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Dispersion of the mean dipole polarizability α(ω) and dipole polarizability anisotropy κ(ω) of bromotrifluoromethane CBrF3
14 February1997
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 265 (1997) 638-642
Dispersion of the mean dipole polarizability a(w) and dipole polarizability anisotropy K(w) of bromotrifluoromethane CBrF3 Dirk Goebel, Uwe Hohm Institut fiir Physikalische und Theoretische Chemie der Technischen Universit~it Braunschweig, Hans-Sommer-Strafle 10, D-38106 Braunschweig, Germany Received 11 November 1996; in final form 4 December 1996
Abstract Dispersive Fourier transform spectroscopy measurements of the refractive index of bmmotrifluoromethane, CBrF3, have been performed at pressures up to l05 Pa in the wavelength range between 900 and 325 nm, in order to determine the frequency dependence of the mean dipole polarizability a(o~). Combining our results with measured depolarization ratios p(to) yields a dispersion formula for the two independent components of the polarizability tensor a_L(to) and all (to). The results are used to get estimates of the dispersion interaction energy constant C6, the second hyperpolarizability y and the mean diamagnetic susceptibility X of CBrF3. @ 1997 Elsevier Science B.V.
1. Introduction Bromotrifluoromethane, CBrF3, also known as refrigerant 13B 1, is of considerable interest in chemical reaction kinetics, photochemistry, molecular spectroscopy, atmospheric chemistry, and semiconductor technologies [ 1-5]. However, the frequency dependence of its electro-optical properties, such as the refractive index and the dipole polarizability, are only poorly known [6-8]. In this Letter we present an experimental study of the dispersion of the refractive index and the mean dipole polarizability a(ro) of CBrF3. Due to the quite large frequency dependence of its depolarization ratio p(to) [9] CBrF3 is also one of the few promising canditates for which reliable values of the frequency dependence of the polarizability anisotropy r ( w ) can be determined. Critical combination of refractive index data and depolarization ratios therefore yield a Kramers-Heisenberg type dispersion formula for the individual tensor components
Ce_L(W) and Ol[l(W). This allows for a calculation of the low-frequency dispersion of a ( w ) and r ( w ) . Additionally, we will show that our results are also useful to obtain the mean diamagnetic susceptibility X in terms of a modern version of the Drude model [ 10], to estimate the static second dipole hyperpolarizability y [ 11 ], and to get a first indication of the dispersion interaction energy constant C6 of CBrF3. Atomic units (au) are used throughout this Letter. Conversion factors into SI units are given in the appendix.
2. Theory For molecules of C3v symmetry, the polarizability tensor consists of only two different elements Old_(W ) -~ Olyy(tO) ---- OlXX(tO) and Olll(w) = Olzz(tO), respectively, where to is the frequency of the measuring light. Outside electronic absorption bands the frequency dependence of the individual components
0009-2614/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. PII S0009-2614(96)01455-8
D. Goebel, U. Hohm/ Chemical Physics Letters 265 (1997) 638-642
a , ( t o ) (77 =2_, II) of the polarizability tensor can be described via the Kramers-Heisenberg formula _fk~ ~"(to)
=
~
togk. - '°2
f.
( 1)
tog. - '°2 '
where took. is the transition frequency and fk,7 the corresponding oscillator strength, f . and too. are effective values, too. being in the order of the ionization potential. If one is interested in the behaviour of a , ( t o ) at optical frequencies where to << w017, the approximation in Eq. (1) can be used. Hence, with this simplification the frequency dependence of the two invariants of the polarizability tensor are given as a ( t o ) = g1 [ 2 c e ±
(to) +
639
in Ref. [ 12]. Two types of measurements have been performed. First, the pressure dependence of the refractivity (n - 1) has been determined in the range between 0 and 105 Pa simultaneously at four different frequencies to using HeNe and HeCd laser radiation sources with tol = 0.071981, o)2 = 0.076693, o)3 = 0.083831, and 0 ) 4 = 0.140139 au, corresponding to vacuum-wavelengths of AI = 632.99, A2 = 594.096, 3,3 = 543.516, and M = 325.13 nm [13]. This yields the density dependence of the refractivity, from which the mean dipole polarizability a(to) of the free noninteracting molecule can be determined via a modified Lorentz-Lorenz formula:
n2(to,p,T) - 1 NA n2( to, p, T) + 2 = g--eo°e(to ) (-;--T )
all ( t o ) ]
1 (2 f± f[I ) \ tog±_ to2 + to0 _ to2
(2)
1
and fll K(to) =Oql (to) - a ± ( t o ) = o9211_ to2
f± 0)2± -- 0)2 •
(3) ce(to) is the mean dipole polarizability, which can be obtained from e.g. refractive index measurements and K(to) is the anisotropy of the dipole polarizability, which is related to the depolarization ratio p(to) of linearly polarized monochromatic light via
p(to) = 3K2(to)/[45oe2(to) + 4Kz(to) ] .
