Structure and Vibrational Spectrum of Some Polycyclic Aromatic

Institute for Materials Science, Department SBG, Limburgs Universitair Centrum ...... by the Office for Science Policy Programming of the Prime Minist...
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15358

J. Phys. Chem. 1996, 100, 15358-15367

Structure and Vibrational Spectrum of Some Polycyclic Aromatic Compounds Studied by Density Functional Theory. 1. Naphthalene, Azulene, Phenanthrene, and Anthracene† Jan M. L. Martin,*,‡,§ Jamal El-Yazal,‡ and Jean-Pierre Franc¸ ois‡ Institute for Materials Science, Department SBG, Limburgs UniVersitair Centrum, UniVersitaire Campus, B-3590 Diepenbeek, Belgium, and Department of Chemistry, Institute for Materials Science, UniVersity of Antwerp, UniVersiteitsplein 1, B-2610 Wilrijk, Belgium ReceiVed: February 27, 1996; In Final Form: May 1, 1996X

The geometry and harmonic frequencies of naphthalene, azulene, phenanthrene, and anthracene have been computed using the Becke three-parameter Lee-Yang-Parr functional (B3LYP) and basis sets of spd and spdf quality. A simple scaling procedure for the harmonic frequencies is proposed that brings computed harmonics to within 10-20 cm-1 or better of experimental fundamentals without resorting to internal coordinate transformations. A complete reassignment of the vibrational spectrum of phenanthrene is proposed, and several reassignments are due for anthracene as well. The assignments of Sellers, Pulay, and Boggs (J. Am. Chem. Soc. 1985, 107, 6487) for naphthalene, and of Kozlowski, Rauhut, and Pulay (J. Chem. Phys. 1995, 103, 5650) for azulene, are largely confirmed.

Introduction Density functional methods1,2 that include exact exchange contributions3 have emerged as a very powerful tool for assigning vibrational spectra (see, for example, refs 4-6). Actually, the vibrational spectrum of even such notoriously problematic systems as carbon clusters is predicted7 in excellent agreement with accurate coupled cluster calculations.8 One group of compounds for which the vibrational assignment is not quite firmly established are the polyacenes (such as phenanthrene, anthracene, tetracene, pentacene, etc.). Besides their obvious importance in organic chemistry, these PAH (polycyclic aromatic hydrocarbon) compounds and their cations have attracted considerable attention in astrophysics (see, for example, refs 9-13). For these reasons, a firm vibrational assignment for the neutral species and their cations would be highly desirable. In the present paper, we will first investigate the parent molecule of this species, naphthalene. From our results, we will derive a simple scaling procedure which avoids the internal coordinate transformations required in the SQM (scaled quantum mechanical) procedure of Pulay and co-workers.14 Our scaling model will be tested for transferability on the related benzene and azulene molecules. Finally, it will be applied to the assignment of the vibrational spectra of phenanthrene and anthracene. Computational Methods All calculations have been carried out using the Gaussian 94 package15 running on a cluster of IBM RS/6000 workstations at the Limburgs Universitair Centrum. The B3LYP exchangecorrelation functional3,16 has been used throughout. This consists of the Lee-Yang-Parr16 correlation functional in conjunction with a hybrid exchange functional first proposed by Becke.3 The latter is a linear combination of the local density approximation, Becke’s gradient correction,17 and the HartreeFock exchange energy based on Kohn-Sham18 orbitals. Using † Dedicated to the memory of Jan Almlo ¨ f (1945-1996). * Present address: Department of Organic Chemistry, Weizmann Institute of Science, 76100 Rehovot, Israel. ‡ Limburgs Universitair Centrum. § University of Antwerp. X Abstract published in AdVance ACS Abstracts, August 15, 1996.

S0022-3654(96)00598-9 CCC: $12.00

the adiabatic connection19 argument, Becke demonstrated that this is the simplest form of the exchange-correlation functional that correctly describes the limiting cases of Hartree-Fock behavior and long distance interaction. Two “correlation consistent” basis sets due to Dunning20 have been used throughout. The first, cc-pVDZ or correlationconsistent polarized valence double-ζ, is a [3s2p1d/2s1p] contraction of a (9s4p1d/4s1p) primitive set. The second, ccpVTZ or correlation-consistent polarized valence triple-ζ, is a [4s3p2d1f] contraction of a (10s5p2d1f/5s2p1d) primitive set. It has conclusively been shown21,22 that the former is the smallest basis set that will yield useful harmonic frequencies in a correlated calculation and that the latter bridges most of the one-particle basis set incompleteness gap, generally yielding harmonic frequencies within 10 cm-1 of experiment if advanced coupled cluster methods are used for electron correlation. In a recent basis set convergence study in DFT calculations,23 it was found that B3LYP/cc-pVDZ harmonic frequencies are often in surprisingly good agreement with experiment and that the improvement upon going from cc-pVDZ to cc-pVTZ is by no means as marked as for accurate conventional electron correlation methods such as coupled cluster theory.24,21 Hence, while further extension of the basis set will generally not lead to further improvement, an appreciable difference between B3LYP/cc-pVDZ and B3LYP/cc-pVTZ for a particular ωi is a reliable indicator for basis set sensitivity. In that same paper,23 however, it has been found that the errors in computed bond distances at the B3LYP level are fairly systematic (for a given basis set) and that using the average overestimates for single, double, and triple bond lengths in a number of reference molecules as empirical corrections leads to bond distances within a few milli-angstroms of experiment. (Of course, the idea of using empirical bond length corrections is a time-honored one: see, for example, refs 25 and 26 and references therein.) Those at the B3LYP/cc-pVTZ level are markedly better than at the B3LYP/cc-pVDZ level, illustrating the greater basis set sensitivity of geometries as opposed to harmonic frequencies (see also ref 22). Encouraging results have previously been obtained23 for the geometry and particularly the harmonic frequencies of benzene, pyrrole, and furan, with the latter usually being within 20-50 cm-1 of experiment. We expect the same level of accuracy for the PAHs under study here. © 1996 American Chemical Society

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J. Phys. Chem., Vol. 100, No. 38, 1996 15359

TABLE 1: Computed and Observed Frequencies of Naphthalene (cm-1; IR Intensities in km/mol Given in Parentheses) B3LYP cc-pVDZ raw

