J. Phys. Chem. 1993,97, 13575-13579
13575
Infrared Fingerprints of Nine c 8 2 Isomers: A Semiempirical Prediction Giorgio Orlandi' and Francesco Zerbetto' Dipartimento di Chimica "G. Ciamician", Universitb di Bologna, Via F. Selmi 2, 40126 Bologna, Italy
Patrick W. Fowler' Department of Chemistry, University of Exeter, Exeter EX4 4QD. U.K. Received: July 19, 1993@
We use a combination of the quantum chemical force field for ?r electrons (QCFF/PI) method and the modified neglect of differential overlap-parametrization method 3 (MNDO-PM3) to predict some properties of the nine isomers of CSZthat satisfy the isolated pentagon rule. All the structural parameters are optimized with both methods. The two calculations furnish a coherent picture for the relative stabilities, energy ordering, and energy differences among the nine clusters. A C2 isomer is the most stable while the two C3"isomers are the least stable. The two methods also agree in the energy trends of the HOMOs and LUMOs. The nine c82 molecules have HOMOs at higher energies and LUMOs at lower energies than c 6 0 and c 7 0 . On a qualitative basis, they are therefore better electron donors and better electron acceptors. The HOMO-LUMO energy gap is smaller for the less stable species, with the energy of the LUMO more strongly affected upon changing the isomer. In an effort to contribute to the identification of the different isomers, we predict the infrared spectra of all nine molecules by a combination of the QCFF/PI force fields and MNDO-PM3 derivatives of the dipole moments. The QCFF/PI vibrational frequencies are scaled by analogy with the analysis of the neutron scattering spectrum of c 7 0 . The normal modes are overlapped to the MNDO-PM3 derivatives of the dipole moments to yield the infrared intensities. The infrared fingerprints of the nine molecules are reasonably different both in patterns and in intensities, with the more stable isomers having less intense infrared spectra. This behavior is rationalized by a simple physical model.
I. Introduction The bulk synthesis of carbon clusters of various masses and shapes1 has led to an impressive effort to isolate, characterize, and find possible applications for these systems. In particular, the two most prominent and easily obtained members of this family, namely Cm and c70, are attracting attention. The same reasons that make c 6 0 and c 7 0 intriguing, namely their shapes, their versatility,and their potential for technologicalapplications, should also apply to higher fullerenes. Work on these higher clusters has, however, been slower both because of their scant availability and because of the fact that they are produced as mixtures of isomers. For example, Kikuchi et a1.2 find that (282, the system studied in this work, is formed as a mixture of three major isomers plus at least three others in smaller proportions. This is quite remarkable if one considers that only nine isomers were found3 to satisfy the isolated pentagon rule (IPR)4 which seems to be the prerequisite for stability in these clusters. The number of isomers of c82 found experimentally makes it possibly the worst case among carbon clusters with up to 84 atoms. Experimental studies report only one isomer of c765(out of two IPR possibilities6),three for C7g2(out of five IPR possibilities4), and two for c842'7 (out of 24 IPR possibilities*). also appears to be unique among the higher fullerenes in that all of its IPR isomers belong to a single StoneWales family: by successiveStone-Wales transformationsit is possible to convert any one of the nine to any of the others.4 On the extreme "thermodynamic" hypothesis first put forward for (278 by the UCLA group: this would imply that only one isomer of c 8 2 should be found in the product. If all the isomers had equal stability, then nine isomers would be observed with mole fractions in inverse proportion to the order of their point groups. A further reason to study c8z is its ability to encapsulate metals such as lanthanium: scandium,IO and yttrium;II an ability that may result in materials with strikingly new properties.I2 e Abstract
published in Aduance ACS Absrracts, November 15, 1993.
0022-3654/93/2091- 13515$04.00/0
In this article, we combine two semiempirical methods to optimize the structural parameters and to predict the infrared spectra of the 9 IPR c 8 2 clusters. This work is organized as follows: in the next section,we discuss the merits of the semiempirical procedures adopted here. In section 111, we present the results of the energy calculations. In section IV, we discuss the predicted infrared spectra, and then we draw some conclusions.
