8782
J. Phys. Chem. 1996, 100, 8782-8785
Density Functional Theory Study of Vibrational Spectra of Fluorene Sang Yeon Lee† and Bong Hyun Boo*,‡ Department of Chemistry, Chungnam National UniVersity, Taejon 305-764, Korea, and Center for Molecular Science, 373-1 Kusung-dong, Yusung-gu, Taejon 305-701, Korea ReceiVed: January 2, 1996X
Ab initio Hartree-Fock and density functional theory calculations using the 6-31G* basis set were carried out to study the molecular structure and vibrational spectrum of fluorene (FR). Two density functionals were employed: the (1) Becke-Lee-Yang-Parr (BLYP) functional and (2) the Becke 3-Lee-Yang-Parr (B3LYP) functional. Comparison of the calculated and experimental vibrational spectra reveals that the B3LYP calculations are quite accurate in predicting the vibrational frequencies and modes. We report the results of new assignments of the fundamental vibrational frequencies obtained on the basis of the calculations. The new assignments show a one-to-one correspondence between the observed and calculated fundamentals.
Introduction
TABLE 1: Equilibrium Geometrical Parameters of Fluorene in the Ground Statea
The electronic-vibrational structures of fluorene (FR) and its van der Waals complexes have been studied extensively by fluorescence excitation1-5 and resonance two-photon ionization6,7 spectroscopy since the FR molecule is well-known to fluoresce efficiently upon excitation of S0 to S1. Also, the ultrafast dynamics of exciton transfer8-10 and excimer formation11 of fluorene derivatives has been studied by time-resolved fluorescence spectroscopy. Of particular importance is the determination of fundamental vibrational frequencies and modes in the ground state in order to provide insight into the vibronic spectra in the electronically excited state. As revealed in the literature, the vibrational progression appended to the 0-0 band origin in FR includes fundamentals, overtones, second overtones, and combination bands as well as fundamentals.1-7 Exact assignments of fundamentals in the ground state are necessitated for the determination of overtone and combination bands in the excited state. In 1969, Bree and Zwarich (BZ) have published a detailed experimental study of the infrared and Raman spectra of FR.12 Several vibrational modes were distinguished by the different polarization ratios in the polarized infrared spectra in crystals. Although the work of BZ is excellent in the assignment of symmetry to the observed vibrational frequencies, there are some missing in the assignment of a2 modes owing to the forbidden infrared selection rule and to the very weak Raman intensities of those modes. Also, there is a little ambiguity in the assignments of fundamental modes of other symmetries due to their low intensities. Thus, a reliable theoretical study is required to predict the frequencies of a2 modes in the vibrational spectra and to distinguish fundamentals and modes among the peaks appearing in the spectra. Recent ab initio calculations of vibrational spectra using density functional theory predict fundamentals of molecules with high accuracy.13-18 It is reported that density functional theory (DFT) reproduces experimental vibrational frequencies with higher accuracy than obtained by the HF and MP2 calculations. Even when a uniform scaling is performed for the computed vibrational frequencies, the DFT study shows better consistency with the experimental frequencies than the HF and MP2 ones.13a,14,15 As an example, we wish to consider a recent the†
