- Email: [email protected]
Depolarized collision-induced light scattering of gaseous CCl4
Chemical Physics Letters 379 (2003) 380–385 www.elsevier.com/locate/cplett
Depolarized collision-induced light scattering of gaseous CCl4 Uwe Hohm
*
Institut f€ur Physikalische und Theoretische Chemie der Technischen Universit€ at Braunschweig, Hans-Sommer-Straße 10, D-38106 Braunschweig, Germany Received 22 July 2003; in final form 19 August 2003 Published online: 16 September 2003
Abstract Depolarized collision-induced light-scattering spectra of CCl4 in the saturated vapour-phase have been measured in the ranges of temperature and density between T ¼ 501 and 561 K and . ¼ 687 and 1524 mol m3 , respectively. The CILS spectra are analyzed by using an isotropic Lennard–Jones type (18.6–6) intermolecular-interaction potential. It is found that the polarizability anisotropy of the CCl4 -dimer contains a significant short-range repulsive term. Taking into account collision-induced rotational Raman scattering (CIRR) the dipole–quadrupole polarizability A, and dipole– octopole polarizability E of CCl4 were obtained as jAj ¼ 3:8ð1:0Þ 1040 m4 and jEj < 4 1049 m5 . 2003 Elsevier B.V. All rights reserved.
1. Introduction Collision-induced phenomena such as absorption (CIA) and light scattering (CILS) of interacting entities are suitable techniques which allow for an experimental determination of higher electrical moments and higher-order polarizabilities [1,2]. These phenomena allow also for a construction [3] and validation [4] of intermolecularinteraction potentials. Most of these studies are, however, limited to atoms and simple molecules of high symmetry. Additionally most of the systems under study are atoms and molecules which are in the gaseous state at ambient temperature and pressure. The only exceptions are mercury [5], *
Fax: +49-531-3914577. E-mail address: [email protected] (U. Hohm).
tetrahedral phosphorous cluster (P4 ) [6], and adamantane (C10 H16 ) [7], for which CILS spectra have been determined at elevated temperatures. In this Letter we report on the depolarized collisioninduced light-scattering spectrum of CCl4 -vapour. The intention of these measurements is to strengthen the existing model of the intermolecular-interaction potential of CCl4 , to obtain the incremental pair polarizability anisotropy of the CCl4 -dimer and to provide values of the dipole– quadrupole and dipole–octopole polarizability of this molecule.
2. Theory In the case of photon-counting detection techniques the double differential depolarized
0009-2614/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.08.067
U. Hohm / Chemical Physics Letters 379 (2003) 380–385
collision-induced light scattering cross section o2 rxz =oXox is given by Z 1 kf3 o2 rxz ¼ k0 expðixtÞ oXox 20p 1 hað2Þ ð0Þ að2Þ ðtÞi dt;
ð1Þ
x ¼ 2pm is the frequency, kf ¼ x=c is the wavevector of the scattered light, k0 is the wave-vector of the incoming radiation, and c is the speed of light in vacuum. The irreducible tensors að2Þ are the rank 2 contributions to the incremental pair polarizability Db a 12 . Relying on the point-multipole approximation the contributions to the depolarized interaction-induced light-scattering spectra of molecules can be split into several distinct parts. First one can separate the contributions of bound and metastable dimers. Beside these contributions, which are concentrated in the vicinity of the Rayleigh line, the major part of the interaction induced spectra results from contributions of collisional complexes. The latter is called a collison-induced light scattering spectrum (CILS). In the case of tetrahedral molecules the CILS intensity can be split into a pure translational spectrum and a collision-induced rotational Raman (CIRR) spectrum. The point-multipole expansion of the incremental pair polarizability yields
where a0 is the volume-polarizability of an isolated molecule at frequency x, C is the second hyperpolarizability, C6 is the dispersion-interaction energy constant of the R6 term in the dispersioninteraction energy, and e0 is the permittivity of free space. If C6 and the hyperpolarizability C are not known, the R6 -term might be approximated with considerable accuracy by [8] ! CC6 3 6a0 þ 2 3a0 ð4pe0 Þ 6a0 ðxÞ½a20 ðxÞ þ a60 1:14Sð4Þ;
ð2Þ
ha ð0Þ a ðtÞi ¼
ð2Þ FDID ðtÞ
þ
ð2Þ FDQ ðtÞ
768p hS3 ðtÞR3 ðtÞi; 49
þ
ð5Þ
880p hS4 ðtÞR4 ðtÞi; 81
ð6Þ
125 824p hS3 ðtÞR4 ðtÞi; 945
ð7Þ
FDQ ¼ A2 a20
ð2Þ FDO ðtÞ
ð2Þ
þ FQQ ðtÞ þ ;
ð2Þ
where the subscripts refer to Ôdipole–induceddipoleÕ, Ôdipole–quadrupoleÕ, Ôdipole–octopoleÕ, and Ôquadrupole–quadrupoleÕ contributions, respecð2Þ tively. The pure translational contribution FDID ðtÞ is known and has been tested against CILS spectra of many different atoms and molecules. Excellent agreement has been achieved between measurements and calculations. A well-known and often tested model for the incremental pair-polarizabilð2Þ ity anisotropy which enters in FDID ðtÞ is DaðRÞ ¼ 6a20 R3 þ
6a30 þ
þ B expðR=R0 Þ;
!
