Density Functional theory Study of Vibrational Spectra. 2. Assignment

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Density Functional Theory Study of Vibrational Spectra. 2. Assignment of Fundamental Vibrational Frequencies of Fulvene Christine J. M. Wheeless, Xuefeng Zhou, and Ruifeng Liu* Department of Chemistry, East Tennessee State University, Johnson City, Tennessee 37614-0695 Received: March 28, 1995; In Final Form: June 9, 1995@

Ab initio restricted Hartree-Fock (RHF) and density function theory calculations using Becke's exchange and Lee-Yang-Parr's correlation functionals (BLYP) were carried out to study the molecular structure and vibrational spectrum of fulvene. Comparison of the calculated and experimental results indicates the density functional BLYP/6-3 lG* method is more accurate in predicting fundamental vibrational frequencies than the scaled Hartree-Fock approach. On the basis of the calculated results, reassignment of some fundamental vibrational modes of fulvene is proposed. This study shows that density functional theory is a very promising method for understanding the vibrational spectra of organic compounds.

using Becke's gradient-corrected exchangeI6 and Lee-YangParr's gradient-corrected correlation f u n ~ t i o n a l ' (BLYP) ~ ~ ' ~ and The thermal and photochemical interconnection between the 6-31G* basis setI9 reproduced the observed fundamental benzene and its isomers has been a subject of extensive vibrational frequencies of ethylene, formaldehyde, butadiene, experimental and theoretical studies.'** Of particular interest acrolein, glyoxal, and their deuterium isotopomers with a mean are the cyclic isomer fulvene and the valence isomers benzvalene absolute deviation of 13.3 cm-' and a standard deviation of (tricyclo[3.1.0.02.6]hex-3-ene),Dewar benzene (bicyclo[2.2.0]17.9 cm-' for non-CH(D) stretching vibrations. Although this hexa-2,5-diene), and prismane (tetracyclo[3.1.0.@~6.02~4]hexane), accuracy is similar to that of the SQM procedure, the density which are believed to be photochemically connected. Knowlfunctional theory appears more promising, because unlike the edge of the structures, spectra, and potential energy surfaces of case of the SQM method, no empirical parameters pertinent to the pertinent isomers is important for understanding this the subject molecules are used in the density functional interesting system. Despite a large number of experimental calculations. In the present paper we report the results of our studies, the vibrational spectra of most benzene isomers are still density functional theory study of the vibrational spectrum of not well understood. A major problem in the experimental study fulvene. For comparison, we also calculated the vibrational of vibrational spectra for a molecule of this size is that the frequencies of benzene and its deuterium isotopomers. number of independent force constants is much larger than the Fulvene has been known since the beginning of this century?O number of observable fundamental vibrational frequencies. but little experimental work, compared to that on benzene, has Unless the molecule has an extremely high symmetry, as does been done on this compound, presumably because of its inherent benzene (D6h), it is very difficult to derive a reliable force field instability. Infrared and ultraviolet spectra were recorded in from experimental information alone. several of the earlier synthetic but no detailed During the past decade, ab initio molecular orbital calculation investigations or positive assignments of the spectra were done of molecular structure and force fields has been quite before 1970. In 1970, Brown, Domaille, and Kent published a su~cessful.~-~ Force constants calculated by the Hartree-Fock detailed experimental study of the infrared and electronic spectra (HF) method with a basis set of double-c quality are qualitatively of fulvene.26 On the basis of gas-phase contours and P-R correct with mainly systematic errors which could be corrected separations, assignments of 14 vibrational modes were proposed. by an empirical scaling procedure. This scaled quantum This study was supplemented by an infrared and Raman mechanical (SQM) force field procedure6 was successfully investigation in which a plausible assignment of nearly all thirty applied to benzene and resulted in a very accurate quadratic fundamental vibrations was published.27 To our knowledge, force field.' We also successfully applied this method to study there have been no theoretical studies, either empirical normal the vibrational spectrum of Dewar benzene, resulting in reascoordinate analysis or quantum mechanical calculations, on this signment of some fundamental vibrational frequencies.8 For molecule to examine the proposed vibrational assignments. some other benzene isomers, however, our study of vibrational These calculations are valuable for providing insight into the spectra with the SQM approach was less successful, presumably vibrational spectrum. due to less systematic errors from neglect of electron correlation and basis set incompleteness. Recently, density functional theory9-" (DFT) has been Calculations accepted by the traditional ab initio quantum chemistry comAll the geometrical parameters of fulvene and benzene were munity and much effort has been devoted to refine the fully optimized by density functional theory using the BLYP methodology and explore the limits of its appli~ability.'*-'~ A functionals and the 6-31G* basis set. The quadratic force fields recent study by Berces and Ziegler13bhas shown that the SQM were calculated by the analytic second derivative method at the force field7 of benzene was the closest to the physical force field. Our own DFT c a l c ~ l a t i o n salso ~ ~ indicate that the selfoptimized structure. For comparison, we also optimized the consistent Kohn-Sham procedurelo of density functional theory structure and calculated the force field, infrared and Raman intensities, and Raman depolarization ratios of fulvene by the restricted Hartree-Fock (RHF) method using the 6-31G* basis @Abstractpublished in Advance ACS Abstracts, July 15, 1995.

