4954
J. Phys. Chem. 1993,97, 4954-4959
How Accurate Are Molecular Mechanics Predictions for Fullerenes? A Benchmark Comparison with Hartree-Foc k Self-consistent Field Results Robert L. Murry, John R. Colt, and Gustavo E. Scuseria*$+ Department of Chemistry and Rice Quantum Institute, Rice University, Houston, Texas 77251- 1892 Received: October 6, 1992
As research into the growing family of fullerene compounds continues to expand, theoreticians making predictions about these large carbon clusters are interested in reliable approaches to reduce the computational expense of calculations. Here we show that an empirical method, molecular mechanics (MM3), can be effectively used to optimize the geometries of fullerenes and consequently reduce the time required for more elaborate quantum mechanical calculations. Equilibrium structures and heats of formation were predicted for 22 fullerenes ranging from C28 to C120 using MM3. The MM3 geometries are found to be in good agreement with those obtained by the minimum basis Hartree-Fock self-consistent field (SCF) method. However, the heats of formation obtained with MM3 and SCF are quite different. At the MM3 optimized geometry, an SCF energy point was calculated for each structure and found to be very close to the fully optimized S C F energy. This procedure yields accurate energy differences between isomers at a fraction of the computational cost. We propose other ways of using MM3 to speed ab initio calculations as well.
Introduction The fullerenes continue to be an extremely active area of research for chemists, physicists, and materials scientists.lV2 Nevertheless, theoretical investigations of the fullerenes are hindered by a common problem: because thermodynamicfactors probably play an important role in fullerene growth, annealing, and destruction, accurate energy estimates of many fullerene isomers are needed for explanations of the mechanisms and details of these processes. Without theories elucidating these processes, it is difficult to design the next generation of experiments to fully exploit the fullerenes’ unique properties. Performing high-level ab initio quantum mechanical calculations on the myriad of interesting fullerene structures would be ideal, but in general it is computationallyinfeasible. The high-quality ab initio methods that include electron correlation effects simply require too much CPU time for investigations to be simultaneously broad in scope and highly accurate. So far, second-order perturbation theory (MP2) is the highest level for which calculations on fullerenes have been reported,3and these have been limited to c60,a uniquely symmetric case. Remarkably, the Hartree-Fock self-consistent field (SCF) method, the starting point for most correlated methods, has been shown to give qualitatively reliable results when applied to f~llerenes.~ However, even with the SCF method, theoretical studiesof the fullerenes have been limited to examining molecules of immediate experimental interest, especially CsOs and Exhaustive ab initio investigations of the isomers of a single-size fullerene to determine the lowest energy structure have been limited to a relatively small number of fullerenes, those for which there are only a few isomers: the small fullerenes up to C3d7 and the isolated-pentagon isomers of c768and C78a9 Although important methodological and computational improvements have aided the application of ab initio methods to the fullerenes, they remain computationally expensive, and this is one of the reasons why empirical and semiempirical methods have been applied to these large systems. One of the first procedures proposed for determining the relative stability of a fullerene was considering the size of its HOMO/LUMO gap, and this gap has been examined for a large number of fullerenes. Unfortunately, however, more detailed calculations have shown that the HOMO/LUMO gap is not always a reliable indicator of ~ t a b i l i t y . ~Other J ~ methods that have been Camille and Henry Dreyfus Teacher-Scholar.