B3LYP cc-pVTZ raw

B3LYP cc-pVDZ scaled

B3LYP cc-pVTZ scaled

expt νi: ref 27 Ii: ref 74

remarks and possible alternatives

SQM SPB28

RHF 3-21G raw29

RHF 6-31G* raw29

RHF 6-31G* scaled29

ag 3197 3171 1623 1484 1417 1172 1048 775 519

3189 3164 1614 1497 1397 1188 1048 775 521

3069 3044 1574 1459 1374 1152 1017 752 510

3077 3053 1574 1460 1362 1158 1022 756 508

3060 3031 1577 1460 1376 1145 1025 758 512

3183 3164 1681 1488 1260 1164 944 516

3175 3157 1671 1496 1275 1175 956 520

3056 3037 1631 1463 1239 1137 928 507

3064 3047 1629 1459 1243 1146 932 507

3092 3060 1624 1438 1239 1158 935 506

3184 (59) 3166 (6) 1648 (4) 1408 (4) 1277 (7) 1144 (5) 805 (0.2) 365 (1)

3176 (57) 3159 (6) 1643 (4) 1425 (4) 1292 (7) 1155 (4) 812 (0.2) 366 (1)

3057 3039 1599 1384 1255 1125 791 359

3065 3048 1602 1389 1260 1126 792 357

3065 (4) 3058 (4) 1595 (0) 1389 (2) 1265 (2) 1125 (2) 753 (0) 359 (0.5)

3196 (44) 3168 (1) 1552 (8) 1403 (1) 1233 (1) 1162 (0.7) 1039 (7) 631 (3)

3188 (43) 3161 (0.85) 1551 (9) 1393 (0.91) 1235 (0.77) 1171 (0.78) 1037 (7) 637 (3)

3068 3041 1505 1361 1196 1142 1008 620

3076 3050 1512 1358 1204 1142 1011 621

3090 (1) 3027 (w) 1506 (2) 1361 (0) 1209 (1) 1138 (0.5) 1008 (3) 618 (0.5)

965 736 398

966 733 398

936 714 386

942 715 388

943 717 386

1008 912 799 484

1004 906 793 482

978 885 775 469

979 883 773 470

980 876 846/770 461

1001 858 637 189

998 856 639 188

971 832 618 183

973 835 623 183

970 841 581 195

982 (2) 805 (77) 492 (15) 176 (2)

983 (4) 803 (107) 491 (19) 173 (2)

953 781 477 171

958 783 479 169

958 (3) 782 (10) 476 (2) 176

116330

3085 3056 1590 1458 1385 1170 1023 757 505

3383 3354 1749 1635 1464 1321 1106 831 567

3387 3361 1783 1626 1487 1287 1117 831 551

3085 3056 1645 1453 1355 1143 1043 780 517

3067 3047 1644 1458 1255 1156 940 512

3365 3345 1811 1626 1399 1293 1057 578

3371 3350 1845 1623 1373 1273 1023 555

3069 3046 1673 1461 1219 1149 917 491

3070 3049 1595 1391 1272 1137 792 354

3368 (54) 3347 (2) 1785 (12) 1567 (4) 1410 (8) 1266 (4) 884 (1) 401 (2)

3374 3353 1806 1543 1390 1245 859 389

3071 (88) 3047 (3) 1654 (8) 1364 (4) 1261 (7) 1130 (3) 781 (2) 364 (2)

3083 3052 1515 1341 1204 1158 1003 626

3381 (46) 3349 (1) 1664 (15) 1481 (3) 1333 (0) 1208 (3) 1070 (3) 706 (4)

3385 3355 1683 1476 1311 1193 1075 674

3083 (73) 3052 (2) 1539 (12) 1321 (2) 1165 (0.1) 1091 (3) 988 (2) 596 (5)

952 705 387

1130 831 442

1080 811 432

968 708 397

987 879 773 471

1185 1041 898 537

1124 993 854 520

985 880 785 478

981 825 622 188

1176 976 718 210

1117 944 685 206

970 824 606 184

969 777 480 172

1154 (6) 915 (168) 548 (33) 192 (3)

1098 887 535 188

977 (4) 791 (125) 485 (19) 172 (2)

b3g

1445,34 14585

b1u

probably not obsd b2u

116332 doubtful

b1g

b2g clearly 770 395,30 implausible au implausible b3u

Results and Discussion Naphthalene. Vibrational Frequencies. A considerable amount of work has been done on the naphthalene molecule. The vibrational assignment of Krainov27 was subjected to revision by Sellers, Pulay, and Boggs (SPB),28 who obtained a scaled quantum mechanical (SQM) force field based on HF/421G calculations and scale factors for benzene. Some further calculations were performed by Pauzat et al.,29 who computed HF/3-21G and HF/6-31G* harmonic frequencies. Computed and observed frequencies can be found in Table 1. As can be seen from comparing the B3LYP/cc-pVDZ and

B3LYP/cc-pVTZ frequencies, basis set effects are quite small on most modes with a few exceptions. In benzene,23 these were found to be mainly the “Kekule´-type” vibration of the ring and a concerted “pinwheel” motion of all the hydrogens. Not surprisingly, significant basis set effects are therefore seen in naphthalene for the 5ag mode (-20 cm-1), which is the “doubleKekule´” vibration, and in the 4b1u (+17 cm-1) and 5b3g (+15 cm-1) modes, which correspond to symmetric and antisymmetric combinations, respectively, of “pinwheel” motions on the two rings. Further significant basis set effects are seen for 6a1g (+16 cm-1; this is a symmetric combination of all four HCCH

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TABLE 2: Computed and Observed Frequencies of Benzene (cm-1 ) estimated B3LYP B3LYP B3LYP B3LYP cc-pVDZ cc-pVTZ cc-pVDZ cc-pVTZ experiment ωi: ref 76 scaled νi: ref 75 (“ωav”) scaled raw raw a1g a1g a2g b2g b2g e2g e2g e2g e2g e1g a2u b1u b1u b2u b2u e2u e2u e1u e1u e1u

1019 3200 1365 723 1022 618 3173 1646 1187 866 691 1014 3163 1358 1163 414 987 1060 1507 3189

1015 3192 1390 727 1021 624 3167 1637 1201 867 690 1031 3157 1335 1177 414 988 1062 1519 3182

988 3072 1342 701 991 607 3046 1597 1167 840 670 997 3036 1317 1143 402 957 1042 1481 3061

990 3080 1355 709 995 608 3056 1596 1171 845 673 1005 3047 1302 1148 404 963 1035 1481 3071

993 3074 1350 707 990 608 3057 1601 1178 847 674 1010 3057 1309 1150 398 967 1038 1484 3064

