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Measurement of the second-order molecular hyperpolarizability of fullerene anions by CARS spectroscopy
26 June 1998
Chemical Physics Letters 290 Ž1998. 117–124
Measurement of the second-order molecular hyperpolarizability of fullerene anions by CARS spectroscopy Robert Lascola, John C. Wright
)
Department of Chemistry, 1101 UniÕersity AÕenue, UniÕersity of Wisconsin, Madison, WI 53706, USA Received 20 January 1998; in final form 25 March 1998
Abstract 2y 3y and C 60 by use of We have determined the second-order molecular hyperpolarizability, g , for the fullerene anions C 60 coherent anti-Stokes Raman spectroscopy ŽCARS. vibrational lineshape analysis. We observe values that are only 2–3-times greater than the corresponding value for the singly charged anion Cy 60. The behavior of g with the addition of charge is explained by symmetry considerations. Comparison with similar results for rare earth-containing endohedral fullerenes suggests that increased charge on the cage, and not asymmetric charge distribution due to metal–cage interactions, is the primary reason for increased nonlinearity compared to neutral fullerenes. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The extended conjugation of fullerenes makes them attractive potential materials for nonlinear optical applications. The conjugation is necessary to support the intramolecular charge separation that is a feature of strongly nonlinear organic polymers w1x. In recent years, it has become clear that despite their conjugation, fullerenes must be chemically modified in order to have nonlinearities that are competitive with already-existing polymers. For example, both experimental w2,3x and theoretical w4,5x work on C 60 find values for the second-order molecular hyperpolarizability, g , that are less than 6 = 10y3 5 esu. This value corresponds to a bulk third-order susceptibility Ž x Ž3. . which is several orders of magnitude less than the ; 10y9 cm3rerg that is deemed necessary for serial all-optical processing applications w6x. Other work on higher fullerenes has shown only small increases in the nonlinearity over that of C 60 w7x. The most common modification has involved introducing charge to the fullerene cage, either through charge-transfer complexes or by chemical reduction w8x. When added to C 60 , the charge introduces new electronic states and reduces the symmetry of the molecule so there are more and potentially stronger coherent four-wave mixing pathways that can contribute to the overall susceptibility. We have previously reported that Cy 60 has a hyperpolarizability ) 65 times larger than neutral C 60 . Somewhat smaller increases have also been seen for charge-transfer complexes.
)
Corresponding author. E-mail: [email protected]
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 7 7 - 1
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R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
ny Ž In this Letter we report the measurement of g 1111 for the fullerene anions C 60 n s 2, 3. by the coherent anti-Stokes Raman spectroscopy ŽCARS. lineshape analysis method of Levenson and Bloembergen w8,9x. The values obtained, 4.0 Ž"1.0. and 7.6 Ž"0.5. = 10y3 3 esu, respectively, are only slightly larger than the value obtained for the y1 anion. This increase can be explained by symmetry considerations. This work also has implications for lanthanide-containing endohedral fullerenes, in which trapped trivalent metal ions transfer charge to the surrounding cage. These materials have several interesting optical features. In addition to the effects intrinsic to the addition of charge, interactions between the cage and the metal ionŽs. result in an off-center position for the metal, which further reduces the symmetry of the molecule w10x and could enhance the nonlinearity by the increased transition probability. Also, the rare-earth ions are protected by the cage from the surrounding medium and thus preserve their unique spectral characteristics w11x. Thus, for example, the low-energy electronic states of Er 3q Ž1.55 mm. could be used to enhance the nonlinearity of materials used in fiber-optics applications, which use light near that wavelength. A recent report w12x indicates a x Ž3. for Er2 C 82 that is more than two orders of magnitude greater than that of C 60 , measured at 1064 nm. The work reported here will allow us to comment on the relative importance of the various effects that give rise to the enhancement of fullerene nonlinearity.
2. Experimental CARS spectral lineshapes were measured by probing the sample with two XeCl excimer-pumped dye lasers Žfrequencies v 1 and v 2 . with identical polarizations. The frequency of laser 1 was fixed at 22100 cmy1 , while that of laser 2 was scanned so that the difference v 1 y v 2 ranged from 1000–800 cmy1 . The angle between the dye laser beams Ž; 0.78. was adjusted to maximize the signal due to the tetrahydrofuran ŽTHF. Raman resonance at 914 cmy1 . Details of the experimental apparatus have been described previously w13,14x. Neutral density filters limited the input laser powers to less than 100 mJ in order to prevent sample degradation Žindicated by the generation of bubbles at the focal region of the lasers.. There was no indication of sample degradation under our experimental conditions since there were no spectral changes over the course of the work. Solutions of anionic C 60 in THF were prepared by chemical reduction of neutral C 60 with naphthalate ion w15x under inert atmospheric conditions in a glove box. The solutions were transferred to a modified quartz cuvette with a Teflon stopcock and an extra arm that allowed solvent to be pumped away through a standard Schlenk line to concentrate the sample. The sample pathlength was 0.81 mm and each window was 1.39 mm. The cuvette was positioned relative to the focused laser beams by maximizing the four-wave mixing signal with the laser frequencies set to excite the THF resonance. The total cuvette thickness is smaller than the confocal region of the lasers Ž; 0.6 cm. so the entire cuvette with its windows contributes to the signal. Sample composition was determined by comparing the near-infrared ŽNIR. spectra with published spectra w16,17x. These spectra indicated that the prepared solutions were often mixtures of the y1 and y2 anions, or the y2 and y3 anions, due to the variability of the naphthalate stoichiometry during the synthesis. The relative concentrations of the two anions were determined by fitting the NIR spectra as combinations of the spectra of the pure anions. The information allowed us to subtract the contributions of the unwanted anion from the total 2y fullerene nonlinearity. The four samples that were used for the measurement of the C 60 nonlinearity usually 2y y 2y y contained a mixture of C 60 and C 60. The concentrations of C 60 and C 60 in the four samples were 0.77 and 0.25, 1.2 and 0.40, 1.6 and 0.0, and 2.3 and 0.0 mM. The three samples that were used in the measurement of 3y 3y 2y the C 60 nonlinearity contained C 60 and C 60 concentrations of 0.13 and 0.59, 0.16 and 0.36, and 0.62 and 0.0 mM. A total of 23 and 13 CARS spectra, respectively, were taken for each solution. Several spectra of pure THF solutions in the cuvette were also taken in order to isolate the contributions of the windows to the FWM signal.