QQ22-3654 193/2O91-4954$O4.OQ/O
used on the fullerenes and their negative curvature analogues include r-system strain arguments,13J4the Tersoff potential,l5J6 tight-binding m o d e l ~ , ~and ~ - Ithe ~ MNDO method.2G22However, none of these methods has been thoroughlybenchmarked against SCF calculationson the fullerenes, whose curvature and peculiar structure could force erroneous extrapolations in the empirical parameters employed by these methods. For instance, the last two methods, while extremely promising, disagree with SCF predictions on the relative energy ordering of the c78 isomer^.^^^^ Nevertheless, fullerenes would seem to be ideal candidates for a well-tested empirical or semiempirical treatment. All the carbons in these structures are sp2carbons, and hence one could expect that an accurate mathematical description of the force field around such carbons should beenough to predict the geometry of the whole fullerene structure. Additionally, carbon is one of the most thoroughly investigated elements, both experimentally and theoretically, and compounds containing networks of sp2 carbons are many and well-studied. It would seem possible to draw upon this great reservoir of data to construct a force field of sufficientqualitytoaccuratelycalculategeometriesand energies for the fullerenes. In this paper we have investigated using the molecular mechanics program MM323-25as a starting point for more sophisticatedcalculationson the fullerenesand compare the results to SCF calculations. We have chosen MM3 for several reasons. It optimizesgeometries of structures with a hundred atoms or so very quickly and, thus, can be used on the numerous structures of theoretical interest. It is based on a large quantity of experimental data, has been used for several years with success. If MM3’s predictions for the fullerenes can be shown to be accurate, it would be an ideal tool for making a dent in the large number of theoretical calculations that we would like to perform while examining the structure and properties of fullerenes. In our view, a procedure to do two things is required of MM3: to calculate reasonable relative energies of large fullerenes and to get good starting geometries to speed SCF optimizations and further investigations. Furthermore, the goals should be achievable at a low computationalcost. For this paper we chose several fullerenes to investigate MM3’s capabilities: the lowest energy isomers of c 2 8 and (230,the six isomers of C32r c60, C70r c72, and C74, the two IPR isomers (that is, isomers satisfying the isolatedpentagon r ~ l e ~of~c76, - ~the~ five ) IPR isomers of c78, and three IPR isomers of Ct20. All structures investigated in this paper 0 1993 American Chemical Society
Molecular Mechanics Predictions for Fullerenes have been optimized in prior studies at the Hartree-Fock selfconsistent field (SCF) STO-3G level of theory.'9.28 We used MM3 to optimize the geometry and calculate the heat of formation for each fullerene and compared those to the results from SCF optimizations. We examined the prospects of using the MM3 geometry as the initial geometry for more elaborate calculationswith SCF energy pointscalculated at MM3 geometries (these energies are hereafter referred to as the S C F a M M 3energies) andcompared thesevaluesto theoptimized energies from the full ab initio SCF optimizations. In this study, we present an overview of MM3's strengths and weaknesses in investigating the wide variety of fullerene compounds.
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 4955 TABLE I: Cbmparison of Experimental Bond Lengths (A) of Cm with hedictioas by MM3, Hartree-Fock SCF,and MP2 Methods bond C=C C-C
MM3 STO-3G
SCF" dz dzP
MP2b T Z P experimentr
1.391 1.452
1.376 1.463
1.368 1.372 1.370 1.406 1.451 1.453 1.448 1.446
av devd 0.008
0.015
0.020 0.017 0.021 0.009
1.401 (10) 1.458 (6)
Reference 5. Reference 3. Reference 33. Experimental error in parentheses. Average deviation from experimental bond lengths.
TABLE II: Comparison of Experimental Bond Lengths (A) of GOwith MM3 and Hartree-Fock SCF Predictions
Computational Details For each MM3 optimization, calculations were done with the MMP3 force field option in place using the MM3-92 program. The MM3 program will optimize even very poor starting geometries of fullerenes; however, to ensure reproducible heats of formation, the MM3 full-matrix optimization method must be run several times (usually two to four) until the total energy and heat of formation converge to about 0.05 kcal/mol. Running the optimization scheme only once does not ensure that themolecules are fully optimized within the MM3 field. For each SCF energy point, the STO-3G basis set,29 the standard minimum basisconsistingof [ b l p ] contracted functions, was used in conjunction with the direct SCF30,31method as implemented in the TURBOMOLE32package. SCF calculations were performed at Rice University on MIPS M-2000, IBM RS/ 6000-550, and IBM RS/6000-220 workstations.