1001 3198 1378 712 1000 610 3182 1623 1185 856 680 1016 3173 1313 1158 402 978 1048 1503 3186

scissorings) and for 5b1u (+15 cm-1), which is essentially an antisymmetric stretching motion of the central CC bond. Scaling Factors. Since here we are dealing with calculations on a fairly closely related class of compounds, it might be possible to absorb the anharmonicities and residual shortcomings of the method at least partially into one or more scaling factors. Like all such procedures, or, for that matter, more sophisticated schemes such as SQM, success implies that no strong Fermi resonances exist. “Strong” is defined here as “of the same or larger order of magnitude than the remaining other sources of error”. Considering the ratios ν(obsd)/ω(B3LYP/cc-pVDZ) for naphthalene, we find that they cluster in three distinct groups: (a) the CH stretches, for which the average ratio is 0.96; (b) the in-plane bends, for which the ratio is 0.983; and (c) the remaining vibrations, for which the average ratio comes to 0.97. (No significant difference was found, interestingly, between CC stretches and out-of-plane bends.) If we consider ν(obsd)/ω(B3LYP/cc-pVTZ instead, the distinction between b and c is no longer required (the average scaling being 0.975), while the high anharmonicity of a requires singling them out, with scale factor 0.965. To test the performance of these scaling factors for related molecules, we have applied them to our earlier calculations23 on benzene. As seen in Table 2, the performance of this simple scaling procedure is nothing short of impressive: with the exception of the CH stretches (for which anharmonic effects are simply too important to be absorbed entirely in a simple scaling), all fundamentals are reproduced to better than 13 cm-1 for the B3LYP/cc-pVDZ calculations and to better than 7 cm-1 for the B3LYP/cc-pVTZ calculations. This kind of accuracy should normally be more than enough to resolve doubtful assignmentssagain, presupposing that no severe Fermi resonances are involved. From our scaled frequencies we can now reconsider the assignments for naphthalene (Table 1). In the ag block, all assignments are confirmed. The suggested alternative assignment30 of 1163 cm-1 for the 6ag band remains a distinct possibility: it lies somewhat closer to the scaled B3LYP/ccpVTZ frequency than the accepted value of 1145 cm-1, but the difference is really too close to call. The suggested alternative assignment31 of 1099 cm-1 for the 6b3g band, however, can be rejected with certainty now, while another32 of 1145 cm-1 remains a possibility. As for the 4b3g band, not only does 1458

cm-1 remain an acceptable alternative for the 1438 cm-1 assignment, as suggested by SPB, but the calculations would actually favor the 1458 cm-1 alternative. In the b1u block, our calculations confirm the Krainov assignment except for 753 cm-1 for the 7b1u band, which is clearly incorrect. The 810 cm-1 assignment would have been plausible but was rejected by SPB because it disappears at low temperatures.33 We concur with SPB that the 7b1u band was probably not observed: it is calculated here to have a very low IR intensity. Our calculations completely confirm the Krainov assignment for the b2u block: while the suggested alternative32 of 1163 cm-1 for 5b2u cannot be rejected with absolute certainty, our calculations clearly favor the 1138 cm-1 of Krainov. As pointed out by SPB, very little actual experimental information exists for the au bands, which are both IR and Raman inactive. Krainov’s assignments are based on crystal bands: those for 1au, 2au, and 4au are probably all correct, while the 3au band is clearly misassigned. There is no disagreement for the b3u block. For the b1g block, we again confirm the Krainov assignment: our calculations support SPB in rejecting the alternative assignments of Behlen et al.,30 who suggested that 2b1g is 620 rather than 717 cm-1 and that 3b2g and 4b2g should be reassigned to 465 and 395 cm-1, respectively. Of the two alternative assignments offered by Krainov for 3b2g, 846 and 770 cm-1, our calculations definitely favor the latter, in agreement with previous suggestions by Hanson and Gee34 and by SPB. In summary, our simple scaling procedure appears to be a powerful tool for resolving these assignments, at least if Fermi resonance is not a major factor in the vibrational spectrum. Geometry. The empirical corrections to B3LYP/cc-pVn Z geometries used in the present work were previously successfully applied to the azabenzenes.6 Specifically,23 they involve subtracting the average overestimate of a bond length for a particular bond order: the mean absolute error on the thus corrected geometries for a number of reference molecules (taken from ref 23) can be used as an approximate error bar. The agreement between corrected B3LYP/cc-pVDZ and B3LYP/ cc-pVTZ distances can be used as an additional gauge for the reliability of our predicted re distances. As generally found in refs 6 and 23, the corrected CH distances are in nearly total agreement between the two levels of theory. Contrary to experimentswhere CH bond distances are the hardest to determine and are indeed absent in the experimental data for naphthalenesCH distances are by far the easiest to compute. For the CC distances, the bond orders create some ambiguity, since some bonds have bond order 4/3 and others 5/3 according to a canonical resonance picture. However, taking this into account did not significantly affect the agreement between corrected B3LYP/cc-pVDZ and B3LYP/cc-pVTZ distances. We have therefore adopted a bond order of 3/2 throughout the present work for determining the empirical correction to such bonds. Agreement with the various crystallographical35,36,37 and electron diffraction38,39 geometries (Table 3) is as good as can reasonably be expected, given the underlying differences between those kinds of structures. The geometric parameters are defined in Figure 1. Azulene. As an additional test, we will consider the azulene molecule, which is an isomer of C10 H8 like naphthalene. Structure and Dipole Moment. Azulene has the same chemical formula (C10H8) and number of π electrons as naphthalene, yet there the similarity ends. It derives its name from its typical blue color. Hu¨ckel theory predicts the molecule

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J. Phys. Chem., Vol. 100, No. 38, 1996 15361

TABLE 3: Computed, Empirically Corrected, and Observed Bond Distances (Å) of Naphthalenea B3LYP corrected B3LYP corrected X-ray e- diffraction SQM a

cc-pVDZ cc-pVDZ cc-pVTZ cc-pVTZ ref 35 ref 36 ref 37 ref 38 ref 39 SPB28

R1

R2

R3

R4

r1

r2

1.4188 1.410(3) 1.4116 1.412(2) 1.415 1.411 1.417 1.412 1.417 1.417

1.3788 1.370(3) 1.3703 1.371(2) 1.364 1.377 1.377 1.371 1.381 1.373

1.4233 1.414(3) 1.4162 1.416(2) 1.421 1.424 1.425 1.422 1.422 1.424

1.4354 1.426(3) 1.4280 1.428(2) 1.418 1.421 1.424 1.420 1.412 1.426

1.0924 1.079(4) 1.0819 1.079(2)

1.0932 1.080(4) 1.0829 1.080(2)

Statistical uncertainties (see the text) for the empirically corrected geometries are indicated in parentheses.