CC6 2
3a0 ð4pe0 Þ
R6 ð3Þ
ð4Þ
where Sð4Þ is the second term in the Cauchy expansion of the frequency dependence P of the mean dipole-polarizability a0 ðxÞ ¼ a30 k Sð2kÞx2k2 (here x is given in atomic units, a0 is the Bohr radius). The approximation equation (4) holds in the case of many small atoms and molecules, including e.g. mercury. For Hg, the lhs yields 3:8 105 a.u., whereas on the rhs 4:1 105 a.u. is calculated [9]. As was shown by Bonechi et al. [10] in some cases damping of the multipole expansion in Eq. (3) at low distances R can approve the agreement between theory and experiment considerably. However, due to the accuracy of our measurements the damping function is neglected in this Letter. The other parts in Eq. (2) are given by [11] ð2Þ
ð2Þ
381
ð2Þ
FDO ¼ E2 a20 ð2Þ
FQQ ¼ A4
A and E are the dipole–quadrupole and dipole– octopole polarizabilities. In the case of Td symmetry they can be represented by a single scalar quantity. hS‘ ðtÞi are the rank ‘ tensor correlation functions and R‘ ðtÞ is the Fourier transform of the free-rotor spectral density R‘ ðxÞ. More details and the explicit formulation of S‘ ðtÞ and R‘ ðtÞ are given by Posch [11–13]. Approximate formulae of the translational broadening functions S‘ ðtÞ which are based on the Birnbaum–Cohen line shape [14] as well as additional terms to Eq. (2) are presented by Bancewizc et al. [2].
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U. Hohm / Chemical Physics Letters 379 (2003) 380–385
In order to calculate the translational broadening functions S‘ ðtÞ the intermolecular-interaction energy U ðRÞ must be known. Here, we will adopt to an isotropic temperature-dependent potential, which is written in the form [15] " n 6 # eðT Þ Rm ðT Þ Rm ðT Þ 6 U ðRÞ ¼ n : n6 R R
ð8Þ
The measurements are carried out with two different polarizations. In the polarized geometry the totally symmetric m1 vibrational band of the gaseous CCl4 at 459 cm1 is recorded for internal intensity calibration and thorough adjustment of the apparatus. The depolarized scattering geometry then allows for a detection of the collisioninduced light-scattering spectrum in the range between 10 and 170 cm1 . In both cases the resolution is set to about 1.5 cm1 .
3. Experimental 4. Evaluation of data The light-scattering experiments are carried out in the usual 90 scattering geometry. An Arþ -laser (model Spectra Physics 2017S) operated at 1.25 W at 514.5 nm is used as radiation source. A Glan laser prism and a polarizer ensure the light to be polarized either in the xy-plane (so-called depolarized scattering geometry, index ÔdÕ) or in the xz-plane (polarized geometry, index ÔpÕ). The light is focussed into a cylindrical scattering cell. The perpendicularly scattered light enters a Coderg T800 triple monochromator and is recorded with a photomultiplier tube cooled down to )20 C. A photon counting technique is employed. The scattering cells are small glass tubes with inner diameter of 4 mm and height of 10 cm. The cells are filled with a small amount of liquid CCl4 , which was distilled before several times. The CCl4 was frozen with liquid nitrogen, the sample cell evacuated down to a pressure of <0.1 Pa and fused off. The so prepared sample cells are placed inside a massive cylindrical stainless steel sample holder. It has four bore holes of 2 mm diameter for the laser beam and the scattered radiation. We found it necessary to use also a fourth bore hole opposite to the entrance slit of the Raman spectrometer. Without this hole spurious scattering from the inside of the sample holder was observed. The sample holder is heated with a heating ribbon to the desired temperature which is in the range between 501 and 561 K. The temperature is recorded with NiCr–Ni thermocouples. From the vapour pressure pðT Þ [16] and second virial coefficient BðT Þ [15] the density . of the CCl4 -vapour is calculated according to . ¼ .0 ð1 BðT Þ.0 Þ, with .0 ¼ p=ðRT Þ. In our studies the density range is between 687 and 1524 mol m3 .
In order to avoid contamination from the signal scattered from bound and metastable dimers, the spectral intensity at wavenumber shift Dm P 20 cm1 (Stokes-side) is used for the evaluation of the molecular quantities. This ensures also that unavoidable depolarized Rayleigh scattering from the sample cell is excluded. First we make sure that only binary collisions contribute to the measured depolarized scattered spectrum. Let Ip ðm1 Þ be the integrated scattered intensity of the m1 -band centred at 459 cm1 measured in the polarized geometry. Id denotes the integrated scattered collision-induced intensity between 20 and 170 cm1 measured in the depolarized geometry. Ip ðm1 Þ and Id are measured under the same thermodynamic conditions of the sample. If the contribution of binary interactions to the polarized m1 -band in our density range is small its intensity is given by Ip ðm1 Þ ¼ Ap .. On the other hand the depolarized collision-induced intensity should vary as Id ¼ Ad .2 if only binary interactions contribute to the observed spectrum. Hence we have the ratio Id =Ip ðm1 Þ ¼ ðAd =Ap Þ .. This linear dependence on the density is shown in Fig. 1 which proves that within the detection limits of our spectrometer the recorded depolarized spectra are caused by binary interactions. The CILS intensity is recorded in counts/s. In order to obtain the absolute scattered intensity this signal must be calibrated. Usually this is done by comparing the scattered intensity to the known depolarized scattering cross sections of e.g. hydrogen. However, our sample preparation does not allow for using gaseous hydrogen as internal standard. Instead of this we record the totally
U. Hohm / Chemical Physics Letters 379 (2003) 380–385
Fig. 1. Ratio of the experimentally observed integrated scattered intensities of the collision-induced depolarized spectrum, Id , and the m1 -band, Ip ðm1 Þ, of CCl4 -vapour. The dotted line is obtained from a linear fit.
