2463
J . Phys. Chem. 1992,96, 2463-2466 A possible mechanism of the electronic charge transfer due to the introduction of alkali metals is illustrated in Figure 8. We have also shown that previous spectroscopic data are satisfactorily reproduced by the MO calculations, and the respective spectral features are well explained in terms of these changes in charge distributions in the Si-0-Si network.
Acknowledgment. We are grateful to the Data Processing Center of Kyoto University for providing the GAUSSIAN 82 program and for the generous permission to use the FACOM M780/30 computer system. We also thank Professor S. Sakka (Kyoto University) and Dr. K. Hirao (Kyoto University) for valuable discussions and useful comments.
Imperfect Fullerene Structures: Isomers of C8,, Krishnan Raghavachari* AT& T Bell Laboratories, Murray Hill, New Jersey 07974
and Celeste McMichael Rohlfing* Sandia National Laboratories, Livermore, California 94551 -0969 (Received: August 15, 1991)
The structures and energies of several alternative isomeric structures of Cmspheroids have been computed with semiempirical (MNDO) and ab initio Hartree-Fock molecular orbital techniques. Unlike the ideal icosahedral structure, these isomers are characterized by the presence of two or more pairs of adjacent pentagonal rings. The energy contributions of these adjacent pentagonal defects are roughly additive, with each defect making the structure less stable by 1 eV. The lowest energy alternative isomer of C60has C2, symmetry and lies -2 eV higher in energy than the icosahedral ground-state structure.
-
Introduction Icosahedral Ca has 12 pentagonal rings and 20 hexagonal rings arranged in a hollow spheroidal geometry.’s2 Its stability arises from the fact that each pentagon in this structure is separated from the remaining pentagons by at least one hexagon. Since 60 atoms are needed to form a closed structure with 12 separated pentagons, the stability of c 6 0 relative to the smaller carbon clusters can be explained in an elegant and simple manner. The icosahedral ( I h )structure (1, Figure 1) is just one isomer which can be formed from 60 carbon atoms. Consideration of alternative possibilities is, however, simplified by the general realization that, at this cluster size, spheroidal structures containing five- and six-membered rings are much more stable than graphitic or other forms.3” Nevertheless, even if we restrict ourselves to only closed spherical geometrical arrangements, many other isomeric structures are Unlike the ideal icosahedral form, these alternative structures have pentagonal rings adjacent to each other (edge-sharing pentagons). This is expected to cause significant local strain which would make all such isomers considerably less stable than the Zh form. However, there has not been any previous quantitative evaluation of the energies of lowenergy alternative structures. In this work, we find that a pair of adjacent pentagonal rings (1) Kroto, H. W.; Heath, J. R.; OBrien, S. C.; Curl, R. F.; Smalley, R.
E. Nature 1985, 318, 162.
(2) Kritschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman. D. R. Nafure 1990, 347, 354. (3) Newton, M. D.; Stanton, R. E. J. A m . Chem. Soc. 1986, 108, 2469. (4) Luthi, H.P.; Almlof, J. Chem. Phys. Len. 1987, 135, 357. (5) Almlof, J.; Luthi, H. P. Supercompuf. Res. Chem. Chem. Eng. ACS Symp. Series 353, 1987, 35. (6) Haufler, R. E.; Chai, Y.;Chibante, L. P. F.; Conceicao, J.; Jin, C.; Wang, L.-S.; Maruyama, S.; Smalley, R. E. Mater. Res. SOC.Symp. Proc., in press. (7) Stone, A. J.; Wales, D. J. Chem. Phys. Leu. 1986, 128, 501. (8) Haymet, A. D. J. J. A m . Chem. SOC.1986, 108, 319. (9) Fowler, P. W.; Woolrich, J. Chem. Phys. Left. 1986, 127, 78. (IO) Schmalz, T. G.; Seitz, W. A,; Klein, D. J.; Hite, G . E. Chem. Phys. Leff.1986, 130, 203. ( 1 1) McKee, M. L.; Herndon, W. C. J . Mol. Strucf. (THEOCHEM) 1987. 153. 75. (12) Schmalz, T. G.; Seitz, W. A.; Klein, D. J.; Hite, G.E. J. A m . Chem. SOC.1988, 110, 1 113. (13) Bakowies, D.; Thiel, W. J. Am. Chem. SOC.1991, 113, 3704. (14) Coulombeau, C.; Rassat, A. J. Chim. Phys. 1991, 88, 173.
