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Raman cross section for the pentagonal-pinch mode in buckminsterfullerene C60
16May 1997
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 270 (1997) 129-134
Raman cross section for the pentagonal-pinch mode in buckminsterfullerene C60 J.D. Lorentzen a, S. Guha a, j. Men6ndez a, p. Giannozzi b, S. B a r o n i c,d a Department o f Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504, USA b Istituto Nazionole di Fisica della Materia and Scuola Normale Superiore, Piazza dei Cavalieri 7, 1-56126 Pisa, Italy c Istituto Nazionale di Fisica della Materia and Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2 - 4 , 1-34014 Trieste, Italy a Centre Europ~en de Calcul Atomique et Mol~culaire, 46 All~e d'ltalie, 69007 Lyon, France
Received 24 February 1997
Abstract
The Raman cross section for the pentagonal pinch mode of C6o at 1469 cm- ~ has been determined at a laser excitation wavelength AL = 752.5 nm. The experimental value, d o ' / d O = (2.09 + 0.29)× 10 - 2 9 cm2/sr, is found to be in good agreement with predictions from an ab initio calculation based on density functional perturbation theory. A simple bond polarizability model with parameters obtained from hydrocarbon Raman measurements yields Raman intensities in C60 within one order of magnitude of the experimental data.
1. Introduction Shortly after C60 became available in macroscopic quantities, its Raman spectrum was investigated by several groups [1]. While the initial emphasis was on a confirmation of group theory predictions for the proposed icosahedral symmetry of the molecule, later Raman studies addressed issues such as photopolymerization [2], crystal dynamics [3], optical transitions [4] and isotope effects [5,6]. The discovery of superconductivity in doped fullerenes [7] generated an additional interest in their Raman spectrum [1], since Raman-active vibrations are believed to play a key role in the superconducting properties of these materials [8]. All previous Raman studies of fullerenes have focused on r e l a t i v e intensities between the different Raman peaks of the molecule. It is well known, however, that cross section measurements are needed
for the determination of electron-phonon coupling parameters [9]. In the case of C60 , a knowledge of the Raman cross section would also provide a definitive test of two theoretical predictions of Raman intensities in C60. In a recent paper, Giannozzi and Baroni [10] applied density functional perturbation theory [11] to the vibrational and dielectric properties of C60. One of the properties that can be calculated within the scope of density functional theory is the off-resonance Raman cross section [12]. The parameter-free Raman spectrum computed by Giannozzi and Baroni is in excellent agreement with the experimental r e l a t i v e Raman intensities in C60 [13]. This result is important in view of the recent interest in density functional methods for the calculation of Raman intensities in small molecules [14-16]. The ab initio calculations provide not only relative cross section predictions but also absolute values, which so far have not been tested experimentally. The Raman
0009-2614/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PH S0009-261 4(97)003 37-0
J.D. Lorentzen et a l . / Chemical Physics Letters 270 (1997) 129-134
130
spectrum of C6o has also been fit using a bond polarizability model [13,17,18]. It was found that the ratios of bond polarizability parameters for the carbon-carbon single and double bonds in C60 are quite similar to their counterparts in hydrocarbons, suggesting that the bond polarizability parameters might be transferable. However, the transferability of individual polarizability parameters can only be tested with measurements of the Raman cross sections in C60. In this Letter, we report the determination of the Raman cross section for the 1469 cm -I A~(2) pentagonal pinch mode of C60. Our measured cross section is in good agreement with first-principles predictions. Our results also suggest that the transferability of bond polarizabilities from hydrocarbons to fullerenes is reasonably justified.
2. Method
The Raman scattering cross section for a molecular vibrational mode is defined as the number of scattered photons per unit time and unit solid angle divided by the incident laser flux (in number of photons per unit time per unit area). For a totally symmetric mode, the expression for the cross section for parallel incident and scattered light polarizations is [19] dtr
toE (,OS 3
dO
C4
h
2tof((n/) + 1)lafl2Z,
(1)
where toL is the incident light angular frequency, tos the scattered light angular frequency, c the speed of light, toy the angular vibrational frequency of the Raman mode, and ( n f ) its mean occupation number. The factor L is a local field correction [19,20]. The key molecular parameter which determines the Raman cross section of a totally symmetric mode is the average a t of the Raman tensor, which in the offresonance limit can be written in terms of polarizability derivatives as
as=
(P/x s + P . . s + Pzz,s) •
(2)
Here P~a.S is the derivative OP~rJadf of the electronic polarizability tensor P,~t3 relative to dr, the
normal coordinate for mode f. The normal coordinate is defined in such a way that the classical Hamiltonian for the vibrational system becomes