polarized m1 ðA1 Þ band of CCl4 under the same thermodynamic conditions for which the depolarized CILS spectra are obtained. For an excitation wavelength of 514.5 nm the absolute differential Raman scattering cross section dr=dX of the m1 fundamental is given in the literature. However, the values vary between 4:41 1030 and 5:27 1030 cm2 sr1 [17]. Due to this uncertainty, it is only possible to estimate the depolarized scattering intensity. As known experimentally derived input data we have used the polarizability a0 ð514:5 nmÞ ¼ 10:594 1030 m3 , the Cauchy coefficient Sð4Þ ¼ 249 a.u. [18], and the rotational constant B0 ¼ 0:0573 cm1 [19]p(note, that B0 in [19] is too ffiffiffi large by a factor of 3). The parameters of the intermolecular-interaction potential U ðRÞ show a very small temperature dependence, which is given in a large temperature range in [15]. At T ¼ 543 K, where most of our measurements are evaluated, they are given as n ¼ 18:6, Rm ¼ 5:5427 1010 m, and e=k ¼ 741:35 K. The CILS spectra, Eq. (2), are now calculated by a careful variation of the mean dipole–quadrupole polarizability A, the mean dipole–octopole polarizability E, see Eqs. (5)–(7), and the short-range parameters B and R0 , Eq. (3). Classical free trajectory calculations of the spectral broadening functions S‘ ðtÞ are performed, which yield the contributions S‘F ðtÞ of the free
383
molecules only. Detailed balance was taken into account by multiplying S‘F ðtÞ with exp½hc Dm=ðkT Þ. Whereas changes of A and E only leads to a simple scaling of the DQ, QQ and DO contributions, variation of B and R0 will lead to a change in the shape of the corresponding translational contrið2Þ bution FDID ðtÞ, too. We compare both, the frequency dependence of the CILS spectra as well as the integrated scattered intensity in the range between 20 and 150 cm1 . This range will definitely exclude contributions from bound and metastable dimers. We have calculated various spectra by systematically changing the variables B and R0 in the ranges 0:4 6 R0 =ð1010 mÞ 6 1:4 and 1:5 1026 6 B=m3 6 5:0 1027 , respectively. These ranges are chosen in order to include the B and R0 values derived from an empirical scaling law given by Barocci et al. [20]. Both, R0 and B have a marked influence on the integrated scattered intensity and on the calculated line shape. The best agreement of the line shape between calculated and measured binary CILS spectra is obtained with the data set jAj ¼ 3:8ð1:0Þ 1040 m4 and jEj < 4 1049 m5 , B ¼ 6:0 1027 m3 and R0 ¼ 0:57 1010 m. The experimental spectrum and the calculations are compared in Fig. 2. Although the molecular parameters are obtained
Fig. 2. Comparison of the experimentally recorded CILS spectrum at T ¼ 543 K and . ¼ 1211 mol m3 (full squares) and the free trajectory calculations (full line) of CCl4 -vapour. Only the experimental spectrum at Dm P 20 cm1 (dotted line) is used for the fit.
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U. Hohm / Chemical Physics Letters 379 (2003) 380–385
by fitting the experimental spectrum in the range Dm P 20 cm1 acceptable agreement is also observed at 10 < Dm=ðcm1 Þ < 20. This indicates that the contribution of bound and metastable dimers is located at Dm < 10 cm1 . With this data set, the integrated scattered intensity in the range between 20 and 150 cm1 is calculated to be 1:08 1060 m5 . The experimentally recorded integrated intensity based on the m1 band of CCl4 is in the range between Id ¼ 0:98 and 1:17 1060 m5 . Close agreement is observed between the measured and calculated integrated scattered intensities which gives further support to our line shape calculations.
5. Discussion In order to check our calculations for consistency the zero moment of the DID-contribution to the scattered spectrum was calculated according to Z 1 1 ð0Þ UDID ¼ DaðRÞ2 R2 exp½U ðRÞ=ðkT Þ dR 15 0 ð0Þ
ð0Þ
ð0Þ
¼ Ubound þ Umeta þ Ufree ;
ð9Þ
where the subscripts ÔboundÕ and ÔmetaÕ refer to bound and metastable dimers, respectively. Following the procedure proposed by Levine [21] we have calculated the dimer contribution after suitable subdivision of the phase space according to Stogryn and Hirschfelder [22]. At T ¼ 543 K we found that the bound and metastable dimers conð0Þ ð0Þ ð0Þ ð0Þ tribute to Ubound =UDID ¼ 36% and Umeta =UDID ¼ 10%, respectively, to the whole scattered intensity. ð0Þ ð0Þ This can be compared with the result Ifree =IDID from the free trajectory calculations of the scattered spectrum. Z þ1 V o2 rxz ð0Þ IDID ¼ dm 3 1 k0 kf oXom ð0Þ
ð0Þ
ð0Þ
¼ Ibound þ Imeta þ Ifree : ð0Þ
ð0Þ
ð10Þ
Note that UDID ¼ IDID for all three parts (ÔboundÕ, ÔmetaÕ, and ÔfreeÕ) in Eqs. (9) and (10). We found ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ 1 ðIfree =IDID Þ ¼ ðIbound þ Imeta Þ=IDID ¼ 43% in reasonable agreement with 46% obtained from the alternative calculation.