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introduces an energy instability of 1 eV. The lowest energy alternative isomer of Cm contains two such pairs of edge-sharing pentagons, and thus lies -2 eV higher in energy than the ground state. We have also studied isomers with three and four pairs of adjacent pentagons, and find that the energy associated with such defects is additive to within -0.1-0.2 eV. Thus, a simple energy parameter can be used to estimate the energies of other isomers containing more of such defects.
Qualitative Considerations The number of possible spheroidal isomers of c 6 0 is obviously very large. However, there is a simple local transformation which connects the ground-state structure with other isomeric forms. This local transformation, first suggested by Stone and Wales; is such that the motion of only two atoms is required to go from one isomer to another. Though restricted to structures containing only pentagonal and hexagonal rings, this transformation provides a simple way of systematically characterizing other higher energy isomers. This local two-atom transformation is schematically shown in Figure 2. Here the local environment of the two atoms (in black) is shown before and after the transformation. Figure 2 illustrates how a defect-free structure (shown at the bottom) is converted to a structure containing two pairs of adjacent pentagons (shown on top). Such a transformation from an icosahedron yields a C , structure containing two pairs of adjacent pentagons. Simple topological arguments prove that it is not possible to get a closed spheroidal isomer containing only one pair of adjacent pentag~ns.’~ Thus,the C , isomer is expected to be the lowest energy alternative (15) In icosahedral C60,each of the 12 pentagons is separated from the remaining pentagons by at least one hexagon. Each atom is at the junction of two hexagons and one pentagon. This ideal geometrical arrangement is disrupted if we have a pair of adjacent pentagons. This puts two atoms at the intersection of two pentagons and one hexagon. In order to have the remaining pentagons isolated from each other, exactly two other atoms have to be at the junction of three hexagons (graphite-like). This can be shown to be topologically impossible. Let us assume that we start from a pair of fused pentagons and attempt to build a cage structure around it without allowing any further pentagons to be adjacent to each other. After adding two layers of atoms around the starting structure, it can be seen (by enumeration of the limited number of possibilities) that there will be at least four atoms which are at the junction of three hexagons. In order to have the remaining atoms satisfy the local bonding requirements, there has to be at least one other pair of fused pentagons.
0 1992 American Chemical Society
2464 The Journal of Physical Chemistry, Vol. 96, No. 6,199'2
Figure 1. The icosahedral C60structure (1) viewed down a twofold axis.
t
,-.
Raghavachari and Rohlfing
Figure 3. T h e C,, isomer of C60(2) viewed down the twofold axis.
-
U
t
c
Figure 4. The local region of interest in the C,, structure 2. HF/3-21G bond lengths are shown in
Figure 2. The local transformation connecting different Cm isomers. The Ih isomer is a t the bottom and the C,, isomer is on top. The two a.toms participating in the transformation a r e shown in black.
isomeric form of Cm based on these qualitative ideas7 It is also possible to arrive at structures containing three or more pairs of adjacent pentagons by a sequence of similar transformations. After the completion of this work, we found the very recent paper by Coulombeau and Rassat.I4 They have generated and qualitatively classified c60 isomers based on the number of such two-atom transformations necessary to reach a given isomer starting from the ground state 1, form. However, to our knowledge, there have not been any previous quantitatiue studies of the relative energies of these alternative isomers of C60. Isomers and Their Energies The isomers of C60have been sorted according t o the number of local transformations necessary to derive them from the ground state 1, ~ t r u c t u r e . ~However, energetically it is much more meaningful to classify them according to the number of pairs of adjacent pentagonal rings. In this paper, we consider isomers containing two, three, and four pairs of adjacent pentagons and show that there is a linear relationship between the number of adjacent pentagons and the associated energy instability. The first isomer we considered is the C,, form (2, Figure 3) containing two pairs of adjacent pentagons. In contrast to the icosahedron 1 which has only 2 structural degrees of freedom, 2 has 46 degrees of freedom. We have performed complete geometry optimizations on 2 with the MNDO semiempirical technique16 (16) Dewar, M. J. S.; Thiel, W. J. A m . Chem. SOC.1977, 99, 4899.