11. The Semiempirical Background Our method of choice to treat carbon clusters if the quantum chemical force field for r electrons (QCFF/PI).I3 We have shown that this procedure yields very satisfactory results for both electronicand vibrational states.14J5Perhaps the most noticeable of the results obtained to date was the prediction,14J5with a standard deviation of less than 30 cm-l, of the vibrational frequencies of neutral c 6 0 several years before it was possible to measure them experimentally. The accuracy of the prediction was later substantiated by a comparison with infrared, Raman, inelastic neutron scattering, and electron energy loss experim e n t ~ .More ~ ~ recently, the QCFF/PI force field of c 7 0 was used to assign its inelastic neutron scattering spectrum.16 It was found that excellent agreement between computational theory and experiment was reached by simply scaling down the frequencies above 135 meV (1089 cm-l) through the simple expression = 0 0 + 0.9(w,,, - 0 0 ) (1) with wo = 135 meV. The normal modes themselves were not modified. The successful simulation of the inelastic neutron scattering spectra of c 6 0 and C70 is a strong indication that the QCFF/PI description of the normal modes is correct. In fact, in this type of experiment, the spectral intensity is a direct function of the vibrational deformation. del,@
0 1993 American Chemical Society
Orlandi et al.
13576 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
The situation is rather more complicated for infrared intensities which depend on the derivative of the dipole moment with respect to the normal mode. In nonplanar molecules, such as the spheroidal carbon clusters of this work, one can envisage such a derivative as arising from the mixing of ?r and u electron clouds upon the vibrational deformation. To calculate infrared intensities, a correct description of the vibrational modes and a method able to predict a realistic dipole moment surface are therefore required. The n electron approximation that is at the basis of the QCFF/PI method seems inadequate to be used for the latter purpose. Toavoid theproblem posed by thelackoftheuelectrons, we decided to resort to calculating the derivative of the dipole moment with an all valence electron MNDO procedure.17 Specifically, the parametrization used is the so-called parametrization method 3, PM3.IS MNDO calculations of the infrared spectrum of C70 gave reasonable re~u1ts.l~ However, the major shortcoming of MNDO-type calculations of infrared spectra is not the intensity of the fundamentals but the values of the vibrational frequencies which are subject to errors of up to 200 cm-I. This is, in our opinion, inadequate for any spectroscopic prediction, though it can be useful for analysis and assignment of existing experimental data. Two operative possibilities were available to us, the first was to use the optimized QCFF/PI geometry and calculate the dipole moment derivatives around this point; the second, and more cumbersome, one was to reoptimize the molecular structure with the MNDO-PM3 model and calculate thederivativesof thedipole moment at this point. We decided to reoptimize the nine IPR Cs2 isomers both to gather more information on their relative stabilities and toavoid the possibility that theQCFF/PI-optimized structure could result in a cusp of the potential energy surface a t the MNDO-PM3 level: this unlikely event would have resulted in a totally nonphysical picture with very large derivatives of the dipole moment for small deformations. To make the presentation of the predicted spectra more agreeable to the eye, we arbitrarily assigned a 5-cm-' Lorentzian line width to each fundamental. This line width is certainly well within the accuracy of any quantum chemical method.
Figure 1. Structures,point group symmetries,and numberingof the nine C82 clusterswhich satisfy the isolated pentagon rule.
EoIpis the Hueckel HOMO-LUMO gap in B units; this is added to help the identification of the isomer.