Postdoctoral fellow at the Center for Molecular Science. Chungnam National University and the Center for Molecular Science. X Abstract published in AdVance ACS Abstracts, May 1, 1996. ‡
S0022-3654(96)00020-2 CCC: $12.00
parameters C1-C2 C1-C10 C2-C3 C3-C4 C4-C11 C9-C10 C10-C11 C11-C12 C1-H C2-H C3-H C4-H C9-H C1-C2-C3 C1-C10-C11 C2-C3-C4 C2-C1-C10 C3-C4-C11 C4-C11-C10 C4-C11-C12 C9-C10-C11 C10-C9-C13 C10-C11-C12 C11-C4-H C2-C1-H C3-C2-H C4-C3-H H-C9-H
HF 1.389 1.381 1.388 1.386 1.385 1.514 1.395 1.475 1.076 1.076 1.076 1.076 1.087 120.5 120.5 120.6 119.0 118.9 120.5 131.0 110.3 102.3 108.5 121.0 120.2 119.7 119.7 107.2
MP2
BLYP
Bond Lengths 1.399 1.410 1.393 1.400 1.401 1.409 1.396 1.406 1.397 1.407 1.511 1.526 1.409 1.423 1.465 1.476 1.089 1.095 1.088 1.094 1.088 1.094 1.089 1.094 1.098 1.107 Bond Angles 120.6 120.5 120.5 120.5 120.7 120.7 118.9 119.1 118.6 118.9 120.7 120.3 130.8 131.0 110.1 109.9 102.7 102.8 108.5 108.7 121.0 120.8 120.2 120.2 119.6 119.8 119.7 119.7 106.8 106.0
B3LYP QCFF/PIb X-rayc 1.400 1.390 1.399 1.396 1.397 1.516 1.411 1.470 1.088 1.087 1.087 1.087 1.099 120.5 120.4 120.6 119.1 118.9 120.4 131.0 110.0 102.8 108.6 120.8 120.2 119.7 119.7 106.3
1.413 1.396 1.411 1.411 1.401 1.520 1.419 1.469
120.9 120.5 120.8 118.5 117.9 121.5 131.0 112.6 99.8 107.5
1.38 1.43 1.38 1.40 1.41 1.47 1.41 1.48
120.59 118.50 122.14 119.24 116.47 122.30 130.36 109.35 105.38 107.34
a Bond lengths in angstroms, and bond angle in degrees. b Reference 20. c Reference 21; standard deviation for X-ray parameters is 0.02 Å.
oretical work for the vibrational assignment of benzene.14,16 MP2 calculations obtained by using the 6-311G(d,p) basis set still underestimate the vibrational frequencies of benzene such as 707, 967, and 990 cm-1, the frequencies corresponding to the CCC out-of-plane bending, and overestimate that of 1309 cm-1, the frequency corresponding to a b2u mode which tends to dissociate into three acetylene molecules. These frequencies are, however, accurately predicted by the DFT calculation using BLYP.14,16 Rauhut and Pulay (RP) applied the Becke-Lee-Yang-Parr (BLYP) functional and Becke 3-Lee-Yang-Parr (B3LYP) functional methods to 20 small molecules such as benzene, ether, and methanol, whose vibrational frequencies are exactly assigned, and then derived uniform scaling factors, 0.995 and 0.963, having root mean square (rms) deviations of 26.2 and © 1996 American Chemical Society
Vibrational Spectra of Fluorene
Figure 1. Comparison of the calculated and X-ray structural parameters of fluorene.
18.5 cm-1, respectively.17 It is reported in the study that the scaling factors are derived by the least squares procedure. When these scaling factors are applied to another 11 molecules, such as aniline, ethanol, and oxetane, the rms deviations turned out to be 26.9 and 19.7 cm-1 for the BLYP and B3LYP methods, respectively. By using the BLYP and B3LYP methods with the scaling factors, we performed the normal vibrational mode analysis of FR to predict the spectral positions of the missing lines and to distinguish fundamentals from the various vibrational frequencies reported previously by BZ. It is found in this study that this theoretical approach is sufficiently powerful to predict fundamentals as well as to estimate the structural parameters of FR in view