© Copyright 1996 by the American Chemical Society
VOLUME 100, NUMBER 12, MARCH 21, 1996
LETTERS Static Third-Order Polarizability Calculations for C60, C70, and C84 Craig E. Moore Space Science Laboratory, George C. Marshall Space Flight Center, National Aeronautics and Space Administration, HuntsVille, Alabama 35812
Beatriz H. Cardelino* Department of Chemistry, Spelman College, Atlanta, Georgia 30314
Xiao-Qian Wang Department of Physics and Center for Theoretical Studies of Physical Systems, Clark Atlanta UniVersity, Atlanta, Georgia 30314 ReceiVed: October 10, 1995X
Valence electron contributions to the static molecular third-order polarizabilities (γ) are calculated for C60, C70, and two stable structures of C84 (D2 and D2d). The method utilized is based on the finite-field approach coupled with semiempirical polarization calculations on all-valence electrons. An increase in the third-order polarizability contributions is observed for molecular structures with a reduction in group symmetry, in agreement with recent experimental observations for these fullerenes. This increase is attributed mainly to the appearance of aromatic structures within the molecules as well as to the increase in molecular volume.
Fullerenes are the subject of intensive research by many groups on several contents. There has also been a great deal of speculation about potential uses for fullerenes and their derivatives, ranging from electrochemistry and electronics (the latter based in part on the superconducting properties of alkalidoped fullerides) to catalysis, hydrogen storage, antiviral agents, and all-optical devices.1 Associated with the delocalized π-conjugated electrons in fullerenes, their large nonlinear optical responses have been the subject of several recent experimental studies.1-10 A wealth of experimental data concerning the thirdorder optical nonlinearities have been accumulated for the fullerenes C60, C70, and C84. The two most abundant fullerenes, C60 and C70, have configurations with Ih and D5h symmetries, respectively.11-14 C84 has been recently synthesized,15,16 and the most stable configurations have been determined to be D2 X
Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-4685$12.00/0
and D2d.1,17 An interesting feature discovered by the experimental study is the increase of the third-order nonlinearity from C60 to C70 and to C84.1 With the advances in the synthesis of macroscopic quantities of fullerenes, a theoretical understanding of the dependence of the nonlinear optical responses on the molecular symmetry and electronic properties is thus of particular interest. Theoretical calculations of third-order nonlinear effects have been carried out using various semiempirical and ab initio methods. π-electron calculations on unsaturated hydrocarbon chains have been used to predict third-order effects.19 Ab initio methods applied to molecules with more than 15 atoms are extremely computer intensive. An alternative method using allvalence electrons and a semiempirical perturbation (sum-overstates approach) has generally been applied to molecules with less than 30 atoms.20,21 In the present work, the procedure © 1996 American Chemical Society
4686 J. Phys. Chem., Vol. 100, No. 12, 1996
Letters
TABLE 1: Nonlinear Polarizability of Carbon Fullerenes: Second-Order Polarizability in Units of 10-51 C m3 V-2, Third-Order Polarizability in Units of 10-61 C m4 V-3
TABLE 2: Third-Order Coefficient (in Units of 10-61 C m3 V-3) in the Polynomial Expansion of Polarization versus Static Electric Field for C60
structure
second-order polarizability
third-order polarizability
polynomial expansion
third-order coefficient
polynomial expansion
third-order coefficient
C60 C70 C84 D2d C84 D2
0.02 ( 0.03 0.27 ( 0.29 0.02 ( 0.01 0.01 ( 0.01
5.09 ( 0.05 11.21 ( 0.03 11.84 ( 0.04 13.09 ( 0.04
4 6 8 10
5.12a 5.07 5.08 5.08
12 14 16
5.07 5.10 5.14
a
developed by Dewar and Stewart22 to calculate linear polarizabilities has been modified to estimate nonlinear terms. An allvalence electron calculation using a semiempirical Hamiltonian (AM1;23 implemented within the MOPAC program24), was used to obtain polarization versus static electric field data. A quasihomogeneous static field was created by means of four collinear charges, placed sufficiently far away from the molecular origin. Following a finite-field approach, the required tensor elements for the hyperpolarizability calculations were obtained using the program HYPER which fits polarization versus static electric field data. The HYPER program was formerly used to calculate second-order polarizabilities.25 The details for calculating thirdorder polarizabilities will be described in a future communication. When a molecule is placed in an electric field, the change in a given component of its polarization can be expressed by the following summation:
Pq ) µq + ∑RqjFj + ∑∑βqjkFjFk + j
j
k
∑j ∑k ∑l γqjklFjFkFl + ...
