J. Phys. Chem. 1995, 99, 4486-4489
4486
Prediction of the Fundamental Vibrational Frequencies for C ~ by O Local Density Functional Theory David A. Dixon,*,t Bruce E. Chase,? George Fitzgerald,* and Nobuyuki Matsuzawao DuPont Central Research and Development,"Experimental Station, P. 0. Box 80328, Wilmington, Delaware 19880-0328, Cray Research, 655E Lone Oak Dr., Eagan, Minnesota 551 21, and SONY Corporation Research Center, I74 Fujitsuka-cho, Hodogaya-ku, Yokohama 240, Japan Received: October 6, 1994; In Final Form: January 12, 1995@
The harmonic vibrational frequencies of c60 have been calculated at the local density functional level by using analytic second derivatives. The calculated values for the observed infrared and Raman transitions are in good agreement with the experimental values and suggest that there are a number of misassigned bands based on thick film infrared and Raman measurements. The calculated transitions have been used to provide tentative assignments to the peaks observed by inelastic neutron scattering and high-resolution electronenergy-loss spectroscopy.
Introduction There have been a number of reported measurements of the vibrational transitions in C60.1-5 There are a total of 46 fundamental vibrational modes for Cm, but due to the high symmetry ( z h ) , there are only four infrared active modes of TI, symmetry and 10 Raman active modes, two of A, symmetry and eight of H, symmetry. The remaining 32 modes are silent as fundamental transitions and can only be observed as weak combination bands or from inelastic neutron scattering measurements where the selection rules are relaxed. These measurements of the weak transitions have been made on thick films and are extremely difficult to interpret. The assignments based on the experimental measurements are given in Table 1. There have also been a number of theoretical studies of the vibrational modes of c60. The only complete studies6s7of the spectrum in which all modes are reported are those at the semiempirical molecular orbital level with either the MND08 or MNDO/PM39 Hamiltonians and a recent calculation10at the local density functional theory (DFT) level." The DFT calculations were done with numerical basis sets and the HedinLundquist potential for the exchange correlation energy. l2 The frequencies were calculated by finite differencing of analytic gradients, and because of the very high I h symmetry of c60, only one carbon atom had to be displaced. calculation^'^.^^ of the infrared and Raman active modes have been reported at the local density functional level based on Car-Parinell~'~or Sankey-Niklewski16 molecular dynamics calculations. In the former, a plane-wave, pseudopotential approach is used, and in the latter, a pseudoatomic orbital (PAO) minimal sp3 basis set is used. In the latter case, the frequencies were also calculated by a finite differencing procedure. Density functional perturbation theory (DFPT)" with pseduopotentials and plane-wave basis sets has been used to calculate the frequencies at the local level.I8 Symmetry was used in this calculation to reduce the size of the response matrix. Only the calculated infrared and Raman active bands were reported in these initial studies. Subsequent improved analysis of the trajectory reported in ref 13a allowed for the abstraction of all of the frequencies for
' DuPont Central Research and Development.
* Cray Research.
Sony Corporation Research Center. I ' DuPont Contribution No. 6996. Abstract published in Advance ACS Abstracts, March 1, 1995. @
C60.13bThe only ab initio molecular orbital calculations of vibrational modes are those for the totally symmetric A, modes at the TZP/MP-2 level with reported values of 1586 and 437 cm-l.19 As shown in Table 1, there is not particularly good agreement among the theoretical results or of the theoretical results with many of the experimental results. The development of analytic methods for predicting first and second derivatives of the electronic energy has proven to be of great benefit to quantum chemistry.