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
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3. Results and discussion Details of the CARS lineshape analysis method have been presented previously w8,9x. The excitation lasers produce a sample nonlinear polarization given by P Ž v4 s 2 v 1 y v 2 . s x Ž3. Ž v4 ; v 1 , yv 2 , v 1 . E12 E2) s F Ý Nbg b Ž v 4 ; v 1 , yv 2 , v 1 . E12 E2) ,
Ž 1.
b
where x Ž3. is the bulk third-order nonlinear susceptibility, Ei is the electric field at frequency v i , Nb is the number density of the species b in the sample and F s Ł i Ž n2i q 1.rŽ3. is the field correction factor for the solution with frequency-dependent refractive indices n i . g b is the orientationally averaged second-order molecular hyperpolarizability for species b, and can be expressed as w18x
g i jk l Ž v4 ; v 1 , yv 2 , v 1 . s
4 KIy4 , 1, 2, 3 h3 y
ž
j mgi l m ll m m mk n m ng
Ý l, m , n Ž vg l y v 4 . Ž v mg y v 1 y v 2 . Ž v ng y v 1 . j mgi m m lmg mgk n m ng
Ý Ž v yv . Ž v yv . Ž v qv . mg 4 ng 1 ng 2
m, n
/
.
Ž 2.
Here, K is a degeneracy factor, Iy4 , 1, 2, 3 is an average over all permutations of the laser frequencies, m ljm is the dipole moment between states l and m with polarization j, v l m is the frequency difference between states l and m, and the summations do not include the ground state. The FWM signal in this experiment is produced both by the sample and the cell windows. The relative contributions of the components depend not only on the nonlinearities, but also on the absorbance of the solution Žwhich potentially reduces the effects of the back window. and the phase-matching Žwhich influences the efficiency of the FWM process.. The intensity of the total signal is described by w19x I
A E4 A
2
ey a 4 l s r2xsŽ3. e i f w
eŽ2 i f syD a l s . y 1 2 Ž yD a q iD k s .
q x wŽ3.
sin f w Dkw
2
exp 2i Ž fs q f w . y D a l s q 1 I12 I2
,
Ž 3.
where s and w indicate sample and window contributions, respectively, D a s Ž2 a 1 q a 2 y a 3 .r2, D k s 2 k 1 y k 2 y k 3 , f s D klr2, l is the thickness of the sample or windows and a i and k i are the absorption coefficient and wavevector of field i. The sample contribution can be described as a sum of the vibrational susceptibility Ž x RŽ3. . from the THF ny Ž Ž3. . Raman resonance and the nonresonant electronic contributions from the THF and the C 60 xB :
xsŽ3. s x RŽ3. q x BŽ3. s
AR e i u R
V R y Ž v1 y v 2 . y i GR
q AB eiu B ,
Ž 4.
where A i and u i represent the magnitude and complex phase angle from electronic contributions and V R and G R are the frequency and linewidth of the Raman transition. As in our previous study of the Cy 60 anion, we assume that the frequency dependence of the electronic contributions is negligible over the small range that the lasers are scanned Ži.e., A w is constant.. ny Each of the C 60 spectra were fit individually to Eqs. Ž3. and Ž4. using a Marquardt least-squares regression analysis w20x. Examples of spectra and fits are shown in Fig. 1. To reduce the number of fitting variables, the absorbances, pathlengths and phase-matching parameters were independently determined. x wŽ3. was determined from fits of spectra of pure THF in the cuvette. ŽThe nonlinearities of THF were determined by comparison with 2y 3y benzene, as reported previously w8x.. The values A R rA B , V R and G R were determined for the C 60 and C 60 samples from the individual spectra.
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
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2y 2y Fig. 1. Dotted lines: normalized CARS vibrational spectra of Ža. pure THF, Žb. 1.6 mM C 60 in THF, Žc. 2.3 mM C 60 in THF. Solid lines: 2y results of spectral fits, as discussed in the text. The changes in lineshape with increasing C 60 concentration are consistent with increases in the electronic susceptibility of the solutions.