There are no measurements or high-level ab initio calculations of the higher-order polarizabilities in the literature to compare with. An unpublished value of Maroulis [23] gives jAj ¼ 4:7 5:5 1040 m4 . This value is calculated for zero frequency and temperature. It lies within the error bounds of our measurements. There are two estimates derived from bond-polarizability models. In this way Posch [13] has obtained for the dipole–quadrupole polarizability jAj ¼ 6:5–11:3 1040 m4 . This value is about 2–3 times higher than jAj ¼ 3:8ð1:0Þ 1040 m4 obtained in this work. In the case of the dipole– octopole polarizability only an upper limit can be deduced from our work. Obviously, the sensitivity of our apparatus in the far wings of the spectrum is too low to detect the contribution of jEj to the scattered signal. Moreover, due to the low rotational constant the collision-induced rotational Raman spectrum should be located in a range, where the DID term still gives a significant contribution to the scattered spectrum. The isotropic intermolecular-interaction potential used for CCl4 was obtained by a simultaneous fit of thermophysical transport and equilibrium data [15]. The functional form has proven to reproduce the existing data on the second pVT -virial coefficient BðT Þ and the viscosity g within their experimental error bounds. In this work we have demonstrated that this potential is also capable of reproducing the collision-induced spectra, thus giving additional confidence to the functional form proposed by Zarkova and Hohm [15]. Any slight deviation of Rm and e=k results in a completely different slope of the calculated scattered spectrum especially at low wavenumber shifts Dm. Additionally the calculated integrated scattered intensity does not fit any more to the measured intensity Id . As a separate test for our used combination of the interaction induced polarizability anisotropy DaðRÞ and the intermolecular-interaction potential U ðRÞ we have calculated the second Kerr-effect virial coefficient BK according to [24] Z 8p2 NA2 1 DaðR; 0Þ DaðR; xÞ BK ¼ 4pe0 405kT 0 U ðRÞ ð11Þ exp R2 dR; kT
U. Hohm / Chemical Physics Letters 379 (2003) 380–385
DaðR; 0Þ is the static limit of the pair polarizability anisotropy, whereas DaðR; xÞ is its analogue at frequency x, both given in m3 . They are calculated according to Eqs. (3) and (4) assuming that the short-range contribution does not depend on x. The polarizabilities at 632.8 nm, 10:476 1030 m3 , and at infinite wavelength (static limit), 10:270 1030 m3 , are again calculated from the refractivity data of Ramaswamy [18]. The calculations are compared to measurements of Bogaard et al. [25] carried out at 632.8 nm. The experimental results for BK are 4.5(8), 5.5(6), and 3.3(4) (all in 1030 m8 C2 J2 mol2 ) at T ¼ 333:7, 349.3, and 365.2 K, respectively. With our best parameter set for the pair polarizability anisotropy we obtain 6.72, 6.01, and 5.41 · 1030 m8 C2 J2 mol2 . Our values are generally higher and show the expected monotonic decrease with increasing temperature. The experimental results show a somewhat irregular temperature dependence which might be due to the rather low pressure applied by Bogaard et al. Only at T ¼ 349:3 K experiment and theory agree within the error bounds. Although the binary CILS spectra can be reproduced with acceptable accuracy it must be stated that the pair polarizability anisotropy and the intermolecularinteraction potential used in the present work are not able to reproduce the measured Kerr-effect virial coefficients. This appreciable disagreement between theory and experiment was also observed by Bogaard et al. However, no reference was given to the model used by these authors [25].
6. Conclusion We have determined the incremental pair polarizability anisotropy and the higher-order dipole polarizabilities A and E of CCl4 from depolarized collision-induced light-scattering spectra in the gas phase . The obtained data set is able to reproduce the experimentally observed spectrum. The intermolecular-interaction potential used does not contradict the measurements. However, only poor
385
agreement is observed with existing Kerr-effect virial coefficients. This observation clearly underlines the need for new density and temperaturedependent measurements of the Kerr-effect of CCl4 vapour.
