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previously used; and (4) the implementation of analytical derivative methods for the calculation of atomic axial tensors. (AATs)4 using gauge-invarian...
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9262

J. Phys. Chem. 1996, 100, 9262-9270

Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra: 6,8-Dioxabicyclo[3.2.1]octane C. S. Ashvar,† F. J. Devlin,† K. L. Bak,‡ P. R. Taylor,§ and P. J. Stephens*,† Department of Chemistry, UniVersity of Southern California, Los Angeles, California 90089-0482; UNI-C, Olof Palmes Alle´ 38, DK-8200 Aarhus N, Denmark; and San Diego Supercomputer Center, P.O. Box 85608, San Diego, California 92186-9784 ReceiVed: December 14, 1995; In Final Form: February 28, 1996X

Predictions of the unpolarized vibrational absorption and vibrational circular dichroism (VCD) spectra of the chiral molecule 6,8-dioxabicyclo[3.2.1]octane (1) are reported. Harmonic force fields and atomic polar tensors are obtained using the density functional theory (DFT), MP2 and SCF methodologies, and the 3-21G and 6-31G* basis sets. Three functionals, LSDA, BLYP, and B3LYP, are used in DFT calculations. Atomic axial tensors are obtained using the Distribution Origin gauge; distributed atomic axial tensors are calculated using gauge-invariant atomic orbitals (GIAOs) at the SCF level of approximation. The quality of the predicted spectra is highly dependent on the methodology and the basis set employed. Spectra calculated using 6-31G* MP2 and DFT/B3LYP force fields are very similar and in excellent agreement with experimental spectra. 6-31G* SCF, DFT/LSDA and DFT/BLYP calculations are in significantly worse agreement with experiment, as are 3-21G MP2 and DFT/B3LYP calculations. When both accuracy and computational effort are considered, at this time, predictions of vibrational absorption and VCD spectra of molecules comparable to 1 in size are optimally performed using 6-31G* or equivalent basis sets and (i) harmonic force fields and atomic polar tensors calculated via DFT and a hybrid density functional; (ii) Distributed Origin gauge atomic axial tensors; and (iii) SCF GIAO-based distributed atomic axial tensors.

Introduction We report ab initio calculations of the unpolarized vibrational absorption and the vibrational circular dichroism (VCD) spectrum of the chiral molecule 6,8-dioxabicyclo[3.2.1]octane (1).

Our calculations take advantage of several important recent developments: (1) the introduction of semidirect analytical derivative methods for the calculation of MP2 harmonic force fields (HFFs) and atomic polar tensors (APTs);1 (2) the development and implementation of direct analytical derivative methods for the calculation of HFFs and APTs via density functional theory (DFT);2 (3) the introduction by Becke3 of hybrid density functionals, of greater accuracy than functionals previously used; and (4) the implementation of analytical derivative methods for the calculation of atomic axial tensors (AATs)4 using gauge-invariant atomic orbitals (GIAOs) at the SCF level of approximation.5 These developments together permit vibrational frequencies, dipole strengths, and rotational strengths to be calculated, within the harmonic level of approximation, more accurately and efficiently than heretofore. The vibrational spectrum of 1 has been studied in detail by Wieser and co-workers.6,7 Unpolarized absorption spectra of 1 in the gas phase and in liquid solutions, VCD spectra of 1 in * Author to whom correspondence should be addressed. † University of Southern California. ‡ UNI-C. § San Diego Supercomputer Center. X Abstract published in AdVance ACS Abstracts, May 1, 1996.