f Truly absolute measurements of Raman cross sections are sensitive to systematic errors due to the required calibration of the entire optical system. In our case, this complication can be partially circumvented by using solutions of C60 in solvents with Raman modes of known cross section. We chose as a reference the 992 c m - l mode in benzene (C6H6), for which very accurate cross section measurements have been reported [21]. However, since carbon disulfide (CS 2) is a better C60 solvent than C6H6, we performed our measurements in two steps. We first prepared solutions of C60 in CS 2 and measured the intensity of the 1469 cm -1 C60 mode relative to the 656 cm -j mode of CS 2. Next we prepared solutions of C S 2 in C6H 6 and measured the Raman intensity of the 656 cm-1 mode in CS 2 relative to the 992 cm -1 mode in C6H 6. By combining the results from these measurements we were able to obtain the ratio of intensities between the 1469 c m mode in C60 and the 992 c m - 1 mode in C6H6, from which we finally obtained the Raman cross section for the C6o vibration. Ab initio predictions of Raman intensities using the method described in Ref. [10] correspond to the off-resonance limit, for which the excitation photon energy is well below the energy of dipole-allowed optical transitions. In the case of molecular C6o, the first allowed optical transition occurs at h -- 408 nm [22], but the Raman cross section of the Ag(2) mode in crystalline C60 displays a strong resonant enhancement around h L = 500 nm [4]. This resonance has been attributed to icosahedral symmetry breaking in the solid state [4]. The 500 nm resonance is not observed in C60/CS 2 solutions at room temperature, but the cross section shows clear evidence of a pre-resonance enhancement [23]. This suggests that the off-resonance range for C60 corresponds to wavelengths much longer than 500 nm. A safe choice is probably the 1064 nm line of a Nd:YAG laser. Experiments by Chase and coworkers [24] using this
131
J.D. Lorentzen et a l l Chemical Physics Letters 270 (1997) 129-134
excitation wavelength show that the relative intensities of the Raman peaks in C60 are in excellent agreement with ab initio calculations [13]. Unfortunately, however, the 1064 nm line has two important disadvantages for Raman cross section measurements. Firstly, the resulting Raman spectrum is outside the useful range of silicon-based charge-coupled-device (CCD) detectors, which are almost indispensable in measuring the weak Raman signal from C60 in solution. In addition, the longest wavelength for which the Raman cross section of the 992 c m - i line of C 6 H 6 has been measured is 656 nm [21]. An empirical functional expression for the cross section can be used to extrapolate the experimental data to longer wavelengths [21], but the reliability of this expression at 1064 nm is unknown. Consequently, we have chosen the 752.5 nm line of a Kr ÷ laser as the excitation wavelength for the experiments reported here. This line provides a reasonable compromise between practical needs and theoretical requirements.
3. Experimental details
4. Results Fig. 1 shows a Raman spectrum of a C60//C52 solution obtained with the 752,5 nm Kr + line. Notice that the relative intensities of the peaks are not identical to the values reported for excitation at 1064 nm [24]. For example, the ratio of integrated intensities between the two totally symmetric modes changes from 0,9 at 1064 nm (Ref. [24]) to 0.7 at 752.5 nm. This suggests that our excitation wavelength is not completely outside the range within which resonance effects are noticeable. From the raw Raman data we computed the integrated 'intensities' (in photonss-1). The ratio of such intensities for the 1469 cm -1 mode in C60, the 656 cm-1 mode in CS 2 and the 992 c m - i mode in C 6 H 6 is shown in Fig. 2(a) and 2(b). The experimental data fit well by a straight line in both cases. From the slopes of the lines we can calculate the corresponding ratios of Raman cross sections using Eq. (1), the molecular weights of C6o, CS 2, and C 6 H 6 , and the densities of the two solvents. Notice that the local field correction cancels out when ratios of Raman cross sections are computed. We obtain
l(Ag(2))/I(Ag(1))
(d t r / d / 2 ) 1469 A C6o/CS 2 solution was prepared by dissolving 0.5 mg of purified C6o powder ( > 99.9% purity, from MER Corporation) in 1.5 ml of spectrophotometric grade CS 2 (Aldrich Chemical Company). The solution was placed in a cuvette, which was sealed with Parafilm ' M ' (American National Can) and a Teflon stopper. Since C6o precipitation is a potential source of systematic error, we verified that the Raman signal was proportional to the C6o concentration by repeating the experiments with increased amounts of CS 2 in the solution. The Raman measurements using the 752.5 nm Kr + laser line - were performed in the backscattering configuration at room temperature. The incident power was 5 0 - 6 0 mW. The scattered light was analyzed with a SPEX 1404 double monochromator and detected with a CCD. Similar experiments were performed using C 5 2 / / C 6 H 6 solutions. A small correction for the spectrometer/CCD response was applied to all Raman data. This correction is necessary due to the different scattering wavelengths for the three Raman modes involved in the experiments.
( d or//d f~)656
= 36.69 + 0.59,
(3)
and (d o'/d~)656
0.439 +_ 0.0095.
(do-/dO)992
(4)
The laser-wavelength dependent Raman cross section for the 992 cm -1 of neat C 6 H 6 has been
A
.
,
.
.
.
.
.
C60 ~
/ / ,
-
,
.
,
.
.
C60 Ag(2)
~_~ ~._.___
=
485
490
495
1465 1470 1475
RAMAN SHIFT (cm"1) Fig. 1. Room temperature Raman spectrum of 6.2 mg of C6o dissolved in 0.8 ml of CS 2. The excitation wavelength is 752.5 rim.
J.D. Lorentzen et a l . / Chemical Physics Letters 270 (1997) 129-134
132 15.0
.
,
.
,
.
,
.