References [1] L. Frommhold, Collision-induced Absorption in Gases, Cambridge University Press, Cambridge, 1993. [2] T. Bancewicz, Y. Le Duff, J.-L. Godet, Adv. Chem. Phys. 119 (2001) 267. [3] M.S.A. El Kader, Chem. Phys. 281 (2002) 49. [4] Y. Le Duff, J.-L. Godet, T. Bancewicz, K. Nowicka, J. Chem. Phys. 118 (2003) 11009. [5] P. Schwerdtfeger, R. Wesendrup, G.E. Moyano, A.J. Sadlej, J. Greif, F. Hensel, J. Chem. Phys. 115 (2001) 7401, Erratum: 117 (2002) 6881. [6] U. Hohm, Chem. Phys. Lett. 311 (1999) 117. [7] G. Maroulis, D. Xenides, U. Hohm, A. Loose, J. Chem. Phys. 115 (2001) 7957. [8] U. Hohm, Chem. Phys. Lett. 211 (1993) 498. [9] D. Goebel, U. Hohm, J. Phys. Chem. 100 (1996) 7710. [10] A. Bonechi, M. Moraldi, L. Frommhold, J. Chem. Phys. 109 (1998) 5880. [11] H.A. Posch, Mol. Phys. 46 (1982) 1213. [12] H.A. Posch, Mol. Phys. 37 (1979) 1059. [13] H.A. Posch, Mol. Phys. 40 (1980) 1137. [14] G. Birnbaum, E.R. Cohen, Can. J. Phys. 54 (1976) 593. [15] L. Zarkova, U. Hohm, J. Phys. Chem. Ref. Data 31 (2002) 183. [16] D.L. Hildenbrand, R.A. McDonald, J. Phys. Chem. 63 (1959) 1521. [17] H.W. Schr€ otter, H.W. Kl€ ockner, in: A. Weber (Ed.), Raman Spectroscopy of Gases and Liquids, SpringerVerlag, Berlin, 1979. [18] K.L. Ramaswamy, Proc. Ind. Acad. Sci. A 4 (1936) 675. [19] M.T. do N. Varella, A.P.P. Natalense, M.H.F. Bettega, M.A.P. Lima, Phys. Rev. A 60 (1999) 3684. [20] F. Barocci, A. Guasti, M. Zoppi, S.M. El Sheikh, G.C. Tabisz, N. Meinander, Phys. Rev. A 39 (1989) 4537. [21] H.B. Levine, J. Chem. Phys. 56 (1972) 2455. [22] D.E. Stogryn, J.O. Hirschfelder, J. Chem. Phys. 31 (1959) 1531. [23] G. Maroulis, private communication, 2002. [24] D.A. Dunmur, D.C. Hunt, N.E. Jessup, Mol. Phys. 37 (1979) 713. [25] M.P. Bogaard, B.J. Orr, A.D. Buckingham, G.L.D. Ritchie, J. Chem. Soc. Faraday Trans. 74 (1978) 1573.
Depolarized collision-induced light scattering of gaseous CCl4 Uwe Hohm
*
Institut f€ur Physikalische und Theoretische Chemie der Technischen Universit€ at Braunschweig, Hans-Sommer-Straße 10, D-38106 Braunschweig, Germany Received 22 July 2003; in final form 19 August 2003 Published online: 16 September 2003
Abstract Depolarized collision-induced light-scattering spectra of CCl4 in the saturated vapour-phase have been measured in the ranges of temperature and density between T ¼ 501 and 561 K and . ¼ 687 and 1524 mol m3 , respectively. The CILS spectra are analyzed by using an isotropic Lennard–Jones type (18.6–6) intermolecular-interaction potential. It is found that the polarizability anisotropy of the CCl4 -dimer contains a significant short-range repulsive term. Taking into account collision-induced rotational Raman scattering (CIRR) the dipole–quadrupole polarizability A, and dipole– octopole polarizability E of CCl4 were obtained as jAj ¼ 3:8ð1:0Þ 1040 m4 and jEj < 4 1049 m5 . 2003 Elsevier B.V. All rights reserved.
1. Introduction Collision-induced phenomena such as absorption (CIA) and light scattering (CILS) of interacting entities are suitable techniques which allow for an experimental determination of higher electrical moments and higher-order polarizabilities [1,2]. These phenomena allow also for a construction [3] and validation [4] of intermolecularinteraction potentials. Most of these studies are, however, limited to atoms and simple molecules of high symmetry. Additionally most of the systems under study are atoms and molecules which are in the gaseous state at ambient temperature and pressure. The only exceptions are mercury [5], *
Fax: +49-531-3914577. E-mail address: [email protected] (U. Hohm).
tetrahedral phosphorous cluster (P4 ) [6], and adamantane (C10 H16 ) [7], for which CILS spectra have been determined at elevated temperatures. In this Letter we report on the depolarized collisioninduced light-scattering spectrum of CCl4 -vapour. The intention of these measurements is to strengthen the existing model of the intermolecular-interaction potential of CCl4 , to obtain the incremental pair polarizability anisotropy of the CCl4 -dimer and to provide values of the dipole– quadrupole and dipole–octopole polarizability of this molecule.
2. Theory In the case of photon-counting detection techniques the double differential depolarized
0009-2614/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.08.067
U. Hohm / Chemical Physics Letters 379 (2003) 380–385
collision-induced light scattering cross section o2 rxz =oXox is given by Z 1 kf3 o2 rxz ¼ k0 expðixtÞ oXox 20p 1 hað2Þ ð0Þ að2Þ ðtÞi dt;
ð1Þ
x ¼ 2pm is the frequency, kf ¼ x=c is the wavevector of the scattered light, k0 is the wave-vector of the incoming radiation, and c is the speed of light in vacuum. The irreducible tensors að2Þ are the rank 2 contributions to the incremental pair polarizability Db a 12 . Relying on the point-multipole approximation the contributions to the depolarized interaction-induced light-scattering spectra of molecules can be split into several distinct parts. First one can separate the contributions of bound and metastable dimers. Beside these contributions, which are concentrated in the vicinity of the Rayleigh line, the major part of the interaction induced spectra results from contributions of collisional complexes. The latter is called a collison-induced light scattering spectrum (CILS). In the case of tetrahedral molecules the CILS intensity can be split into a pure translational spectrum and a collision-induced rotational Raman (CIRR) spectrum. The point-multipole expansion of the incremental pair polarizability yields
where a0 is the volume-polarizability of an isolated molecule at frequency x, C is the second hyperpolarizability, C6 is the dispersion-interaction energy constant of the R6 term in the dispersioninteraction energy, and e0 is the permittivity of free space. If C6 and the hyperpolarizability C are not known, the R6 -term might be approximated with considerable accuracy by [8] ! CC6 3 6a0 þ 2 3a0 ð4pe0 Þ 6a0 ðxÞ½a20 ðxÞ þ a60 1:14Sð4Þ;
ð2Þ
ha ð0Þ a ðtÞi ¼
ð2Þ FDID ðtÞ
þ
ð2Þ FDQ ðtÞ
768p hS3 ðtÞR3 ðtÞi; 49
þ
ð5Þ
880p hS4 ðtÞR4 ðtÞi; 81
ð6Þ
125 824p hS3 ðtÞR4 ðtÞi; 945
ð7Þ
FDQ ¼ A2 a20
ð2Þ FDO ðtÞ
ð2Þ
þ FQQ ðtÞ þ ;
ð2Þ
where the subscripts refer to Ôdipole–induceddipoleÕ, Ôdipole–quadrupoleÕ, Ôdipole–octopoleÕ, and Ôquadrupole–quadrupoleÕ contributions, respecð2Þ tively. The pure translational contribution FDID ðtÞ is known and has been tested against CILS spectra of many different atoms and molecules. Excellent agreement has been achieved between measurements and calculations. A well-known and often tested model for the incremental pair-polarizabilð2Þ ity anisotropy which enters in FDID ðtÞ is DaðRÞ ¼ 6a20 R3 þ
6a30 þ
þ B expðR=R0 Þ;
!
