned potentials, the integrated intensities of the dipole-quadrupole and dipole-octopole contributions are altered by less than 20%, and the corresponding changes in the dipole-multipole values are less than 10%.³ This may be checked in Table II, where we report fitted values of (A8^,E8) for these potentials.^9–11 Moreover, except for the purely translational DID contribution, the choice of potential affects both theoretical depolarized and isotropic intensities in a similar manner. Therefore, in the 30– 80 cm²¹ frequency range, it could not allow simulta- neous ‘‘decreasing’’ of the theoretical depolarized spectrum and ‘‘increasing’’ of the theoretical isotropic spectrum in order to make them closer to depolarized and isotropic experiments, respectively. With regard to the influence of the ro- tovibrational coupling of the n1 Q line on its width at half intensity, it may be equal to several cm²¹ at a few amagat.

However, the effect of coupling in the wing of the n1 line decreases with density and with frequency. Therefore, we may estimate that it is negligible in the 30– 80 cm²¹ frequency region for the density range used in these experi-

ments. A fortiori, it cannot explain the discrepancies beyond 80 cm²¹, which are observed for both the isotropic and the depolarized spectra. The discrepancies observed might mainly be seen as results of short range effects ~such as overlap and exchange, or molecular frame distortion; the nonpoint-like size of the CF₄molecule may also play a role, particularly in the case of the trace-induced isotropic spectrum²⁴!. The dipole-octopole ~DO! mechanism is the shortest range effect considered here. It may explain why our fitting procedure leads to overestimated values of E8^{in com-} parison to Maroulis’ value. In Figs. 3 and 4, overestimation of the DO contributions makes up for short-range effects, not taken into account in our theoretical model. We use E8 518 Å⁴ in these figures, but it is rather an order of magnitude and an upper limit than a real estimate @in Ref. 3, we only wrote that E8,28 Å⁴#. On the other hand, our measure- ment of A8 ~A8'5 Å³; in Ref. 3, we found A8'5.3 Å³! is close to the ab initio calculation of Maroulis^5,6 (A8 54.09 Å³). We assume that it is due to two factors:~i!the dipole-quadrupole~DQ!mechanism is predominant from 30 up to 80 cm²¹for both depolarized and isotropic spectra;~ii! our theoretical model is sufficient in this frequency range.

The measurement of the isotropicn1-Raman band thus con- firms the conclusion of our previous paper on the depolarized spectrum.³At the same time, the lack of theoretical isotropic intensity not only concerns the frequencies beyond 80 cm²¹ like in the depolarized case, but all of the frequency range scanned. This has shed new light on the relative importance of short-range effects on the trace and on the anisotropy of the CF₄pair polarizability tensor.

V. CONCLUSION

In the present work, the isotropic intensities of the n1-Raman band measured in absolute units have been reported for the first time in the 10– 110 cm²¹frequency range.

Comparison between experiment and theoretical predictions as well as comparison with previous measurements of the depolarized n1-Raman band³ have shown that CIS Raman experiments on gaseous CF₄are a good way to measure the dipole-quadrupole derivative A8. The value A8^'^{5 Å}³ ^that we have deduced from fitting both depolarized and isotropic Raman spectra is close to the quantum-mechanical computed A854.09 Å³ of Maroulis.^5,6 Most certainly, short-range effects modify depolarized and, more significantly, isotropic spectral line shapes at high frequencies. This leads to an overestimated value of the dipole-octopole derivative E8^. However, it does provide a good order of magnitude of E8^. In conclusion, the Raman CIS experiment on both isotropic and depolarized spectra is a suitable technique by which to measure at least the dipole-quadrupole Raman-polarizability tensor in the case of a globular molecule such as CF₄. More- over, measurement and theoretical analysis of isotropic Ra- man spectra has provided us with new information on mechanisms contributing to the trace of the Raman pair polarizability tensor.

FIG. 4. The same as in Fig. 2, except that the theoretical curves computed here are for the multipolarizability setM8given in Table II.

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ACKNOWLEDGMENTS

This work has been supported in part by the University of Angers, and in part by Grant No. 98086 of French-Polish Scientific cooperation program POLONIUM.

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