Introduction

0022-3654/95/2099-12488$09.00/0 Q 1995 American Chemical Society

Fundamental Vibrational Frequencies of Fulvene

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J. Phys. Chem., Vol. 99, No. 33, 1995 12489

..

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H12

H11 BLY P/6-31G* HF/6*310* Microwave

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Figure 1. Comparison of the calculated and microwave structural parameters of fulvene. set. All calculations were carried out using the Gaussian921 DFT program.28

Results and Discussion Structure. The CC and CH bond distances of benzene calculated with the BLYP/6-3lG* basis set are 1.407 and 1.094 A, respectively. The experimental results are 1.3964(6) and 1.0831(40), as determined by an IR (vibration-rotation) and 1.399(1) and 1.101(5) A, as determined by electron diffraction studies.30 The former values are perhaps closer to those of the re structure. Compared to the experimental results, it appears the BLYP/6-31G* calculation overestimates the re bond distances slightly. The structural parameters of fulvene calculated by the BLYP/6-31G* and HF/6-31G* methods are compared with those of the microwave structure of Baron et aL3' in Figure 1. In this figure, the bond lengths are given in angstroms, and angles are given in degrees. Compared to the experimental results, the BLYP/6-3lG* calculation also slightly overestimates all the bond distances while the RHF/6-3 lG* calculation underestimates all the C--C double-bond distances. All the bond angles calculated by both methods are in good agreement with the experimental results. It has been well documented that the RHF/6-3 lG* calculation gives reliable structural parameters for most hydrocarbon compounds.32 The good agreement between the BLYP and RHF results and the good agreement between the calculated and experimental structural parameters indicate the BLYP/6-3 lG* structures of simple hydrocarbon compounds are reliable with perhaps slight overestimation of the bond distances. Vibrational Spectra. I . Benzene. Benzene is the most thoroughly studied aromatic compound. Its vibrational spectrum has been extensively studied and is well understood. Comparison between the calculated and observed results for benzene serves as a check of the reliability of theoretical methods. Previous DFT calculations' 3b,33 using different functionals and basis sets have shown that DFT frequencies of benzene are in better agreement with experiment than the ab initio HartreeFock and the much more expensive MP2 results. In the present study, we calculated the quadratic force field and vibrational frequencies of benzene with the same functional and basis set we use to study fulvene and other benzene isomers. The calculated harmonic frequencies of ben~ene-hs,benxne-ds, and s-benxne-d3 are compared with the observed fundamental