Figure 1. Definition of geometric parameters.

to have a very high dipole moment of 6.9 D,40 yet experimental values are 1.08 D in benzene solution41 and 0.79 ( 0.01 D (Stark effect) in the gas phase.42 The computed dipole moment is very sensitive to the electron correlation treatment: in a recent study by Grimme,43 µ drops from 1.52 D at the HF/6-31G* level to 1.04 D at the MP2/6-31G* level and to 0.93 D in a π-electron multireference CI calculation. Grimme also finds that, at the SCF level, a symmetry-broken planar structure is found to be the global energy minimum (Cs symmetry only), which corresponds to a localization of one of the two equivalent canonical resonance structures. C2V symmetry was assumed in deriving the various experimental structures44-46 of the carbon skeleton of azulene. At the B3LYP/cc-pVDZ and B3LYP/cc-pVTZ levels, the computed dipole moment are 1.05 and 1.01 D, respectively. The remaining difference with the Stark effect value is probably at least partly due to anharmonicity:

()

µ0 ) µe + ∑〈xi - xi,e〉 i

∂µ

∂xi

+ ... e

The computed structures are all presented in Table 4. Both B3LYP/cc-pVDZ and B3LYP/cc-pVTZ calculations clearly find C2V symmetric ground-state structures. This is consistent with the known fact (see, for example, refs 7 and 8 for some very clear illustrations involving carbon clusters) that SCF tends to be underestimate the delocalization energy, while MP2 and other

low-level electron correlation methods tend to exaggerate it and more sophisticated electron correlation methods are required to obtain the correct result. It was also seen in ref 7 that B3LYP consistently predicts the qualitatively correct structure even in cyclic carbon clusters, which are notorious8 for qualitatively incorrect structures at all but the best levels of theory. Negri and Zgierski (NZ)47 found an imaginary 13b2 frequencysimplying a Cs structure corresponding to a localized canonical resonance structuresat the MP2/STO-3G level (Table 5). This is not surprising, since a widespread consensus exists48-50 that “the smallest acceptable basis sets for electron correlation will be of split valence plus polarization [...] or DZP type”.50 NZ do find a real frequency, albeit anomalously low, at the HF/6-31G level: it is recalled that at the HF/6-31G* level the frequency is again imaginary.43 A quantum consistent force field (QCFF51)-CI calculation52 by NZ, however, finds a frequency of 1039 cm-1, in quite good agreement with the experimental fundamental of 960 cm-1. As seen in Table 5, both B3LYP/cc-pVDZ and B3LYP/ccpVTZ compute this band in excellent agreement with experiment, namely 971 and 968 cm-1, respectively. We can therefore safely dismiss the idea of a Cs structure. Using the empirical corrections proposed in ref 23, we can again provide estimates for the re geometry (Table 4). As usual, the corrected CH distances are in nearly total agreement between corrected cc-pVDZ and cc-pVTZ geometries: no experimental data are available. Among the CC distances, the common bond between the two rings and the one on the seven-ring that is in resonance position to it exhibit significant differences between cc-pVDZ and cc-pVTZ basis sets. Obviously, the ones with the larger basis set are to be preferred since the empirical correction terms for them are much smaller. Our best re geometry compares quite well with the X-ray diffraction structure of Hanson,44 but not with the older values of Robertson et al.45 Deviations from electron diffraction results46 are such as can be expected for an re-rg comparison. Vibrational Frequencies. Infrared spectra of four different isotopic species of azulene were published by van Tets and Gu¨nthard (TG),53 while a recent paper by Bree, Pal, and Taliani (BPT)54 covers both the IR and Raman spectra. The vibrational assignment has very recently been revised by Kozlowski, Rauhut, and Pulay (KRP)5 on the basis of an SQM (scaled quantum mechanical) B3LYP/6-31G* calculation. An overview of the results is given in Table 5. In the a1 block, our calculations are entirely consistent with the KRP assignment: residuals are particularly small with the cc-pVTZ basis set. (Note that no elaborate internal coordinate definition or scaling of the force constant matrix is involved here.) The largest residual in the b2 block, apart from the CH stretches, is found in 13b2, which corresponds to distortion to one of the localized canonical resonance structures. Since this mode can legitimately be considered to have appreciable

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Martin et al. multireference character and it is quite sensitive to the level of theory, it testifies to the power of the B3LYP method that the residual is as small as it is. No problems at all are seen in the b1 block. While very little experimental data are available for the a2 block, we can state that the assignment of the 425 cm-1 band to 5a2 is probably correct and that, for the 6a2 band, the assignment of 165 cm-1 from BPT appears to be clearly favored by our calculations over the 189 cm-1 proposed by TG.

TABLE 4: Computed, Empirically Corrected, and Observed Bond Distances (Å) of Azulenea electron B3LYP B3LYP corrected corrected X-ray X-ray diffraction cc-pVDZ cc-pVTZ cc-pVDZ cc-pVTZ ref 45 ref 44 ref 46 1.4073 1.4086 1.5013 1.3943 1.4002 1.4001 1.0909 1.0892 1.0958 1.0936 1.0950

R1 R2 R3 R4 R5 R6 r1 r2 r3 r4 r5

1.3996 1.4011 1.4958 1.3862 1.3962 1.3922 1.0801 1.0787 1.0856 1.0832 1.0844

1.398(3) 1.400(3) 1.492(3) 1.385(3) 1.391(3) 1.391(3) 1.077(4) 1.076(4) 1.082(4) 1.080(4) 1.081(4)

1.400(2) 1.401(2) 1.496(2) 1.386(2) 1.396(2) 1.392(2) 1.078(2) 1.076(2) 1.083(2) 1.081(2) 1.082(2)

1.391 1.413 1.483 1.383 1.401 1.385

1.394 1.398 1.498 1.391 1.400 1.392

1.399 1.418 1.501 1.383 1.406 1.403

From comparing the B3LYP/cc-pVDZ and B3LYP/cc-pVTZ harmonic frequencies (Table 5), we see that the basis set sensitivity is really moderate: the only significant effects seen are on 6a1 (-17 cm-1), 5b2 (-14 cm-1), 9b2 (+14 cm-1), and 10b2 (+12 cm-1). Of these, the first is a symmetric seven-ring “compression” mode, the second resembles a Kekule´-type

a Statistical uncertainties (see the text) for the empirically corrected geometries are indicated in parentheses.