Simultaneously, Ž u R y u B . is determined from a global fit of the concentration dependence of the nonresonant susceptibility. u B is concentration-dependent and is related to the molecular phase angle u F Žrepresenting the ratio of real to imaginary character for the fullerene nonlinearity.:
u B s tany1
ž
NF Fg F sin u F NF Fg F cos u F q NS Fg S
/
,
Ž 5.
where the subscripts F and S refer to the fullerene and solvent, respectively. The solvent contribution is assumed to be real. u F is related to the total nonlinearity by Rm y Rn y Rs s
(Ž N Fg F
F
2
2
cos u F q NS Fg S . q Ž NF Fg F sin u F . ,
Ž 6.
where R m , R s and R n are, respectively, ratios of the total background nonlinearity, the THF nonlinearity and 2y Ž . Ž the Cy 60 or C 60 nonlinearity as appropriate to the THF Raman nonlinearity i.e., R m s x B rx R , R s s xsrx R 2y y Ž y. and R n s xnrx R .. For solutions containing Cy 60 and C 60 , we determined R n C 60 from the concentration of C 60 y y3 3 and from our previously reported value for g ŽC 60 ., g s 2.4 = 10 esu, which was measured at the same 2y . wavelengths used in these experiments. After the value for g ŽC 60 was determined, we used that value in a 2y 3y similar way to determine that species’ contribution to the nonlinearity in the solutions containing C 60 and C 60 . The mean values for the samples are reported in Table 1. The reported uncertainties reflect the deviations of these parameters over the replicate determinations for each concentration. The resulting concentration dependence of R m is presented in Fig. 2. The molecular hyperpolarizabilities required for the fitting are 4.0Ž"1.0. = 3y 2y 10y3 3 esu for the C 60 and 7.6Ž"0.5. = 10y3 3 esu for C 60 . The magnitudes of g are well determined for the anions. However, we find large uncertainties for u F . This uncertainty reflects an insufficient sensitivity at these small phase angles for determination of the imaginary part of the susceptibility. Note that at the wavelengths used in this experiment, the exciting lasers are partially absorbed, and thus are close to electronic state resonances. We would thus expect a non-zero imaginary component. ny , n s 0–3. Our results are With this work, we have determined the nonlinearities for the series C 60 summarized in Table 2. Two features are immediately apparent: a very large increase in susceptibility Ž) 65 = . upon the initial addition of charge, and a subsequent small increase as further charge is added to the cage Ž; 2–3 = .. These features can be explained in terms of the symmetry loss associated with the addition of charge, as well as the creation of new electronic transitions.
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
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Table 1 ny Averaged fit results for C 60 solutions
uR yu B Ž8. THF
a
1.2 Ž6.3.
V Žcmy1 .
G Žcmy1 .
AR r A B
Rb
913.0 Ž0.8.
6.5 Ž0.2.
11.9 Ž0.1.
0.54 Ž0.03.
2y Ž C 60 mM. 0.77 1.2 1.6 2.3
9 12 16 19
912.6 Ž0.2. 911.6 Ž0.9. 912.7 Ž0.4. 914.1 Ž0.5.
6.0 Ž0.2. 5.7 Ž0.7. 6.6 Ž0.3. 5.6 Ž1.0.
10.2 Ž0.1. 7.9 Ž0.2. 7.4 Ž0.1. 4.2 Ž0.2.
0.59 Ž0.05. c 0.72 Ž0.07. d 0.89 Ž0.06. 1.33 Ž0.11.
3y Ž C 60 mM. 0.135 0.156 0.62
2 3 7
911.8 Ž0.6. 910.9 Ž0.2. 911.6 Ž0.3.
6.5 Ž0.5. 6.8 Ž0.2. 5.6 Ž0.5.
10.8 Ž0.1. 10.6 Ž0.1. 6.1 Ž0.1.
0.60 Ž0.06. e 0.64 Ž0.02. f 0.92 Ž0.04.
a
2y 3y For anions, u R y u B are determined by the global parameter u F Žsee Eq. Ž5... For, C 60 u F s 40"30; for C 60 , u F s15"20. 2y b Rs R s for the pure THF solution, R m for the C 60 solutions. Values indicated by footnotes c–f are corrected for contributions of the secondary anion Ž R n ; see text.. c R n s 0.04. d R n s 0.07. e R n s 0.12. f R n s 0.07.
It is well known that neutral C 60 belongs to the highly symmetric I h point group, and that the ground electronic state belongs to the symmetric A g representation w21x. These factors combine to severely restrict the number of symmetry allowed four-wave mixing pathways, as shown in Fig. 3a. Only coherences which involve states with the symmetries shown will give non-zero contributions in the sum-over-states representation of g Ži.e., Eq. Ž2... However, adding an electron to the lowest unoccupied molecular orbital changes the ground state to the T1u representation and increases the number of allowed pathways, as shown by the bold lines in Fig. 3b. This effect is magnified by the symmetry reduction associated with Jahn–Teller interactions Žindicated by the w x thin lines in Fig. 3b.. In polar media, Cy 60 is thought to have D 2h symmetry 22 , which causes a slight splitting of degenerate states and further relaxes the requirements for allowed transitions. ŽSince the Jahn–Teller interactions are weak, the electronic states still retain considerable I h character. Therefore, in Fig. 3b the original I h symmetry state labels are preserved for simplicity.. There are 116 symmetry allowed pathways for the anion Žof which 100 are introduced by the Jahn–Teller interactions., compared to 3 for the neutral. This ; 39-fold increase does not imply a specific increase of the susceptibility, since it ignores important factors such as the detunings from resonance and actual dipole moments of the newly allowed transitions, as well as interferences Ži.e., positive and negative contributions. that can occur as the individual pathways are added together w23x. Indeed, the presence of strongly allowed transitions in the near-infrared for the y1 anion will greatly enhance the susceptibility, especially at longer wavelengths. But these symmetry changes illustrate how substantially more electronic states are involved in the four-wave mixing process and can lead to the observed large enhancement. Similar effects due to symmetry reduction have been observed in calculations of the third-order susceptibilities of several different isomers of C 78 w24x. These symmetry effects can also explain the relatively small increases associated with subsequent addition of 2y charge. The singlet and triplet states of C 60 are calculated to be very close in energy w25x and to have either D 2h 3y or D 3d symmetry, respectively. The same calculations predict that C 60 has C i symmetry. In terms of the number of symmetry allowed pathways, there will be little change for either anion. Note that parity dependences will still be conserved, limiting the total number of states that can be coupled. Thus there are few additional contributions to the susceptibility associated with newly allowed transitions. In addition, the frequencies of NIR and UV electronic states do not change substantially compared to the laser frequencies of this experiment. Therefore we do not expect to see large changes in the susceptibilities of these anions.