(4)
Amos [ 10] has recently demonstrated the capabilities of a modern version of the Drude model for obtaining electro-optical properties. In this context we mention that the mean diamagnetic susceptibility X of CBrF3 is given as [ 10]
-- X ~
1=0
I 1 (fx + fl') 4to0 / ~ ~ too± to011
(5) '
where N is the total number of electrons and toot are the respective transition frequencies.
where NA is Avogadro's constant, e0 the permittivity of vacuum, and B*(to, T) = bR(to, T) - B ( T ) , with bR (to, T) being the reduced second refractivity virial coefficient and B(T) the second (p, V, T) virial coefficient. In the second type of measurement a white light radiation source is used and isobaric dispersive Fourier transform spectra were recorded in the frequency range between 0.051 and 0.096 au (corresponding to a range of 475 to 900 nm) with a resolution of about 0.00077 au (see Refs. 112,141 ). In order to obtain the dispersion of the mean dipole polarizability or(to), the resulting quasi-continuous refractive index spectrum is reduced to ideal behaviour by means of the measured B* value. The CBrF3 sample has a purity of better than 99.9% and is used without further purification (CBrF3 supplied by Union Carbide, USA, and analyzed via GCMS). The individual error sources and uncertainties are evaluated in Refs. [ 12,13]. In the context of this work it is important to note that the relative error in c~(to) is less than 0.07%, while the uncertainty in B*(to,T) is less than I% .
4. Results and discussion 3. Experimental The measurements of the gas-phase refractive index of CBrF3 were performed with an evacuated Michelson twin-interferometer, which is described in detail
Our pressure dependent refractivity measurements yield a value of B*(wI,T) = 341.1(2.3) x 10 -6 m3mol - I at 298 K. This value of B*, which should be nearly independent of frequency, allows for
640
D. Goebel, U. Hohm/Chemical Physics Letters 265 (1997) 638-642
a reduction of all measured refractivities in order to obtain the dipole polarizability of the free molecule according to Eq. (6). a(tO) obtained in this way can be fitted to a one-term Kramers-Heisenberg dispersion formula with an r.m.s, error of Aa = 0.001 au in the whole frequency range between 0.05 and 0.14 au. The aim of this Letter is, however, to obtain the frequency dependence of the individual components a±(tO) and Otll(to) of the polarizability tensor. To this end we have first calculated x(tO) from the two depolarization ratios of P(tOl) = 0.0059(2) and P(tO5 = 0.088559 au) = 0.0066(2) given by Baas and van den Hout [9] using an interpolated ot(tOs) value of our measurements. This yields the two anisotropies K(tol) = 11.54(20) and K(tos) = 12.33(20) au. All measured mean polarizabilities (60 values) and the two polarizability anisotropies are now subject to a non-linear fit from which the parameters f ± = 24.83(40), tOo± = 0.8480(68) au, fll = 6.302(25) and tO011 = 0.37581(70) au (r.m.s. errors in parentheses) are obtained. The r.m.s, error of the fit is Aa = 0.0006 au. Quantitative results of a(tO) and K(tO) can now be obtained with Eqs. (2) and (3). Extrapolation to zero frequency gives the electronic part of a ( 0 ) and K(0) as d E = 37.89(53) au and KE = 10.09(82) au, respectively. In order to obtain the total static polarizability or(0) = dE(0) + a A(0) and its anisotropy K(0) = rE(0) + KA(0) we have to add the atomic ('vibrational') contributions a~_ and a~, respectively, to the individual tensor elements a~_ and a~. a~_ and ot~ can be evaluated with the help of absolute transition moments of infra-red active bands which have been tabulated by Bishop and Cheung [ 15]. For CBrF3 a~ = 3.95(40) au (vibrations with symmetry a t ) and a ~ = 3 . 0 4 ( 3 0 ) au (vibrations with symmetry e), leading to the final estimates of the mean static dipole polarizability o~(0) = 41.23(60) au and its anisotropy K(0) = 11.00(96) au. In order to support the numerical results of the frequency dependence of the individual tensor elements of the polarizability of CBrF3, we compare tOo± and tO011 with transition frequencies and ionization potentials recorded with far-UV and photoelectron spectroscopy, respectively [ 5,16 ]. Generally, the spectra of CF3X molecules (X = CI, Br, I) can be split into two regions. First the frequency range below tO = 0.56 au (15 eV), which is rather sensitive to the substituent X. However, the range above 0.56 au is dominated