TABLE 5: Computed and Observed Frequencies of Azulene (cm-1; Infrared Intensities in km/mol Given in Parentheses) B3LYP/ cc-pVDZ raw

B3LYP/ cc-pVTZ raw

B3LYP/ cc-pVDZ scaled

B3LYP/ cc-pVTZ scaled

KRP assignment ref 5

3231 (11) 3203 (7) 3178 (22) 3149 (7) 3139 (0.2) 1648 (60) 1582 (7) 1487 (28) 1420 (76) 1302 (0.7) 1238 (0.3) 1074 (7) 965 (2) 910 (2) 830 (5) 682 (1) 411 (1)

3224 (10) 3197 (7) 3168 (22) 3140 (6) 3130 (0.2) 1631 (59) 1583 (6) 1490 (16) 1421 (94) 1302 (3) 1245 (0.51) 1084 (6) 963 (3) 920 (3) 833 (6) 681 (1) 413 (2)

3102 3075 3051 3023 3013 1599 1555 1462 1387 1263 1209 1049 936 896 810 662 405

3111 3085 3057 3030 3020 1590 1543 1453 1385 1269 1214 1057 939 897 812 664 403

a1 3089 3080 3063 3032 3022 1578 1534 1453 1392 1264 1212 1054 939 897 821 676 404

3222 (13) 3169 (32) 3141 (13) 1650 (4) 1545 (9) 1482 (6) 1424 (0.6) 1331 (1) 1311 (4) 1232 (7) 1182 (0.2) 1060 (0.03) 1023 (13) 746 (0.4) 498 (2) 337 (0.7)

3216 (12) 3159 (31) 3131 (13) 1644 (4) 1531 (9) 1493 (8) 1432 (0.04) 1339 (1) 1325 (2) 1244 (7) 1024 (13) 1188 (0.39) 1066 (0.57) 747 (0.39) 499 (2) 339 (0.82)

3093 3042 3015 1601 1509 1447 1391 1300 1289 1211 1147 1028 992 728 486 331

3103 3048 3021 1603 1493 1456 1396 1306 1292 1213 1158 1039 998 728 487 331

b2 3071 3058 3042 1586 1476 1453 1392 1303 1288 1207 1153 1034 966 708 480 332

1017 (10-4) 983 (4) 945 (0.03) 789 (80) 746 (3) 613 (1) 576 (7) 326 (8) 172 (1)

1016 (0.07) 982 (6) 942 (0.05) 787 (110) 745 (3) 615 (1) 577 (9) 323 (10) 171 (2)

986 954 917 765 724 595 559 316 167

991 957 918 767 726 600 563 315 167

1001 884 805 740 432 167

1000 881 801 732 432 164

971 857 781 718 419 162

975 859 781 714 421 160

SQM B3LYP/ 6-31G* 5

HF/6-31G scaled × 0.9 ref 47

MP2/STO-3G scaled × 0.9 ref 47

QCFF/PI CISD ref 47

3109 3083 3054 3027 3017 1582 1536 1449 1384 1262 1207 1051 929 901 811 655 405

1621 (120) 1562 (5) 1470 (17) 1408 (199) 1274 (4) 1234 (1) 1066 (5) 937 (2) 914 (4) 824 (13) 660 (1) 408 (2)

1577 (48) 1559 (0) 1477 (12) 1428 (61) 1287 (9) 1216 (1) 1064 (2) 928 (3) 902 (1) 822 (4) 649 (1) 395 (1)

1571 1560 1472 1467 1277 1182 1087 992 957 808 714 458

3101 3045 3019 1598 1489 1449 1390 1302 1286 1209 1155 1050 (0) 1002 732 487 331

1618 (9) 1495 (21) 1476 (15) 1408 (1) 1314 (1) 1291 (8) 1212 (3) 1136 (0) 1038 (2) 57 (24) 860 (10) 536 (0) 336 (1)

1606 (0) 1506 (8) 1477 (6) 1412 (0) 1298 (2) 1283 (0) 1208 (0) 1127 (0) 1075 548i (116) 808 (9) 516 (3) 321 (0)

1597 1498 1473 1402 1312 1265 1222 1150 1039 726 491 411

975 949 908 765 726 598 562 316 169

1067 (0) 1018 (10) 990 (0) 806 (177) 755 (5) 619 (5) 586 (12) 320 (13) 171 (1)

1002 (0) 955 (3) 925 (0) 782 (27) 728 (7) 598 (2) 561 (1) 302 (4) 163 (1)

1084 1000 935 812 737 612 519 334 182

966 857 776 716 423 164

1044 (0) 905 (0) 833 (0) 717 (0) 430 (0) 165 (0)

980 (0) 861 (0) 808 (0) 701 (0) 418 (0) 159 (0)

987 967 835 658 419 186

b1 954 757 728 595 560 317 171 a2

a

[908] [795] [744] 425 165a

Following BPT.54 Assignment of 189 cm-1 from van Tets and Gu¨nthard53 is probably incorrect.

Polycyclic Aromatic Compounds

J. Phys. Chem., Vol. 100, No. 38, 1996 15363

TABLE 6: Computed and Observed Frequencies of Phenanthrene (cm-1; Infrared Intensities in km/mol Given in Parentheses) B3LYP/cc-pVDZ raw scaled

ref 61

experiment ref 60

ref 55

Cyvin et al.56 assignment

remarks and alternatives

SQM MNDO57 a

a1 3214 (21) 3195 (1) 3184 (35) 3179 (14) 3167 (4) 1671 (0.3) 1652 (3) 1567 (1.5) 1471 (2.5) 1453 (2.3) 1388 (0.9) 1325 (2.4) 1260 (9) 1229 (3) 1178 (0.3) 1165 (0.05) 1111 (2) 1065 (1) 841 (0.06) 723 (0.07) 555 (0.3) 411 (0.5) 248 (0.4)

3085 3067 3057 3052 3040 1621 1602 1520 1436 1419 1355 1294 1230 1192 1150 1145 1085 1033 827 706 546 401 244

... 3075 3067 3056 3020 1608 1567 1524 1441 1417 1365 1303 1244 1200 1160 1142 1100 1038 830 710 540 406 250

3086 3063 3046 3038 3006 1602 1522 1441 1420 1350 1295 1245 1202 1165 1142 1093 1037 865 831 711 615 547 407

3082? 3072 3057 3037 3002? 1626 1602 1526 1443 1431 1352 1304 1247 1203 1163? 1144 1094 1038 832 711 548 408 247

3203 (13) 3193 (44) 3178 (0.04) 3168 (2) 3164 (0.3) 1666 (0.3) 1618 (0.1) 1538 (6) 1486 (12) 1438 (0.09) 1388 (0.24) 1293 (0.0003) 1237 (0.5) 1177 (0.4) 1158 (2) 1061 (6) 1015 (1.1) 886 (1.6) 725 (2) 629 (5) 504 (0.8) 446 (2)

3075 3065 3051 3041 3037 1616 1580 1502 1451 1404 1355 1263 1208 1149 1131 1036 991 865 708 618 495 438

3102 3064 3024 1670 1548 1500 1458 1430 1303 1220 1148 1039 1002 618 -

3100 3071 3056 3021 3017 1622 1565 1500 1457 1430 1352 1303 1280 1223 1206? 1147 1001 980 618 442 398 240

3094 3064 3047 s 3019 1616 1572 1502 1458 1340? 1282 1227 1173? 1144 1040 1001 876 712 619 536 441

1003 990 962 912 824 777 603 554 404 246 100

973 960 933 885 799 754 585 537 392 239 97

928 880 761 594 513 352 123

944 811 765 -

1159? 969 946 791? 763 761 594 395? -

1004 (0.004) 970 (2) 894 (8) 839 (33) 756 (59) 738 (0.04) 515 (3) 441 (6) 233 (2.8) 102 (0.6)

974 941 867 814 733 716 500 428 226 99

951 874 819 735 713 494 441 427 233 -

950 870 818 793? 733 714 497 428 -

1149 950 871 817 732 715 495 426 234 124

3092 3084 3080 3076 3073 1629 1605 1536 1463 1429 1338 1290 1279 1205 1183 1123 1106 1072 812 744 543 402 247 b2

1148 prob. better

implausible: 513 cm-1 (w)?