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
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Fig. 2. Anion concentration dependence of the background nonlinearity, for the 2y Žtop. and 3y Žbottom. anions. R m y R s y R n represents corrections for contributions of the THF Ž R s . and other C 60 anions Ž R n . to the background Žsee text..
It is interesting to compare our results with the nonlinearity measured for endohedral fullerenes. Heflin et al. w12x measured x Ž3. for a film of Er2 @C 82 by degenerate four-wave mixing at 1064 nm. To our knowledge this work is the only reported measurement of the third-order nonlinearity of any endohedral fullerene. There are several important differences that prevent a quantitative comparison. The measurements are taken at different wavelengths and thus resonant contributions to the electronic susceptibility are likely to differ in the two cases. In particular, the higher frequency in our experiments can cause an increased contribution from two-photon Table 2 ny Second hyperpolarizabilities Žg . for C 60 , ns 0–3 n
g Ž10y3 3 esu. Ref.
0
1
2
3
F 0.037 w2x
2.4 Ž1.0. w8x
4.0 Ž1.0. this Letter
7.6 Ž0.5. this Letter
All values measured by CARS lineshape analysis; v 1 s 22100 cmy1 . Uncertainties Žin parentheses. represent 2 s .
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
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Ž3. allowed transitions. In addition, our measurement provides the spatially averaged value for x 1111 , whereas their Ž3. measurement reports x 1221. Since both experiments probably include electronic resonances, we cannot invoke Ž3. Ž3. Kleinman symmetry Žwhich predicts x 1111 s 3 x 1221 for a given material w26x. to establish relative values for the two quantities. Finally, the structures of the fullerene cages are different; C 82 , like all higher fullerenes, has a larger nonlinearity than C 60 . Nonetheless, there are several broad characteristics that deserve comment. First, the susceptibility increase observed by Heflin et al. is similar to that observed in our experiment for 3y C 60 . This similarity suggests that the increased charge on the cage may be the dominant factor for the larger endohedral fullerene nonlinearity and other factors such as the asymmetric charge distribution caused by cage–metal interaction may not cause a significantly larger nonlinearity. Therefore, it is not clear that endohedral fullerenes will provide larger nonlinearities than other fullerene materials with excess charge Žsuch as charge-transfer polymers or salts.. Second, Heflin et al. observed the endohedral fullerenes have a negative x Ž3. and our work shows the fullerene anions have a positive x Ž3.. This difference suggests the nonlinearity of charged fullerenes is highly dispersive, and can even approach zero at some frequencies. This effect can be seen from the sum-over-states representation of the nonlinearity ŽEq. Ž2... There are two terms in this expression, the first representing four-wave mixing pathways that involve two-photon transitions to or between excited electronic states and the second which describes pathways involving transitions that return to the ground state w27x. The final value for x Ž3. depends on the interference between these terms. x Ž3. is generally positive when the first term dominates ny and negative when the second term dominates. C 60 and Er2 @C 82 have similar electronic structures because
Fig. 3. Representations of symmetry-allowed FWM pathways for neutral and anionic C 60 . Ža. C 60 has I h symmetry, and the ground electronic state is A g . Since all FWM processes for CARS are parametric, the pathways must terminate in the ground state. For the I h point group, m x, y, z ™ T1u . Žb. Bold lines represent allowed transitions assuming that the anion retains I h symmetry; thin lines represent couplings allowed under D 2h symmetry. For the D 2h point group, m x ™B 3u , m y ™ B 2u , m z ™ B1u . Correlations from I h to D 2h are: A g ™ A g ; T1g , T2g ™B1g qB 2g qB 3g ; Gg ™ A g qB1g qB 2g qB 3g ; H g ™ 2A g qB1g qB 2g qB 3g Žsubstitute u for g for odd parity states.. This analysis assumes that ‘mixed polarization’ excitations are allowed.
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
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there are strong one-photon allowed transitions in both the near-infrared and bluernear-UV regions. In the blue region of the spectrum, the first term is large due to allowed two-photon transitions between excited electronic states. The second term is smaller, since the strong one-photon allowed transition of the NIR transitions are not near resonance, and the total susceptibility is positive. The opposite situation holds in the near-infrared. The first term is smaller because the two-photon transitions are further from resonance but the second term is larger because the one-photon resonance in the NIR is nearer resonance. The net result is a negative x Ž3.. Since the two terms have opposite frequency dependences, it is likely that there is some frequency at which the two terms will destructively interfere and yield a very small susceptibility.