'
'
'
'
'
'
"
'
'
0.6
' ~
,../1oo 0.4
5.0
o.o 0 . 0
0.2
'
'.
01
'
i
0 2
'
i
0 3
'
Cone. (mg C60/ml CS2)
0.4
o.o 0 . 0
0.2
0.4
0.6
0.8
1.0
Conc.(ml CS2/ml C6H6)
Fig. 2. (a) Relative Raman intensities of the 1469 c m - ~ C6o vibration and the 656 c m - ~ vibration in CS z as measured in C6o//CS2 solutions for an excitation wavelength of 752.5 n m . The straight line is a linear fit to the experimental points, which yields a slope of (3.090 5z 0.052) × 10 3 ml C S 2 / m g C6o. (b) Relative Raman intensities of the 656 c m - i vibration in CS 2 and the 992 e m - i vibration in
C6H6 as measured in CS2/C6H 6 solutions for the same excitation wavelength. The straight line is a linear fit to the experimental points, which yields a slope of (0.648 +_0.014) ml C6H6/ml CS2.
investigated in detail by Schomacker et al. [21]. Extrapolating their results to excitation at 752.5 nm, and performing a local field correction (as described in Ref. [21]) to obtain the Raman cross section for an isolated C 6 H6 molecule, we find ((d t r ) / ( d ~ ))992 = (1.30_+ 0.13) X 10 -30 cmE/sr. Inserting this cross section value in Eqs. (3) and (4), we finally obtain for the totally symmetric pentagonal pinch mode of an isolated C60 molecule (dtr)
= (2.09_+ 0 . 2 9 ) x 10-29 c m 2 / s r 1469
(at a L = 752.5 n m ) .
(5)
This result can be combined with Eq. (1) to compute the square of the average Raman tensor:
lasl 2 = (3.21 __+0.45) × 10 -7 c m g / g (at a L = 752.5 n m ) .
(6)
The correctness of our approach can be verified by computing the Raman cross section for the 656 c m - 1 line of CS 2 using Eq. (4). If we apply the local field correction from Ref. [20], we obtain (dtr/d~O)656 = (3.4 + 0.4) X 10 -30 c m 2 / s r (liquid phase). This is in very good agreement with the value ( d o t / d O ) 6 5 6 = 3.7 X 10 -30 cm2/sr, which we obtain from extrapolation to 752.5 nm of earlier cross section measurements by Kato and Takuma
[251.
5. Discussion Giannozzi and Baroni [10] calculated the vibrational frequencies, dynamical electric charges, and Raman intensities for a face-centered cubic C60 lattice. Details of these calculations are given in Ref. [10] For the calculation of Raman intensities, they compute the derivatives of the electric susceptibility relative to the vibrational normal coordinates. These quantities can be related to the derivatives of the molecular polarizability which appear in Eq. (2) by application of the Clausius-Mossotti formula. The relative Raman intensities predicted by Giannozzi and Baroni are in very good agreement [13] with experimental data obtained with IR excitation at 1064 nm [24]. Absolute intensities were not included in Ref. [10] because no experimental data were available at that time. Assuming Eq. (2) to be valid, the method of Ref. [10] yields [otf[ 2 = 7.6 × 10 -8 c m g / g .
(7)
This is in good agreement with the measured average Raman tensor (Eq. (6)), particularly if one considers that the experimental value is likely to be somewhat affected by pre-resonance effects and that the calculated quantity (the polarizability derivative) differs from the corresponding experimental value by a factor of only 2. The Raman cross section can also be predicted from a bond polarizability model [17], which as-
J.D. Lorentzen et al. / Chemical Physics Letters 270 (1997) 129-134
sumes that the molecular polarizability can be written as a sum of individual bond polarizabilities. In the simplest version of such a model, one makes the additional assumption that the components of the bond polarizability tensor are functions of the bond lengths R only. Furthermore, the bond is assumed to have cylindrical symmetry, so that the polarizability tensor has only two independent components, denoted as all(R) and a±(R). Using such a model it is straightforward to write the polarizability derivatives in Eq. (2) as [13]
+ 2ot'± ( B) )Ro( I, B ) "x(llf)~ + [ C~il(B) -- c~'±( B ) ] [/~o~(l, 1
B)Rot3(l, B)
^
- ~I~ ] Ro( I, B ) " x( llI)
+(,
(8) )
× (1~o~(l, B ) x ~ ( l , f ) + Ro~(l, B) X X~(l,f) -
×[Ro(l,B).x(llf)])].