3091 3083 3075 3073 3068 1613 1571 1505 1436 1381 1284 1266 1219 1165 1158 1095 1020 862 695 600 498 437

a2 -

prob. not observed 969 946 880 791?

clearly 880 clearly 791

513 352 -

536 more plausible implausible; 395 more plausible

950 871 817 732 715 495 426 234 124

prob. not observed

1011 954 943 911 821 785 599 562 384 238 78

b1

a

Based on naphthalene scale factors.

973 935 904 835 764 731 520 453 229 94

15364 J. Phys. Chem., Vol. 100, No. 38, 1996 vibration, and the latter two correspond to pinwheel-type motions on the seven-ring and five-ring, respectively. Phenanthrene. For phenanthrene, a set of vibrational frequencies (IR and Raman) is available from the work of Bree, Solven, and Vilkos (BSV).55 An assignment using a fiveparameter model for the force field has been performed by Cyvin et al.56 Rougeau et al.57 determined an SQM (scaled quantum mechanical14) force field based on MNDO (modified neglect of diatomic overlap58) calculations: the various scaling factors involved in the SQM treatment were obtained from comparing computed and observed frequencies for benzene, naphthalene, and their completely deuterated isotopomers (see refs 57 and 59 for details of the procedure). Given the size of the system, we have only performed B3LYP/cc-pVDZ calculations in this case. The results are given in Table 6. It is seen there that very good agreement between computed and observed frequencies has again been achieved: any deviations are consistent with the expected anharmonic contributions. In the a1 symmetry block, no mismatches are found with the experimental assignment of BSV. There are apparently serious qualitative problems with the five-parameter force field of Cyvin et al.,56 which is not surprising given its simplicity. Agreement between computed B3LYP and observed frequencies is particularly good for the lower bending modes. Of the two sets of SQM frequencies, the one based on naphthalene scaling factors appears to be the better one (as expected). Both sets, however, appear inferior even to the uncorrected B3LYP results which, on present-day fast computers, can be obtained in a matter of days. It should be noted that the observed CH stretching fundamentals span a considerably wider range than the computed CH stretching harmonics, which illustrates the importance of anharmonicity for these bands. Our assignment for the b2 block largely agrees with that of BSV. For the 11b2 mode, however, both the tentative assignment to 1340 cm-1 of BSV and the 1352 cm-1 value proposed in ref 60 would appear to be possibilities; however, the latter band was assigned as a1 based on the depolarization ratio by BSV, and its frequency is indeed nearly “spot on” our scaled 11a1 frequency. For 14b2, our calculations prefer the 1148 cm-1 assignment in ref 61. This latter bands was assigned to 1b1 by BSV, but this is almost certainly incorrect in view of the scaled computed b1 frequency of 974 cm-1. While the assignment of 21b2 in ref 60 is definitely incorrect, so is probably the one by BSV. Two bands are missing experimentally and have a low computed intensity: they are calculated at 1355 and 1404 cm-1 (scaled). The conspicuously large difference with the SQM values for the former mode (which is 71 cm-1 lower for the naphthalene-based model) illustrates the shortcomings of the MNDO method even if subjected to the SQM procedure. Again, problems with the Cyvin et al. five-parameter force field are numerous. In the a2 block, our calculations by and large confirm the Cyvin et al.56 assignment, except for 9a2, where it appears to be incorrect, and the suggestion of BSV (395 cm-1) is almost spot on the calculation. In addition, the error on 8a2 (537 cm-1) seems somewhat out of character. If we however swap assignments with the 21b2 band, both 8a2 and 21b2 fit in quite nicely. The Cyvin et al. assignment for the b1 block likewise appears to be largely correct, although the relatiVe deviation between computed and observed frequency for the lowest band (10b1) is rather out of character, and this assignment seems questionable. It should be noted that the calculated frequency is in the same range as the lattice modes: it is not impossible that the

Martin et al. TABLE 7: Computed, Empirically Corrected, and Observed Bond Distances (Å) of Phenantrenea

R1 R2 R3 R4 R5 R6 R7 R8 R9 r1 r2 r3 r4 r5

B3LYP cc-pVDZ

corrected cc-pVDZ

X-ray ref 62

1.4169 1.3853 1.4095 1.3830 1.4168 1.4287 1.4373 1.4593 1.3622 1.0902 1.0922 1.0922 1.0931 1.0930

1.408(3) 1.376(3) 1.401(3) 1.374(3) 1.408(3) 1.420(3) 1.428(3) 1.450(3) 1.353(3) 1.077(4) 1.079(4) 1.079(4) 1.080(4) 1.079(4)

1.405 1.383 1.391 1.381 1.457 1.404 1.395 1.448 1.372

a Statistical uncertainties (see the text) for the empirically corrected geometries are indicated in parentheses.