CC6 2
3a0 ð4pe0 Þ
R6 ð3Þ
ð4Þ
where Sð4Þ is the second term in the Cauchy expansion of the frequency dependence P of the mean dipole-polarizability a0 ðxÞ ¼ a30 k Sð2kÞx2k2 (here x is given in atomic units, a0 is the Bohr radius). The approximation equation (4) holds in the case of many small atoms and molecules, including e.g. mercury. For Hg, the lhs yields 3:8 105 a.u., whereas on the rhs 4:1 105 a.u. is calculated [9]. As was shown by Bonechi et al. [10] in some cases damping of the multipole expansion in Eq. (3) at low distances R can approve the agreement between theory and experiment considerably. However, due to the accuracy of our measurements the damping function is neglected in this Letter. The other parts in Eq. (2) are given by [11] ð2Þ
ð2Þ
381
ð2Þ
FDO ¼ E2 a20 ð2Þ
FQQ ¼ A4
A and E are the dipole–quadrupole and dipole– octopole polarizabilities. In the case of Td symmetry they can be represented by a single scalar quantity. hS‘ ðtÞi are the rank ‘ tensor correlation functions and R‘ ðtÞ is the Fourier transform of the free-rotor spectral density R‘ ðxÞ. More details and the explicit formulation of S‘ ðtÞ and R‘ ðtÞ are given by Posch [11–13]. Approximate formulae of the translational broadening functions S‘ ðtÞ which are based on the Birnbaum–Cohen line shape [14] as well as additional terms to Eq. (2) are presented by Bancewizc et al. [2].
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U. Hohm / Chemical Physics Letters 379 (2003) 380–385
In order to calculate the translational broadening functions S‘ ðtÞ the intermolecular-interaction energy U ðRÞ must be known. Here, we will adopt to an isotropic temperature-dependent potential, which is written in the form [15] " n 6 # eðT Þ Rm ðT Þ Rm ðT Þ 6 U ðRÞ ¼ n : n6 R R
ð8Þ
The measurements are carried out with two different polarizations. In the polarized geometry the totally symmetric m1 vibrational band of the gaseous CCl4 at 459 cm1 is recorded for internal intensity calibration and thorough adjustment of the apparatus. The depolarized scattering geometry then allows for a detection of the collisioninduced light-scattering spectrum in the range between 10 and 170 cm1 . In both cases the resolution is set to about 1.5 cm1 .
3. Experimental 4. Evaluation of data The light-scattering experiments are carried out in the usual 90 scattering geometry. An Arþ -laser (model Spectra Physics 2017S) operated at 1.25 W at 514.5 nm is used as radiation source. A Glan laser prism and a polarizer ensure the light to be polarized either in the xy-plane (so-called depolarized scattering geometry, index ÔdÕ) or in the xz-plane (polarized geometry, index ÔpÕ). The light is focussed into a cylindrical scattering cell. The perpendicularly scattered light enters a Coderg T800 triple monochromator and is recorded with a photomultiplier tube cooled down to )20 C. A photon counting technique is employed. The scattering cells are small glass tubes with inner diameter of 4 mm and height of 10 cm. The cells are filled with a small amount of liquid CCl4 , which was distilled before several times. The CCl4 was frozen with liquid nitrogen, the sample cell evacuated down to a pressure of <0.1 Pa and fused off. The so prepared sample cells are placed inside a massive cylindrical stainless steel sample holder. It has four bore holes of 2 mm diameter for the laser beam and the scattered radiation. We found it necessary to use also a fourth bore hole opposite to the entrance slit of the Raman spectrometer. Without this hole spurious scattering from the inside of the sample holder was observed. The sample holder is heated with a heating ribbon to the desired temperature which is in the range between 501 and 561 K. The temperature is recorded with NiCr–Ni thermocouples. From the vapour pressure pðT Þ [16] and second virial coefficient BðT Þ [15] the density . of the CCl4 -vapour is calculated according to . ¼ .0 ð1 BðT Þ.0 Þ, with .0 ¼ p=ðRT Þ. In our studies the density range is between 687 and 1524 mol m3 .