molecular species with the same type of bonds, the hydrocarbon parameters can be used to compute the static dielectric properties and the Raman spectrum of fullerenes. The agreement with experiment is excellent for the static polarizabilities and reasonably good for the Raman spectrum [13]. In the latter case, however, the comparison with experiment has been limited to ratios of bond polarizability parameters due to the lack of experimental cross sections for the Raman-active vibrations in fullerenes. Our cross section measurements for the Ag(2) mode in C60 make it possible to test the transferability of individual polarizability parameters. The second and third terms in Eq. (8) are zero for the totally symmetric Ag(2) vibration of C60 at 1469 cm - I . Therefore, the polarizability derivatives depend only on the parameters [c~il(s)+ 2c~'± (s)] and [all(d) + 2a'± (d)] for single and double bonds, respectively. Using [ otil(s) + 2 c~'. ( s)] = 3.13 × 10-16 cm 2 (Ref. [27]), [all(d) + 2 a ' ± ( d ) ] = 6.50 × 10 -16 cm 2 (Ref. [28]), and the product Ro(l, B). x(llf) from ab initio calculations, we obtain Icrfl 2 = 2.8 × 10 -8 cm4//g,
2/?0~ ( l, B)/~0~ ( l, B) (8)
Here the sum over B extends over all bonds connected to site l. The primes denote radial derivatives and the carets denote dimensionless unit vectors. Ro(I,B) is the equilibrium bond vector from atom l to its neighbor B and x(llf) is the mode eigenvector, obtained from the 3N × 3N eigenvalue problem ( ~ - t o ~ M ) g ( f ) = 0 and subject to the orthonormality condition
~ x~( llf ) x~( llf')mt=- ~fff , lct
where m t is the mass of atom l. The notation used is the same as in Ref. [13], but Eq. (8) corrects for several typographic errors in Eq. (4) of Ref. [13]. Using expressions equivalent to Eq. (8), the bond polarizability parameters for carbon-carbon single and double bonds have been determined for a number of hydrocarbons [26-28]. Assuming the bond polarizabilities to be transferable among different
133
(9)
which is within one order of magnitude of the experimental value. This result is consistent with the partial agreement found earlier between the bond polarizability model predictions and the experimental relative Raman intensities. While the absolute intensities are found here to differ from experiment by a factor of 12, the predicted relative Raman intensities were shown in Ref. [ 13] to differ from experiment by a factor as high as 6. It is interesting to point out that the overall sign of the components of the Raman tensor cannot be determined from light scattering experiments, since the Raman intensities are proportional to the square of these quantities. However, for a given choice of eigenvectors the sign of the Raman tensor components has a clear physical meaning: it represents the change in the electronic polarizability for a displacement of the atoms according to the eigenvector pattern. This sign can be obtained from theoretical calculations of the polarizability derivatives. The sign of the bond polarizability parameters in Refs. [27] and [28] was chosen so that they agree with the sign of the Raman tensor obtained from ab initio
134
J.D. Lorentzen et al.// Chemical Physics Letters 270 (1997) 129-134
calculations for hydrocarbons. When these polarizability parameters are transferred to C60 , we find from Eq.(8) that the average Raman tensor for the Ag(2) mode is negative if the eigenvectors correspond to an expansion of the pentagons. This agrees with the sign we obtain from our own ab initio calculation for C60.
6. Conclusion
We have measured the cross section for inelastic Raman scattering by the pentagonal pinch mode of C6o at 1469 cm -J. Our measured value compares well with a prediction from ab initio calculations using density functional perturbation theory. Our measurements are also consistent with a simple bond polarizability model using parameters transferred from hydrocarbons. This suggests that such a model might be sufficient to predict the observable Raman peaks in higher fullerenes and nanotubes, for which Raman spectroscopy is an important characterization technique.
Acknowledgements This work was partially supported by the US National Science Foundation under Grants DMR9058343 and DMR 9624102. One of us (JM) thanks John B. Page for numerous discussions and constant encouragement.
References [1] See, for example, M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, J. Raman Spectrosc. 27 (1996) 351, and references therein. [2] A.M. Rao, P. Zhou, K.A. Wang, G.T. Hager, J.M. Holden, Y. Wang, W.T. Lee, X.X. Bi, P.C. Eklund, D.S. Comet, M.A. Duncan, l.J. Amster, Science 259 (1993) 955. [3] P.J. Horoyski, M.L.W. Thewalt, Phys. Rev. B 48 (1993) 11446.