TABLE 1: Comparison of the Calculateda and Observed6 Vibrational Frequencies of Benzene&, Benzened6, and Benzene43 benzene-hb benzene-ds benzene-d, v sym expt calc expt calc expt calc 1 Aig 993 987 945 942 956 947 2 AI, 3073 3122 2303 2315 3065 3105 3 A2* 1350 1349 1059 1049 1259 1278 4 Bzg 707 695 599 598 697 691 5 Bzg 990 970 829 797 917 893 6 E2, 606 606 579 578 592 592 7 Ezg 3056 3096 2274 2283 2282 2294 8 1599 1594 1557 1551 1580 1574 9 E2, 1178 1177 869 861 1101 1097 10 Elg 846 836 660 649 708 691 11 A2, 673 672 496 494 531 529 12 BI, 1010 993 970 954 1004 991 13 BI, 3057 3086 2249 2272 2294 2293 14 Bzu 1309 1329 1282 1322 1321 1331 15 B2, 1146 1159 824 824 912 909 16 E2, 398 402 349 350 368 375 17 E2, 967 927 787 752 924 891 18 El, 1037 1037 814 812 833 833 19 El, 1482 1487 1333 1334 1414 1422 20 Elu 3064 3111 2288 2303 3063 3103 a Calculated with the BLYP/6-31G* basis set, in cm-l. Reference 34. vibrational f r e q u e n c i e ~in~ ~Table 1. Most of the observed frequencies of non-CH(D) stretching modes are reproduced very well by the DFT calculations. Larger deviations were found for the C-H(D) stretching modes. These modes are strongly affected by anharmonicity and are less important than modes of the fingerprint region for chemical analysis. Compared with the observed fundamental frequencies, the BLYP/6-3 lG* harmonic C-H(D) stretching frequencies are slightly higher. However, the true harmonic C-H(D) stretching frequencies are perhaps -100 cm-' higher than the observed fundamentals. Therefore, the BLYP/6-3 1G* calculations perhaps underestimate the harmonic C-H(D) stretching frequencies slightly. In principle, one should compare the calculated harmonic frequencies with "experimental" harmonic frequencies, or the calculated anharmonic frequencies with the observed fundamentals. However, neither calculation of anharmonic vibrational energy levels nor reliable derivation of harmonic frequencies from the observed fundamentals is routinely feasible for polyatomic molecules. Fortunately anharmonicity has a very small effect on most non-CH(D) stretching modes and, due to error cancellation, the BLYP/6-3 lG* harmonic frequencies are very close to the observed fundamentals. For the results presented in Table 1, the mean absolute deviation between the calculated and observed frequencies of non-CH@) stretching modes is only 9.4 cm-', indicating BLYP/6-31G* results are accurate enough for identifying spectral features due to the subject molecules. 2. Fulvene. Under CzV symmetry, the 30 fundamental vibrational modes of fulvene are distributed among the various symmetry species as follows: lla,

+ 4% + 5b1 + lob,

We have followed Mulliken's convention in designating symmetry planes of a CzVmolecule such that the b, species refers to modes antisymmetric with respect to the molecular plane, while the bl modes are symmetric with respect to the molecular plane. Table 2 compares the calculated frequencies and infrared and Raman intensities with the experimental assignments of Domaille, Kent, and O ' D ~ y e r . In ~ ~this table, most of the experimental frequencies are infrared frequencies. Raman

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TABLE 2: Comparison of Observed and Calculated Fundamental Vibrational Frequencies of Fulvene exptc BLYP/6-31G* scaled RHF/6-31G* symb mode IR Ra" freq ZIR freqd ZIR IRA D P new assignment' 3088 4.3 1 3083 10.75 234.92 0.11 3169 3088 m 3083 W,V 9.89 70.72 0.7 1 3051 17.38 3061 3145 3051 w 3052 vw',p 86.10 0.11 3008 8.85 3003 14.04 308 1 3008 s 1632 9.44 255.04 0.28 2.3 1 1690 1640 s,p 1652 1632 vw 86.03 0.12 1486 20.42 1536 41.72 1494 1486 s 1485 s,p 0.32 54.04 0.56 1449 1435 0.32 1436 1391 w 1342 3.00 0.14 1372 vw 1368 vw,p 19.95 1352 22.73 1348 1082 43.62 0.39 1.14 1.29 1087 1342 w,p 1086 1342 s 986 950 9.10 0.19 3.27 3.00 1082 w 1080 ms,p 978 894 8.80 0.67 869 13.94 886 5.63 989 mw,p 986 m 641 12.74 0.30 650 658 0.00 0.09 894 m 895 w,p 3.40 0.75 0.00 963 895 0.00 788 9.32 0.75 0.00 750 0.00 666 698 0.08 0.75 0.00 668 0.00 666w 665m 1.32 0.75 0.00 494 487 472 0.00 . 494m 490w 926 33.98 984 48.85 9.04 0.75 926 s 930 w,dp 913 3.12 907 0.69 0.75 1.80 953 881 907 m 915 vvw,dp 11.92 0.75 37.96 769 791 63.65 776 769 s 613 9.32 0.75 20.8 1 611 3 1.45 630 613 vs 620 w,dp 0.10 0.75 3.04 2.69 202 205 350m 346mw 24.58 3077 37.06 2.86 0.75 3164 3104 91.71 0.75 0.68 3104 w 3159 10.36 3075 3075 5.17 3052 89.23 0.75 2.56 3075mw 3075 w 3134 1.14 0.75 1550 0.01 1617 0.03 1550 vvw,dp 1574 1314 1328 0.15 0.75 0.05 1449 m 1450 vvw,dp 1303 0.09 1232 1228 1237 0.05 0.09 0.75 0.01 1410 mw,dp 1077 1088 1090 7.22 10.28 0.75 5.29 1130w 945 940 1077 m 952 2.78 0.20 0.75 2.85 777 1.25 2.54 0.75 3.17 779 952m 956w 795 33 1 350 0.29 2.54 0.75 0.40 330 795m 795w