*O Such methods allow one to find various stationary points on the potential energy surface and then to identify whether the stationary points are minima, transition states, or saddle points. The matrix of second derivatives can also be converted to a set of force constants which then yield a set of vibrational frequencies in the harmonic approximation. Although one could numerically differentiate the gradient to find the force constants, this can be extremely time-consuming, especially as the size of the molecule grows. For c60, using a two-point differencing scheme would require 360 gradient evaluations on the assumption that no symmetry is present. The presence of symmetry would obviously drastically reduce the number of finite difference calculations for c ~ , but for most molecules with 60 atoms there will be little or no symmetry, and we, are in general, interested in such asymmetric systems far more than in highly symmetric ones. Until very recently, analytic second derivative calculations were not possible for more than 250-300 basis functions at the HartreeFocWself consistent field (HF/SCF) level or 150-200 basis functions at the MP-2 level. Beyond 250 basis functions, the calculations are quite time-consuming, and few, if any, have been carried out in the absence of symmetry. Recently, Komornicki and FitzgeraldZ1derived expressions for molecular Hessians (second derivatives) when the energy is evaluated from local density functional theory. An additional advantage of this formalism is that infrared intensities can be calculated with very little additional computational effort. These expressions have been implemented into the DGauss computer programz2 Below, we report the results of an analytic calculation without using symmetry of the vibrational frequencies of c60 with a basis set of polarized double zeta quality which yields 900 basis functions. Calculations All calculations were done with the program DGauss which is part of the UniChem program system.** The calculations were
0022-3654/9S/2099-4486$09.00/0 Q 1995 American Chemical Society
Prediction of Vibrational Frequencies for
J. Phys. Chem., Vol. 99, No. 13, 1995 4487
c60
TABLE 1: Fundamental Vibrational Transitions for LDFT LDFT'3a LDFPb LDFT/ LDFT/ MNDOl sym (this work) expt5 expe plane-wave plane-wave DFPT1* PAO14 numericallo MND06 PM3' Svmmetrv Allowed Transitions Raman
1525 499 1618 1475 1297 1128 788 727 43 1 26 1 1486 1224 591 535
1470 495 1576, 1567 1425,1418 1252 1102 775,778 709 43 1 272,267 1429,1433 1183 577 526
1318 830 579 1393 839 804 55 1 1548 1347 1122 788 573 484
1479, I484 IR, Ra, Neutron 973 IR, Ra, Neutron 568 IR, Ra, Neutron IR, Ra, Neutron I544 1214 IR, Neutron 764 IR, Ra, Neutron 541 IR, Ra, Neutron 1596 Neutron 1345 IR, Ra, Neutron 1330 IR, Neutron IR, Ra, Neutron 1199 IR, Ra, (Neutron) 961 485 IR, Ra, Neutron
972 1571 1234 996 726 342 1480 1359 984 830 762 350 1611 1389 1248 762 67 1 541 40 1
1122 I313 1038 797 712 353
1526 1290
I080 753 739
402 1600
1242 1222
828 664,668
579 342
Raman Raman Raman Raman Raman Raman Raman Raman Raman IR IR IR IR
IR, Neutron IR, Neutron IR, Neutron IR, Ra, (Neutron) IR, Ra, (Neutron) IR, Ra, Neutron IR, Ra, Neutron IR, Ra, Neutron IR, Ra, Neutron IR, Neutron IR, Ra, Neutron IR, Ra, Neutron Neutron IR, Neutron IR, Neutron IR, Neutron IR, Neutron Ra, (Neutron) IR, Ra, Neutron
i470 498 1578 1426 1251 1101 775 71 1 432 273 1429 1183 576 526
*
1368 454 1500 1314 1110 688 405 246 1345 1105 555 530
1365 455 1484 1315 1140 1036 73 1 689 410 246 1320 1092 534 510
1504 495 1578 1450 1281 1120 783 711 425 259 1462 1218 586 527
Silent Gerade Transitions 1358 1211 976 799 502 547 1360
914 865 566 1524 1356 1076
806 621 486
1186 770 744 527 1395 1240 988 737 500 455
Silent Ungerade - Transitions 1143 881 1576 1450 1201 1026
680 356 1446 1310 970
924 760
400 1559 1385
1117 801 696 563 342
Values in bold italics need to be reassigned. done at the local density functional theory (LDFT) level with the exchange-correlation potential of Vosko, Wilk, and Nuair.^^ The basis set is of polarized double zeta quality (DZVP) and was specifically optimized at the LDFT The geometry was optimized by analytic gradient methods. The calculations up to this point were done on a single processor of a Cray C90 supercomputer. As discussed by Komomicki and Fitzgerald,21the most time-consuming step in the calculations of the analytic second derivatives is the solution of the response equations from coupled, perturbed local density functional theory. This also requires a significant amount of computer storage capability in terms of memory. Thus, this calculation was done in parallel processing mode on a dedicated 16processor C90 supercomputer. Important information about the computational requirements is given in Table 2. The computational requirements show that most of the time is spent in the solution of the coupled perturbed Kohn-Sham (CPKS) equations. This is quite different from traditional ab initio molecular orbital calculations at the Hartree-Fock level for much smaller systems where the integral evaluation dominates the computa-
1045 833 730 32 1 1316 1216 886 751 720 337 1458 1240 1130 725 634 509 381
1680 537 1726 1624 1453 1209 845 68 1 413 249 1641 1358 643 494
1529 484 1598 1469 1282 1111 778 713 432 263 1485 1214 548 533
1667 610 1722 1596 1407 1261 924 77 1 447 263 1628 1353 719 577