109 cm-1 band (assigned by Bree et al. to the highest lattice vibration) and the 124 cm-1 band should have their assignments exchanged. On the other hand, significant coupling could exist. Also, the 10b1 band could lie in the “ill-defined region of scattering” noted by BSV around 99 cm-1. As a final note, BSV's tentative suggestions of 1159 cm-1 for the 1a2 and 1149 cm-1 for the 1b1 bands (the latter of which is calculated to have an extremely low IR intensity) both appear to be incorrect. The 1149 cm-1 band appears to belong to 14b2 instead; the 1b1 band probably has not been observed. Summarizing, we can state that the B3LYP method again gives a good description of the vibrational frequencies of phenanthrene and has helped resolve some assignment issues. The computed and corrected geometries (Table 7; see Figure 1 for the definition of the parameters) are only in semiquantitative agreement with the crystallographic work of Trotter.62 In the present case, given the quite significant bond length alternation, a B3LYP/cc-pVTZ geometry calculation would probably have been desirable. Anthracene. Because of the higher symmetry (D2h), vibrational assignments are comparatively easier in this molecule. A complete vibrational spectrum has been assigned by Bakke et al.63 for the in-plane, and Neerland et al.64 for the out-ofplane, frequencies: see also refs 65-68 for measurements. Again, SQM calculations were performed by Rougeau et al.57 based on MNDO force constant matrices and scaling factors derived from benzene and naphthalene. The electronic spectrum was studied ab initio and experimentally by Zilberg, Haas, and Shaik (ZHS),69 who also computed vibrational frequencies for ground and excited states at the HF/3-21G and CIS/3-21G levels, respectively. HF/3-21G ground-state frequencies and infrared intensities were also computed by Szczepanski et al.,70 who in addition published the observed IR spectrum in a noble gas matrix, including intensities. Computed and observed frequencies are given in Table 8. Again, very good agreement between computed B3LYP/ccpVDZ harmonics and observed fundamentals is found, except for such differences as are reasonable for anharmonic effects. Not surprisingly, the agreement with experiment of even the unscaled B3LYP frequencies is much superior to that for the HF/3-21G frequencies scaled by the customary factor of 0.89. The notable exception here is for the CH stretches, where the factor of 0.89 is supposed to absorb the anharmonic effects to some extent. No obvious mismatches are seen between B3LYP/cc-pVDZ and experimental assignments in the ag block. In the b3g block,

Polycyclic Aromatic Compounds

J. Phys. Chem., Vol. 100, No. 38, 1996 15365

TABLE 8: Computed and Observed Frequencies of Anthracene (cm-1; Infrared Intensities in km/mol Given in Parentheses) B3LYP/cc-pVDZ raw scaled

ZHS69 a

expt refs 63 and 64

remarks and alternatives

matrix IR ref 70

SQM MNDO57

HF/3-21Gb

ag 3197 3172 3164 1604 1517 1445 1294 1177 1034 765 638 398

3069 3045 3037 1556 1471 1402 1272 1157 1003 742 627 391

3062?d 3048 3027 1556 1480 1412 1264 1164 1007 754 625 397

3072 3048 3027 1556 1480 1412 (s)/1400 (vs) 1264 1164 1007 754 625 397

3185 3166 1679 1631 1404 1282 1200 1125 924 533 395

3058 3039 1629 1582 1380 1260 1180 1106 908 524 388

3054 3005 1627 1574 1433 1273 1187 1102 903 521 397

3054 3017/3005 1632/1627 1574 1433 1273 1187 1102??e 903 521 397 (ag?)

3185 (63) 3167 (12) 3162 (10) 1679 (6) 1486 (1) 1337 (6) 1281 (6) 1165 (7) 913 (2) 659 (1) 235 (1)

3058 3040 3036 1629 1441 1297 1259 1145 897 648 231

3084 3053 3007 1620 1448 1317 1272 1147 906 653 234

3084 3053 3007 1620 1448 1317 1272 1147 906 653 234

3197 (57) 3171 (0) 1587 (5) 1477 (1) 1427 (2) 1386 (3) 1182 (2) 1155 (2) 1030 (6) 825 (0) 614 (8)

3069 3044 1539 1433 1403 1362 1162 1120 999 800 604

3048 3021 1534 1495 1397 1346 1162 1124 998 809?? 601

3048 3021 1690 1534 1495 1397 1167 1124 998 809 601

976 778 491 239

947 755 476 232

956 747 477 242

956 760 479 244

1005 932 858 796 594 273

975 904 832 772 576 265

975 916 896 771 577 284

977 916 896 773 580 287

1004 851 774 512 123

974 825 751 497 119

958? 858 743 552 137

988 858 743 ... ...

981 (4) 908 (38) 749 (56) 485 (17) 390 (0) 93 (1)

952 881 727 470 378 90

952 892 732 504 383 96

956 883 737 474 380 106

3011 2984 2976 1541 1455 1366 1214 1182 955 733 636 383

3090 3074 3069 1562 1492 1433 1321 1186 1079 809 617 388

2997 2978 1620 1571 1391 1281 1193 1074 914 531 392

3083 3072 1639 1599 1345 1303 1214 1124 856 522 401

2997 (56) 2979 (7) 2974 (4) 1620 (16) 1450 (6.4) 1346 (2.0) 1272 (6.9) 1149, 1151 (5.3) 908 (3.2) 652 (1.3) ... (...)

3082 3072 3067 1637 1435 (1) 1300 (1) 1257 (8) 1156 (6) 908 (4) 640 (2) 229 (2)

1463 1302 1264 1152 847 631 241

b2u 307968 304868 clearly 1534; 153368 1450?; 146268 clearly 1397; 139868 clearly 1346 116968 106272 implausiblec

3068, 3067 (59.5) ... 1542, 1540 (5.3) 1460 (5.3) 1400 (2.7) 1318 (17.0) 1167, 1169 (4.8)

809

1001 (9.6) ... 603 (20.0)

3010 (62) 2982 (0) 1521 (7) 1443 (6) 1364 (1) 1269 (7) 1156 (1) 1044 (2) 940 (2) 756 (0) 605 (9)

3090 3074 1553 1448 1413 1237 1180 1078 1036 884 580

974 746 463 220

935 790 517 238

1003 923 850 760 564 254

1009 952

1001 845 733 479 116

977 905 783 510 108

980 (13) 899 (87) 718 (127) 456 (38) 365 (0) 86 (2)

950 896 759 518 356 95

39071 b3g

implausible; 137665

b1u 3062, 3055 (35.2) 3032 (7.3) 3022, 3017 (8.2) 1627 (16.3) 1460? prob. correct

b1g

b2g clearly wrong

824

781 552 249

au

clearly wrong b3u 955, 958 (10.1) 878.5 (95.8) 729, 726 (139.9) 470, 468 (40.5) ... (...) ... (...)

a Assignment compiled by these authors from refs 67, 66, 65, and 63. b Intensities from ref 70; scaled frequencies from ZHS.69 anthracene crystal.77 d ? means possible typo. e ?? means queried in original reference.

c

1068 cm-1 in

15366 J. Phys. Chem., Vol. 100, No. 38, 1996

Martin et al.