S0022-3654(95)03738-5 CCC: $12.00

liquid solutions, and Raman spectra of molten 1 have been measured. Vibrational frequencies, dipole strengths, rotational strengths, Raman intensities, and polarization ratios have been obtained. Our purpose is to evaluate the accuracy of state-ofthe-art quantum chemical methodologies in reproducing the vibrational properties of 1. A number of derivatives of 1 are important natural products.8 As a result, the unpolarized absorption and VCD spectra of a range of derivatives of 1, including the natural products (+)(1R,5S)-frontalin (1,5-dimethyl-6,8-dioxabicyclo[3.2.1]octane) and (-)-(1S,5R,7S)-exo-brevicomin (5-methyl, 7-exo-ethyl-6,8dioxabicyclo[3.2.1]octane) have also been studied by Wieser and co-workers.9 Our study of 1, reported here, is preliminary to the theoretical study of methyl and dimethyl derivatives of 6,8-dioxabicyclo[3.2.1]octane. Methods Harmonic force fields were calculated ab initio at the SCF, MP2, and DFT levels of approximation, using analytical derivative techniques via GAUSSIAN 92 and GAUSSIAN 92/ DFT.10 The accuracy of DFT calculations is dependent on the functional used. Functionals can be grouped into three classes: local, nonlocal, and hybrid. Local functionals were the first to be used and are the simplest. Nonlocal functionals include gradient terms and are more accurate. Hybrid functionals have recently been introduced by Becke3 and contain mixtures of exchange and correlation functionals, weighted by coefficients whose values were determined semiempirically, based on fitting to experimental data.3 Three density functionals were employed in this study: the local spin density approximation (LSDA) functional, the nonlocal Becke-Lee-Yang-Parr (BLYP) functional, and the hybrid Becke 3-Lee-Yang-Parr (B3LYP) functional. They have been defined in detail previously.11 λ 12 , of nucleus λ (R, β The atomic polar tensors (APTs), PRβ ) x, y, z) were calculated simultaneously with the HFFs. Atomic axial tensors (AATs)4,12,13 were calculated using the © 1996 American Chemical Society

Spectra of 6,8-Dioxabicyclo[3.2.1]octane

J. Phys. Chem., Vol. 100, No. 22, 1996 9263

Figure 1. Unpolarized vibrational absorption spectra of 1: (a) experimental spectrum;6,7 (b-f) calculated spectra. HFFs and APTs are calculated using the (b) DFT/B3LYP, (c) DFT/BLYP, (d) DFT/ LSDA, (e) MP2, and (f) SCF methodologies. The basis set is 6-31G*. Predicted spectra use Lorentzian band shapes with γ ) 4 cm-1 for all bands. Fundamentals are numbered.

Figure 2. Unpolarized vibrational absorption spectra of 1: (a) experimental spectrum;7 (b-f) calculated spectra. HFFs and APTs are calculated using the (b) DFT/B3LYP, (c) DFT/BLYP, (d) DFT/LSDA, (e) MP2, and (f) SCF methodologies. The basis set is 6-31G*. Predicted spectra use Lorentzian band shapes with γ ) 4 cm-1 for all bands. Fundamentals are numbered.

Distributed Origin (DO) with origins at nuclei gauge,4,12 in λ O ) , with respect to origin O which the AATs of nucleus λ, (MRβ are given by

λ 〈0|(µmag)β|1〉i ) - 2p3ωi∑MRβ SλR,i

λ O ) (MRβ

)

0 λ B (IRβ )Rλ

+

i

∑

4pc γδ

0 λ βγδRλγ PRδ

(1)

x

where the SλR,i matrix interrelates Cartesian displacement coordinates XλR and normal coordinates, Qi:

XλR ) ∑SλR,iQi

Di(0 f 1) ) ∑|〈0|(µel)β|1〉i|2

(2)

β

Ri(0 f 1) ) ∑Im[〈0|(µel)β|1〉i〈1|(µmag)β|0〉i] β

where the electric and magnetic transition moments of fundamental transition i, of energy pωi, are given by

〈0|(µel)β|1〉i )

x( )∑ p

2ωi

λ,R

λ PRβ SλR,i

(3)

(4)

i

B R0λ

is the equilibrium position of nucleus λ and where 0 λ B )Rλ is the AAT calculated with the origin at B R0λ. The (IRβ λ B Rλ0 distributed AATs (DAATs), (IRβ) were calculated at the SCF level of approximation using analytical derivative methods and gauge-invariant atomic orbitals5 via the SIRIUS/ABACUS program suite.14 The use of correlated APTs provides partially correlated AATs. Distributed origin gauge AATs calculated using MP2 and DFT APTs are referred to as semi-MP2 and semi-DFT, respectively. Dipole and rotational strengths, D and R, are calculated within the harmonic approximation using the equation13