A.
and ab initio Hartree-Fock (HF) theory using a valence-double-f 3-21G basis The C , isomer 2 was found to be less stable than the 1, form 1 by 2.12 eV at the HF/3-21G level and 1.89 eV at the MNDO level. Thus the lowest alternative structure of c 6 0 lies -2 eV higher in energy than 1. MNDO vibrational analysis confirms that 2 is a local minimum. The structural changes in 2 relative to 1 are significant. To illustrate this, we show the local region of interest for the C,, structure 2 in Figure 4. The dominant resonance configuration (in terms of single and double bonds) is shown to contrast to the I h form 1. The optimized 3-21G bond lengths are also given. The first observation is that there is a larger bond length range in 2 relative to 1. Thus, while 1 has two distinct bond lengths (1.37 and 1.45 A at the 3-21G level), 2 has bond lengths ranging from 1.32 to 1.49 A. The short bond length of 1.32 A is between the two atoms in the center of the figure, and corresponds to a nearly typical double bond. Secondly, the bond lengths within the pentagons vary between 1.37 and 1.49 A. This is in contrast to the 1, form where the pentagonal bonds are all equivalent, illustrating clearly that the nature of the dominant resonance configuration between the two structures differs. We have previously considered the strains and hybridizations in fullerene systems by means of the POAV (pi orbital axis vector) analysis, originally proposed by Haddon and Scott.I9 In this scheme, the u-bonds are assumed to lie along the internuclear axes, and orbital orthogonality relationships are used to solve for the (17) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (18) All the calculations reported in this work were performed using GAUSSIAN 90 computer program: Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foreman, J. 9.;Schlegel, H. B.; Raghavachari, K.; Robb, M.; Binkley, J. S . ; Gonzalez, C.; Defrees, D. J.; Fox, D. J.; Whiteside, R. A,; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. Gaussian, Inc., Pittsburgh, PA, 1990. (19) Haddon, R. C.; Scott, L. T. Pure Appl. Chem. 1986, 58, 137.
The Journal of Physical Chemistry, Vol. 96, No. 6, 1992 2465
Imperfect Fullerene Structures: Isomers of C60
TABLE I:
UT
Interorbital Angle for the Ct,Isomer 2"
structure
atom type
CbQ ( C d
c, c2 c3
c r
angle
101.0 102.4 105.0 98.5 102.8 100.8 101.5 102.6
c4
c5 c6
Cl CS
'For the numbering system see Figure 4.
Figure 6. The D2h isomer of
c60
(4) viewed down a twofold axis.
Figure 7. The Du isomer of
c60
(5) viewed down a twofold axis.
Figure 5. The D, isomer of CbQ(3) viewed down a threefold axis.
*-orbital hybridization and direction. In particular, the u-r interorbital angle is a good measure of the local curvature of the spheroid, and has a value of 101.6' for C60(compared to 90' for graphite and 109.5' for diamond).20s2' Consideration of the angle for the different carbon atoms in the C2, form shows a variation ranging from 98.5' to 105.0'. The atoms with the smallest u-?T angle (closest to graphite), labeled 4 in Figure 4, are at the junction of three hexagons. The atoms with the largest u-?T angle, labeled 3 in Figure 4, are situated at the junction of two pentagons and one hexagon. This is consistent with simple ideas of local strain energies at the fused pentagonal positions. The u-r angles at the remaining atoms in Figure 4 are intermediate and are listed in Table I. The atoms in the other hemisphere which are far from the defect have bond lengths and e~ angles resembling those in 1, and are not shown. In a recent paper on non-planar conjugated molecules,22Haddon has noted that the highest u-r angle in any structurally characterized molecule is 103.0' (in 9,9',10,1O'-tetrahydrodianthracene). The u-T angle of 105.0' at the fused pentagonal positions in 2 is significantly larger, suggesting that these carbon atoms should be highly reactive. Although a structure with two pairs of adjacent pentagons as in 2 is unique, several possibilities exist for isomers containing three or more such pairs. An isomer with three pairs of adjacent pentagons is expected to be the next lowest in energy. Here we only consider a structure 3, with D3symmetry, which can be obtained by three successive local transformations starting from the 1, structure l.I4 This form is shown in Figure 5 and has the three local defects distributed around the spheroid uniformly. 3 has 30 structural degrees of freedom which have been completely optimized with the MNDO and the HF/3-21G techniques. 3 is 3.28 eV less stable than 1 at the 3-21G level (2.98 eV with MNDO). MNDO vibrational analysis confirms that 3 is a local minimum. Next we consider isomers with four pairs of adjacent pentagons. Among the different possibilities, we have chosen two high-symmetry forms. The first of these (4, Figure 6) has D2* symmetry ~~~~~
~
~
(20) Haddon, R. C.; Brus, L. E.; Raghavachari, K. Chem. Phys. Left. 1986, 125.459; 1986, 131, 165. (21) Raghavachari, K.; Rohlfing, C. M.J . Phys. Chem. 1991,95, 5768. (22) Haddon, R. C. J . Am. Chem. SOC.1990, 1 1 2 , 3385.