TABLE 9: Computed, Empirically Corrected, and Observed Bond Distances (Å) of Anthracenea

R1 R2 R3 R4 R5 r1 r2 r3

B3LYP cc-pVDZ

corrected cc-pVDZ

1.4282 1.3721 1.4322 1.4467 1.4029 1.0924 1.0931 1.0938

1.419(3) 1.363(3) 1.423(3) 1.438(3) 1.394(3) 1.079(4) 1.079(4) 1.079(4)

electron diffraction ref 73 1.422 1.397 1.437 1.437 1.392

a Statistical uncertainties (see the text) for the empirically corrected geometries are indicated in parentheses.

the experimental value of 1433 cm-1 for 5b3g seems an obvious misassignment; the 1376 cm-1 proposed by Ra¨sa¨nen et al.65 agrees quite well with the scaled frequency and seems to be the obvious alternative. Bakke et al. assigned the 397 cm-1 line to the lowest vibrations of both ag and b3g symmetry: Krainov27 assigned a band at 390 cm-1 in the fluorescence spectrum to 12ag. While this may well actually be the same vibration as the 397 cm-1 band in azulene crystal, it might be a different band and constitute an alternative 12ag assignment. All Bakke et al.63 assignments in the b1u block appear to be correct. In the b2u block, however, their assignments of 3b2u through 6b2u appear to be spurious: the 1690 cm-1 band is obviously not a b2u fundamental, leaving the 1534, 1495, and 1397 cm-1 bands to be 3b2u through 5b2u, while 6b2u clearly belongs to a band observed at 1346 cm-1 as suggested in ZHS. Contrary to the scaled HF/3-21G values, the present calculations do support the uncertain assignment of the weak 809 cm-1 band to 10b2u: the B3LYP calculation predicts a very low IR intensity. Note that the alternative assignment of Neto et al.68 of 3079 and 3048 cm-1 for 1b2u and 2b2u, respectively, fits our scaled computed values of 3069 and 3044 cm-1 substantially better than the Bakke et al. assignment of 3048 and 3021 cm-1, respectively; however, given the importance of anharmonicity in this region it is hard to say conclusively which is the correct assignment. On the basis of their scaled HF/3-21G calculations, Szczepanski et al. swapped the assignments of the 1346 and 1317 cm-1 bands to 6b1u and 6b2u, respectively. The present calculations, however, find these bands at 1297 and 1362 cm-1, respectively. Allowing for anharmonicity, these results fit the unswapped assignment69 quite well but are incompatible with the suggestion of Szczepanski et al. An alternative assignment72 of 1068 cm-1 for the 8b2u band can safely be rejected on the basis of the present calculations: this was not so clear from the HF/3-21G results. As for the out-of-plane modes, the b3u and b1g assignments67,64 all appear to be correct. In the au block, the error in 2au looks a bit on the large side, suggesting that a misassignment might be involved. The assignment of 552 cm-1 proposed67 for 4au can be rejected with confidence. The b2g frequencies are probably all correctly assigned, with the clear exception of 3b2g. In summary, the B3LYP calculations were found to be of considerable value in verifying the experimental assignment. As for the geometry (Table 9; see Figure 1 for the definition of the geometric parameters), our calculations are only in semiquantitative agreement with the electron diffraction results of Ketkar et al.,73 but since no significant bond order alternation exists here, there is little reason to question the reliability of even the corrected B3LYP/cc-pVDZ results. Again, no bond distances or angles involving the hydrogen atoms are available, and our calculations probably represent the most accurate results available.

Conclusions In our study, we have been able to establish the following: (1) an alternative for the SQM procedure is to run a B3LYP/ cc-pVTZ frequency calculation and scale all CH stretching frequencies by 0.965 and all others by 0.975; (2) if only a B3LYP/cc-pVDZ calculation is feasible, all CH stretches should be scaled by 0.960, all in-plane bends by 0.983, and all remaining vibrations by 0.970; (3) the resulting model reproduces the fundamental frequencies for benzene and naphthalene to better than 10 cm-1, on average, with some exceptions for the CH stretches; (4) the model is somewhat transferable to aromatic hydrocarbons other than polyacenes; (5) a completely revised vibrational assignment of phenanthrene is proposed; (6) several reassignments in the vibrational spectrum of anthracene have been carried out; (7) basis set sensitivity, as measured by the difference between B3LYP/spd and B3LYP/spdf frequencies, is quite modest except for specific types of modes such as Kekule´ vibrations (or linear combinations thereof) and “pinwheel-type” concerted hydrogen motion; (8) estimated re geometries are proposed for naphthalene, azulene, phenanthrene, and anthracene. The power of the B3LYP method, with appropriate basis sets, for assigning and elucidating vibrational spectra is therewith once more confirmed. Note Added in Proof: After completion and submission of the present work, a paper was published by S. R. Langhoff (J. Phys. Chem. 1996, 100, 2819) concerning B3LYP frequency calculations with double-ζ basis sets on the infrared active bands of several polyacenes, including naphthalene and anthracene, and polyacene cations. His results for naphthalene and anthracene are in very good agreement with those in the present work. Acknowledgment. J.M.L.M. is an NFWO/FNRS (National Science Foundation of Belgium) Senior Research Associate (“Onderzoeksleider”). J.E.Y. acknowledges a graduate fellowship from the “Stimuleringsfonds” of the LUC. The authors would like to thank Dr. Shmu’el Zilberg and Prof. Peter Pulay for helpful discussions. This work constitutes part of the research results of project IUAP 48 (Characterization of Materials) sponsored by the Office for Science Policy Programming of the Prime Minister’s Cabinet. Supporting Information Available: B3LYP/cc-pVDZ and B3LYP/cc-pVTZ optimized geometries in Cartesian coordinates are available on the Internet only. Access information is given on any current masthead page. References and Notes (1) Parr, R. G.; Yang, W. Density functional theory of atoms and molecules; Oxford University Press: Oxford, U.K., 1989. (2) Handy, N. C. In Lecture Notes in Quantum Chemistry II. Lecture Notes in Chemistry; Roos, B. O., Ed.; Springer: Berlin, 1994: Vol. 64. (3) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (4) Rauhut, G.; Pulay, P. J. Phys. Chem. 1995, 99, 3093. (5) Kozlowski, P. M.; Rauhut, G.; Pulay, P. J. Chem. Phys. 1995, 103, 5650. (6) Martin, J. M. L.; Van Alsenoy, C. J. Phys. Chem. 1996, 100, 6973. (7) Martin, J. M. L.; El-Yazal, J.; Franc¸ ois, J. P. Chem. Phys. Lett. 1995, 242, 570. (8) Martin, J. M. L.; Taylor, P. R. J. Phys. Chem. 1996, 100, 6047. (9) Allamandola, L. J. Topics Current Chem. 1990, 1. Allamandola, L. J.; Tielens, A. G. G. M.; Barker, J. R. Astrophys. J. Suppl. Ser. 1989, 71, 733. Le´ger, A.; Puget, J. L. Astron. Astrophys. 1984, 137, L5. DeFrees, D. J.; Miller, M. D.; Talbi, D.; Pauzat, F.; Ellinger, Y. Astrophys. J. 1993, 408, 503 and references therein.

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