by the CF3 group and quite insensitive to the halogen X. For our parallel component Crll we have obtained to011 = 0.37581 au (10.23 eV), which corresponds very nicely to the absorption maximum of CBrF3 in this region located at to = 0.387 au (10.54 eV). In view of our analysis of the components of the polarizability tensor, this low absorption energy must be due to the C - B r unit, which lies on the z-axis of the molecule. Sandorfy et al. [ 5] assigned this transition frequency to a 5s ~ (C-Br) Rydberg transition. The transition frequency to0_t = 0.8480 au (23.08 eV) of the perpendicular component should be dominated by the presence of the CF3 group. The photoelectron spectra of both CF3Br and CF3I show intense absorption peaks at energies of about 0.72 au (19.5 eV), 0.76 au (20.8 eV), and 0.87 au (23.7 eV) [ 16]. The last one is in good accordance with too±, but, this coincidence might be fortuitous. If we apply the same analysis to ~r(to) and p(~o) to other trifluoromethanes, we should get a value of o~0±, which is also located in the absorption region of the CF3 group. Due to lack of experimental data, we only can get a rough estimate of too± = 0.71 au (19.3 eV) for CC1F3 (four a ( t o ) and two p(to) as input from Refs. [7,9] ) and tOo± = 0.77 au (20.9 eV) for CHF3 (two a(tO) and two p(tO) as input from Ref. [9]). Again, too± can be attributed to electronic transitions within the CF3 group. In our analysis of the frequency dependence of a(tO) we have obtained two effective transition frequencies and the corresponding oscillator strengths. In the framework of the simple Drude model we can apply Eq. (5) to obtain an estimate of the mean diamagnetic susceptibility X of CBrF3. The capabilities of this rather crude model of electric and magnetic properties have recently been demonstrated by Amos [ 10]. With our parameters of f and too we obtain a mean X = -11.51 au, which agrees fairly well with 11.68 au obtained from the Pascal group contribution model [ 17] and the two values of - 1 1 . 0 7 and - 1 1 . 3 9 au, calculated with semi-empirical methods [ 18]. There is no experimental value of X of CBrF3 to compare with. The Kramers-Heisenberg-type representation Eqs. (2), (3) of the mean polarizability or(tO) and anisotropy K(tO) is suitable for obtaining isotropic and anisotropic dispersion interaction energy constants C6. They are defined in terms of integrals over -
D. Goebel, U. Hohm/ Chemical Physics Letters 265 (1997) 638-642 the polarizability at imaginary frequencies, that is C6 = 3 f a ( i t o ) 2 dto/zr, C; = f a ( i w ) J c ( i t o ) dw/Tr, and C~' = f K(ito) z d t o / ( 3 ~ ) . For the relation of C6, C~, and C~' to the dispersion interaction energy UDisp see e.g. Ref. [ 19]. With our values o f the oscillator strengths and transition frequencies we obtain for the isotropic term C6 = 667 au, and for the two anisotropic contributions C~ = - 3 9 au and C~' = 13 au. There are no other values for CBrF3 to compare with. However, a crude comparison can be made with the isotropic part C6 o f cyclo-C3H6, because this molecule shows a similar frequency dependence of its mean polarizability a ( t o ) [13]. In that case a theoretical value obtained with density functional methods is 638 au [20], which is quite close to our findings for CBrF3, Last, it is worth mentioning, that the Cauchy moments S ( - 2 k ) o f the series expansion a ( t o ) = S ( - 2 ) + S ( - 4 ) t o 2 -k- S ( - 6 ) t o 4 give an estimate o f the static second hyperpolarizability via y(0;0,0,0) ~ 27.4S(-3)S(-4)/S(-2), S(-3) [ S ( - 2 ) S ( - 4 ) ] (1/2) [ 11 ]. For the mean dipole polarizability or(to) our refractive index measurements yield S ( - 2 ) = 3 7 . 8 9 0 ( 1 ) , S ( - 4 ) = 137.3(1.2), and S ( - 6 ) = 9 3 3 ( 5 0 ) , which give an approximation of the second hyperpolarizability o f y ( 0 ; 0 , 0 , 0) = 7160 au. This result compares with y ( -2o9; 0, to, to) = 7445 au (to = 0.06565 au) measured with electric field-induced second harmonic generation by Ward and Bigio [21].