and can be obtained by two consecutive local transformations on opposite sides of l.14It has 24 structural degrees of freedom. The second structure (5, Figure 7) has DU symmetry and can be related to 1 by three successive local tran~formations.'~It has 23 structural degrees of freedom. Both forms have been optimized at the 3-21G level and lie 4.23 and 4.26 eV higher in energy than 1, respectively. The corresponding energy differences a t the MNDO level are 3.74 and 3.88 eV, respectively. MNDO vibrational analysis confirms that both are local minima. As mentioned earlier, it is convenient to classify the different isomers according to the number of pairs of adjacent pentagons. This can be referred to as the defect index of the cluster. The roughly comparable stabilities of 4 and 5 with defect indices of 4 suggest that there is a fairly constant energy associated with the presence of such defects. We can now quantify this idea by considering isomers 2, 3, 4, and 5 with defect indices of 2, 3.4, and 4, respectively. The energy cost per defect can be calculated from the relative energies of each isomer with respect to defect-free 1. At the HF/3-21 G level, we derive values of 1.06, 1.09, I .06, and 1.06 eV for this quantity from the relative energies of the four different isomers. At the MNDO level, the corresponding energies per defect are 0.95, 0.94,0.97, and 0.99 eV, respectively. Thus, each pair of adjacent pentagons makes a given structure less stable by 1 eV, due to the local strain in the spheroid associated with the presence of adjacent pentagonal rings. Highly imperfect Cs0isomers have recently been seen in computer simulations. For example, in a molecular dynamics study of 60 carbon atoms, Chelikowsky23 eventually arrives a t a spheroidal geometry containing many pairs of fused pentagons. Quantitative ideas about the local strain energies in such imperfect structures may be useful in determining their energies relative to the zh form. In the future, we plan to study other possible isomers with different defects such as heptagonal rings, etc. However, since each heptagonal ring also introduces a pentagonal ring to cause closure, such defects are expected to be much less favorable energetically. (23) Chelikowsky, J. R., to be published.
J . Phys. Chem. 1992, 96, 2466-2470
2466
Acknowledgment. We are indebted to R. C. Haddon for many stimulating discussions. This work was supported in part by the US.Department of Energy, Office of Basic Energy Sciences, Division of Materials Science.
Supplementary Material Available: Tables giving the Cartesian coordinates of all the atoms in the four defective isomers 2, 3, 4, and 5 (8 pages). Ordering information is given on any current masthead page.
Internal Rotation Potential Function for Anisole in Solution: A Liquid Crystal NMR Study Giorgio Celebre, Giuseppina De Luca, Marcello Longeri, Dipartimento di Chimica Universitd della Calabria. Arcavacata di Rende 87036, Italy
and James W. Emsley* Department of Chemistry, University of Southampton. Southampton SO9 5NH, U.K. (Received: August 22, 1991)
The 'Hand I3C NMR spectra of ani~ole-methyl-'~Cdissolved in a nematic solvent ZLI 1115 have been analyzed. The set of dipolar couplings obtained is used to test various models for the internal rotations about the ring-0 and O-CH, bonds. It is shown that the data are not consistent with the simplest models for this motion: a 2-fold jump about the ring-0 bond between planar structures or a four-site jump between nonplanar structures, in both cases with a three-site jump model for motion about the O-CH3 bond. Two models are consistent with the data: a continuous potential for rotation through an angle 4 about the ring-0 bond, V ( 4 )= 1/2V2(1 - cos 24) + '/2V4( 1 - cos 44), while retaining the three-site jump motion about the O-CH3 bond or a continuous rotation about both bonds such that the barrier for rotation through I9 about the O-CH, bond is a maximum when 4 = 0' and decreases to zero for 4 = 90'. This concerted motion is described by the potential V(4,e) = 1/zV2(l- cos 24) + '/ZV3,4 cos2 @(l - cos (30 + 24)), and the values obtained for the coefficients V2 and V3,4are in excellent agreement with previously reported molecular orbital calculations. It is concluded that V ( @ ) is closer to reality.