1752 623 1814 1670 1437 1291 942 758 440 266 1709 1378 734 558
1292 825 566 1360 795 77 1 550 1528 1326 1117 713 5 14 484
1410 865 627 1483 919 784 591 1650 1404 1235 856 579 49 1
1440 867 613 1518 913 817 586 1730 1445 1244 85 1 593 489
947 1558 1227 987 717 344 1467 1339 977 784 752 356 1588 1360 1230 742 663 534 396
972 1687 1314 1134 776 348 1587 1436 1110 914 750 362 1709 1467 1344 822 706 546 403
978 1777 1336 1158 763 354 1656 1476 1125 909 783 357 1800 1501 1379 816 701 547 407
tion. The assignment of the modes to the gerade or ungerade irreducible representations was made by visual inspection of the atomic motions for each of the 172 normal modes by using the UniChem interface on a Silicon Graphics Indigo work station. The degeneracy was evaluated by inspection of the calculated frequencies and of the normal mode motions. Results and Discussion The only two parameters that need to be optimized for c60 are the short C-C double bond and the long C-C single bond. The results of the geometry optimization are shown in Table 3 where they are compared to other r e s ~ l t s . ~The ~ - ~computa~ tional results for the short C=C bond cluster are about 1.381.41 8, and are similar to the experimental value^^^^^^ which are also in this range. We note that the X-ray diffraction value seems to be too short.29 Our LDFT value is about 0.01 8, shorter than the MP-2 value19 for this bond length and is in good agreement with the gas phase experimental value of 1.398 The computational results are all near 1.45 8, for the long C-C
4488
J. Phys. Chem., Vol. 99, No. 13, 1995
Dixon et al.
TABLE 2: Computational Parameters and Timings computational parameters DZVP Basis: 62114111 total no. of basis functions: 900 A1 fitting basis: 71313 total no. of fitting basis functions: 2040 default grid: 76 700 points property timings (s) property timings (s) SCF 1619 solution of CPKS 194 576 derivative integrals" 10 567 equations exchange corelation 24 535 total calculation' 261 OOO construct RHS of CPKS 7403 equationsb Includes time for 2-electron second-derivative integrals and time for 2-electron fist-derivativecontributions to derivative Fock matrices. b R H S = right hand side, CPKS = coupled perturbed Kohn Sham. Includes other terms not listed individually. Total wall-clock time = 9 h. TABLE 3: Molecular Geometries for Cw method R(C=C) R(C-C) LDFTDZVP 1.395 1.445 MNDO 1.400 1.474 1.384 1.458 MNDOPM3 1.406 1.446 MP-2iTZP LDFTPWIQM 1.39 1.45 LDAIPAOIQMD 1.398 1.442 HFiTZP 1.370 1.448 LDA 1.382 1.444 LDFTIDNP 1.386 1.442 NMR 1.40 1.45 X-ray 1.355(9) 1.467(21) electron diffraction 1.398(10) 1.436(6)
ref
this work
6 7 19 13 14 25 26 27 28 29 30
bond except for the MNDO results6 which give too long a bond length. The computational results for the long bond are all close to the NMRZ8and electron diffraction30values whereas the X-ray diffraction value is apparently slightly too long.29 Thus, as has been previously established, theoretical methods are well able to predict the molecular geometry parameters for c 6 0 . The calculated vibrational frequencies are shown in Table 1. Because exact icosahedral symmetry was not imposed or used in the calculations, the degenerate frequencies computed in this study only agree to within 5-10 cm-'. Thus, it was critical to identify the modes by visual analysis of all of the atomic motions. An important result of the calculations is that there are a number of modes that are very close to each other. This is important information for analyzing the experimental results as it is often difficult to distinguish the number of bands in a region when they overlap closely and when some are allowed while others are not. The most congested region is between 750 and 790 cm-' where the H, and G, bands overlap near 788 cm-', and there is an overlap of H, and G, bands near 760 cm-'. First, we compare our calculated results to the observed, allowed frequencies. It is important to remember that the calculated vibrational transitions correspond to harmonic frequencies and the true vibrational modes are anharmonic. Thus, the calculated values should be greater than the observed values. For transitions above 1200 cm-', the calculated values are uniformly high by about 50 cm-'. For transitions above 500 cm-' and below 1200 cm-', the calculated values are 10-25 cm-' above the observed values. For the two lowest observed transitions (of H, symmetry), the calculated value is the same as the experimental value for the higher frequency transition (431 cm-') and is 6-10 cm-' below the lowest energy transition. This comparison suggests that we are able to predict the frequencies of c 6 0 with good reliability.