(1)
where Pq represents the qth component of the polarization, Fj the jth component of the applied electric field, R the polarizability, β the second-order polarizability, γ the third-order polarizability, etc. The orientationally averaged third-order polarizability can be expressed as
γ ) (γxxxx + γyyyy + γzzzz + 2γxxyy + 2γyyzz + 2γxxzz)/5 (2) To reduce and evaluate the numerical instabilities of the calculations of the third-order polarizability terms, the polynomial fittings have been performed under the following conditions: (a) 29 values of polarization versus external field strength for each expansion; (b) a range of field strengths from -0.7 to +0.7 V/Å with the charges that create the field placed no closer than 400 Å away; (c) external field increments of 0.05 V/Å; (d) variable order of polynomial expansions from 4 to 16; (e) no preassumption of Kleinman symmetry involving interchange of the first index (e.g., the terms γiijj and γjjii were calculated from different expansions, the first one from an expansion on the polarization along the i direction and the second from an expansion on the polarization along the j direction). The molecular coordinates for the four structures studied were obtained from first-principle methods using density-functional theory within the local-density approximation.17 Table 1 summarizes the results obtained. The values of second-order polarizabilities and valence contribution to third-order polarizabilities are average values obtained from polynomial expansions of orders 4-16. The details on the calculation of the secondorder polarizabilities using the HYPER program are given in ref 25. The errors reported in these two quantities correspond to half of the range of values obtained for these terms in the expansions of orders 4-16.
Average γ from valence contribution ) 5.09 ( 0.05.
TABLE 3: Third-Order Terms in the Polynomial Expansions of Polarization versus Static Electric Field for C60, Related through Kleinman Symmetry by Interchange of the First Index polynomial expansion 4 6 8 10 12 14 16
third-order tensor elements γxxyy ) γyxxy γxxzz ) γzxxz γyyzz ) γzxxz +1.73 +1.63 +1.67 +1.67 +1.66 +1.67 +1.66 +1.68 +1.66 +1.66 +1.61 +1.66 +1.69 +1.73
+1.63 + 1.73 +1.67 +1.67 +1.66 +1.67 +1.67 +1.67 +1.68 +1.69 +1.62 +1.67 +1.67 +1.69
+1.74 +1.64 +1.68 +1.68 +1.68 +1.69 +1.70 +1.68 +1.68 +1.69 +1.67 +1.68 +1.70 +1.70
Stability of the Calculations. The calculations have three indicators of numerical stability: (1) All of these molecules contain one or more improper rotation axes; thus, their secondorder polarizabilities should be zero. Table 1 shows that the magnitude of these terms is very small and comparable to their error. (2) The errors of the valence contributions to the thirdorder polarizabilities are smaller than 1%. As an example, Table 2 shows the values of the third-order term obtained for C60, for the expansions of orders 4-16. (3) The third-order terms in the polynomial expansions of polarization versus static electric field are reasonably in concord with the Kleinman symmetry involving interchange of the first index. In the calculation of the tensor elements, no Kleinman symmetry involving the first index has been preassumed. For example, the term γxxyy was obtained from an expansion of the polarization along the x direction (with static electric fields along the xy plane), and the term γyyxx was obtained from an expansion of the polarization along the y direction (also with static electric fields along the xy plane). Under the condition of a static field, these two terms must be equal. As seen in Table 3, where the corresponding values for C60 are listed as an example, the deviations from the Kleinman symmetry are within 6%. Valence Contributions to the Third-Order Polarizability. Table 1 shows that C70 and the two stable structures of C84 have valence contributions to the third-order polarizability more than twice that of C60. Table 1 also shows that the valence contribution for the D2 conformation of C84 is about 10% larger than for the D2d conformation. Thus, an increase in valence contribution occurs with symmetry lowering. To understand the origin of the polarization in the four molecular structures studied, it is useful to look at their bond distances. Since fullerenes consist of five- and six-member carbon rings, polarization may occur due to the two possible geometrical structures: conjugation and aromaticity.26 From the standpoint of the valence bond approach, one may consider the single bonds as sharing two electrons and the double bonds as sharing four electrons (two π electrons). To have an aromatic structure, the molecule must contain several contiguous bonds of intermediate distance between a double and a single bond and should satisfy Hu¨ckel’s rule of 4n + 2 (where n is the number of π electrons). In this case, each bond may be considered to be sharing three electrons (one π electron). If
Letters
J. Phys. Chem., Vol. 100, No. 12, 1996 4687
TABLE 4: Single, Aromatic, and Double Bond Distributions (Ls, La, and Ld) in Carbon Fullerenes (All in Units of Å), together with the Corresponding Numbers of Single, Aromatic, and Double Bonds (ns, na, and nd) molecule C60 C70 C84 D2 C84 D2d
Ls
La
Ld
ns
na
nd
N
1.44 1.39 60 0 30 90 1.44-1.46 1.41-1.43 1.39 55 30 20 105 1.43-1.46 1.40-1.42 1.36-1.37 52 64 10 126 1.43-1.46 1.40-1.42 1.36-1.37 52 64 10 126
Figure 1. Bonding patterns for (a) C60, (b) C70, (c) strained aromatic ring in C84, (d) C84 D2d, and (e) C84 D2. Bonds are classified as (1) long single bond (single dashed line); (2) normal single bond (single solid line); (3) “strained” aromatic bond (dash plus solid line); (4) aromatic bond (straight line plus a circle); (5) double bond (two parallel solid lines).