of accuracy and computational effort. Calculations The molecular geometries are optimized at HF, MP2, BLYP, and B3LYP levels of theory with the 6-31G* basis set by using the Gaussian 94 program.19 Vibrational frequencies are computed with the HF, BLYP, and B3LYP methods and then scaled by 0.8929, 0.995, and 0.963, respectively. The frequencies and normal modes for FR were determined by diagonalizing the mass-weighted force constant matrix. Results and Discussion Geometrical Structures. The FR molecule is shown to have C2V symmetry in the gas phase. The numbering of the atoms is depicted in Figure 1. The optimized bond lengths and angles of FR calculated with the HF, MP2, BLYP, and B3LYP methods
J. Phys. Chem., Vol. 100, No. 21, 1996 8783 are displayed in Table 1. Those from quantum chemical force field/π electron (QCFF/PI)20 and X-ray study21,22 are also included for comparison. Some of the structural parameters are also represented in Figure 1. It is well-known that the HF method underestimates bond lengths, and inclusion of electron correlation at the MP2 level makes them approach the real experimental values. This theoretical pattern is also found for the FR molecule. As discussed by Johnson et al,18 the BLYP functional theory predicts bond lengths too long, particularly the C-H bond length. This overestimation is also verified in our calculation as represented in Table 1. By contrast, the B3LYP computed results are very close to the corresponding MP2 ones and are in good agreement with the reported X-ray data.21 The calculated bond angles are similarly accurate to within a few tenths of a degree with the methods used except the QCFF/PI method,20 which underestimates the C10-C9-C13 angle in particular. But the X-ray C9-C10 bond length is unusually shorter than all the optimized values. The origin of this difference may be the crystal packing force. Assignment of Fundamentals. The FR molecule involves 63 fundamentals having the various symmetries of 22a1 + 10a2 + 11b1 + 20b2. We seek to determine fundamentals making a one-to-one correspondence between the observed frequencies and the calculated ones. Fundamentals in the region ν < 2920 cm-1 are well separated, and thus the assignment is straightforward. The scaled vibrational frequencies and modes calculated with the BLYP, B3LYP, and HF methods are presented in Table 2. Our frequencies scaled with the reported factors by RP are in excellent agreement with our new assigned fundamentals and are found to have quite low values of root mean square deviations of 16.5, 14.6, and 35.7 cm-1 for BLYP, B3LYP, and HF, respectively. By contrast, the frequencies extracted by BZ as fundamentals deviate very much, being 60.3, 59.5, and 60.6 cm-1 for BLYP, B3LYP, and HF, respectively. Therefore, we conclude that the new fundamentals assigned here are well correlated with the calculated fundamentals, almost without exception, and that among the methods the B3LYP calculations are quite reliable in predicting the fundamentals, and moreover the B3LYP calculation predicts more reliable infrared spectral intensity than the HF/6-31G* calculation. However, the assignments of fundamentals 57-59 are less straightforward due to the slight band separations and/or probably to band overlapping. Combination of the vibrational frequencies in the C-H stretching region (ν > 2905 cm-1) from the frequencies extracted from the spectra by BZ and of our