The MP-2 calculation^'^ predict the high energy A, mode to be too high by 116 cm-' and the low-energy A, mode to be low by 61 cm-', clearly showing worse agreement with experiment than our results at the LDFT level, although the authors do note that numerical errors could have contaminated their results. The LDFTPW results13 tend to predict frequencies that are at least 100 cm-' too low for the high-frequency stretches, and the difference is somewhat smaller at lower frequencies although we do note a continuation of a tendency to underestimate the size of the modes. The LDFTPAO calculations14were done by performing molecular dynamics (MD) in a symmetryconstrained set of motions for the H, and A, modes and by finite displacements (FD). The agreement for the higher frequencies calculated by the MD and FD methods in the LDFT/ P A 0 calculations is within 1% and for the lower frequencies, about 4% or about 15 cm-' of each other. The frequencies obtained with the minimal PA0 basis set are significantly higher than our values for frequencies above 700 cm-'. For example, the highest frequency stretches are 200 cm-' above the A, and TI, experimental values and 150 cm-' above the H, experimental value. The lower frequency values tend to scatter around the experimental values and do not show a specific pattern. The LDFT results1° obtained by finite differencing and a numerical basis set are in reasonable agreement with our calculated values except that these values tend to be -20 cm-' lower than our calculated values. In a number of cases, the agreement is worse, notably for the TI, band which we predict to be at 591 cm-' as compared to the finite difference value of 548 cm-' and the TI, band which we predict to be at 1393 cm-' as compared to the finite difference value at 1360 cm-'. Similarly, the allowed transitions calculated at the DFPT level18 tend to be smaller than our calculated values. The largest difference, 40 cm-', between our values and the DFPT values is for the highest energy H, stretch. The semiempirical MO values are in general higher than our calculated values except for some of the H, modes. The MNDOPM3 values' are in worse agreement with experiment and our LDFT values than are the values6 obtained with the MNDO parameterization. For example, for the highest energy A, mode, the MNDOPM3 value is almost 300 cm-' above experiment whereas MNDO predicts a value about 200 cm-' above experiment. A similar comparison can be made for the highest TI,stretching frequency. Even the low energy A, modes are predicted to be about 120 cm-' too high at the semiempirical level. Our calculated infrared intensities for the allowed transitions are 41 M m o l (1486 cm-'), 27 M m o l (1224 cm-'), 49 M mol (591 cm-'), and 72 M m o l (535 cm-l). These are the only absolute values that have been reported. Infrared intensities relative to the most intense transition have been reported at the DFPT level,18 and the reported ratios are 0.57, 0.36, 0.63, and 1.0 in the order of decreasing frequency as compared to our comparable ratios of 0.57, 0.38, 0.68, and 1.0. The agreement between the two calculations is quite reasonable for the ratios of the infrared intensities. It is clear from the above discussion that the LDFT calculations provide the best theoretical treatment of the observed Raman and infrared bands, and the quality of the agreement is such that we should be able to reliably predict the frequencies of the remaining "silent" vibrational transitions across the complete spectral range. In the following discussion, we use our LDFT values. We first compare to the assigned "silent" gerade transitions. We mark what we consider to be misassigned bands by bold italicizing the values in Table 1. In general we suggest that a band is misassigned if it is greater than our
Prediction of Vibrational Frequencies for
J. Phys. Chem., Vol. 99, No. 13, 1995 4489