In order to avoid contamination from the signal scattered from bound and metastable dimers, the spectral intensity at wavenumber shift Dm P 20 cm1 (Stokes-side) is used for the evaluation of the molecular quantities. This ensures also that unavoidable depolarized Rayleigh scattering from the sample cell is excluded. First we make sure that only binary collisions contribute to the measured depolarized scattered spectrum. Let Ip ðm1 Þ be the integrated scattered intensity of the m1 -band centred at 459 cm1 measured in the polarized geometry. Id denotes the integrated scattered collision-induced intensity between 20 and 170 cm1 measured in the depolarized geometry. Ip ðm1 Þ and Id are measured under the same thermodynamic conditions of the sample. If the contribution of binary interactions to the polarized m1 -band in our density range is small its intensity is given by Ip ðm1 Þ ¼ Ap .. On the other hand the depolarized collision-induced intensity should vary as Id ¼ Ad .2 if only binary interactions contribute to the observed spectrum. Hence we have the ratio Id =Ip ðm1 Þ ¼ ðAd =Ap Þ .. This linear dependence on the density is shown in Fig. 1 which proves that within the detection limits of our spectrometer the recorded depolarized spectra are caused by binary interactions. The CILS intensity is recorded in counts/s. In order to obtain the absolute scattered intensity this signal must be calibrated. Usually this is done by comparing the scattered intensity to the known depolarized scattering cross sections of e.g. hydrogen. However, our sample preparation does not allow for using gaseous hydrogen as internal standard. Instead of this we record the totally
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Fig. 1. Ratio of the experimentally observed integrated scattered intensities of the collision-induced depolarized spectrum, Id , and the m1 -band, Ip ðm1 Þ, of CCl4 -vapour. The dotted line is obtained from a linear fit.
polarized m1 ðA1 Þ band of CCl4 under the same thermodynamic conditions for which the depolarized CILS spectra are obtained. For an excitation wavelength of 514.5 nm the absolute differential Raman scattering cross section dr=dX of the m1 fundamental is given in the literature. However, the values vary between 4:41 1030 and 5:27 1030 cm2 sr1 [17]. Due to this uncertainty, it is only possible to estimate the depolarized scattering intensity. As known experimentally derived input data we have used the polarizability a0 ð514:5 nmÞ ¼ 10:594 1030 m3 , the Cauchy coefficient Sð4Þ ¼ 249 a.u. [18], and the rotational constant B0 ¼ 0:0573 cm1 [19]p(note, that B0 in [19] is too ffiffiffi large by a factor of 3). The parameters of the intermolecular-interaction potential U ðRÞ show a very small temperature dependence, which is given in a large temperature range in [15]. At T ¼ 543 K, where most of our measurements are evaluated, they are given as n ¼ 18:6, Rm ¼ 5:5427 1010 m, and e=k ¼ 741:35 K. The CILS spectra, Eq. (2), are now calculated by a careful variation of the mean dipole–quadrupole polarizability A, the mean dipole–octopole polarizability E, see Eqs. (5)–(7), and the short-range parameters B and R0 , Eq. (3). Classical free trajectory calculations of the spectral broadening functions S‘ ðtÞ are performed, which yield the contributions S‘F ðtÞ of the free
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molecules only. Detailed balance was taken into account by multiplying S‘F ðtÞ with exp½hc Dm=ðkT Þ. Whereas changes of A and E only leads to a simple scaling of the DQ, QQ and DO contributions, variation of B and R0 will lead to a change in the shape of the corresponding translational contrið2Þ bution FDID ðtÞ, too. We compare both, the frequency dependence of the CILS spectra as well as the integrated scattered intensity in the range between 20 and 150 cm1 . This range will definitely exclude contributions from bound and metastable dimers. We have calculated various spectra by systematically changing the variables B and R0 in the ranges 0:4 6 R0 =ð1010 mÞ 6 1:4 and 1:5 1026 6 B=m3 6 5:0 1027 , respectively. These ranges are chosen in order to include the B and R0 values derived from an empirical scaling law given by Barocci et al. [20]. Both, R0 and B have a marked influence on the integrated scattered intensity and on the calculated line shape. The best agreement of the line shape between calculated and measured binary CILS spectra is obtained with the data set jAj ¼ 3:8ð1:0Þ 1040 m4 and jEj < 4 1049 m5 , B ¼ 6:0 1027 m3 and R0 ¼ 0:57 1010 m. The experimental spectrum and the calculations are compared in Fig. 2. Although the molecular parameters are obtained
Fig. 2. Comparison of the experimentally recorded CILS spectrum at T ¼ 543 K and . ¼ 1211 mol m3 (full squares) and the free trajectory calculations (full line) of CCl4 -vapour. Only the experimental spectrum at Dm P 20 cm1 (dotted line) is used for the fit.
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by fitting the experimental spectrum in the range Dm P 20 cm1 acceptable agreement is also observed at 10 < Dm=ðcm1 Þ < 20. This indicates that the contribution of bound and metastable dimers is located at Dm < 10 cm1 . With this data set, the integrated scattered intensity in the range between 20 and 150 cm1 is calculated to be 1:08 1060 m5 . The experimentally recorded integrated intensity based on the m1 band of CCl4 is in the range between Id ¼ 0:98 and 1:17 1060 m5 . Close agreement is observed between the measured and calculated integrated scattered intensities which gives further support to our line shape calculations.