TZP
SCFb bondu
MM3
STO-3G
dz
dzP
experimentC
1 2 3 4 5 6 7 8
1.380 1.392 1.416 1.422 1.448 1.452 1.459 1.465
1.364 1.379 1.414 1.417 1.455 1.461 1.470 1.486
1.357 1.370 1.402 1.413 1.445 1.450 1.455 1.474
1.361 1.375 1.407 1.415 1.446 1.451 1.457 1.475
1.370 (4) 1.380 (4) 1.407 (7) 1.430(4) 1.459 (5) 1.458 (6) 1.460 (4) 1.476 (5)
avdev"
0.009
0.007
0.009
0.007
Bond labels from Figure 1. Reference 6. Reference 34. Experimental error in parentheses. Averagedeviation from experimental bond lengths.
Results and Discussion a. Time Savings. As expected, the MM3 calculations demanded a fraction of the computational time that the complete SCFoptimizations had taken. TheMM3 geometryoptimizations, which were done without imposing symmetry on the molecules (see section d), were very fast, typically 30-45 min for a complete optimization of a large fullerene on our -6 Mflops IBM RS/ 6000-220 workstations. For comparison, a single SCF energy point on the same size molecule would depend heavily on the symmetry imposed but could easily take many hours. Therefore a complete SCF optimization (involving multiple energy point and gradient calculations) frequently takes days or even weeks on these machines. As we will show later in this paper, a single STO-3G/SCF energy point at the MM3 geometry suffices for many of the applications in which we are interested, and the energy obtained by this procedure is very close to the SCF fully optimized energy. Although this requires running both MM3 and an SCF energy point, using this S C F a M M 3 energy saves the cost of performing the full SCF optimization which involves many such energy points and gradient calculations. Even if the ab initio accuracy is needed, using MM3 may be beneficial. Frequently, the initial geometry used for an SCF calculationon a fullereneis an equal-bond-length geometry, where all the bond lengths of the cage are set to some value around 1.42 A, the carbon-carbon bond length in graphite. By starting with an MM3-optimized geometry instead of an equal-bond-length geometry, we found that the number of SCF cycles required for geometry optimization was typically cut in half. Obviously, any means of cutting the required number of SCFoptimization cycles is significant. b. Geometry Comparisons. As mentioned above, one of our primary goals for MM3 was to obtain reasonably good initial geometries for further SCFcalculations. Ideally, we would want the MM3 optimized geometry to be very close to the STO-3G/ SCF geometry and to experimental data where available. In
Figure 1. C ~ bond J labels.
practice, we were pleasantly surprised that the MM3 geometry and SCF predictions for bond lengths were quite similar in most cases. The bond lengths of CSOcalculated by MM3 and ab initio methods,3,5are compared with experiment33in Table I. The MM3 predictions are in good agreement with experiment and are remarkably close to the MP2 results.3 MM3 also agrees well with SCF calculations using several basis sets.5 For instance, the average deviation between the MM3 predicted bond lengths and the STO-3G/SCF values is 0.013 A. Table I1 shows the comparison betwe& the MM3 predicted bond lengths for C70 (see Figure 1 for bond labels), those predicted by SCF with various basis sets,6 and e~periment.3~ Again the
4956 The Journal of Physical Chemistry, Vol. 97, No. 19, 1993
TABLE III: Com risoa of MM3 Calculated Bond Lengths
(A)for C78 (ahIIrwith STO-3G/SCF Predictions bonda
MM3
STO-3Gb
Arc
1 9 13 3 6 8 10 2 12 5 11 7 4
1.360 1.317 1.393 1.409 1.413 1.437 1.443 1.444 1.444 1.457 1.457 1.459 1.465
1.341 1.361 1.380 1.397 1.405 1.443 1.454 1.452 1.450 1.476 1.465 1.465 1.482
0.019 0.016 0.013 0.012 0.008 -0.006 -0.01 1 -0.008 -0.006 -0.019 -0.008 -0.006 -0.017
av dev
0.01 1
Bond labels from Figure 2. Reference 9. Difference between columns 2 and 3.