the bonds are classified into double, aromatic, and single bonds as defined above, the following two relations should be satisfied:
4nd + 3na + 2ns ) N
(3)
nd + na + ns ) 3/2M ) 3/2(N/4)
(4)
where nd, na, and ns are the number of double, aromatic and single bonds, respectively, N the total number of valence electrons and M the total number of carbon atoms. Equations 3 and 4 consist of a system of two equations and three unknowns. Thus, to classify the type of bonds, an additional consideration was made: all structures show a definite break between double bonds and the other types of bonds whereas the distinction between aromatic and single bonds is somewhat vague. Table 4 summarizes the information. C60 has only single and double bonds in a ration of 2:1, in accordance with full conjugation. The shared bond between two six-member rings is always a double bond. This pattern is depicted in Figure 1a. Almost one third of C70’s bonds are of an aromatic nature. C70 contains five aromatic six-member rings surrounding its equator (the largest cross-sectional circle), which are separated from each other by the five longest bonds in this molecule. All double and single bonds of this molecule follow the same arrangement as C60 of full conjugation. The five aromatic six-member rings are “conjugated” among themselves and with the rest of the structure through double bonds. Thus, C70 resembles C60 except for the presence of a belt consisting of five “conjugated” aromatic six-member rings (see Figure 1b). The two isomers of C84 present eight aromatic six-member rings connected into groups through single bonds or through “conjugation”. In addition, they have four six-member rings
that contain four intermediate bonds separated into pairs by a double bond and a single bond (“strained” aromatic; Figure 1c). Unlike C70, all double bonds in C84 serve to “conjugate” aromatic rings. The four strained aromatic rings separate groups of “conjugated” aromatic rings. Finally, the two isomers of C84 present a different pattern in the organization of the main groups of “conjugated” aromatic rings. In the D2d isomer (see Figure 1d), there are two groups that resemble tetraphenylethylene, with the two phenyls attached to a single “ethylene” carbon atom bonded together and with the cis phenyls “conjugated” through a double bond. In D2 (see Figure 1e), there are four groups that resemble triphenylethylenes, with four phenyls being shared by two triphenylethylenes. Also in D2, the two phenyls attached to a single “ethylene” carbon atom are bonded together, and the cis phenyls are conjugated through a double bond. Among the four structures studied, C60 is the only one that presents no aromaticity. The inclusion of aromatic sections in the molecule more than doubles the value of their third-order polarizability. With the exclusion of C60, C70 is the only other structure that has substantial “normal” conjugation, i.e., alternation of single and double bonds. In the C84 structures, double bonds always serve as conjugation bridges between aromatic structures. The larger number of aromatic rings that C84 D2d has with respect to C70 (eight versus five) is followed by only a 5% increase in the polarizability. Thus, the presence of aromaticity is an important contribution to high third-order polarizability; however, an increase of aromaticity results in only a slight increase in the property. A second consideration is the overall dimension of the molecules, since electron transfer within a larger volume favors polarizability. The largest interatomic distances in C60, C70, and C84 D2d and C84 D2 are 7.06, 8.21, 8.44, and 8.61 Å, respectively. These values are highly correlated to third-order polarizabilities shown in Table 1 (coefficient of correlation ) 0.998 from a least-squares regression). The relative valence contributions for C70 and C84 D2d and D2 with respect to C60 are 2.2, 2.3, and 2.6, respectively. These values are in good agreement with experimentally observed relative values of γ1111 for C70 and C84,1 which are 2.9 and 3.3, respectively. The calculated ratios follow the general trend of the values. The difference may be due to resonance enhancement1 since the experimental measurements were taken at 647 nm, which is closer to the significant absorptions of C70 and C84 than of C60. In summary, we have calculated, for the first time, the valence contribution to the third-order polarizability of C60, C70, and the two most stable structures of C84. Our results reveal the same trend as experimental determinations of γ1111. A detailed analysis shows that the appearance of aromatic structures, as well as the dimension of the molecules, are important for the third-order nonlinear optical response. The computational method utilized shows good numerical stability and can be applied to other extractable carbon fullerenes. Acknowledgment. This work was partially supported by the NSF under Grant HRD9450386, by the NASA Alliance for Nonlinear Optics (Grant MAGW-4078), by NASA Cooperative Agreement NCC8-71, and by Department of Defense under Contract DAAH04-95-1-0651. References and Notes (1) Sun, F.; Zhang, S.; Xia, Z.; Zou, Y. H.; Chen, X.; Qiang, D.; Zhou, X.; Wu, Y. Phys. ReV. B 1995, 51, 4614. (2) Bezel, I. V.; Chekalin, S. V.; Matveets, Yu. A.; Stepanov, A. G.; Yartsev, A. P.; Letokhov, V. S. Chem. Phys. Lett. 1994, 218, 475.