5. Discussion In order to check our calculations for consistency the zero moment of the DID-contribution to the scattered spectrum was calculated according to Z 1 1 ð0Þ UDID ¼ DaðRÞ2 R2 exp½U ðRÞ=ðkT Þ dR 15 0 ð0Þ
ð0Þ
ð0Þ
¼ Ubound þ Umeta þ Ufree ;
ð9Þ
where the subscripts ÔboundÕ and ÔmetaÕ refer to bound and metastable dimers, respectively. Following the procedure proposed by Levine [21] we have calculated the dimer contribution after suitable subdivision of the phase space according to Stogryn and Hirschfelder [22]. At T ¼ 543 K we found that the bound and metastable dimers conð0Þ ð0Þ ð0Þ ð0Þ tribute to Ubound =UDID ¼ 36% and Umeta =UDID ¼ 10%, respectively, to the whole scattered intensity. ð0Þ ð0Þ This can be compared with the result Ifree =IDID from the free trajectory calculations of the scattered spectrum. Z þ1 V o2 rxz ð0Þ IDID ¼ dm 3 1 k0 kf oXom ð0Þ
ð0Þ
ð0Þ
¼ Ibound þ Imeta þ Ifree : ð0Þ
ð0Þ
ð10Þ
Note that UDID ¼ IDID for all three parts (ÔboundÕ, ÔmetaÕ, and ÔfreeÕ) in Eqs. (9) and (10). We found ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ 1 ðIfree =IDID Þ ¼ ðIbound þ Imeta Þ=IDID ¼ 43% in reasonable agreement with 46% obtained from the alternative calculation.
There are no measurements or high-level ab initio calculations of the higher-order polarizabilities in the literature to compare with. An unpublished value of Maroulis [23] gives jAj ¼ 4:7 5:5 1040 m4 . This value is calculated for zero frequency and temperature. It lies within the error bounds of our measurements. There are two estimates derived from bond-polarizability models. In this way Posch [13] has obtained for the dipole–quadrupole polarizability jAj ¼ 6:5–11:3 1040 m4 . This value is about 2–3 times higher than jAj ¼ 3:8ð1:0Þ 1040 m4 obtained in this work. In the case of the dipole– octopole polarizability only an upper limit can be deduced from our work. Obviously, the sensitivity of our apparatus in the far wings of the spectrum is too low to detect the contribution of jEj to the scattered signal. Moreover, due to the low rotational constant the collision-induced rotational Raman spectrum should be located in a range, where the DID term still gives a significant contribution to the scattered spectrum. The isotropic intermolecular-interaction potential used for CCl4 was obtained by a simultaneous fit of thermophysical transport and equilibrium data [15]. The functional form has proven to reproduce the existing data on the second pVT -virial coefficient BðT Þ and the viscosity g within their experimental error bounds. In this work we have demonstrated that this potential is also capable of reproducing the collision-induced spectra, thus giving additional confidence to the functional form proposed by Zarkova and Hohm [15]. Any slight deviation of Rm and e=k results in a completely different slope of the calculated scattered spectrum especially at low wavenumber shifts Dm. Additionally the calculated integrated scattered intensity does not fit any more to the measured intensity Id . As a separate test for our used combination of the interaction induced polarizability anisotropy DaðRÞ and the intermolecular-interaction potential U ðRÞ we have calculated the second Kerr-effect virial coefficient BK according to [24] Z 8p2 NA2 1 DaðR; 0Þ DaðR; xÞ BK ¼ 4pe0 405kT 0 U ðRÞ ð11Þ exp R2 dR; kT
U. Hohm / Chemical Physics Letters 379 (2003) 380–385
DaðR; 0Þ is the static limit of the pair polarizability anisotropy, whereas DaðR; xÞ is its analogue at frequency x, both given in m3 . They are calculated according to Eqs. (3) and (4) assuming that the short-range contribution does not depend on x. The polarizabilities at 632.8 nm, 10:476 1030 m3 , and at infinite wavelength (static limit), 10:270 1030 m3 , are again calculated from the refractivity data of Ramaswamy [18]. The calculations are compared to measurements of Bogaard et al. [25] carried out at 632.8 nm. The experimental results for BK are 4.5(8), 5.5(6), and 3.3(4) (all in 1030 m8 C2 J2 mol2 ) at T ¼ 333:7, 349.3, and 365.2 K, respectively. With our best parameter set for the pair polarizability anisotropy we obtain 6.72, 6.01, and 5.41 · 1030 m8 C2 J2 mol2 . Our values are generally higher and show the expected monotonic decrease with increasing temperature. The experimental results show a somewhat irregular temperature dependence which might be due to the rather low pressure applied by Bogaard et al. Only at T ¼ 349:3 K experiment and theory agree within the error bounds. Although the binary CILS spectra can be reproduced with acceptable accuracy it must be stated that the pair polarizability anisotropy and the intermolecularinteraction potential used in the present work are not able to reproduce the measured Kerr-effect virial coefficients. This appreciable disagreement between theory and experiment was also observed by Bogaard et al. However, no reference was given to the model used by these authors [25].
6. Conclusion We have determined the incremental pair polarizability anisotropy and the higher-order dipole polarizabilities A and E of CCl4 from depolarized collision-induced light-scattering spectra in the gas phase . The obtained data set is able to reproduce the experimentally observed spectrum. The intermolecular-interaction potential used does not contradict the measurements. However, only poor
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agreement is observed with existing Kerr-effect virial coefficients. This observation clearly underlines the need for new density and temperaturedependent measurements of the Kerr-effect of CCl4 vapour.
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