Figure 2. C78 (D3h 11) bond labels.
agreement is good between MM3 and experiment. MM3 deviates by an average of 0.009 A from experiment, while the best SCF calculation to date using the dzP basis set6deviates by an average of 0.007 A. MM3 performs well relative to STO-3G/SCF, for the average deviation between the two methods is 0.01 1 A. For comparison purposes, Froimowitz,’* who used MM2-89 (a previous version of the molecular mechanics force field) to calculate the equilibrium geometries of c 6 0 and C70,obtained deviations,in the bond lengths from experiment of 0.01 1 and 0.015 A, respectively. Evidently, MM3 predictsequilibrium bond lengths for these molecules significantly better than MM2. In order to show in greater detail the results of an MM3 optimization, we present a comparison of the MM3 and STO3G/SCF equilibrium bond lengths of the DU(II) isomer of C789 in Table 111. The bond labels for this isomer are in Figure 2. The results are typical of MM3 predictions on IPR fullerenes. The MM3 calculated bond lengths are in nearly the same order as STO-3G/SCF predictions, and overall, the correlation between the two methods is fairly good. The differences between the MM3 and STO-3G/SCF bond lengths presented in the last column of Table I11 exhibit a definite trend: MM3 predicts the short bonds (by SCF theory) to be a little too long, and it predicts the long bonds to be a little too short. That is, MM3 predicts the bond lengths closer to the graphitic value of 1.42 A than STO-3G/SCF does. MM3 calculates the equilibrium values of
Murry et al. these bond lengths from the bond orders, which come from a variable electronegativity self-consistent field (VESCF) calculation;25 MM3 appears to be predicting less deviation from the average (and graphitic) bond order of 1.5 and consequently predicts a smaller range of bond length deviations from 1.42 A. This trend was present to some extent in all fullerenes examined. The average deviations between the MM3 and the SCF geometries for all symmetry-distinctbond lengthsof the fullerenes investigated in this work are shown in the last column of Table IV. These differences are relatively small, showing good agreement between the two methods. It is important to note that MM3 agrees with STO-3G/SCF predictions better for the lessstrained IPR fullerenes. There is an additional convenience in using MM3 on the fullerenes. The final geometry that MM3 predicts does not depend on the quality of the initial geometry if the minimization scheme described above is used. As long as the connectivity list (MM3’s way of knowing which carbon is connected to which) is correct, MM3 quickly brings a poor guess at the initial geometry to a much better one. Hence, unlike more computationally expensive methods where reducing the number of optimizationcycles greatly speeds calculations, using MM3 does not require accurate initial coordinates before beginning geometry optimizations. c. Energy Comparisons. Table IV also presents the MM3 heats of formation and those predicted by STO-3G/SCF.36 In this table, an SCF heat of formation indicates the difference in energy between the fullerene isomer and an ‘idealized” graphitic sheet with an equal number of carbons. To quantify this graphitic sheet, different sizes of monolayer graphite were optimized using SCF, and from these a SCF total energy per carbon of graphite was e ~ t r a p o l a t e da, ~procedure ~ originally proposed by Almi3f and Liithi.3’ Predicting energies for the fullerenes (and, indeed, interpreting those predictions) is a difficult task. Experimental values for the heats of formation and other thermodynamic propertiesarescarce for the fullerenes. Only for c 6 0 has the experimental heat of formation been measured, and three groups have reported a range of values: 581,601, and 635 kcal/m01.