[4] K. Sinha, J. Men6ndez, R.C. Hanson, G.B. Adams, J.B. Page, O.F. Sankey, L.D. Lamb, D.R. Huffman, Chem. Phys. Lett. 186 (1991) 287. [5] S. Guha, J. Men6ndez, J.B. Page, G.B. Adams, G.S. Spencer, J.P. Lehman, P. Giannozzi, S. Baroni, Phys. Rev. Lett. 72 (1994) 3359. [6] P.J. Horoyski, M.L. Thewalt, T.R. Anthony, Phys. Rev. Lett. 74 (1995) 194. [7] A.F. Hebard, M.J. Rosseinsky, R.C. Haddon, D.W. Murphy, S.H. Glarum, T.T.M. Palstra, A.P. Ramirez, A.R. Kortan, Nature 350 (1991) 350. [8] C.M. Varma, J. Zaanen, K. Raghavachari, Science 254 (1991) 989. [9] P.M. Champion, A.C. Albrecht, Ann. Rev. Phys. Chem. 33 (1982) 353. [10] P. Giannozzi, S. Baroni, J. Chem. Phys. 100 (1994) 8537. [11] S. Baroni, P. Gainnozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [12] S. Baroni, R. Resta, Phys. Rev. B 33 (1986) 5969. [13] S. Guha, J. Men6ndez, J.B. Page, G.B. Adams, Phys. Rev. B 53 (1996) 13106. [14] B.G. Johnson, J. Fori~in, Chem. Phys. Lett. 247 (1995) 120. [15] A. Stirling, J. Chem. Phys. 104 (1996) 1254. [16] D. Porezag, M.R. Pderson, Phys. Rev. B 54 (1996) 7830. [17] D.W. Snoke, M. Cardona, Solid State Commun. 87 (1993) 121. [18] D.W. Snoke, M. Cardona, S. Sanguinetti, G. Benedek, Phys. Rev. B 53 (1996) 12641. [19] H.W. Schrbtter, H.W. Kl~chner, in Raman Spectroscopy of Gases and Liquids, Vol. 11, A. Weber (Ed.), Springer-Verlag, Berlin, 1979, p. 123. [20] J.R. Nestor, E.R. Lippincott, J. Raman Spectrosc. 1 (1973) 305. [21] K.T. Schomacker, J.K. Delaney, P.M. Champion, J. Appl. Phys. 85 (1986) 4241. [22] S. Leach, M. Vervloet, A. Despr~s, E. Br~heret, J.P. Hare, T.J. Dennis, H.W. Kroto, R. Taylor, D.R.M. Walton, Chem. Phys. 160 (1992) 451. [23] S. Guha, J. Loretnzen, K. Sinha, J. Men~ndez, G.B. Adams, J.B. Page, O.F. Sankey, Mol. Cryst. Liq. Cryst. 256 (1994) 391. [24] B. Chase, N. Hcrron, E. Holler, J. Phys. Chem. 96 (1992) 4262. [25] Y. Kato, H. Takuma, J. Chem. Phys. 54 (1971) 5398. [26] F. Orduna, C. Domingo, S. Montero, W.F. Murphy, Mol. Phys. 45 (1982) 65. [27] J. Martin, S. Montero, J. Chem. Phys. Phys. 80 (1984) 4610. [28] M.F. Orduna, A.d. Olmo, C. Domingo, S. Montero, J. Mol. Struc. 142 (1986) 201.
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 270 (1997) 129-134
Raman cross section for the pentagonal-pinch mode in buckminsterfullerene C60 J.D. Lorentzen a, S. Guha a, j. Men6ndez a, p. Giannozzi b, S. B a r o n i c,d a Department o f Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504, USA b Istituto Nazionole di Fisica della Materia and Scuola Normale Superiore, Piazza dei Cavalieri 7, 1-56126 Pisa, Italy c Istituto Nazionale di Fisica della Materia and Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2 - 4 , 1-34014 Trieste, Italy a Centre Europ~en de Calcul Atomique et Mol~culaire, 46 All~e d'ltalie, 69007 Lyon, France
Received 24 February 1997
Abstract
The Raman cross section for the pentagonal pinch mode of C6o at 1469 cm- ~ has been determined at a laser excitation wavelength AL = 752.5 nm. The experimental value, d o ' / d O = (2.09 + 0.29)× 10 - 2 9 cm2/sr, is found to be in good agreement with predictions from an ab initio calculation based on density functional perturbation theory. A simple bond polarizability model with parameters obtained from hydrocarbon Raman measurements yields Raman intensities in C60 within one order of magnitude of the experimental data.
1. Introduction Shortly after C60 became available in macroscopic quantities, its Raman spectrum was investigated by several groups [1]. While the initial emphasis was on a confirmation of group theory predictions for the proposed icosahedral symmetry of the molecule, later Raman studies addressed issues such as photopolymerization [2], crystal dynamics [3], optical transitions [4] and isotope effects [5,6]. The discovery of superconductivity in doped fullerenes [7] generated an additional interest in their Raman spectrum [1], since Raman-active vibrations are believed to play a key role in the superconducting properties of these materials [8]. All previous Raman studies of fullerenes have focused on r e l a t i v e intensities between the different Raman peaks of the molecule. It is well known, however, that cross section measurements are needed
for the determination of electron-phonon coupling parameters [9]. In the case of C60 , a knowledge of the Raman cross section would also provide a definitive test of two theoretical predictions of Raman intensities in C60. In a recent paper, Giannozzi and Baroni [10] applied density functional perturbation theory [11] to the vibrational and dielectric properties of C60. One of the properties that can be calculated within the scope of density functional theory is the off-resonance Raman cross section [12]. The parameter-free Raman spectrum computed by Giannozzi and Baroni is in excellent agreement with the experimental r e l a t i v e Raman intensities in C60 [13]. This result is important in view of the recent interest in density functional methods for the calculation of Raman intensities in small molecules [14-16]. The ab initio calculations provide not only relative cross section predictions but also absolute values, which so far have not been tested experimentally. The Raman
0009-2614/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PH S0009-261 4(97)003 37-0
J.D. Lorentzen et a l . / Chemical Physics Letters 270 (1997) 129-134
130
spectrum of C6o has also been fit using a bond polarizability model [13,17,18]. It was found that the ratios of bond polarizability parameters for the carbon-carbon single and double bonds in C60 are quite similar to their counterparts in hydrocarbons, suggesting that the bond polarizability parameters might be transferable. However, the transferability of individual polarizability parameters can only be tested with measurements of the Raman cross sections in C60. In this Letter, we report the determination of the Raman cross section for the 1469 cm -I A~(2) pentagonal pinch mode of C60. Our measured cross section is in good agreement with first-principles predictions. Our results also suggest that the transferability of bond polarizabilities from hydrocarbons to fullerenes is reasonably justified.