Acknowledgements This research is supported by the National Science Foundation under grant DMR-9632293. The authors acknowledge helpful discussions and assistance from Dr. Wei Zhao, Professor Robert West and Professor Baocheng Han.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x
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Chemical Physics Letters 290 Ž1998. 117–124
Measurement of the second-order molecular hyperpolarizability of fullerene anions by CARS spectroscopy Robert Lascola, John C. Wright
)
Department of Chemistry, 1101 UniÕersity AÕenue, UniÕersity of Wisconsin, Madison, WI 53706, USA Received 20 January 1998; in final form 25 March 1998
Abstract 2y 3y and C 60 by use of We have determined the second-order molecular hyperpolarizability, g , for the fullerene anions C 60 coherent anti-Stokes Raman spectroscopy ŽCARS. vibrational lineshape analysis. We observe values that are only 2–3-times greater than the corresponding value for the singly charged anion Cy 60. The behavior of g with the addition of charge is explained by symmetry considerations. Comparison with similar results for rare earth-containing endohedral fullerenes suggests that increased charge on the cage, and not asymmetric charge distribution due to metal–cage interactions, is the primary reason for increased nonlinearity compared to neutral fullerenes. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The extended conjugation of fullerenes makes them attractive potential materials for nonlinear optical applications. The conjugation is necessary to support the intramolecular charge separation that is a feature of strongly nonlinear organic polymers w1x. In recent years, it has become clear that despite their conjugation, fullerenes must be chemically modified in order to have nonlinearities that are competitive with already-existing polymers. For example, both experimental w2,3x and theoretical w4,5x work on C 60 find values for the second-order molecular hyperpolarizability, g , that are less than 6 = 10y3 5 esu. This value corresponds to a bulk third-order susceptibility Ž x Ž3. . which is several orders of magnitude less than the ; 10y9 cm3rerg that is deemed necessary for serial all-optical processing applications w6x. Other work on higher fullerenes has shown only small increases in the nonlinearity over that of C 60 w7x. The most common modification has involved introducing charge to the fullerene cage, either through charge-transfer complexes or by chemical reduction w8x. When added to C 60 , the charge introduces new electronic states and reduces the symmetry of the molecule so there are more and potentially stronger coherent four-wave mixing pathways that can contribute to the overall susceptibility. We have previously reported that Cy 60 has a hyperpolarizability ) 65 times larger than neutral C 60 . Somewhat smaller increases have also been seen for charge-transfer complexes.
)
Corresponding author. E-mail: [email protected]
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 7 7 - 1
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R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
ny Ž In this Letter we report the measurement of g 1111 for the fullerene anions C 60 n s 2, 3. by the coherent anti-Stokes Raman spectroscopy ŽCARS. lineshape analysis method of Levenson and Bloembergen w8,9x. The values obtained, 4.0 Ž"1.0. and 7.6 Ž"0.5. = 10y3 3 esu, respectively, are only slightly larger than the value obtained for the y1 anion. This increase can be explained by symmetry considerations. This work also has implications for lanthanide-containing endohedral fullerenes, in which trapped trivalent metal ions transfer charge to the surrounding cage. These materials have several interesting optical features. In addition to the effects intrinsic to the addition of charge, interactions between the cage and the metal ionŽs. result in an off-center position for the metal, which further reduces the symmetry of the molecule w10x and could enhance the nonlinearity by the increased transition probability. Also, the rare-earth ions are protected by the cage from the surrounding medium and thus preserve their unique spectral characteristics w11x. Thus, for example, the low-energy electronic states of Er 3q Ž1.55 mm. could be used to enhance the nonlinearity of materials used in fiber-optics applications, which use light near that wavelength. A recent report w12x indicates a x Ž3. for Er2 C 82 that is more than two orders of magnitude greater than that of C 60 , measured at 1064 nm. The work reported here will allow us to comment on the relative importance of the various effects that give rise to the enhancement of fullerene nonlinearity.
2. Experimental CARS spectral lineshapes were measured by probing the sample with two XeCl excimer-pumped dye lasers Žfrequencies v 1 and v 2 . with identical polarizations. The frequency of laser 1 was fixed at 22100 cmy1 , while that of laser 2 was scanned so that the difference v 1 y v 2 ranged from 1000–800 cmy1 . The angle between the dye laser beams Ž; 0.78. was adjusted to maximize the signal due to the tetrahydrofuran ŽTHF. Raman resonance at 914 cmy1 . Details of the experimental apparatus have been described previously w13,14x. Neutral density filters limited the input laser powers to less than 100 mJ in order to prevent sample degradation Žindicated by the generation of bubbles at the focal region of the lasers.. There was no indication of sample degradation under our experimental conditions since there were no spectral changes over the course of the work. Solutions of anionic C 60 in THF were prepared by chemical reduction of neutral C 60 with naphthalate ion w15x under inert atmospheric conditions in a glove box. The solutions were transferred to a modified quartz cuvette with a Teflon stopcock and an extra arm that allowed solvent to be pumped away through a standard Schlenk line to concentrate the sample. The sample pathlength was 0.81 mm and each window was 1.39 mm. The cuvette was positioned relative to the focused laser beams by maximizing the four-wave mixing signal with the laser frequencies set to excite the THF resonance. The total cuvette thickness is smaller than the confocal region of the lasers Ž; 0.6 cm. so the entire cuvette with its windows contributes to the signal. Sample composition was determined by comparing the near-infrared ŽNIR. spectra with published spectra w16,17x. These spectra indicated that the prepared solutions were often mixtures of the y1 and y2 anions, or the y2 and y3 anions, due to the variability of the naphthalate stoichiometry during the synthesis. The relative concentrations of the two anions were determined by fitting the NIR spectra as combinations of the spectra of the pure anions. The information allowed us to subtract the contributions of the unwanted anion from the total 2y fullerene nonlinearity. The four samples that were used for the measurement of the C 60 nonlinearity usually 2y y 2y y contained a mixture of C 60 and C 60. The concentrations of C 60 and C 60 in the four samples were 0.77 and 0.25, 1.2 and 0.40, 1.6 and 0.0, and 2.3 and 0.0 mM. The three samples that were used in the measurement of 3y 3y 2y the C 60 nonlinearity contained C 60 and C 60 concentrations of 0.13 and 0.59, 0.16 and 0.36, and 0.62 and 0.0 mM. A total of 23 and 13 CARS spectra, respectively, were taken for each solution. Several spectra of pure THF solutions in the cuvette were also taken in order to isolate the contributions of the windows to the FWM signal.