5. Conclusion We have combined our precise measurements of the dispersion of the mean polarizability a ( t o ) with literature results of the depolarization ratio p(to) in order to obtain the frequency dependence o f the individual components a ± (to) and Otll(to) o f the polarizability tensor ~ ( t o ) o f CBrF3. This allows for the calculation o f the frequency dependence of the polarizability anisotropy K(to). Despite its importance in various fields o f electro-optics and intermolecular interactions [19,20] experimental values of K(to) are only sparsely known in the literature and are obtained by combining values o f a ( t o ) and p(to) [ 9 , 1 9 , 2 2 - 2 4 ] . It is desirable that accurate time dependent ab initio calculations (see e.g. Refs. [ 2 5 , 2 6 ] ) and time dependent density functional schemes (see
641
e.g. Refs. [20,27-29] ) give additional information on tr(to) and x ( t o ) , although at present the apparent accuracy o f D F T calculations seems to result from cancellation o f different artifacts [ 30].
Appendix A Conversion factors from atomic (au) to SI units: Energy:
1Eh = 4.3597482 x 10 -18 J;
to: 1 E h / h = 4 . 1 3 4 1 3 7 3 x 1016 s -1 ; C 6 : 1 a6Eh = 9.573448 × 10 -80 J m 6 ;
ct, K: l e2a2E~ I = 1.648778 × 10 -41 C 2 m 2 J - I ; y: 1 eaa4Eh 3 = 6.235378 × 10 -65 C 4 m 4 J -3 ; X: 1 e2a~me I = 7.98104 × 10 -29 J T -2 .
References 111 M.-A. Thelen and P. Felder, Chem. Phys. 204 (1996) 135. 121 M. Alagia, N. Balucani, P. Casavecchia, A. Lagana, G.O. de Aspuru, E.H. Van Kleef, G.G. Volpi and G. Lendvay, Chem. Phys. Lett. 258 (1996) 323. 13] C. Gerke and H. Harder, Chem. Phys. Lett. 255 (1996) 287. [41 T. Underwood-Lemons,T.J. Gergel and J.H. Moore, J. Chem. Phys. 102 (1995) 119. I51 J. Doucet, R. Gilbert, P. Sauvageau and C. Sandorfy, J. Chem. Phys. 62 (1975) 366. [61 N. Davidson, J. Am. Chem. Soc. 73 (1951) 467. 17] W.A. Gault and G.G. Shepherd, Appl. Opt. 12 (1973) 1739. 181 W. Herkt, G. MUller and R. Winkler, Opt. Acta 22 (1975) 667. [9] F. Baas and K.D. van den Hout, Physica A 95 (1979) 597. [ 101 A.T. Amos, Int. J. Quant. Chem. 60 (1996) 67. I111 U. Hohm, Chem. Phys. Lett. 183 (1991) 304. [ 12] D. Goebel and U. Hohm, J. Phys. D: Appl Phys. (1996). in press. [131 U. Hohm, Mol. Phys. 78 (1993) 929. [ 141 D. Goebek U. Hohm, K. Kerl, U. Triimper and G. Maroulis, J. Phys. Chem. 98 (1994) 13123. [ 151 D.M. Bishop and L.M. Cheung, J. Phys. Chem. Ref. Data 11 (1982) 119. 116] T. Cvita~, H. GiJsten, L. Klasine, 1. Novadj and H. Van~ik, Z. Naturforsch. 33a (1978) 1528. [171 R.R. Gupta, in: Landolt-BOrnstein, Numerical data and functional relationships in science and technology, New Series, Group 1I/16, eds. K.-H. Hellwege and A.M. Hellwege (Springer, Berlin, 1986). 1181 J.A. Beran and L. Kevan, J. Phys. Chem. 73 (1969) 3860. 119] U. Hohm, Chem. Phys. 179 (1994) 533. [201 S.J.A. van Gisbergen, J.G. Snijders and E.J. Baerends, J. Chem. Phys. 103 (1995) 9347. 121 ] J.E Ward and I.J. Bigio, Phys. Rev. A 11 (1975) 60.