new assigned frequencies except the C-H stretching region (ν < 2905 cm-1) leads to only slightly larger rms deviations of 20.7, 17.4, and 33.5 cm-1 for BLYP, B3LYP, and HF. Almost all the modes are delocalized over the whole molecule and thus cannot be assigned to several local bonds. This is a characteristic feature of cyclic compounds, particularly aromatic compounds.23 Therefore, we represent in Table 2 only the approximate mode descriptions. a1 Symmetry. On the basis of our calculations and the reported infrared and Raman spectra by BZ, we made a reliable one-to-one correspondence between our fundamentals and any of our frequencies calculated with the BLYP, B3LYP, and HF methods. Twenty-two frequencies which we identify as a1 fundamentals are 217, 421, 628, 743, 857, 1016, 1089, 1143, 1186, 1231, 1291, 1349, 1397, 1440, 1480, 1570, 1612, 2920, 3048, 3064, 3072, and 3094 cm-1. It is noticed that two values, 1319 and 3018 cm-1, identified as a1 fundamentals by BZ are replaced by fundamentals at 1143 (ν36) and 3072 cm-1 (ν61). The peaks at 1143 and 3072 cm-1 are actually observed in the polarized infrared spectra and are identified as a1 symmetry.12 Two
8784 J. Phys. Chem., Vol. 100, No. 21, 1996
Lee and Boo
TABLE 2: Comparison of the Observed and Calculated Vibrational Spectraa exptlc species modeb a1
ν3 ν6 ν14 ν17 ν21 ν30 ν32 ν36 ν38 ν40 ν41 ν44 ν45 ν46 ν48 ν50 ν52 ν54
a2
b1
b2
ν57 ν59 ν61 ν63 ν2 ν5 ν8 ν12 ν16 ν19 ν23 ν25 ν27 ν34 ν1 ν4 ν7 ν9 ν15 ν18 ν22 ν24 ν26 ν28 ν55 ν10 ν11 ν13
BZd
ν42 ν43 ν47 ν49 ν51 ν53 ν56 ν58 ν60 ν62
HF/6-31G*
freq
IIR
freq
IIR
freq
IIR
IRA
DPf
approx mode descptg
217 (mw) 421(-)h 628(m) 743(-)h 857(w) 1016(vw) 1089(w) 1143(vw)i 1186(m) 1231(w) 1291(w)i
210 406 623 731 824 1017 1084 1154 1170 1220 1286
0.18 0.34 0.21 0.03 0.12 0.43 1.11 0.01 1.40 4.31 0.07
209 405 620 729 822 1016 1081 1147 1169 1218 1282
0.18 0.34 0.27 0.03 0.11 0.51 1.40 0.00 2.12 3.37 0.10
208 398 614 721 813 1005 1066 1104 1161 1183 1254
0.18 0.27 0.40 0.03 0.00 0.35 0.74 0.24 2.96 0.61 2.18
0.37 7.03 0.65 22.10 27.43 54.22 8.36 2.42 5.57 61.58 9.95
0.49 0.24 0.20 0.14 0.17 0.15 0.24 0.32 0.22 0.21 0.46
ip(CCC) + sci(CH2) ip(CCC and sci(CH2) ip(CCC and CH) + sci(CH2) ip(CCC and CH) + sci(CH2) sci(CH2) + ip(CCC and CH) ip(CCC and CH) ip(CCC and CH) + sci(CH2) ip(CH) ip(C1-H, C2-H, and C4-H) + ip(CCC) + sci(CH2) ip(CH and CCC) + sci(CH2) ip(CH and CCC)
1349(-)i 1397(s) 1440(s) 1480(-)h 1570(w) 1612(-)h 2920(m)
1354 1431 1443 1466 1567 1590 2933
0.11 10.29 2.07 0.04 1.24 0.05 19.68
1341 1426 1440 1469 1576 1604 2922
0.23 7.42 8.22 0.00 1.17 0.09 6.51
1287 1436 1440 1477 1585 1621 2862
3094
3048(m) 3064(s) 3072(sh) 3094(vw)
287
287(-)h
788
788(-)h
119 260
119(m) 260(vs) 410(s) 470(m) 693(mw) 735(vs) 841(w)i 873(w)i 910(mw)i 950(s)
3073 3080 3091 3103 135 270 429 556 718 773 847 894 932 1128 97 237 411 467 686 731 840 886 934 943
11.19 6.04 61.53 8.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.24 4.79 3.01 0.86 1.43 77.12 0.59 0.22 0.14 2.93
3059 3066 3076 3088 134 269 427 556 718 772 852 905 943 1126 96 236 411 465 686 732 844 894 942 947
8.90 4.83 1.41 6.86 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.32 5.32 3.53 0.92 1.71 9.60 0.69 0.15 2.38 1.17