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3. Results and discussion Details of the CARS lineshape analysis method have been presented previously w8,9x. The excitation lasers produce a sample nonlinear polarization given by P Ž v4 s 2 v 1 y v 2 . s x Ž3. Ž v4 ; v 1 , yv 2 , v 1 . E12 E2) s F Ý Nbg b Ž v 4 ; v 1 , yv 2 , v 1 . E12 E2) ,
Ž 1.
b
where x Ž3. is the bulk third-order nonlinear susceptibility, Ei is the electric field at frequency v i , Nb is the number density of the species b in the sample and F s Ł i Ž n2i q 1.rŽ3. is the field correction factor for the solution with frequency-dependent refractive indices n i . g b is the orientationally averaged second-order molecular hyperpolarizability for species b, and can be expressed as w18x
g i jk l Ž v4 ; v 1 , yv 2 , v 1 . s
4 KIy4 , 1, 2, 3 h3 y
ž
j mgi l m ll m m mk n m ng
Ý l, m , n Ž vg l y v 4 . Ž v mg y v 1 y v 2 . Ž v ng y v 1 . j mgi m m lmg mgk n m ng
Ý Ž v yv . Ž v yv . Ž v qv . mg 4 ng 1 ng 2
m, n
/
.
Ž 2.
Here, K is a degeneracy factor, Iy4 , 1, 2, 3 is an average over all permutations of the laser frequencies, m ljm is the dipole moment between states l and m with polarization j, v l m is the frequency difference between states l and m, and the summations do not include the ground state. The FWM signal in this experiment is produced both by the sample and the cell windows. The relative contributions of the components depend not only on the nonlinearities, but also on the absorbance of the solution Žwhich potentially reduces the effects of the back window. and the phase-matching Žwhich influences the efficiency of the FWM process.. The intensity of the total signal is described by w19x I
A E4 A
2
ey a 4 l s r2xsŽ3. e i f w
eŽ2 i f syD a l s . y 1 2 Ž yD a q iD k s .
q x wŽ3.
sin f w Dkw
2
exp 2i Ž fs q f w . y D a l s q 1 I12 I2
,
Ž 3.
where s and w indicate sample and window contributions, respectively, D a s Ž2 a 1 q a 2 y a 3 .r2, D k s 2 k 1 y k 2 y k 3 , f s D klr2, l is the thickness of the sample or windows and a i and k i are the absorption coefficient and wavevector of field i. The sample contribution can be described as a sum of the vibrational susceptibility Ž x RŽ3. . from the THF ny Ž Ž3. . Raman resonance and the nonresonant electronic contributions from the THF and the C 60 xB :
xsŽ3. s x RŽ3. q x BŽ3. s
AR e i u R
V R y Ž v1 y v 2 . y i GR
q AB eiu B ,
Ž 4.
where A i and u i represent the magnitude and complex phase angle from electronic contributions and V R and G R are the frequency and linewidth of the Raman transition. As in our previous study of the Cy 60 anion, we assume that the frequency dependence of the electronic contributions is negligible over the small range that the lasers are scanned Ži.e., A w is constant.. ny Each of the C 60 spectra were fit individually to Eqs. Ž3. and Ž4. using a Marquardt least-squares regression analysis w20x. Examples of spectra and fits are shown in Fig. 1. To reduce the number of fitting variables, the absorbances, pathlengths and phase-matching parameters were independently determined. x wŽ3. was determined from fits of spectra of pure THF in the cuvette. ŽThe nonlinearities of THF were determined by comparison with 2y 3y benzene, as reported previously w8x.. The values A R rA B , V R and G R were determined for the C 60 and C 60 samples from the individual spectra.
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
120
2y 2y Fig. 1. Dotted lines: normalized CARS vibrational spectra of Ža. pure THF, Žb. 1.6 mM C 60 in THF, Žc. 2.3 mM C 60 in THF. Solid lines: 2y results of spectral fits, as discussed in the text. The changes in lineshape with increasing C 60 concentration are consistent with increases in the electronic susceptibility of the solutions.