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[22] G.R. Alms, A.K. Burnham and W.H. Flygare, J. Chem. Phys. 63 (1975) 3321. [23] M.P. Bogaard, A.D. Buckingham, R.K. Pierens and A.H. White, J. Chem. Soc. Faraday Trans. I, 74 (1978) 3008. [24] I.E. Craven, M.R. Hesling and G.UD. Ritchie, Chem. Phys. Lett. 185 (1991) 371. [25] C. Htittig and B.A. Hess, J. Phys. Chem. 100 (1996) 6243. [26] R. Kobayashi, R.D. Amos, H. Koch and P. Jcrgensen, Chem. Phys. Lett. 253 (1996) 373.
[27] C. Jamorski, M.E. Casida and D.R. Salahub, J. Chem. Phys. 104 (1996) 5134. [281 S.J.A. van Gisbergen, V.E Osinga, O.V. Gritsenko, R. van Leeuwen J.G. Snijders and E.J. Baerends, J. Chem. Phys. 105 (1996) 3142. [291 S.J.A. van Gisbergen, J.G. Snijders and E.J. Baerends, Chem. Phys. Lett. 259 (1996) 599. [30] S.A.C. McDoweU, R.D. Amos and N.C. Handy, Chem. Phys. Lett. 235 (1995) 1.
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 265 (1997) 638-642
Dispersion of the mean dipole polarizability a(w) and dipole polarizability anisotropy K(w) of bromotrifluoromethane CBrF3 Dirk Goebel, Uwe Hohm Institut fiir Physikalische und Theoretische Chemie der Technischen Universit~it Braunschweig, Hans-Sommer-Strafle 10, D-38106 Braunschweig, Germany Received 11 November 1996; in final form 4 December 1996
Abstract Dispersive Fourier transform spectroscopy measurements of the refractive index of bmmotrifluoromethane, CBrF3, have been performed at pressures up to l05 Pa in the wavelength range between 900 and 325 nm, in order to determine the frequency dependence of the mean dipole polarizability a(o~). Combining our results with measured depolarization ratios p(to) yields a dispersion formula for the two independent components of the polarizability tensor a_L(to) and all (to). The results are used to get estimates of the dispersion interaction energy constant C6, the second hyperpolarizability y and the mean diamagnetic susceptibility X of CBrF3. @ 1997 Elsevier Science B.V.
1. Introduction Bromotrifluoromethane, CBrF3, also known as refrigerant 13B 1, is of considerable interest in chemical reaction kinetics, photochemistry, molecular spectroscopy, atmospheric chemistry, and semiconductor technologies [ 1-5]. However, the frequency dependence of its electro-optical properties, such as the refractive index and the dipole polarizability, are only poorly known [6-8]. In this Letter we present an experimental study of the dispersion of the refractive index and the mean dipole polarizability a(ro) of CBrF3. Due to the quite large frequency dependence of its depolarization ratio p(to) [9] CBrF3 is also one of the few promising canditates for which reliable values of the frequency dependence of the polarizability anisotropy r ( w ) can be determined. Critical combination of refractive index data and depolarization ratios therefore yield a Kramers-Heisenberg type dispersion formula for the individual tensor components
Ce_L(W) and Ol[l(W). This allows for a calculation of the low-frequency dispersion of a ( w ) and r ( w ) . Additionally, we will show that our results are also useful to obtain the mean diamagnetic susceptibility X in terms of a modern version of the Drude model [ 10], to estimate the static second dipole hyperpolarizability y [ 11 ], and to get a first indication of the dispersion interaction energy constant C6 of CBrF3. Atomic units (au) are used throughout this Letter. Conversion factors into SI units are given in the appendix.