~~Note that these values must be compared with theory only in the gas phase, so the heat of formation of C ~(56 O kcal/mol) has been added to the first two experimental numbers (which were reported in the crystalline solid phase) according to the procedure outlined in ref 38c. Previous SCF calculations by S ~ h u l m a npredicted 3~ the c 6 0 heat of formation to be 672, while the STO-3G/SCF value from Table I11 is 625.0 and that predicted by MM3 is 573.1 kcal/mol. One would not normally expect the minimum STO-3G basis set to predict accurate heats of formation at the SCF level of theory, and it is well-known that STO-3G calculations poorly predict strain energies of carbon compounds.40 Evidently, there is a fortuitous cancellation of errors between basis set and correlation effects, at least for C60, and hopefully for the other fullerenes as well. Nevertheless, the STO-3G/SCF numbers are the only ab initio predictions that are currently available for a large number of fullerenes,and it is for this reason that we compare them to MM3. On the other hand, MM3’s sp2carbon potential was developed for small, relatively flat ?r systems and might not be appropriate for accurate energy predictions of fullerenes. Both methods should be viewed suspiciously, at least until further experimental data are available. It is interesting to note that the MM3 and STO-3G/SCF calculations do not agree even qualitatively for the heats of formation over the wide range of fullerenes examined in this work. There is, however, a reasonable explanation of the differences between the SCF heats of formation and those calculated by MM3. One of the most important factors affecting the SCF energy of a fullerene is the strain of the cage; there seems to be a direct correlationbetween the curvature of a fullerene cage and the energy per atom of that structure. It appears that
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 4957
Molecular Mechanics Predictions for Fullerenes
TABLE IV A Comparison of Fullerene Heats of Formation, Total Energies, and Bond Lengths As Predicted by MM3 and the STO-X/SCF Method. AHr fullerene c 2 8 (Td)
1) C x (D3) c 3 2 (c2 1) C32 (02) c 3 2 (c2 11) c 3 2 (D3h) c 3 0 (C2c
(cr)
c 3 2 c 6 0 (Ih)
c70 c 7 2 (D6d) ci4 (D3d C7h (02) C76 (D2d) C78 (C2c 1) C78 (C2r 11) C78 (03) c7w (D3h 1) c7x (D3h 11) c 1 2 0 (Td) c I 20 (&d) CI 20 (Dsd-vf)
total energy
MM3 (kcal/ mol)
SCFb (kcal/mol)
SCF@MM3c (hartree)
S C F optimized (hartree)
457.0 459.3 470.1 473.6 475.6 491.4 484.9 503.5 573.1 639.6 665.2 695.1 683.2 707.4 693.9 695.0 697.1 697.6 701.3 900.3 920.5 959.6
820.5 772.1 750.1 783.7 827.3 833.9 846.2 879.8 625.0 666.8 710.5 739.3 702.6 748.0 704.1 705.8 712.5 714.8 721.3 769.8 801.5 877.5
-1 046.4058 -1121.3292 -1 196.2240 -1 196.1552 -1 196.0866 -1 196.0755 -1 196.046 1 -1 196.0218 -2244.21 16 -261 8.3467 -2693.1060 -2767.9 186 -2842.8064 -2842.7312 -29 17.6429 -291 7.6401 -2917.6298 -2917.6254 -291 7.61 57 -4489.1897 -4489.1 327 -4489.0241
-1046.4605 -1121.3781 -1 196.2538 -1 196.2002 -1 196.1308 -1 196.1203 -1 196.1006 -1 196.0471 -2244.221 2 -2618.3574 -2693.1283 -2767.9300 -2842.8220 -2842.7497 -2917.6603 -2917.6575 -29 17.6469 -29 17.6432 -29 17.6329 -4489.2075 -4489.157 1 -4489.0360
AEd
(hartree) 0.0547 0.0489 0.0298 0.0450 0.0442 0.0448 0.0545 0.0253 0.0096 0.0107 0.0223 0.01 14 0.01 56 0.0185 0.0174 0.0174 0.0171 0.0178 0.0172 0.0178 0.0244 0.01 19
AF (4 0.020 0.029 0.019 0.026 0.024 0.027 0.032 0.0 18 0.013 0.01 1 0.016 0.009 0.01 1 0.012 0.01 1 0.012 0.012 0.01 1 0.01 1 0.0 12 0.01 1 0.0 12
STO-3G/SCF results for fullerenes at optimized geometries obtained from refs 7-1 2 and 24. Reference 34. STO-3G/SCF energy point at MM3 optimized geometry (this work). Difference between columns 4 and 5. e Average deviation of symmetry-distinct bond lengths between MM3 and STO-3G/SCF equilibrium geometries.