TABLE I: QCFF/PI and MNDO-PM3 Relative Stabilities, E (kcai/mol), HOMO, EH,and LUMO,EL,Energies (ev),
and HOMO-LUMO Energy Gaps, AE (eV), of the Nine Clusters MNDO-PM3
QCFFJPI
111. Relative Stabilities of the Nine IPR Satisfying C82
Clusters QCFF/PI and MNDO-PM3 are twosides of thesemiempirical coin. QCFF/PI considers quantum chemically only ?r electrons and parametrizes the u core in terms of two-, three-, and fourbody interactions, that is bond stretches described as Morse oscillators, in-plane bendings, out of plane torsions, and fouratom pyramidalizationsdescribedas springs ofvarious complexity. MNDO-PM3 is an all valence electron Hamiltonian which considers explicitly all the 2s and 2p electrons of carbon. Both procedures treat the electron-electron repulsion explicitly and parametrize it through functions of the atom-atom distance. Becauseof thedrasticdifferencesof thetwomodels, it isunlikely that an agreement of the two methods in the trend of the relative stabilities of the nine IPR c82 isomers would be due to error cancellation or to chance. It is therefore remarkable in the results summarized in Table I that there are so many similarities between those furnished by the two methods. Even more pleasing is the further agreement with another quantum chemical procedure, a tight binding model.20 Both methods used here find that nine isomers span slightly more than 1 eV in stability and that isomer 3 of Figure 1 is the most stable. This isomer is of C2 symmetry and has the second largest HOMO-LUMO energy gap of the nine clusters. The two procedures differ in the identity of the second most stable species. However, they do find the same four molecules within a range of 6 kcal/mol and also concur in finding that the higher symmetry species with C3uand CzUsymmetry are the least stable.
1 Ct 2 Cs 3 C2 4 Cs 5 Ct 6 Cs
I C30 84 0 9Cb
E
EH
EL
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5.1 6.1 0.0 3.8 6.9 9.1 25.1 23.1 13.0
-1.29 -1.39 -1.42 -1.41 -7.37 -1.35 -1.19 -1.30 -1.29
-2.44 -2.19 -2.33 -2.55 -2.82 -2.91 -3.09 -3.16 -3.08
4.85 5.20 5.09 4.86 4.55 4.38 4.10 4.14 4.21
4.1 5.2 0.0
6.4 11.8 16.6 29.4 35.2 22.1
EH
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AI%
-8.59 -8.69 -8.12 -8.15 -8.12 -8.13 -8.55 -8.68 -8.66
-3.13 -3.49 -3.58 -3.18 4.02 4.18 -4.29 4.40 4.30
4.86 5.20 5.14 4.91 4.70 4.55 4.26 4.28 4.36
This stability ordering, the fact that Kikuchi et a1.2found C2: C Z ~ :production C~~ ratios of 8:l:l (plus a t least three more C, or C2 isomers), and the further report2I that c82 is not formed a t all in certain experimental conditions may indicate that its formation is kinetically driven. This is at odds with c84 which is formed only in the two more stable isomer^.^^^^^ A possible explanation of this behavior can be given in terms of dipole moments: the point group symmetries of the nine c82 isomers allow a nonzero dipole moment. In the MNDO-PM3 calculations these are found to be 0.04 D for isomer 1,0.02 D for isomer 2, 0.03 D for isomer 3, 0.09 D for isomer 4, 0.15 D for isomer 5 , 0.18 D for isomer 6, 0.16 D for isomer 7, 0.24 D for isomer 8, and 0.22 D for isomer 9. One possibility that we present very tentatively is that small differences or fluctuations of the dipole moments interacting with the electric field present in the apparatus in which the clusters are formed can lead to stabilization of a particular isomer or of one isomer with respect to another. This, in turn, would make the relative yields of formation dependent on the method of preparation. Such dependence is observed for c 7 8 . It is also conceivable that dipole moment interactions with
Infrared Fingerprints of Nine Cs2 Isomers the solvent or the substrate used in the process of separation may modify the abundance of an isomer. Notice, however, that stabilization through dipole moment interactions cannot occur for c84 since the more stable species belong to D2 and D u point group symmetries in which no permanent dipole moment can exist. Also of some interest are the energies of the HOMO and the LUMO. The two methods calculate the energies of these two molecular orbitals rather differently,although there is an excellent agreement when the energy trend through the series is considered. We even notice that it is generally though no invariably the case that the more stable molecules have lower HOMO and higher LUMO energies. Also noteworthy is the near coincidence of the HOMO-LUMO energy gaps for the two methods. These gaps are smaller than in c 6 0 (QCFF/PI gives 6.60 eVZ4),c 7 0 (QCFF/ PI gives 5.65 eVZ4),and c 7 6 (QCFF/PI gives 5.19 eVZ5). Note that it is the energy of the LUMO that is changed the most along the series. On the basis of these molecular orbital energies, all the CSZclusters considered in this work seem to be better electron donors and better electron acceptors than the lower fullerenes. The QCFF/PI energy of the HOMO of c 7 0 is -7.65 eV while that of c 6 0 is -8.12 eV.24 They should also have lower electronically excited states and hence enhanced nonlinear optical properties. As pointed out b e f ~ r e in , ~the limit to which the propensity of a fullerene cage to accept metal atoms and form endohedral complexes is driven by electron transfer, the energy of the LUMO can be used to assess which complex will be more easily formed. According to the present calculation, the more promising species for this purpose are those labeled 6-9, with isomer 8 of C3" symmetry the most likely to encapsulate metals. This concurs with a previous study performed at the Hueckel level.3 Another factor that may bear on the ability of C S to~ wrap around metal atoms is the volume of the cavity inside the cluster. We have calculated the volumes of the isomers of CSZthrough a numerical procedure26and found them to be, on average, slightly more than 60% larger than in the case of (260. The variation between the largest and the smallest cavity of the nine isomers was about 3%. One can conclude that steric interactions inside the cavity are unlikely to make one isomer a better host than another.