2. Theory For molecules of C3v symmetry, the polarizability tensor consists of only two different elements Old_(W ) -~ Olyy(tO) ---- OlXX(tO) and Olll(w) = Olzz(tO), respectively, where to is the frequency of the measuring light. Outside electronic absorption bands the frequency dependence of the individual components
0009-2614/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. PII S0009-2614(96)01455-8
D. Goebel, U. Hohm/ Chemical Physics Letters 265 (1997) 638-642
a , ( t o ) (77 =2_, II) of the polarizability tensor can be described via the Kramers-Heisenberg formula _fk~ ~"(to)
=
~
togk. - '°2
f.
( 1)
tog. - '°2 '
where took. is the transition frequency and fk,7 the corresponding oscillator strength, f . and too. are effective values, too. being in the order of the ionization potential. If one is interested in the behaviour of a , ( t o ) at optical frequencies where to << w017, the approximation in Eq. (1) can be used. Hence, with this simplification the frequency dependence of the two invariants of the polarizability tensor are given as a ( t o ) = g1 [ 2 c e ±
(to) +
639
in Ref. [ 12]. Two types of measurements have been performed. First, the pressure dependence of the refractivity (n - 1) has been determined in the range between 0 and 105 Pa simultaneously at four different frequencies to using HeNe and HeCd laser radiation sources with tol = 0.071981, o)2 = 0.076693, o)3 = 0.083831, and 0 ) 4 = 0.140139 au, corresponding to vacuum-wavelengths of AI = 632.99, A2 = 594.096, 3,3 = 543.516, and M = 325.13 nm [13]. This yields the density dependence of the refractivity, from which the mean dipole polarizability a(to) of the free noninteracting molecule can be determined via a modified Lorentz-Lorenz formula:
n2(to,p,T) - 1 NA n2( to, p, T) + 2 = g--eo°e(to ) (-;--T )
all ( t o ) ]
1 (2 f± f[I ) \ tog±_ to2 + to0 _ to2
(2)
1
and fll K(to) =Oql (to) - a ± ( t o ) = o9211_ to2
f± 0)2± -- 0)2 •
(3) ce(to) is the mean dipole polarizability, which can be obtained from e.g. refractive index measurements and K(to) is the anisotropy of the dipole polarizability, which is related to the depolarization ratio p(to) of linearly polarized monochromatic light via
p(to) = 3K2(to)/[45oe2(to) + 4Kz(to) ] .
(4)
Amos [ 10] has recently demonstrated the capabilities of a modern version of the Drude model for obtaining electro-optical properties. In this context we mention that the mean diamagnetic susceptibility X of CBrF3 is given as [ 10]
-- X ~
1=0
I 1 (fx + fl') 4to0 / ~ ~ too± to011
(5) '
where N is the total number of electrons and toot are the respective transition frequencies.
where NA is Avogadro's constant, e0 the permittivity of vacuum, and B*(to, T) = bR(to, T) - B ( T ) , with bR (to, T) being the reduced second refractivity virial coefficient and B(T) the second (p, V, T) virial coefficient. In the second type of measurement a white light radiation source is used and isobaric dispersive Fourier transform spectra were recorded in the frequency range between 0.051 and 0.096 au (corresponding to a range of 475 to 900 nm) with a resolution of about 0.00077 au (see Refs. 112,141 ). In order to obtain the dispersion of the mean dipole polarizability or(to), the resulting quasi-continuous refractive index spectrum is reduced to ideal behaviour by means of the measured B* value. The CBrF3 sample has a purity of better than 99.9% and is used without further purification (CBrF3 supplied by Union Carbide, USA, and analyzed via GCMS). The individual error sources and uncertainties are evaluated in Refs. [ 12,13]. In the context of this work it is important to note that the relative error in c~(to) is less than 0.07%, while the uncertainty in B*(to,T) is less than I% .