2992 7.23 73.37 0.57 str(CH) 3000 11.78 62.39 0.58 str(CH) 3011 54.75 99.86 0.61 str(CH) 3023 11.28 532.24 0.11 tot sym str(CH) 132 0.00 0.00 0.75 oop(CCC) + twi(CH2) 272 0.00 11.36 0.75 oop(CCC) + twi(CH2) 428 0.00 1.82 0.75 oop(CCC) + twi(CH2) 559 0.00 0.24 0.75 oop(CH + CCC) 730 0.00 1.85 0.75 oop(CH) + twi(CH2) 782 0.00 0.02 0.75 oop(CH) + twi(CH2) 881 0.00 0.86 0.75 oop(CH and CCC) + twi(CH2) 953 0.00 1.73 0.75 oop(CH and CCC) 993 0.00 1.14 0.75 oop(CH and CCC) 1144 0.00 8.24 0.75 twi(CH2) 95 0.57 0.26 0.75 oop(CCC) + roc(CH2) 237 6.29 1.60 0.75 roc(CH2) + oop(CCC) 415 5.09 0.01 0.75 oop(CCC) + roc(CH2) 470 1.01 0.23 0.75 roc(CH2) + oop(CCC) 692 1.36 0.17 0.75 roc(CH2) + oop(CCC) 750 126.74 0.09 0.75 oop(CH) 872 0.73 4.45 0.75 oop(CH and CCC) + roc(CH2) 930 0.05 0.12 0.75 oop(CH and CCC) + roc(CH2) 972 3.48 0.67 0.75 roc(CH2) + oop(CH) 994 0.02 0.07 0.75 oop(CH) + roc(CH2)
2905(m) 487(mw) 542(m)i 618(ms)i
2955 481 534 616
14.98 0.18 0.20 7.59
2948 478 532 613
2.99 0.23 0.20 7.91
2889 473 527 609
21.24 0.38 0.11 8.42
786
0.26
786
0.24
777
0.70
994(m) 1023(w) 1103(s)i 1146(-)h 1172(-)h 1188(m)i
989 1022 1098 1148 1163 1189
1.64 3.23 0.32 0.44 3.25 0.62
987 1020 1097 1143 1158 1187
1.41 3.92 0.20 0.08 3.35 1.67
980 1011 1072 1093 1150 1174
1.76 5.09 17.10 1.27 2.44 1.84
0.64 0.02 27.06 10.30 0.11 4.45
0.75 0.75 0.75 0.75 0.75 0.75
str(CC) ip(CH and CCC) + wag(CH2) ip(CH and CCC) ip(CH) + wag(CH2) wag(CH2) + ip(C2-H and C3-H) wag(CH2) + ip(C1-H and C4-H)
1303(m) 1336(w) 1440(s) 1471(s)
1296 1324 1447 1471
0.03 10.18 16.01 5.69
1292 1312 1446 1471
0.59 0.90 9.80 7.23
1231 1312 1448 1476
3.06 2.04 29.30 13.23
2.36 0.04 0.52 6.97
0.75 0.75 0.75 0.75
ip(C-H) + wag(CH2) wag(CH2) + ip(CH and CCC) ip(CH and CCC) + wag(CH2) ip(CH and CCC)
1582(vw)i 1568 1635(w)i 1592
0.05 4.13
1580 1604
0.23 3.45
1594 1618
1.51 2.17
3.77 2.87 8.90 77.93
3058 3064 3075 3087
3.19 2.66 7.79 4.28
2991 2998 3009 3021
217 421 628 743 857 1016 1089 1186 1231 1291 1319(w)i 1349 1397 1440 1480 1570 1612 2920 3018(m) 3048 3064
470 693 735 841 873 910 950 1153(w) 2905 487 542 618 722(w)
ν20 ν29 ν31 ν33 ν35 ν37 ν39
BLYP/6-31G* B3LYP/6-31G*
new(IIR)e
773(w) 904(mw)h 994 1023 1103 1146 1188 1215(mw) 1303 1336 1440 1471 1521(m)i
3006?(m)i 3040 3040(s)i 3062 3062(s)i ? 3084 3084(sh)i
3073 3078 3089 3102
0.26 127.51 0.25 ip(CCC and CH) + sci(CH2) 0.08 21.40 0.37 sci(CH2) 18.06 2.04 0.51 sci(CH2) + ip(CH and CCC) 0.01 61.65 0.34 sci(CH2) + ip(CCC and CH) 1.05 37.53 0.57 ip(CCC and CH) + sci(CH2) 0.18 352.10 0.41 ip(CCC and CH) + str(C1-C10 and C3-C4) 23.92 147.34 0.10 tot sym str(CH2)
95.85 0.02 10.02 0.42
0.75 0.75 0.75 0.75
asym str(CH2) wag(CH2) + ip(CCC) wag(CH2) + ip(CCC and CH) ip(CCC and CH) + str(CC) + wag(CH2)
0.04 0.75 wag(CH2) + ip(CH and CCC)
0.16 0.75 ip(CH and CCC) + str(CC) + wag(CH2) 8.67 0.75 ip(CH and CCC) + str(C1-C10 and C4-C11)
2.28 5.98 0.75 str(CH) 4.37 119.59 0.75 str(CH) 13.82 73.39 0.75 str(CH) 73.74 5.75 0.75 str(CH)
a Vibrational frequencies in cm-1. b Mode numbers are extracted from the output result of the B3LYP calculation. c Italicized numbers indicate the previous assignment by Bree and Zwarich; bold numbers refer to our new assignment. d Bree and Zwarich (ref 12). e Our vibrational frequency assignment on the basis of the ab initio calculations. f Depolarization ratio. g sci, scissoring; twi, twisting; roc, rocking; ip, in-plane; oop, out-ofplane; the modes assigned are sorted in the order of their contributions to the vibrational motions. h Not observed in IR, but observed in Raman. i Relative intensity in the crystal spectrum. The peak is not observed in a cyclohexane solution.