modedescriptions CHI)~ t r CHg str CH2 str

cI=c2 str c2=c5 str

H7CHs bend CHII in-plane bend CHI]in-plane bend c3-c4 c2-c5

str str

ring in-plane def CH 0-0-p bend CH2 tor + CH2 out CH out + CH2 tor ring tor CH2 ring 0-0-p bend CH 0-0-p bend CH 0-0-p bend CH2 0-0-p bend ring tor CHii str CH2 str CH9 str C3=C5 str CHll bend + CC str CH2 in-plane bend CH9 in-plane bend CICZC~ in-plane bend ring def C-C str ClCzC5 in-plane bend

+

a Frequencies are given in cm-]; calculated infrared intensities (ZIR)and Raman intensities (IRA) are in km/mol and in A4/amu, respectively. Symmetry species under Czvpoint group. Experimental frequencies and their assignments are taken from refs 26 and 27. Abbreviations used to describe spectral features are as follows: s, strong; w, weak; m, medium; vw, very weak; p, polarized; dp, depolarized; pp, partially polarized. RHF frequencies are uniformly scaled by 0.9. e Polarization ratio of Raman band calculated by RHF/6-31G*. For polarized bands, DP < 0.75; for depolarized bands, DP = 0.75. f Assignments of the fundamental vibrational frequencies of the present study. 8 Mode descriptions are based on BLYP/6-3lG* vibrational energy decomposition analysis. Abbreviations used: str, stretching; bend, bending; def, deformation; tor, torsion; 0-0-p, out-of-plane.

frequencies were listed for modes not observed in the infrared spectra. The columns under BLYP are frequencies and infrared intensities calculated with the BLYP/6-31G* basis set. The columns under RHF are infrared and Raman intensities, depolarization ratios of the Raman bands, and scaled vibrational frequencies. The RHF frequencies have been scaled by 0.9 to account for systematic errors due to neglecting electron correlation. As we can see, most of the scaled RHF frequencies are in good agreement with the BLYP results. Relatively large deviations between the BLYP and scaled RHF frequencies of non-CH stretching modes are found for v4, VS, v12, vl6, v l 7 , and ~ 2 4 . Three of these (v4, v5, and v24) are mainly CC doublebond stretching vibrations. Hartree-Fock results for these vibrations are too high due to electron correlation effects. This is corroborated by the fact that the Hartree-Fock C=C bond lengths are too short compared to the experimental results. The other three modes are mainly CH out-of-plane bending vibrations which give even larger deviations. It is shown in Table 2 that, for non-CH stretching modes with relatively large deviations between the BLYP and scaled Hartree-Fock frequencies, the BLYP results are apparently in much better agreement with the experimental results. A comparison between the experimental vapor-phase infrared spectrum and the theoretical infrared spectrum of fulvene is given in Figure 2. The theoretical spectrum was simulated from BLYP frequencies and infrared intensities by summing the Lorentzian profiles.35 The agreement between the experimental and theoretical infrared spectral features indicates not only the vibrational frequencies but also the infrared intensities are

I

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la00

1600

1400

1200

1000

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600

400

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2000

1800

1600

1400

1200

1000

800

800

400

200

Figure 2. Comparison of the vapor-phase infrared spectrum (upper,

from ref 26) and the theoretical infrared spectrum simulated from BLYP/6-3lG* frequencies and infrared intensities (lower). predicted satisfactorily by BLYP/6-3 lG* calculations. Although the DlT CH stretching frequencies are not in satisfactory agreement with the observed results, these vibrations are much higher in energy than non-CH stretching vibrations and do not couple with the latter. Experimentally these modes are easy to identify, and they do not pose a problem for correctly interpreting the observed spectral features. The following is a detailed discussion, based on comparison between the calculated and