for MM3, while this strain is important, other factors may play a larger role in the calculation of the total energy. Thus for the small fullerenes (CZS-C~~), where the strain is very great and outweighs other factors affecting the energy, SCF predicts high heats of formation relative to MM3 because SCF penalizes this strain very heavily. For the larger and less-strained fullerenes, where perhaps other factors become more important, MM3's heats of formation are higher than the SCF values, which are more dependent on the (now lower) strain. Further evidence of these differing treatments of strain can be seen by comparing the isomers of a single fullerene. The three C120isomers investigated are an example, and they are shown in Figure 3. Both methods predict the spheroidal (i.e. lowest strain) molecule as the most stable, and the two remaining isomers in order of increasingstrain. The differences between the isomers, however are crucial. STO3G/SCF estimates the most strained structure, the Dspvf, to be over 100 kcal/mol higher in energy than the least strained structure. MM3, predicting the same effects of strain, calculates the differences between isomers at 59 kcal/mol. Thus, SCF predictions for heats of formations show a strong correlation to the amount of strain in the molecule, while MM3 predictions do not show as strong a dependence. This difference has several consequences for using the MM3 heats of formation to compare the fullerenes. While general trends in per carbon energy will be predicted similarly to SCF theory, MM3 will not penalize a highly strained structure (for instance, a small isomer or a non-IPR cage) as much as SCF would. Hence, if one were looking at the annealing of CSO,MM3 would predict the 60-atom isomers with adjacent pentagons (sources of great local strain) much closer in energy to buckminsterfullerene than SCF would. Furthermore, when investigating the IPR isomers of a given size fullerene, the differences in relative energy between isomers that MM3 predicts will be smaller than STO-3G/SCF predictions, and the total difference between highest and lowest energy isomers will also be less. In consequence, using the MM3 heats of formation to compare fullerenes should be done with these considerations in mind. There is, however, a method which uses MM3 and predicts the total energy close to the STO-3G/SCF values. Encouraged by the similarity between the MM3 and SCF geometries, we ran an
Figure 3. Three isomers of C ~ Z O .
STO-3G/SCF energy point at the MM3 optimized geometry (denoted as S C F a M M 3 energy). Results comparing the fully optimized SCF total energy with that of the S C F a M M 3 procedure are included in Table IV. Particularly encouraging is the column showing the differences between the S C F a MM3 energy point and the SCF fully optimized energy. The differences aresmal1,andfor thelargeIPRfullerenes theSCFaMM3energy consistently predicts the energy about 0.016 h 0.005 hartree (10 f 3 kcal/mol) higher than the SCF fully optimized energy. Thus, this S C F a M M 3 energy can be used to estimate the STO-3G/ SCF fully optimized energy and is ideal for applications where
Murry et al.
4958 The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 900-
m-
700
-
600-
500-
Figure 4. Comparison of heats of formation of the C32 isomers as predicted by STO-3G/SCF and MM3. -1196.0
1
Figure 5. Comparison of total energies of the Cj2 isomers as predicted by STO-3G/SCF and SCF@MM3.
more accuracy than an empirical method is desired, but full SCF calculations are too expensive. d. Isomer Resolution. If SCF optimized energies can be estimated accurately with an energy point at the MM3 geometry, one may examine the isomers of a fullerene at a fraction of the SCF computational cost. We have done such studies on C32 and C78 and compared then with SCF predictions. Energy data are again included in Table IV. To better understand the results, a comparison of heats of formation of the C ~ isomers Z calculated by MM3 and SCF is presented in Figure 4. MM3 and STO-3G/SCF predict very different heats of formation for these structures because of the reasons outline above. Notice, for instance, the wider spread of the SCF heats of formations compared to the MM3 values. Both methods agree on the three lowest and the highest energy isomers, but MM3 heats of formation predict two of the middle isomers in a different order than SCF. A comparison of the S C F a M M 3 energy with the SCF fully optimized energy is shown in Figure 5 , and this correlation is much better. Here MM3 has predicted the geometries close to the SCF geometries, and thus the S C F a M M 3 energies predict the isomers in the same order as SCF fully optimized energies. The relative spacing between isomers is also predicted fairly well by the S C F a M M 3 energy. Figures 6 and 7 show corresponding results for C78. For these IPR isomers, the results are exceptionallygood. The MM3 heats of formation in Figure 6 are again at a different energy range than the SCF heats of formation, but the relative energy levels look almost the same. Figure 7 shows a near perfect correlation between the energies of the two methods. Such a diagram of S C F a MM3 geometries yields the same predictions about the
690
J
Figure 6. Comparison of heats of formation of the predicted by STO-3G/SCF and MM3. -2917.61
-2917.62
1-
-2917.63
-
-2917.64
-
-2917.65
-
-2917.66
-
c 7 8
IPR isomers as
-2917.67
Figure 7. Comparison of total energies of the C78 IPR isomers as predicted by STO-3G/SCF and S C F a M M 3 .