Vibrational Spectra of Fluorene fundamentals at 1143 and 3072 cm-1 are well correlated with the calculated values of 1147 and 3076 cm-1. It is shown in the B3LYP calculation that most of the fundamentals more or less involve scissoring vibrations of the CH2 and in-plane CH and CCC bending vibrations of the phenyl ring. a2 Symmetry. Under C2V symmetry, a2 modes are infrared inactive and Raman active. Only two fundamentals, at 287 and 788 cm-1, are observed in the Raman spectrum.12 Straightforward identification of these a2 fundamentals is impossible due to the infrared inactive vibrations and to the low intensities in the Raman spectra. Our B3LYP calculations predict 10 fundamentals identifiable as a2, 135, 270, 429, 556, 718, 773, 847, 894, 932, and 1128 cm-1. Most of the fundamentals more or less include twisting vibrations of the CH2 and out-of-plane CH and CCC bending vibrations of the phenyl ring. b1 Symmetry. Eleven fundamentals attributable to b1 fundamentals are 119, 260, 410, 470, 693, 735, 841, 873, 910, 950, and 2905 cm-1. As seen in Table 2, the fundamental at 410 cm-1 is a new member instead of a peak at 1153 cm-1, the frequencies assigned as b1 fundamentals by BZ. It is noticed that the peak at 410 cm-1 actually appears in the crystal spectra and is observed in a cyclohexane solution with reasonable intensities.12 We are also convinced by the fact that the peak at 410 cm-1 has also been reported as belonging to b1 fundamentals by Witt.24 The 11 fundamentals are in excellent agreement with our calculations except the fundamental at 2905 cm-1 (ν55), which we match to the calculated frequency of 2948 cm-1 due to the asymmetric CH2 vibration. The prominent deviation is the largest in this study and may arise from the crystal packing force or from the inadequacy of the B3LYP method in predicting the vibrational frequency. Considering the good agreement between the observed frequencies and the computed ones for the other fundamentals, the deviation seems to originate from the different environment in the gas phase and crystal. This discrepancy may be correlated with the shorter bond length of C9-C10 in the crystal in comparison with the calculated bond length, as we discussed above. Fundamentals 22, 24, 26, and 28 obtained with the HF calculation deviate very much from the experimental values and also from those obtained with the B3LYP method. Almost all the vibrations include the out-of-plane bending modes, and some of them involve the CH2 rocking vibrational modes. b2 Symmetry. Only 19 of 20 modes are identified as b2; these are 487, 542, 618, 773, 994, 1023, 1103, 1146, 1172, 1188, 1303, 1336, 1440, 1471, 1582, 1635, 3040, 3062, and 3084 cm-1. Among these, fundamentals at 773, 1172, 1582, and 1635 cm-1 are new members for the b2 group. Actually, fundamentals at 773, 1582, and 1635 cm-1 are observed in the polarized infrared spectra with weak intensities, and the fundamental at 1172 cm-1 is observed in the Raman spectrum.8 These low intensities may lead to the wrong identification in the assign-