4. Results and discussion 3. Experimental The measurements of the gas-phase refractive index of CBrF3 were performed with an evacuated Michelson twin-interferometer, which is described in detail
Our pressure dependent refractivity measurements yield a value of B*(wI,T) = 341.1(2.3) x 10 -6 m3mol - I at 298 K. This value of B*, which should be nearly independent of frequency, allows for
640
D. Goebel, U. Hohm/Chemical Physics Letters 265 (1997) 638-642
a reduction of all measured refractivities in order to obtain the dipole polarizability of the free molecule according to Eq. (6). a(tO) obtained in this way can be fitted to a one-term Kramers-Heisenberg dispersion formula with an r.m.s, error of Aa = 0.001 au in the whole frequency range between 0.05 and 0.14 au. The aim of this Letter is, however, to obtain the frequency dependence of the individual components a±(tO) and Otll(to) of the polarizability tensor. To this end we have first calculated x(tO) from the two depolarization ratios of P(tOl) = 0.0059(2) and P(tO5 = 0.088559 au) = 0.0066(2) given by Baas and van den Hout [9] using an interpolated ot(tOs) value of our measurements. This yields the two anisotropies K(tol) = 11.54(20) and K(tos) = 12.33(20) au. All measured mean polarizabilities (60 values) and the two polarizability anisotropies are now subject to a non-linear fit from which the parameters f ± = 24.83(40), tOo± = 0.8480(68) au, fll = 6.302(25) and tO011 = 0.37581(70) au (r.m.s. errors in parentheses) are obtained. The r.m.s, error of the fit is Aa = 0.0006 au. Quantitative results of a(tO) and K(tO) can now be obtained with Eqs. (2) and (3). Extrapolation to zero frequency gives the electronic part of a ( 0 ) and K(0) as d E = 37.89(53) au and KE = 10.09(82) au, respectively. In order to obtain the total static polarizability or(0) = dE(0) + a A(0) and its anisotropy K(0) = rE(0) + KA(0) we have to add the atomic ('vibrational') contributions a~_ and a~, respectively, to the individual tensor elements a~_ and a~. a~_ and ot~ can be evaluated with the help of absolute transition moments of infra-red active bands which have been tabulated by Bishop and Cheung [ 15]. For CBrF3 a~ = 3.95(40) au (vibrations with symmetry a t ) and a ~ = 3 . 0 4 ( 3 0 ) au (vibrations with symmetry e), leading to the final estimates of the mean static dipole polarizability o~(0) = 41.23(60) au and its anisotropy K(0) = 11.00(96) au. In order to support the numerical results of the frequency dependence of the individual tensor elements of the polarizability of CBrF3, we compare tOo± and tO011 with transition frequencies and ionization potentials recorded with far-UV and photoelectron spectroscopy, respectively [ 5,16 ]. Generally, the spectra of CF3X molecules (X = CI, Br, I) can be split into two regions. First the frequency range below tO = 0.56 au (15 eV), which is rather sensitive to the substituent X. However, the range above 0.56 au is dominated
by the CF3 group and quite insensitive to the halogen X. For our parallel component Crll we have obtained to011 = 0.37581 au (10.23 eV), which corresponds very nicely to the absorption maximum of CBrF3 in this region located at to = 0.387 au (10.54 eV). In view of our analysis of the components of the polarizability tensor, this low absorption energy must be due to the C - B r unit, which lies on the z-axis of the molecule. Sandorfy et al. [ 5] assigned this transition frequency to a 5s ~ (C-Br) Rydberg transition. The transition frequency to0_t = 0.8480 au (23.08 eV) of the perpendicular component should be dominated by the presence of the CF3 group. The photoelectron spectra of both CF3Br and CF3I show intense absorption peaks at energies of about 0.72 au (19.5 eV), 0.76 au (20.8 eV), and 0.87 au (23.7 eV) [ 16]. The last one is in good accordance with too±, but, this coincidence might be fortuitous. If we apply the same analysis to ~r(to) and p(~o) to other trifluoromethanes, we should get a value of o~0±, which is also located in the absorption region of the CF3 group. Due to lack of experimental data, we only can get a rough estimate of too± = 0.71 au (19.3 eV) for CC1F3 (four a ( t o ) and two p(to) as input from