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13577
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I
I000
800
1
m Figure 3. Infrared spectrum of isomer 2 of Caz, C, symmetry, predicted by the QCFF/PI plus MNDO-PM3 simulation. For details, see the caption of Figure 2. 800
IV. Prediction of the Infrared Spectra of the Nine Cs2 Clusters The calculations presented in this article produced a number of large files. If one considers only the Cartesian coordinates of the nine isomers, they consist of 82 (atoms) X 3 (coordinates) X 9 (isomers) X 2 (computational procedures) = 4428 numbers. This made for a slightly small number of calculated vibrational frequencies, Le., 4320, to which one would have to add the corresponding intensities (actually, some of these intensities are zero by symmetry, although not many). If one considers the possible scaling of the frequencies,or the description of the normal modes, the amount of data becomes overwhelming. It is therefore necessary to extract as much information and condense it in as little space as possible. This was done by generating Figures 2-10 which show our best prediction of the infrared spectra of the nine isomers. Before discussing the simulated spectra, weshould point out that we found in theQCFF/ PI and the MNDO-PM3 normal modes to be rather different and it was impossible to make a one to one correspondencebetween the two sets of vectors. The reader can probably find his or her vantage point for identifying the infrared fingerprints of each and every cluster. In the following, we discuss some of the possibilities that arise from the spectra. There is a feature, common to all the spectra, located slightly above 800 cm-I. This is the first intense band starting from the low-energy end and could be compared with the intensity of bands
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at higher energy such as that found in the 1050-cm-I region. This band can be labeled as weak in isomers 1 and 2, giving a mean to characterize these two isomers. Another band that appears to be useful for labeling purposes is found at 900 cm-I. The ratio of its intensity to that of the 800-cm-' band is close to unity in isomers 8 and 9, and slightly less in isomer 7. Another feature that one may want to consider is the position of the most intense band in the infrared spectrum of each c8Z cluster: for the more stable isomers, the intensity tends to have
Orlandi et al.
13578 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 "O0
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Figure 9. Infrared spectrum of isomer 8 of Csz, C3, symmetry, predicted by the QCFF/PI plus MNDO-PM3 simulation. For details, see the caption of Figure 2.
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Figure 6. Infrared spectrum of isomer 5 of CSZ,C2 symmetry, predicted by the QCFF/PI plus MNDO-PM3 simulation. For details, see the caption of Figure 2. 3500
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its maximum below 1400 cm-1, while for the less stable species, the most intense band is above 1500 cm-I. The best way of comparing the calculated spectrum of a given isomer with those that may be recorded in some future experiment is probably to use intensity ratios between pairs of bands rather than absolute intensities of single bands. An interesting aspect of the simulated spectra is their overall intensity. As a rule of thumb, we find that the more stable the speciesthe less intense its infrared spectrum. This behavior calls for some comment. In fact it can be simply related to the decreasing HOMO-LUMO gap of the less stable species. Let us start from a zero-order picture. The HOMO-LUMO gap can be related to the excitation energy between the ground
Figure 10. Infrared spectrumof isomer 9 of Caz, Cbsymmetry, predicted by the QCFF/PI plus MNDO-PM3 simulation. For details, scc the caption of Figure 2.