2. Method
The Raman scattering cross section for a molecular vibrational mode is defined as the number of scattered photons per unit time and unit solid angle divided by the incident laser flux (in number of photons per unit time per unit area). For a totally symmetric mode, the expression for the cross section for parallel incident and scattered light polarizations is [19] dtr
toE (,OS 3
dO
C4
h
2tof((n/) + 1)lafl2Z,
(1)
where toL is the incident light angular frequency, tos the scattered light angular frequency, c the speed of light, toy the angular vibrational frequency of the Raman mode, and ( n f ) its mean occupation number. The factor L is a local field correction [19,20]. The key molecular parameter which determines the Raman cross section of a totally symmetric mode is the average a t of the Raman tensor, which in the offresonance limit can be written in terms of polarizability derivatives as
as=
(P/x s + P . . s + Pzz,s) •
(2)
Here P~a.S is the derivative OP~rJadf of the electronic polarizability tensor P,~t3 relative to dr, the
normal coordinate for mode f. The normal coordinate is defined in such a way that the classical Hamiltonian for the vibrational system becomes
f Truly absolute measurements of Raman cross sections are sensitive to systematic errors due to the required calibration of the entire optical system. In our case, this complication can be partially circumvented by using solutions of C60 in solvents with Raman modes of known cross section. We chose as a reference the 992 c m - l mode in benzene (C6H6), for which very accurate cross section measurements have been reported [21]. However, since carbon disulfide (CS 2) is a better C60 solvent than C6H6, we performed our measurements in two steps. We first prepared solutions of C60 in CS 2 and measured the intensity of the 1469 cm -1 C60 mode relative to the 656 cm -j mode of CS 2. Next we prepared solutions of C S 2 in C6H 6 and measured the Raman intensity of the 656 cm-1 mode in CS 2 relative to the 992 cm -1 mode in C6H 6. By combining the results from these measurements we were able to obtain the ratio of intensities between the 1469 c m mode in C60 and the 992 c m - 1 mode in C6H6, from which we finally obtained the Raman cross section for the C6o vibration. Ab initio predictions of Raman intensities using the method described in Ref. [10] correspond to the off-resonance limit, for which the excitation photon energy is well below the energy of dipole-allowed optical transitions. In the case of molecular C6o, the first allowed optical transition occurs at h -- 408 nm [22], but the Raman cross section of the Ag(2) mode in crystalline C60 displays a strong resonant enhancement around h L = 500 nm [4]. This resonance has been attributed to icosahedral symmetry breaking in the solid state [4]. The 500 nm resonance is not observed in C60/CS 2 solutions at room temperature, but the cross section shows clear evidence of a pre-resonance enhancement [23]. This suggests that the off-resonance range for C60 corresponds to wavelengths much longer than 500 nm. A safe choice is probably the 1064 nm line of a Nd:YAG laser. Experiments by Chase and coworkers [24] using this
131
J.D. Lorentzen et a l l Chemical Physics Letters 270 (1997) 129-134
excitation wavelength show that the relative intensities of the Raman peaks in C60 are in excellent agreement with ab initio calculations [13]. Unfortunately, however, the 1064 nm line has two important disadvantages for Raman cross section measurements. Firstly, the resulting Raman spectrum is outside the useful range of silicon-based charge-coupled-device (CCD) detectors, which are almost indispensable in measuring the weak Raman signal from C60 in solution. In addition, the longest wavelength for which the Raman cross section of the 992 c m - i line of C 6 H 6 has been measured is 656 nm [21]. An empirical functional expression for the cross section can be used to extrapolate the experimental data to longer wavelengths [21], but the reliability of this expression at 1064 nm is unknown. Consequently, we have chosen the 752.5 nm line of a Kr ÷ laser as the excitation wavelength for the experiments reported here. This line provides a reasonable compromise between practical needs and theoretical requirements.