Simultaneously, Ž u R y u B . is determined from a global fit of the concentration dependence of the nonresonant susceptibility. u B is concentration-dependent and is related to the molecular phase angle u F Žrepresenting the ratio of real to imaginary character for the fullerene nonlinearity.:
u B s tany1
ž
NF Fg F sin u F NF Fg F cos u F q NS Fg S
/
,
Ž 5.
where the subscripts F and S refer to the fullerene and solvent, respectively. The solvent contribution is assumed to be real. u F is related to the total nonlinearity by Rm y Rn y Rs s
(Ž N Fg F
F
2
2
cos u F q NS Fg S . q Ž NF Fg F sin u F . ,
Ž 6.
where R m , R s and R n are, respectively, ratios of the total background nonlinearity, the THF nonlinearity and 2y Ž . Ž the Cy 60 or C 60 nonlinearity as appropriate to the THF Raman nonlinearity i.e., R m s x B rx R , R s s xsrx R 2y y Ž y. and R n s xnrx R .. For solutions containing Cy 60 and C 60 , we determined R n C 60 from the concentration of C 60 y y3 3 and from our previously reported value for g ŽC 60 ., g s 2.4 = 10 esu, which was measured at the same 2y . wavelengths used in these experiments. After the value for g ŽC 60 was determined, we used that value in a 2y 3y similar way to determine that species’ contribution to the nonlinearity in the solutions containing C 60 and C 60 . The mean values for the samples are reported in Table 1. The reported uncertainties reflect the deviations of these parameters over the replicate determinations for each concentration. The resulting concentration dependence of R m is presented in Fig. 2. The molecular hyperpolarizabilities required for the fitting are 4.0Ž"1.0. = 3y 2y 10y3 3 esu for the C 60 and 7.6Ž"0.5. = 10y3 3 esu for C 60 . The magnitudes of g are well determined for the anions. However, we find large uncertainties for u F . This uncertainty reflects an insufficient sensitivity at these small phase angles for determination of the imaginary part of the susceptibility. Note that at the wavelengths used in this experiment, the exciting lasers are partially absorbed, and thus are close to electronic state resonances. We would thus expect a non-zero imaginary component. ny , n s 0–3. Our results are With this work, we have determined the nonlinearities for the series C 60 summarized in Table 2. Two features are immediately apparent: a very large increase in susceptibility Ž) 65 = . upon the initial addition of charge, and a subsequent small increase as further charge is added to the cage Ž; 2–3 = .. These features can be explained in terms of the symmetry loss associated with the addition of charge, as well as the creation of new electronic transitions.
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Table 1 ny Averaged fit results for C 60 solutions
uR yu B Ž8. THF
a
1.2 Ž6.3.
V Žcmy1 .
G Žcmy1 .
AR r A B
Rb
913.0 Ž0.8.
6.5 Ž0.2.
11.9 Ž0.1.
0.54 Ž0.03.
2y Ž C 60 mM. 0.77 1.2 1.6 2.3
9 12 16 19
912.6 Ž0.2. 911.6 Ž0.9. 912.7 Ž0.4. 914.1 Ž0.5.
6.0 Ž0.2. 5.7 Ž0.7. 6.6 Ž0.3. 5.6 Ž1.0.
10.2 Ž0.1. 7.9 Ž0.2. 7.4 Ž0.1. 4.2 Ž0.2.
0.59 Ž0.05. c 0.72 Ž0.07. d 0.89 Ž0.06. 1.33 Ž0.11.
3y Ž C 60 mM. 0.135 0.156 0.62
2 3 7
911.8 Ž0.6. 910.9 Ž0.2. 911.6 Ž0.3.
6.5 Ž0.5. 6.8 Ž0.2. 5.6 Ž0.5.
10.8 Ž0.1. 10.6 Ž0.1. 6.1 Ž0.1.
0.60 Ž0.06. e 0.64 Ž0.02. f 0.92 Ž0.04.
a
2y 3y For anions, u R y u B are determined by the global parameter u F Žsee Eq. Ž5... For, C 60 u F s 40"30; for C 60 , u F s15"20. 2y b Rs R s for the pure THF solution, R m for the C 60 solutions. Values indicated by footnotes c–f are corrected for contributions of the secondary anion Ž R n ; see text.. c R n s 0.04. d R n s 0.07. e R n s 0.12. f R n s 0.07.
It is well known that neutral C 60 belongs to the highly symmetric I h point group, and that the ground electronic state belongs to the symmetric A g representation w21x. These factors combine to severely restrict the number of symmetry allowed four-wave mixing pathways, as shown in Fig. 3a. Only coherences which involve states with the symmetries shown will give non-zero contributions in the sum-over-states representation of g Ži.e., Eq. Ž2... However, adding an electron to the lowest unoccupied molecular orbital changes the ground state to the T1u representation and increases the number of allowed pathways, as shown by the bold lines in Fig. 3b. This effect is magnified by the symmetry reduction associated with Jahn–Teller interactions Žindicated by the w x thin lines in Fig. 3b.. In polar media, Cy 60 is thought to have D 2h symmetry 22 , which causes a slight splitting of degenerate states and further relaxes the requirements for allowed transitions. ŽSince the Jahn–Teller interactions are weak, the electronic states still retain considerable I h character. Therefore, in Fig. 3b the original I h symmetry state labels are preserved for simplicity.. There are 116 symmetry allowed pathways for the anion Žof which 100 are introduced by the Jahn–Teller interactions., compared to 3 for the neutral. This ; 39-fold increase does not imply a specific increase of the susceptibility, since it ignores important factors such as the detunings from resonance and actual dipole moments of the newly allowed transitions, as well as interferences Ži.e., positive and negative contributions. that can occur as the individual pathways are added together w23x. Indeed, the presence of strongly allowed transitions in the near-infrared for the y1 anion will greatly enhance the susceptibility, especially at longer wavelengths. But these symmetry changes illustrate how substantially more electronic states are involved in the four-wave mixing process and can lead to the observed large enhancement. Similar effects due to symmetry reduction have been observed in calculations of the third-order susceptibilities of several different isomers of C 78 w24x. These symmetry effects can also explain the relatively small increases associated with subsequent addition of 2y charge. The singlet and triplet states of C 60 are calculated to be very close in energy w25x and to have either D 2h 3y or D 3d symmetry, respectively. The same calculations predict that C 60 has C i symmetry. In terms of the number of symmetry allowed pathways, there will be little change for either anion. Note that parity dependences will still be conserved, limiting the total number of states that can be coupled. Thus there are few additional contributions to the susceptibility associated with newly allowed transitions. In addition, the frequencies of NIR and UV electronic states do not change substantially compared to the laser frequencies of this experiment. Therefore we do not expect to see large changes in the susceptibilities of these anions.