c60
TABLE 4: Reassigned INSz and HREELSJ1Peaks
References and Notes
spectrral symmetry spectrral position position (cm-') assignment (cm-l)
(1) Hare, J.; Dennis, J.; Kroto, H.; Taylor, R.; Allaf, A,; Balm, S.; Walton, D. Chem. Commun. 1991,412; Bethune, D. S.; Meijer, G.; Tang, W. C.; Rosen, H. Chem. Phys. Lett. 1990, 174, 219. (2) Coloumbeau, C.; Jobic, H.; Bernier, P.; Fabre, C.; Schutz, D.; Rassat, A. J. Phys. Chem. 1992, 96, 22. (3) Chase, B.; Fagan, P. J. J. Am. Chem. SOC.1992, 114, 2252. (b) Chase, B.; Herron, N.; Holler, E. J. Phys. Chem. 1992, 96, 4262. (4) Wang, K.-A.; Rao, A. M.; Eklund, P. C.; Dresselhaus, M. S.; Dresselhaus, G. Phys. Rev. B. 1993,48, 11375. (b) Dong, Z.-H.; Zhou, P.; Holden, J. M.; Eklund, P. C.; Dresselhaus, M. S.; Dresselhaus, G. Phys. Rev. B. 1993, 48, 2862. (5) Martin, M. C.; Du, X.; Kwon, J.; Mihaly, L. Phys. Rev., submitted. (6) Stanton, R. E.; Newton, M. D. J. Phys. Chem. 1988, 92, 2141. (7) Stewart, J. J. P.; Coolidge, M. B. J. Comput. Chem. 1991,12, 1157. (8) Dewar, M. J. S.; Thiel, W. J. Am. Chem. SOC.1977, 99, 4899. (9) Stewart, J. J. P. J. Comput. Chem. 1989, 10, 221. (10) Wang, X. Q.: Wang, C. Z.; Ho, K. M. Phys. Rev. B 1993, 48, 1884. (11) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University: New York, 1989. (b) Salahub, D. R. In Ab Initio Methods in Quantum Chemistry-I1 ed.; Lawley, K. P. J., Ed.; Wiley & Sons: New York, 1987; p 447. (c) Wimmer, E.; Freeman, A. J.; Fu, C.-L.; Cao, P.-L.; Chou, S.-H.; Delley, B. in Supercomputer Research in Eds.; Chemistry and Chemical Engineering; Jensen, K. F., Truhlar, D. G., ACS Symposium Series; American Chemical Society: Washington, D. C. 1987; p 49. (d) Jones, R. 0.; Gunnarsson, 0. Rev. Mod. Phys. 1989, 61, 689. (e) Zeigler, T. Chem. Rev. 1991, 91, 651. (12) Delley, B. J. Chem. Phys. 1990, 92, 508. DMol is available commercially from BIOSYM Technologies, San Diego, CA. (b) Delley, B. In Density Functional Methods in Chemistry; Labanowski, J., Andzelm, J., Eds.; Springer-Verlag: New York, 1991; p 101. (c) Delley, B. J. Chem. Phys. 1991, 94, 7245. (13) Feuston, B. P.; Andreoni, W.; Paninello, M.; Clementi, E. Phys. Rev. B 1991,44,4056. (b) Kohanoff, J.; Andreoni, W.; Paninello, M. Phys. Rev. B 1992, 46, 4371. (14) Adams, G. B.; Page, J. B.; Sankey, 0. F.; Sinha, K.; MenCndez, J.; Huffman, D. R. Phys. Rev. B 1991, 44, 4052. (15) Car, R.; Paninello, M. In Proceedings of NATO AAARW: Simple Molecular Systems at Very High Density; NATO Advanced Studies Institutes Series, Les Houches, France, 1988; Plenum: New York, 1988; p. 455. (16) Sankey, 0. F.; Niklewski, D. J. Phys. Rev. B 1989, 40, 3979. (17) Baroni, S.; Giannozzi, P.; Testa, A. Phys. Rev. Lett. 1987,58,1861. (b) Giannozzi, P.; de Gironcolli, S.; Pavone, P.; Baroni, S. Phys. Rev. B 1990, 43, 7231. (18) Giannozzi, P.; Baroni, S. J. Chem. Phys. 1994, 100, 8537. (19) Haser, M.; Almlof, A,; Scuseria, G. E. Chem. Phys. Lett. 1991, 181, 497. (20) Pulay, P. In Applications of Electronic Structure Theory; Schaefer, H. F., m, Ed.; Plenum: New York, 1977, p 153. (a) King, H. F.; Komomicki, A. J. Chem. Phys. 1986,84,5465. (b) King, H. F.; Komomicki, A. In Geometrical Derivatives of Energy Surfaces and Molecular Properties; Jorgenson, P., Simons, J., Eds.; NATO AS1 Series C. Vol. 166; D. Reidel: Dordrecht, 1986; p 207. (21) Komomicki, A.; Fitzgerald, G. J. J. Chem. Phys. 1993, 98, 1398. (22) Andzelm, J.; Wimmer, E.; Salahub, D. R. In The Challenge of d and f Electrons: Theory and Computation; Salahub, D. R., Zerner, M. C., ACS Symposium Series, No. 394; American Chemical Society: Washington, DC 1989; p 228. (b) Andzelm, J. In Density Functional Methods in Chemistry; Labanowski, J., Andzelm, J., Eds.; Springer-Verlag: New York, 1991; p 101. Andzelm, J. W.; Wimmer, E. J. Chem. Phys. 1992, 96, 1280. DGauss is a density functional program which is part of the UniChem software package and is available from Cray Research, Eagan, MN. (23) Vosko, S . J.