electronicstate and the electronically excited states. Each of the electronically excited states has a transition dipole moment from the ground state, @'(ground). Some are zero by symmetry but others are nonzero, and in general they can be written as ( @o0/ r/@i'),wheres = x , y , z . Alltheelectronicstatesformacomplete set, {@io). Let us now introduce the vibrational deformation Q. Its effect is to mix the electronic states of the complete set so that the ground state acquires some excited-state character. We can write this in a simple way:
:
~
Infrared Fingerprints of Nine C82 Isomers where His the molecular Hamiltonian and E electronic energies. In this picture, the derivative of the dipole moment becomes proportional to ( @oo/~/ECi@io).If, in analogy with polyenic one assumes that the quantity ( @ o o / ~ H / a Q / @ iiso ) quite similar for all the isomers, it is readily seen that the overall infrared intensity depends only on the energy denominator. This denominator is, to first approximation,a function of the HOMOLUMO gap. If this picture holds also for other carbon cluster isomers, the infrared intensity could be used to gauge the stability of the different isomers for a fixed value of carbon atoms.
V. Conclusion The relative stability of the nine IPR satisfying clusters has been studied by QCFF/PI and MNDO-PM3. Both models agree with a previous study at a tight binding level that the C2 isomer labeled 3 in Figure 1 is the most stable and the higher symmetry isomers are the least stable. All the species have a nonzero dipole moment which may account for experimental differences in production rates and in abundance. On average, the size of the cavity inside the nine clusters is 60% larger than for c 6 0 with minor differences between the isomers. The qualitative factors bearing on the ability of these molecules to host metals are therefore the energy of the LUMO and the dipole moment: both suggest that isomer 8 is the most likely to wrap around metal atoms. Qualitative arguments also predict that these moleculesshould have lower electronicallyexcited states, lower ionization potentials, and greater electron affinitiesthan CSO,C70, and c76. This should, in turn, make them more easily doped, and hence, they should be better conductors than the smaller fullerenes. However, they cannot be used to dope fullerites consisting mainly of c 6 0 or other lower fullerenes because they would effectively act as traps. It is therefore important to ascertain their absence in electron transport experiments. The simulated infrared spectra are sufficiently distinct to be useful in the isomer assignment when the experimental spectra become available. The overall intensity of the infrared spectra is related to the HOMO-LUMO gap through a simple model. If one takes the HOMO-LUMO gap as a measure of the cluster stability, the model shows that the more unstable the isomer the more intense its infrared spectrum.
Supplementary Material Available: Listings of the QCFF/ PI-optimizedcoordinatesof the nine isomers (9 pages). Ordering information is given on any current masthead page. References and Notes (1) Kraetschmer, W.; Lamb, L. D.; Fostiropoulos, K.;Huffman, D. R. Nature 1990, 347, 354.