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Fig. 2. Anion concentration dependence of the background nonlinearity, for the 2y Žtop. and 3y Žbottom. anions. R m y R s y R n represents corrections for contributions of the THF Ž R s . and other C 60 anions Ž R n . to the background Žsee text..
It is interesting to compare our results with the nonlinearity measured for endohedral fullerenes. Heflin et al. w12x measured x Ž3. for a film of Er2 @C 82 by degenerate four-wave mixing at 1064 nm. To our knowledge this work is the only reported measurement of the third-order nonlinearity of any endohedral fullerene. There are several important differences that prevent a quantitative comparison. The measurements are taken at different wavelengths and thus resonant contributions to the electronic susceptibility are likely to differ in the two cases. In particular, the higher frequency in our experiments can cause an increased contribution from two-photon Table 2 ny Second hyperpolarizabilities Žg . for C 60 , ns 0–3 n
g Ž10y3 3 esu. Ref.
0
1
2
3
F 0.037 w2x
2.4 Ž1.0. w8x
4.0 Ž1.0. this Letter
7.6 Ž0.5. this Letter
All values measured by CARS lineshape analysis; v 1 s 22100 cmy1 . Uncertainties Žin parentheses. represent 2 s .
R. Lascola, J.C. Wrightr Chemical Physics Letters 290 (1998) 117–124
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Ž3. allowed transitions. In addition, our measurement provides the spatially averaged value for x 1111 , whereas their Ž3. measurement reports x 1221. Since both experiments probably include electronic resonances, we cannot invoke Ž3. Ž3. Kleinman symmetry Žwhich predicts x 1111 s 3 x 1221 for a given material w26x. to establish relative values for the two quantities. Finally, the structures of the fullerene cages are different; C 82 , like all higher fullerenes, has a larger nonlinearity than C 60 . Nonetheless, there are several broad characteristics that deserve comment. First, the susceptibility increase observed by Heflin et al. is similar to that observed in our experiment for 3y C 60 . This similarity suggests that the increased charge on the cage may be the dominant factor for the larger endohedral fullerene nonlinearity and other factors such as the asymmetric charge distribution caused by cage–metal interaction may not cause a significantly larger nonlinearity. Therefore, it is not clear that endohedral fullerenes will provide larger nonlinearities than other fullerene materials with excess charge Žsuch as charge-transfer polymers or salts.. Second, Heflin et al. observed the endohedral fullerenes have a negative x Ž3. and our work shows the fullerene anions have a positive x Ž3.. This difference suggests the nonlinearity of charged fullerenes is highly dispersive, and can even approach zero at some frequencies. This effect can be seen from the sum-over-states representation of the nonlinearity ŽEq. Ž2... There are two terms in this expression, the first representing four-wave mixing pathways that involve two-photon transitions to or between excited electronic states and the second which describes pathways involving transitions that return to the ground state w27x. The final value for x Ž3. depends on the interference between these terms. x Ž3. is generally positive when the first term dominates ny and negative when the second term dominates. C 60 and Er2 @C 82 have similar electronic structures because
Fig. 3. Representations of symmetry-allowed FWM pathways for neutral and anionic C 60 . Ža. C 60 has I h symmetry, and the ground electronic state is A g . Since all FWM processes for CARS are parametric, the pathways must terminate in the ground state. For the I h point group, m x, y, z ™ T1u . Žb. Bold lines represent allowed transitions assuming that the anion retains I h symmetry; thin lines represent couplings allowed under D 2h symmetry. For the D 2h point group, m x ™B 3u , m y ™ B 2u , m z ™ B1u . Correlations from I h to D 2h are: A g ™ A g ; T1g , T2g ™B1g qB 2g qB 3g ; Gg ™ A g qB1g qB 2g qB 3g ; H g ™ 2A g qB1g qB 2g qB 3g Žsubstitute u for g for odd parity states.. This analysis assumes that ‘mixed polarization’ excitations are allowed.
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there are strong one-photon allowed transitions in both the near-infrared and bluernear-UV regions. In the blue region of the spectrum, the first term is large due to allowed two-photon transitions between excited electronic states. The second term is smaller, since the strong one-photon allowed transition of the NIR transitions are not near resonance, and the total susceptibility is positive. The opposite situation holds in the near-infrared. The first term is smaller because the two-photon transitions are further from resonance but the second term is larger because the one-photon resonance in the NIR is nearer resonance. The net result is a negative x Ž3.. Since the two terms have opposite frequency dependences, it is likely that there is some frequency at which the two terms will destructively interfere and yield a very small susceptibility.
Acknowledgements This research is supported by the National Science Foundation under grant DMR-9632293. The authors acknowledge helpful discussions and assistance from Dr. Wei Zhao, Professor Robert West and Professor Baocheng Han.
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