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (24) Godbout, N.; Salahub, D. R.; Andzelm, J.; Wimmer, E. Can. J. Chem. 1992, 70, 560. (25) Scuseria, G. Chem. Phys. Lett. 1991, 176, 423. (26) Martins, J. L.; Troullier, N.; Weaver, J. H. Chem. Phys. Lett. 1991, 180, 457. (27) Matsuzawa, N.; Dixon, D. A. J. Phys. Chem. 1992, 96, 6872. (28) Yannoni, C. S.; Bemier, P. P.; Bethune, D. S.; Meijer, G.; Salem, J. R. J. Am. Chem. Sac. 1991, 113, 3190. (29) Liu, S.; Lu, Y.-J.; Kappes, M. M.; Ibers, J. A. Science 1991, 254, 408. (30) Hedberg, K.; Hedberg, L.; Bethune, D. S.; Brown, C. A.; Dorn, H. C.; Johnson, R. D.; DeVries, M. Science 1991, 254, 410. (3 1) Gensterblum, G.; Pireaux, J. J.; Thky, P. A.; Caudano, R.; Vigneron, J. P.; Lambin, Ph.; Lucas, A. A.; Kratschmer, W. Phys. Rev. Lett. 1991, 67, 2171.
symmetry assignment
INS 264,271 344,355 303 432 488 526,536 563,576 673 715 765
H, G, H" H, A,+G, H,+Ti, Ti,+G, H" H, H,or G,
840 97 1 1044 1089, 1122 1217 1327 1448 1520 1563 1603
813
Gu
1702
355 444 686 758
Tzu H, H,orTzu H, G, or Gu + Hu
TZ, Au not fundamental H, G, Tzuor H, G, or G, G, or H, or Tzu Tzu or G, H, unlikely fundamental, only possibility is H,or H, not fundamental
+
HREELS
+
968 1097 1258 1565
A, H, or G, H, H, or H,
calculated values by more than 30 cm-' as suggested by comparison of the calculated and experimental values for the allowed transitions. Observed frequencies should occur at lower frequencies than the calculated values as the latter are harmonic values. In general the misassigned experimental transitions are higher than the predicted values, in some cases, by as much as 150 cm-'. In fact, for one experimental assignment (second highest Tzg)the value is high by almost 400 cm-'. For the silent gerade transitions, we note that the lowest TI, mode assigned by Eklund et aL4 appears to be too low. Although it is likely that Eklund et al.'s G, transitions at 806 and 621 cm-' are too high, they are not in too great an error as compared to the predicted values. For the silent ungerade transitions, there are misassigned transitions that are too high and too low in contrast to the case for the gerade transitions. For example, for the highest energy Tzu mode, Martin et aL5 assign the transition to be -200 cm-' too low whereas Eklund et aL4 assign the transition to probably be 50 cm-' too high. In both cases the Au transition is assigned to be too high by at least 150 cm-'. It is likely that the lowest energy G, and H, lowest energy transitions need to be switched based on the calculations. We used our theoretical values with appropriate scaling to make some tentative symmetry assignments for the inelastic neutron scattering (INS)2 and high-resolution electron-energyloss spectra (HREELS) ~pectra.~'These are given in Table 4. We were able to assign all of the HREELS bands and all but two of the INS bands to some fundamental transition although in a number of cases, due to the large overlap in the calculated bands and the width of the experimental peaks, we were unable to give a definitive symmetry assignment. The INS bands at 1044 and 1703 cm-' clearly do not correspond to fundamental transitions, and it is unlikely that the band at 1603 cm-' corresponds to a fundamental transition. In conclusion, we have calculated the normal modes of c60 by using analytic second derivatives at the local density functional level. The computational requirements for such a calculation have been described. The calculated frequencies suggest that a number of the assigned infrared and Raman silent bands have not been correctly assigned, and we suggest that our values with appropriate shifts be used to reassign the thick film spectra. On the basis of our calculated spectra, we were able t o assign most of the transitions found in the INS and HREELS experiments.
JP942726L