3. Experimental details
4. Results Fig. 1 shows a Raman spectrum of a C60//C52 solution obtained with the 752,5 nm Kr + line. Notice that the relative intensities of the peaks are not identical to the values reported for excitation at 1064 nm [24]. For example, the ratio of integrated intensities between the two totally symmetric modes changes from 0,9 at 1064 nm (Ref. [24]) to 0.7 at 752.5 nm. This suggests that our excitation wavelength is not completely outside the range within which resonance effects are noticeable. From the raw Raman data we computed the integrated 'intensities' (in photonss-1). The ratio of such intensities for the 1469 cm -1 mode in C60, the 656 cm-1 mode in CS 2 and the 992 c m - i mode in C 6 H 6 is shown in Fig. 2(a) and 2(b). The experimental data fit well by a straight line in both cases. From the slopes of the lines we can calculate the corresponding ratios of Raman cross sections using Eq. (1), the molecular weights of C6o, CS 2, and C 6 H 6 , and the densities of the two solvents. Notice that the local field correction cancels out when ratios of Raman cross sections are computed. We obtain
l(Ag(2))/I(Ag(1))
(d t r / d / 2 ) 1469 A C6o/CS 2 solution was prepared by dissolving 0.5 mg of purified C6o powder ( > 99.9% purity, from MER Corporation) in 1.5 ml of spectrophotometric grade CS 2 (Aldrich Chemical Company). The solution was placed in a cuvette, which was sealed with Parafilm ' M ' (American National Can) and a Teflon stopper. Since C6o precipitation is a potential source of systematic error, we verified that the Raman signal was proportional to the C6o concentration by repeating the experiments with increased amounts of CS 2 in the solution. The Raman measurements using the 752.5 nm Kr + laser line - were performed in the backscattering configuration at room temperature. The incident power was 5 0 - 6 0 mW. The scattered light was analyzed with a SPEX 1404 double monochromator and detected with a CCD. Similar experiments were performed using C 5 2 / / C 6 H 6 solutions. A small correction for the spectrometer/CCD response was applied to all Raman data. This correction is necessary due to the different scattering wavelengths for the three Raman modes involved in the experiments.
( d or//d f~)656
= 36.69 + 0.59,
(3)
and (d o'/d~)656
0.439 +_ 0.0095.
(do-/dO)992
(4)
The laser-wavelength dependent Raman cross section for the 992 cm -1 of neat C 6 H 6 has been
A
.
,
.
.
.
.
.
C60 ~
/ / ,
-
,
.
,
.
.
C60 Ag(2)
~_~ ~._.___
=
485
490
495
1465 1470 1475
RAMAN SHIFT (cm"1) Fig. 1. Room temperature Raman spectrum of 6.2 mg of C6o dissolved in 0.8 ml of CS 2. The excitation wavelength is 752.5 rim.
J.D. Lorentzen et a l . / Chemical Physics Letters 270 (1997) 129-134
132 15.0
.
,
.
,
.
,
.
'
'
'
'
'
'
"
'
'
0.6
' ~
,../1oo 0.4
5.0
o.o 0 . 0
0.2
'
'.
01
'
i
0 2
'
i
0 3
'
Cone. (mg C60/ml CS2)
0.4
o.o 0 . 0
0.2
0.4
0.6
0.8
1.0
Conc.(ml CS2/ml C6H6)
Fig. 2. (a) Relative Raman intensities of the 1469 c m - ~ C6o vibration and the 656 c m - ~ vibration in CS z as measured in C6o//CS2 solutions for an excitation wavelength of 752.5 n m . The straight line is a linear fit to the experimental points, which yields a slope of (3.090 5z 0.052) × 10 3 ml C S 2 / m g C6o. (b) Relative Raman intensities of the 656 c m - i vibration in CS 2 and the 992 e m - i vibration in
C6H6 as measured in CS2/C6H 6 solutions for the same excitation wavelength. The straight line is a linear fit to the experimental points, which yields a slope of (0.648 +_0.014) ml C6H6/ml CS2.
investigated in detail by Schomacker et al. [21]. Extrapolating their results to excitation at 752.5 nm, and performing a local field correction (as described in Ref. [21]) to obtain the Raman cross section for an isolated C 6 H6 molecule, we find ((d t r ) / ( d ~ ))992 = (1.30_+ 0.13) X 10 -30 cmE/sr. Inserting this cross section value in Eqs. (3) and (4), we finally obtain for the totally symmetric pentagonal pinch mode of an isolated C60 molecule (dtr)
= (2.09_+ 0 . 2 9 ) x 10-29 c m 2 / s r 1469
(at a L = 752.5 n m ) .
(5)
This result can be combined with Eq. (1) to compute the square of the average Raman tensor:
lasl 2 = (3.21 __+0.45) × 10 -7 c m g / g (at a L = 752.5 n m ) .
(6)
The correctness of our approach can be verified by computing the Raman cross section for the 656 c m - 1 line of CS 2 using Eq. (4). If we apply the local field correction from Ref. [20], we obtain (dtr/d~O)656 = (3.4 + 0.4) X 10 -30 c m 2 / s r (liquid phase). This is in very good agreement with the value ( d o t / d O ) 6 5 6 = 3.7 X 10 -30 cm2/sr, which we obtain from extrapolation to 752.5 nm of earlier cross section measurements by Kato and Takuma
[251.
5. Discussion Giannozzi and Baroni [10] calculated the vibrational frequencies, dynamical electric charges, and Raman intensities for a face-centered cubic C60 lattice. Details of these calculations are given in Ref. [10] For the calculation of Raman intensities, they compute the derivatives of the electric susceptibility relative to the vibrational normal coordinates. These quantities can be related to the derivatives of the molecular polarizability which appear in Eq. (2) by application of the Clausius-Mossotti formula. The relative Raman intensities predicted by Giannozzi and Baroni are in very good agreement [13] with experimental data obtained with IR excitation at 1064 nm [24]. Absolute intensities were not included in Ref. [10] because no experimental data were available at that time. Assuming Eq. (2) to be valid, the method of Ref. [10] yields [otf[ 2 = 7.6 × 10 -8 c m g / g .