4688 J. Phys. Chem., Vol. 100, No. 12, 1996 (3) Liu, H.; Taheri, B.; Jia, W. Phys. ReV. B 1994, 49, 10166. (4) Gong, Q.; Yuan, P.; Xia, Z.; Zou, Y. H.; Qiang, D.; Fei, L.; Gu, Z.; Zhou, X.; Chen, H. Proceedings of the First International Symposium on Laser and Optoelectronics Technology & Applications; Chong, T. C., Lu, L. F., Eds.; National University of Singapore: Singapore, 1993; p 368. (5) Rosker, M. J.; Marcy, H. O.; Chang, T. Y.; Khoury, J. T.; Hansen, K.; Whetten, R. L. Chem. Phys. Lett. 1992, 196, 427. (6) Zhang, Z.; Wang, D.; Ye, P.; Li, Y.; Wu, P.; Zhu, D. Opt. Lett. 1992, 17, 973. (7) Gong, Q.; Sun, Y.; Xia, Z.; Zou, Y. H. J. Appl. Phys. 1992, 71, 3025. (8) Kafafi, Z. H.; Bartoli, F. J.; Lindle, J. R.; Pong, R. G. S. Phys. ReV. Lett. 1992, 68, 2705. (9) Kafafi, Z. H.; Lindle, J. R.; Pong, R. G. S.; Bartoli, F. J.; Lingg, L. J.; Milliken, J. Chem. Phys. Lett. 1992, 188, 492. (10) Blau, W. J.; Byrne, H. J.; Cardin, D. J.; Dennis, T. J.; Hare, J. P.; Kroto, H. W.; Taylor, R.; Walton, D. R. M. Phys. ReV. Lett. 1991, 67, 1423. (11) Kratschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Nature (London) 1990, 347, 354. (12) Weaver, J. H.; Martins, J. L.; Komeda, T.; Chen, Y.; Ohno, T. R.; Kroll, G. H.; Troullier, N.; Haufler, R. E.; Smalley, R. E. Phys. ReV. Lett. 1991, 66, 1741. (13) Johnson, R. D.; Meijer, G.; Bethune, D. S. J. Am. Chem. Soc. 1990, 112, 8983. (14) Taylor, R.; Hare, J. R.; Abdule-Sada, A. K.; Kroto, H. J. J. Chem. Soc., Chem. Commun. 1990, 20, 1423.
Letters (15) Diederich, F.; Ettl, R.; Rubin, Y.; Whetten, R. R.; Beck, R.; Alvarez, M.; Anz, S.; Sensharma, D.; Wudl, F.; Khemani, K. C.; Koch, A. Science 1991, 252, 548. (16) Kikuchi, K.; Nakahara, N.; Honda, M.; Suzuki, S.; Saito, K.; Shiromaru, H.; Yamauchi, K.; Ikemoto, I.; Kuramochi, T.; Hino, S.; Achiba, Y. Chem. Lett. 1991, 1607. (17) Wang, X-Q.; Wang, C. Z.; Zhang, B. L.; Ho, K. M. Phys. ReV. Lett. 1992, 69, 69. (18) Prasad, P. N.; Williams, D. J. Introduction to Nonlinear Optical Effects in Molecules and Polymers; Wiley: New York, 1991; p 147. (19) Beratan, D. N.; Onuchic, J. N.; Perry, J. W. J. Phys. Chem. 1987, 91, 2696. (20) Shelton, D. P.; Rice, J. E. Chem. ReV. 1994, 94, 3. (21) Pierce, B. M. J. Chem. Phys. 1989, 91, 791. (22) Dewar, M. J. S.; Stewart, J. J. P. Chem. Phys. Lett. 1984, 111, 416. (23) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902. (24) MOPAC, Quantum Chemistry Program Exchange; No. 455, Version 6.1, 1990. (25) Cardelino, B. H.; Moore, C. E.; Stickel, R. E. J. Phys. Chem. 1991, 95, 8645. (26) Kajzar, F.; Messier, J. Nonlinear Optical Properties on Organic Molecules and Crystals; Chemla, D. S., Zyss, J., Eds.; Academic Press: New York, 1987; Vol. 2, p 51.
JP9530257