Refs. [7,9] ) and tOo± = 0.77 au (20.9 eV) for CHF3 (two a(tO) and two p(tO) as input from Ref. [9]). Again, too± can be attributed to electronic transitions within the CF3 group. In our analysis of the frequency dependence of a(tO) we have obtained two effective transition frequencies and the corresponding oscillator strengths. In the framework of the simple Drude model we can apply Eq. (5) to obtain an estimate of the mean diamagnetic susceptibility X of CBrF3. The capabilities of this rather crude model of electric and magnetic properties have recently been demonstrated by Amos [ 10]. With our parameters of f and too we obtain a mean X = -11.51 au, which agrees fairly well with 11.68 au obtained from the Pascal group contribution model [ 17] and the two values of - 1 1 . 0 7 and - 1 1 . 3 9 au, calculated with semi-empirical methods [ 18]. There is no experimental value of X of CBrF3 to compare with. The Kramers-Heisenberg-type representation Eqs. (2), (3) of the mean polarizability or(tO) and anisotropy K(tO) is suitable for obtaining isotropic and anisotropic dispersion interaction energy constants C6. They are defined in terms of integrals over -
D. Goebel, U. Hohm/ Chemical Physics Letters 265 (1997) 638-642 the polarizability at imaginary frequencies, that is C6 = 3 f a ( i t o ) 2 dto/zr, C; = f a ( i w ) J c ( i t o ) dw/Tr, and C~' = f K(ito) z d t o / ( 3 ~ ) . For the relation of C6, C~, and C~' to the dispersion interaction energy UDisp see e.g. Ref. [ 19]. With our values o f the oscillator strengths and transition frequencies we obtain for the isotropic term C6 = 667 au, and for the two anisotropic contributions C~ = - 3 9 au and C~' = 13 au. There are no other values for CBrF3 to compare with. However, a crude comparison can be made with the isotropic part C6 o f cyclo-C3H6, because this molecule shows a similar frequency dependence of its mean polarizability a ( t o ) [13]. In that case a theoretical value obtained with density functional methods is 638 au [20], which is quite close to our findings for CBrF3, Last, it is worth mentioning, that the Cauchy moments S ( - 2 k ) o f the series expansion a ( t o ) = S ( - 2 ) + S ( - 4 ) t o 2 -k- S ( - 6 ) t o 4 give an estimate o f the static second hyperpolarizability via y(0;0,0,0) ~ 27.4S(-3)S(-4)/S(-2), S(-3) [ S ( - 2 ) S ( - 4 ) ] (1/2) [ 11 ]. For the mean dipole polarizability or(to) our refractive index measurements yield S ( - 2 ) = 3 7 . 8 9 0 ( 1 ) , S ( - 4 ) = 137.3(1.2), and S ( - 6 ) = 9 3 3 ( 5 0 ) , which give an approximation of the second hyperpolarizability o f y ( 0 ; 0 , 0 , 0) = 7160 au. This result compares with y ( -2o9; 0, to, to) = 7445 au (to = 0.06565 au) measured with electric field-induced second harmonic generation by Ward and Bigio [21].
5. Conclusion We have combined our precise measurements of the dispersion of the mean polarizability a ( t o ) with literature results of the depolarization ratio p(to) in order to obtain the frequency dependence o f the individual components a ± (to) and Otll(to) o f the polarizability tensor ~ ( t o ) o f CBrF3. This allows for the calculation o f the frequency dependence of the polarizability anisotropy K(to). Despite its importance in various fields o f electro-optics and intermolecular interactions [19,20] experimental values of K(to) are only sparsely known in the literature and are obtained by combining values o f a ( t o ) and p(to) [ 9 , 1 9 , 2 2 - 2 4 ] . It is desirable that accurate time dependent ab initio calculations (see e.g. Refs. [ 2 5 , 2 6 ] ) and time dependent density functional schemes (see
641
e.g. Refs. [20,27-29] ) give additional information on tr(to) and x ( t o ) , although at present the apparent accuracy o f D F T calculations seems to result from cancellation o f different artifacts [ 30].
Appendix A Conversion factors from atomic (au) to SI units: Energy:
1Eh = 4.3597482 x 10 -18 J;
to: 1 E h / h = 4 . 1 3 4 1 3 7 3 x 1016 s -1 ; C 6 : 1 a6Eh = 9.573448 × 10 -80 J m 6 ;
ct, K: l e2a2E~ I = 1.648778 × 10 -41 C 2 m 2 J - I ; y: 1 eaa4Eh 3 = 6.235378 × 10 -65 C 4 m 4 J -3 ; X: 1 e2a~me I = 7.98104 × 10 -29 J T -2 .
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