(7)
This is in good agreement with the measured average Raman tensor (Eq. (6)), particularly if one considers that the experimental value is likely to be somewhat affected by pre-resonance effects and that the calculated quantity (the polarizability derivative) differs from the corresponding experimental value by a factor of only 2. The Raman cross section can also be predicted from a bond polarizability model [17], which as-
J.D. Lorentzen et al. / Chemical Physics Letters 270 (1997) 129-134
sumes that the molecular polarizability can be written as a sum of individual bond polarizabilities. In the simplest version of such a model, one makes the additional assumption that the components of the bond polarizability tensor are functions of the bond lengths R only. Furthermore, the bond is assumed to have cylindrical symmetry, so that the polarizability tensor has only two independent components, denoted as all(R) and a±(R). Using such a model it is straightforward to write the polarizability derivatives in Eq. (2) as [13]
+ 2ot'± ( B) )Ro( I, B ) "x(llf)~ + [ C~il(B) -- c~'±( B ) ] [/~o~(l, 1
B)Rot3(l, B)
^
- ~I~ ] Ro( I, B ) " x( llI)
+(,
(8) )
× (1~o~(l, B ) x ~ ( l , f ) + Ro~(l, B) X X~(l,f) -
×[Ro(l,B).x(llf)])].
molecular species with the same type of bonds, the hydrocarbon parameters can be used to compute the static dielectric properties and the Raman spectrum of fullerenes. The agreement with experiment is excellent for the static polarizabilities and reasonably good for the Raman spectrum [13]. In the latter case, however, the comparison with experiment has been limited to ratios of bond polarizability parameters due to the lack of experimental cross sections for the Raman-active vibrations in fullerenes. Our cross section measurements for the Ag(2) mode in C60 make it possible to test the transferability of individual polarizability parameters. The second and third terms in Eq. (8) are zero for the totally symmetric Ag(2) vibration of C60 at 1469 cm - I . Therefore, the polarizability derivatives depend only on the parameters [c~il(s)+ 2c~'± (s)] and [all(d) + 2a'± (d)] for single and double bonds, respectively. Using [ otil(s) + 2 c~'. ( s)] = 3.13 × 10-16 cm 2 (Ref. [27]), [all(d) + 2 a ' ± ( d ) ] = 6.50 × 10 -16 cm 2 (Ref. [28]), and the product Ro(l, B). x(llf) from ab initio calculations, we obtain Icrfl 2 = 2.8 × 10 -8 cm4//g,
2/?0~ ( l, B)/~0~ ( l, B) (8)
Here the sum over B extends over all bonds connected to site l. The primes denote radial derivatives and the carets denote dimensionless unit vectors. Ro(I,B) is the equilibrium bond vector from atom l to its neighbor B and x(llf) is the mode eigenvector, obtained from the 3N × 3N eigenvalue problem ( ~ - t o ~ M ) g ( f ) = 0 and subject to the orthonormality condition
~ x~( llf ) x~( llf')mt=- ~fff , lct
where m t is the mass of atom l. The notation used is the same as in Ref. [13], but Eq. (8) corrects for several typographic errors in Eq. (4) of Ref. [13]. Using expressions equivalent to Eq. (8), the bond polarizability parameters for carbon-carbon single and double bonds have been determined for a number of hydrocarbons [26-28]. Assuming the bond polarizabilities to be transferable among different
133
(9)
which is within one order of magnitude of the experimental value. This result is consistent with the partial agreement found earlier between the bond polarizability model predictions and the experimental relative Raman intensities. While the absolute intensities are found here to differ from experiment by a factor of 12, the predicted relative Raman intensities were shown in Ref. [ 13] to differ from experiment by a factor as high as 6. It is interesting to point out that the overall sign of the components of the Raman tensor cannot be determined from light scattering experiments, since the Raman intensities are proportional to the square of these quantities. However, for a given choice of eigenvectors the sign of the Raman tensor components has a clear physical meaning: it represents the change in the electronic polarizability for a displacement of the atoms according to the eigenvector pattern. This sign can be obtained from theoretical calculations of the polarizability derivatives. The sign of the bond polarizability parameters in Refs. [27] and [28] was chosen so that they agree with the sign of the Raman tensor obtained from ab initio
134
J.D. Lorentzen et al.// Chemical Physics Letters 270 (1997) 129-134
calculations for hydrocarbons. When these polarizability parameters are transferred to C60 , we find from Eq.(8) that the average Raman tensor for the Ag(2) mode is negative if the eigenvectors correspond to an expansion of the pentagons. This agrees with the sign we obtain from our own ab initio calculation for C60.
6. Conclusion
We have measured the cross section for inelastic Raman scattering by the pentagonal pinch mode of C6o at 1469 cm -J. Our measured value compares well with a prediction from ab initio calculations using density functional perturbation theory. Our measurements are also consistent with a simple bond polarizability model using parameters transferred from hydrocarbons. This suggests that such a model might be sufficient to predict the observable Raman peaks in higher fullerenes and nanotubes, for which Raman spectroscopy is an important characterization technique.
Acknowledgements This work was partially supported by the US National Science Foundation under Grants DMR9058343 and DMR 9624102. One of us (JM) thanks John B. Page for numerous discussions and constant encouragement.
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