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13579 (2) Kikuchi. K.; Nakahara, N.; Wakabayashi, T.;Suzuki,S.; Shiromaru, H.; Miyake, Y.; Saito, K.; Ikemoto, I.; Kainosho, M.; Achiba, Y. Nature 1992, 357, 142. (3) Manolopoulos, D. E.; Fowler, P.W. Chem. Phys. Lett. 1991, 187, 1. (4) Diederich, F.; Whetten, R. L.; Thilgen, C.; Ettl, R.; Chao, I.; Alvarez. M. Science 1992, 254, 1768. Fowler, P. W.;Batten, R. C.; Manolopoulos, D. E. J. Chem. SOC. Faraday Trans. 1991, 87, 3103. Fowler, P. W.; Manolopoulos, D. E.; Ryan, R. P. Carbon 1992, 30, 1235. (5) Ettl, R.; Chao, I.; Diederich, F.; Whetten, R. L. Nature 1991, 353, 149. (6) Manolopoulos, D. E. J . Chem. SOC.Faraday Trans. 1991,87,2861. (7) Diederich, F.;Ettl,R.; Rubin,Y.; Whetten, R. L.;Beck, R.;Alvarez, M.; Sensharma, S.;Wudl, F.; Khemani, K. C.; Koch, A. Science 1991,252, 548. (8) Manolopoulos, D. E.; Fowler, P. W. J . Chem. Phys. 1992,96,7603. (9) Johnson, R.; de Vries, M. S.;Salem, J.; Bethune, D. S.; Yannoni, C. S.Nature 1992,355,239. Chai, Y . ;Guo,T.;Jin, C.; Haufler, R. E.;Chibante, L. P. F.; Fure, J.; Wang, L.; Alford, J. M.; Smalley, R. E. J. Phys. Chem. 1991, 95, 7564. (10) Yannoni, C. S.; Hoinkis, M.; de Vries, M. S.;Bethune, D. S.; Salem, J. R.; Crowder, M. S.; Johnson, R. D. Science 1992, 256, 119. Shinohara,
H.;Sato,H.;Ohkohchi,M.;Ando,Y.;Kodoma,T.;Shida,T.;Kato,T.;Siato, Y. Nature 1992, 357, 52. (1 1) Shinohara, H.; Sato, H.; Saito, Y.; Ohkokchi, M.; Ando, Y. J . Phys. Chem. 1992,96,3571. Weaver, J. H.; Chai, Y.; Krool, G.H.; Jin, C.; Ohno, T. R.; Haufler, R. E.; Guo,T.; Alford, J. M.; Conccicao, J.; Chibante, 0. P. F.; Jain, A.; Palmer, G.;Smalley, R. E. Chem. Phys. Lett. 1992, 190, 460. (12) Laasonen, K.; Andreoni, W.; Parrinello, M.Science 1992,258,1916. (13) Warshel, A.; Karplus, M. J. Am. Chem. Soc. 1972, 94, 5612. (14) Negri, F.; Orlandi, G.;Zerbetto, F. Chem. Phys. Lett. 1988,114.31. (15) Negri, F.; Orlandi, G.;Zerbetto, F. Chem. Phys. Lett. 1992, 190, 174. (16) Negri, F.; Orlandi, G.; Zerbetto, F. J. Am. Chem. SOC.1991, 113, 6037. Christides, C.; Nikolaev, A. V.; Dennis, T.J. S.; Prassides, K.; Negri, F.; Orlandi, G.;Zerbetto, F. J . Phys. Chem. 1993, 97, 3641. (17) Dewar, M. J. S.; Thiel, W. J . Am. Chem. SOC.1977, 99, 4899. (18) Stewart, J. J. P.J. Comput. Chem. 1989, 10, 209. (19) Raghavachari, K.; McMichael Rohlfing, C. J . Phys. Chem. 1991, 95. 5768. (20) Zhang, B. L.; Wang, C. Z.; Ho, K. M.; Xu,C. H.; Chan, C. T.J . Chem. Phys. 1993, 98, 3095. (21) Diederich, F., Whetten, R. L. Acc. Chem. Res. 1992, 25, 119. W.;Taylor,R.;Kroto,H. W.;Walton, (22) Manolopoulos,D.E.;Fowler,P. D. R. W. J. Chem. SOC.Faraday Trans. 1992,88, 31 17. (23) Zhang, B. L.; Wang,C. L.; Ho, K. M. J. Chem. Phys. 1992,96,7183. Raghavachari, K. Chem. Phys. Lett. 1992, 190, 397. (24) Negri, F.; Orlandi, G.;Zerbetto, F. Chem. Phys. Lett. 1992, 189. 495. (25) Orlandi, G.;Zerbetto, F.; Fowler, P. W.; Manolopoulos, D. E. Chem. Phys. Lett. 1993, 208, 441. (26) The numerical procedure to calculate the volume of the clusters was carried out in three steps. The first step defined a cube which contained the cluster. The second filled up the cube was a highly dense three-dimensional grid of points. Finally, a scan through all the points accepted and summed up only the points located inside the cluster. (27) Orlandi, G.; Zerbetto, F.; Zgierski, M. Z. Chem. Reo. 199L91.867.