Fundamental Vibrational Frequencies of Fulvene experimental results, of the assignment of the non-CH stretching vibrational modes of fulvene. a1 Modes. The a1 modes are relatively easy to identify in an experiment, as they give polarized Raman bands and A-type contours in an infrared spectrum. Indeed, experimental assignments of most of the al modes are in good agreement with the calculated results with exceptions for V6 and V I I. Both of them are calculated to have very low infrared intensity and therefore are expected to be very weak in the infrared spectrum. Experimentally, a medium intensity infrared band at 894 cm-I was assigned to the lowest al mode, V I I . Our BLYP calculation indeed predicted a medium intensity infrared band of a1 symmetry at 886 cm-'. However, the calculations indicate this mode is V I Oinstead of V I1. The latter is predicted to be at 658 cm-I. The difference between the BLYP frequency and the experimental assignment of V I 1, -240 cm-', is much larger than the mean absolute deviation between the BLYP and the observed frequencies of benzene. Inspection of the experimental spectra of Brown, Domaille, and Kent26 shows there is indeed a very weak band at 641 cm-I in their solid state infrared spectrum, which was tentatively assigned to a combination of bands at 350 and 298 cm-I. This assignment is probably mainly based on the very low infrared intensity, as experimental studies tend to search for strong bands as candidates for fundamental vibrations. This assignment is not convincing, because the weak infrared band at 298 cm-I is not due to a fundamental vibration of the subject molecule. On the basis of the calculated results, it is likely that the very weak infrared band at 641 cm-I is due to V II , the in-plane ring deformation mode. The experimental assignment of V6 was less well defined. It was assigned by Domaille, Kent, and 0'Dwyer2' to a weak infrared band at 1391 cm-I without a Raman counterpart. The same infrared band was found earlier by Brown, Domaille, and Kent26 to give a B-type contour and was assigned to bl symmetry. According to the RHF/6-31G* results, v6 should have significant Raman activity and therefore should be observable in a Raman spectrum. On the basis of the BLYP results, this mode should be in the neighborhood of 1436 cm-'. There is indeed a medium infrared band with a very weak Raman counterpart at 1449 cm-', which was assigned by Domaille, Kent, and O ' D ~ y e to r ~the ~ b2 mode v25. The latter was calculated to be very weak in both infrared and Raman spectra and should be at around 1303 cm-l. According to the calculated results, the band at 1449 cm-' is most likely the al mode V6. Discrepancy between the calculated and observed infrared intensities may be explained as a result of band overlap between the band at 1449 cm-I and the nearby strong infrared bands. a2 Modes. Under C2" symmetry, a2 modes are infrared inactive. Our calculations also indicate that all of them have very low Raman intensities. As a result, only two of the four a2 modes were positively identified experimentally. The experimental assignments of v14 and VI5 are in good agreement with the calculated results. The CH out-of-plane bending modes v12 and V I 3 are predicted at 895 and 750 cm-l, respectively. bl Modes. The calculated frequencies of the bl modes agree very well with the experimental assignment with only one exception, the out-of-plane ring torsional mode v20. The calculated frequency of this mode is 205 cm-I with low infrared and Raman intensities. This mode was assigned to a band at 350 cm-' in the solid state infrared spectrum of f ~ l v e n eThe .~~ difference between the calculated frequency and the experimental assignment appears too large. In our calculated results, there is indeed a fundamental vibration at 330 cm-', but it is the b2 symmetry methylene group in-plane bending mode ~ 3 0 .