relative energies of the five IPR isomers of C78 as the STO3G/SCF ab initio method. Thus it appears that while the MM3 heats of formation can very roughly approximate the SCFvalues, using an SCF energy point at the MM3 geometry is an effective approach to predict the order and relative energies of IPR fullerenes accurately. A further advantage of empirical programs like MM3 is that they do not require the use of symmetry to lower the computational cost of calculations. This led to an interesting result for the C32 (Ci) isomer. Although the equal-bond-length model has D j d symmetry, MM3 predicted a Ci symmetry for the molecule. We optimized the isomer in D j d and in Ci with STO-3G/SCF, and this symmetry lowering decreased the energy of the isomer by 16.0 kcal/mol.
Conclusions While MM3 fils a valuable rolein certain kinds of examinations of fullerenes, there are problems which do not lend themselves easily to a treatment with MM3. Metallofullerenes,for example, cannot currently be calculated using MM3 because it requires a well-defined connectivity between all atoms involved. MM3 cannot predict to which carbon(s) an endohedral metal atom is bonded, because it actually needs that information as input for the program. Additionally, most of the metals experimentally observed in endohedral fullerenes are relatively rare, and MM3's parameter list does not include them because of lack of experimental data for fitting. Transition states also cannot be calculated with MM3 because they are saddle points in the
Molecular Mechanics Predictions for Fullerenes potential energy surface, and MM3 is equipped to model local minima. But the number of fullerene structures for which MM3 is suitable is considerable. In summary, we have shown that MM3, because of its speed and accuracy, can be a valuable tool for optimizing the geometries of fullerenes and for speeding ab initio calculations. The MM3 geometries compare well with minimum basis set SCF optimized geometries and with experiment (when available). An MM3 geometry alsooffers a better starting point for further calculations than does an equal-bond-length geometry. MM3 might prove valuable in searching for fullerenes which break symmetry as well. The MM3 predicted heats of formation do not agree with the SCFpredictions, but there is not enoughexperimentalevidence to firmly support either method for the fullerenes. Finally, a single SCF energy point at the MM3 geometry is very close to the SCF fully optimized energy and can be used in place of the more computationally expensivevalue in many applications. The S C F a M M 3 energy agrees with SCF predictions even better for large, IPR fullerenes, where an inexpensive computational technique is needed most. Thus, using MM3 on the fullerenes might open calculational avenues that were previously too expensive to pursue.
Acknowledgment. Acknowledgment is made to the Petroleum Research Fund, administered by the American Chemical Society, for the support of this research. We thank Dr. Jenn-Huei Lii and Dr. N. L. Allinger for their help in using the MM3 program. References and Notes (1) See, for example, the March special issue on fullerenes of Acc. Chem. Res. 1992, 25. (2) Kroto, H. W.; Allaf, A. W.; Balm, S . P. Chem. Rev. 1991,91,1213. (3) HPser, M.; Almldf, J.; Scuseria, G. E. Chem. Phys. Lett. 1991,181, 497. (4) Scuseria, G. E. In Buckminsterfullerenes; Billups, W. E., Ciufolini, M. A., Eds.; VCH: New York, in press. (5) Scuseria, G. E. Chem. Phys. Lett. 1991, 176, 423. (6) Scuseria, G. E. Chem. Phys. Letr. 1991, 180, 451. (7) Delabroy, L. P.; Scuseria, G. E. To be submitted for publication. (8) Colt, J. R.; Scuseria, G. E.J. Phys. Chem. 1992, 96, 10265. (9) Colt, J. R.; Scuseria, G. E. Chem. Phys. Lett. 1992, 199, 505. (10) Liu, X.;Schmalz, T. G.; Klein, D. J. Chem. Phys. Lett. 1992,188, 550.
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