Introduction The shape and height of the potential function governing rotation about the c,@ bond in anisole has attracted considerable interest both theoretically and experimentally. This is in part because substituted anisoles and polymethoxylated aromatics are natural products and are of considerable importance as therapeutic agents but also because this molecule is amenable to detailed study both experimentally and theoretically. Extensive reviews have appeared recently of both the theoretical and experimental studies.'x2 Most of the work on anisole agrees in locating the absolute minimum in the potential such that the heavy atoms are coplanar (4 = 0'; see Figure 1). There is some evidence that there is a subsidiary minimum at 4 = 90°, so that V ( 4 )can be represented as the truncated Fourier series V(4) = V2(1 -COS 24)/2 Vd(1 -COS 44)/2 (1) with V2>> V4. The main evidence for the magnitude of V, comes from ab initio molecular orbital calculations, which refer to isolated molecules. This raises the question of how the potential changes on going to a condensed medium. The barrier to rotation in the solid is higher (-50 kJ mol-') than in the gas phase (8 kJ mol-I), and it might be thought, therefore, that an intermediate value would occur in a liquid. However, a recent study of anisole by molecular mechanics calculations predicts a lower barrier in the liquid than in the gas phase,2 which is in agreement with the conclusion of Schaefer et based on the values of long-range JH,CH, N M R coupling constants, together with the assumption that the V4 term is of appreciable magnitude. We report here a study of internal rotation in anisole in a liquid phase by comparing observed with calculated dipolar couplings,
+
( I ) Schaefer, T.;Sebastian, R. Can. J . Chem. 1989, 67, 1148. (2) Spellmeyer, D. C.; Grootenhuis, P. D. J.; Miller, M. D.; Kuyper, 1 . F.; Kollman, P. A. J. Phys. Chem. 1990, 94, 4483. (3) Schaefer, T.; Laatikainen, R.; Wildman, T. A,; Peeling, J.; Penner, G. H.; Baleja, J.; Marat, K. Can. J . Chem. 1984, 62, 1592. (4) Schaefer, T.;Penner, G. Can. J . Chem. 1988, 66, 1635.
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D,. The Dij are obtained by analyzing the N M R spectrum of anisole dissolved in a liquid crystalline solvent. An analysis of the proton spectrum of anisole dissolved in a liquid crystalline solvent was reported in 1977 by Diehl et al.5 They concluded that the data is consistent with a planar minimum energy structure but that the shape and height of the barrier to rotation could not be determined, except that for a purely 2-fold potential the barrier must be greater than about 25 kJ mol-'. Their model for averaging the dipolar couplings over the rotational motion did not allow for the dependence of the orientational order of the solute on the internal rotation, which is now known to be an important factor. We shall reexamine their data using the additive potential (AP) model for describing the orientational order of flexible molecules,bs which does allow for the dependence of orientational order on molecular shape, but as we shall see, the conclusions reached are essentially the same as those found by Diehl et al. To enlarge the set of experimental D,, we have recorded and analyzed the proton spectrum of a sample of ani~ole-methyI-'~C dissolved in a liquid crystalline (LC) solvent with the prospect that the increase in the number of dipolar couplings will allow a choice to be made between different forms for the potential governing internal rotation. There are two problems associated with applying the LCNMR method to investigate the structure of nonrigid molecules. The fmt concerns the analysis of the spectra, which increase in complexity with the number of interacting nuclei. Computer programs now exist which in principle enable an iterative analysis to be made provided that the starting set of dipolar and scalar couplings and chemical shifts are quite close to the true values. (5) Diehl, P.; Huber, H.; Kunwar, A. C.; Reinhold, M. Org. Magn. Reson. 1911, 6, 374. ( 6 ) Emsley, J. W.; Luckhurst, G. R.; Stockley, C. P. Proc. R. Soc. London 1982, A381, 117. ( 7 ) Celebre, G.; Longeri. M.; Emsley, J. W. J . Chem. Soc., Faraday Trans. 2 1988, 84, 1041. (8) Celebre, G.; de Luca, G.; Longeri, M.; Catalano, D.; Veracini, C. A.; Emsley, J. W. J. Chem. SOC.,Faraday Trans. 1991, 87, 2623.
0 1992 American Chemical Societv