J. Phys. Chem., Vol. 100, No. 21, 1996 8785 ments of fundamentals.12 Serious disagreement is observed between the peak positions of the experimental frequencies and the HF values. As seen in Table 2, the HF calculation underestimates fundamentals 35, 42, 56, 58, 60, and 62. Most of the vibrations include out-of-plane bending modes and the wagging CH2 vibrational modes. Acknowledgment. The present studies were supported in part by NON DIRECTED FUND, Korea Research Foundation, 1995-1996, Project No. 01-D-0307, and by the Basic Science Research Institute Program, Ministry of Education, Korea, 1995-1996, Project No. BSRI-95-3432. B.H.B. is grateful to the Center for Molecular Science (CMS) for partial financial support. S.Y.L. thanks CMS for a postdoctoral fellowship. References and Notes (1) Kauffman, J. F.; Coˆte´, M. J.; Smith, P. G.; McDonald, J. D. J. Chem. Phys. 1989, 90, 2874. (2) Saigusa, H.; Itoh, M. J. Phys. Chem. 1985, 89, 5486. (3) Amirav, A.; Even, U.; Jortner, J. Chem. Phys. 1982, 67, 1. (4) Boo, B. H.; Choi, Y. S.; Kim, T.-S.; Kang, S. K.; Kang, Y. H.; Lee, S. Y. J. Mol. Struct., in press. (5) Bree, A.; Zwarich, R. J. Chem. Phys. 1969, 51, 903. (6) Amirav, A.; Even, U.; Jortner, J. J. Chem. Phys. 1981, 75, 3151. (7) Leutwyler, S.; Even, U.; Jortner, J. J. Chem. Phys. 1983, 79, 5769. (8) Kim, Y. R.; Share, P.; Pereira, M.; Sarisky, M.; Hochstrasser, R. M. J. Chem. Phys. 1989, 91, 7557. (9) Labhart, H.; Pantke, E. R.; Seibold, K. HelV. Chim. Acta 1972, 55, 658. (10) Lee, M.; Hochstrasser, R. M. Chem. Phys. Lett. 1988, 153, 1. (11) Chung, Y. B.; Jang, D.-J.; Kim, D.; Lee, M.; Kim, H. S.; Boo, B. H. Chem. Phys. Lett. 1991, 176, 453. (12) Bree, A.; Zwarich, R. J. Chem. Phys. 1969, 51, 912. (13) (a) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. (b) Devlin, F. J.; Finley, J. W.; Stephens, P. J.; Frisch, M. J. J. Phys. Chem. 1995, 99, 16883. (14) (a) Handy, N. C.; Maslen, P. E.; Amos, R. D.; Andrews, J. S.; Murray, C. W.; Laming, G. J. Chem. Phys. Lett. 1992, 197, 506. (b) Handy, N. C.; Murray, C. W.; Amos, R. D. J. Phys. Chem. 1993, 97, 4392. (15) El-Azhary, A. A.; Suter, H. U. J. Phys. Chem. 1995, 99, 12751. (16) Wheeless, C. J. M.; Zhou, X.; Liu, R. J. Phys. Chem. 1995, 99, 12488. (17) Rauhut, G.; Pulay, P. J. Phys. Chem. 1995, 99, 3093. (18) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J. Chem. Phys. 1993, 98, 5612. (19) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; K. Raghavachari, Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chem, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, Revision B.2; Gaussian, Inc.: Pittsburgh, PA, 1995. (20) Negri, F.; Zgierski, M. Z. J. Chem. Phys. 1992, 97, 7124. (21) Burns, D. M.; Iball, J. Proc. R. Soc. (London) 1955, A227, 200. (22) The unit cell of the fluorene single crystal is also presented in ref 5. (23) Szczepanski, J.; Vala, M.; Talbi, O.; Parisel, O.; Ellinger, Y. J. Chem. Phys. 1993, 98, 4494. (24) Witt, K. Spectrochim. Acta 1968, 24A, 1115.
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