J. Phys. Chem., Vol. 99, No. 33, 1995 12491 It is most likely that the infrared band at 350 cm-I is due to the b2 mode V30, while the bl symmetry mode v20 is in the neighborhood of 205 cm-' but was not observed due to its low spectral intensities or to the low-frequency region was not being as thoroughly probed in the experimental studies. bz Modes. Significant differences between the calculated frequencies and the empirical assignments of b2 modes are found for V25, v26, and ~ 3 0 . Domaille, Kent, and O'Dwyer assigned an infrared band at 795 cm-I to the lowest b2 mode "30, which is about 350 cm-' higher than the calculated results. On the other hand, the band at 795 cm-I agrees very well with the calculated ring deformation mode V29 (779 cm-I). Our reassignment of the infrared band at 350 cm-' to V30 is corroborated by the fact that, with this reassignment, the calculated results and experimental assignments for V27, V28, and V29 are in good agreement. Both v25 and v26 are calculated to be very weak in both infrared and Raman spectra. The assignments of Domaille, Kent, and O'Dwyer, 1410 and 1449 cm-', are 182 and 146 cm-' higher than the calculated results, 1228 and 1303 cm-l, respectively. We have pointed out that the band at 1449 cm-I is more likely the a1 fundamental vg. In the solid state infrared spectrum of Brown, Domaille, and Kent,26there is a very weak band at 1314 cm-' unassigned. On the basis of the calculated results, this band is likely the b2 fundamental ~ 2 5 . In the solid state infrared spectrum, there is also a weak band at 1232 cm-I, which was assigned by Brown, Domaille, and Kent to the overtone of v19 (2 x 613 = 1226 cm-I). According to the calculated results, this is likely fundamental V26. The band at 1410 cm-', on the other hand, can be explained as a combination of VI5 with either v17 (494 907 = 1401 cm-I) or VI6 (494 926 = 1420 cm-I). With the reassignments of the fundamental frequencies of fulvene, the mean absolute deviation between the BLYP/6-3 lG* results and the observed non-CH stretching frequencies is 13.2 cm-I. This mean deviation is in line with the results of our previous density functional theory calculations of molecules whose vibrational spectra have been well under~tood.'~The mean absolute deviation between the scaled RHF/6-31G* results and observed non-CH stretching frequencies is 24.9 cm-'. Reassignments of the fundamental vibrational frequencies of this study are given in Table 2 in the column "new assignment". The mode descriptions given in this table are based on BLYP/ 6-31G* vibrational energy decomposition analysis.

+

+

Conclusion Comparison of the observed fundamental vibrational frequencies of fulvene and the results calculated by density functional BLYP and Hartree-Fock methods indicates BLYP is superior to the scaled Hartree-Fock approach for molecular vibrational problems. On the basis of the calculated results, assignment of the fundamental vibrational frequencies given by Domaille, Kent, and O'Dwyer has been examined, and reassignment of some vibrational modes are proposed. The good agreement between the frequencies calculated by BLYP/6-3 lG* and experimental results indicates the density functional method is reliable and provides valuable information for understanding the vibrational spectra of fulvene. Acknowledgment. We are grateful to the Department of Computer and Information Sciences of East Tennessee State University for the use of an IBM RS6000 workstation and to Mr. Luke Pargiter for technical assistance. References and Notes (1) Scott, L. T.;Jones, M., Jr. Chem. Rev. 1972,72, 181.

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Wheeless et al. (23) Meuche, D.; Neuenschwander, M.; Schaltegger, H. Helv. Chim. Acta 1965, 48, 955. (24) Sturm, E.; Hafner, K. Angew. Chem., Znr. Ed. Engl. 1964, 3, 749. (25) Neuenschwander, M.; Neuche, D.; Schaltegger,H. Helv. Chim. Acta 1964,47, 1022. (26) Brown, R. D.; Domaille, P. J.; Kent, J. E. Ausf.J . Chem. 1970, 23, 1707. (27) Domaille, P. J.; Kent, J. E.; O'Dwyer, M. F. Aust. J . Chem. 1974, 27, 2463. (28) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Wong, M. W.; Foresman, J. B.; Robb, M. A,; Head-Gordon, M.; Replogle, E. S.;Gomperts, R.; Andres, J. L.; Raghavachai, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92/DFT, Revision F.3; Gaussian, Inc.: Pittsburgh, PA, 1993. (29) Cabana, A.; Bachand, J.; Giguere, J. Can. J . Phys. 1974,52, 1949. (30) Bastiansen, 0.;Femholt, L.; Seip, H. M.; Kambara, H.; Kuchitsu, K. J. Mol. Sfruct. 1973, 18, 163. Tamagawa, K.; Iijima, T.; Kimura, M. J . Mol. Sfrucf.1976, 30, 243. (31) Baron, P. A.; Brown, R. D.; Burden, F. R.; Domaille, P. J.; Kent, J. E. J. Mol. Spectrosc. 1972, 43, 401. (32) Hehre, W. J.; Radom, K.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (33) Handy, N. C.; Murray, C. M.; Amos, R. D. J . Phys. Chem. 1993, 97, 4392. (34) Brodersen, S.; Langseth, A. K.Dan. Vidensk.Selsk., Mat.-Fys. Skr. 1956, I, 1. (35) Wertz, J. E.; Bolton, J. R. Electron Spin Resonance: Elementary Theory and Pracfical Applicafions; McGraw-Hill: New York, 1972; p 33. JP950873T