λ,R

The use of the DO gauge guarantees origin-independent rotational strengths.4 Unpolarized absorption and circular dichroism spectra were generated using Lorentzian band shapes.15 In the absence of experimental line widths, constant line widths γ of 4 and 10 cm-1 were used in the mid-IR and the C-H stretching regions, respectively. 3-21G and 6-31G*[16] basis sets were used. Results Vibrational spectra of 1 have been reported by Wieser and co-workers.6,7 The room temperature liquid solution unpolarized infrared absorption spectrum of 1 over the spectral range 4001550 cm-1 at 1 cm-1 resolution is reproduced in Figures 1 and 2. The solvent was CCl4 except in the range 700-840 cm-1 where CS2 was used. The VCD spectrum of 1 over the spectral range 800-1500 cm-1 at 4 cm-1 resolution is reproduced in Figure 3. The solvent was CS2, except in the range 14001500 cm-1 where CCl4 was used. VCD was measured for both enantiomers of 1; that of (+)-1 shown in Figure 3 was obtained by halving the difference of the VCD spectra of (+)-1 and (-)1, after normalization to 100% enantiomeric excess. The (+)enantiomer of 1 has the (1R,5S) absolute configuration.6,9b,17

9264 J. Phys. Chem., Vol. 100, No. 22, 1996

Figure 3. VCD spectra of (+)-(1R,5S)-1: (a) experimental spectrum;6 (b-f) calculated spectra. HFFs and APTs are calculated as for Figure 1; AATs are (b) semi-DFT/B3LYP, (c) semi-DFT/BLYP, (d) semiDFT/LSDA, (e) semi-MP2; and (f) SCF. All calculations use SCF GIAO-based DAATs. The basis set is 6-31G*. Predicted spectra use Lorentzian band shapes with γ ) 4 cm-1 for all bands. Fundamentals are numbered.

Frequencies and dipole strengths of the most stable, chair conformer6,7,9b of 1 calculated using the 6-31G* basis set at the SCF, MP2, DFT/LSDA, DFT/BLYP, and DFT/B3LYP levels are given in Table 1. Absorption spectra obtained from these results for the mid-IR spectral region are displayed in Figures 1 and 2. There is a large variation in the predicted spectra. The MP2 and DFT/B3LYP spectra are the most similar. The SCF and DFT/LSDA spectra are very different from the MP2 and DFT/B3LYP spectra, and from each other. The DFT/BLYP spectrum is more similar to the DFT/B3LYP spectrum than to the DFT/LSDA spectrum. Allowing for the overall shift to higher frequency of the calculated spectra relative to the experimental spectrum, the MP2 and DFT/B3LYP spectra are in obvious one-to-one correspondence with the experimental spectrum for almost all fundamentals. The similarity to the experimental spectrum is less for the DFT/BLYP spectrum. The SCF and DFT/LSDA spectra bear little resemblance to the experimental spectrum above 700 cm-1. Thus, the MP2 and DFT/B3LYP spectra permit straightforward assignment of almost all of the fundamentals of 1 by comparison to experiment. The SCF, DFT/LSDA, and DFT/BLYP spectra do not. Band frequencies and dipole strengths were obtained from the solution absorption spectra of 1 assuming Lorentzian band shapes.6,7 The values obtained are given in Table 1. The assignment of these bands to fundamentals 5-38, resulting from the comparison of the MP2 spectrum to experiment, is detailed in Figure 1 and Table 1. The assignment of fundamentals 5-21 and 23-27 is unambiguous. Fundamental 22 is assigned to the two bands at 1119 and 1124 cm-1; both were reported to be strong. It is possible that fundamental 22 is in strong Fermi resonance with a nonfundamental resulting in two strong bands.

Ashvar et al. It is also possible that the deconvolution of these bands is incorrect and that one band is a weak shoulder to a single strong band. Eight bands were reported in the range 1300-1400 cm-1. Those at 1310, 1315, and 1331 cm-1 are assigned to fundamentals 28, 29, and 30. Those at 1362 and 1365 cm-1 are assigned to fundamentals 33 and 34. It is difficult to assign the bands at 1338, 1343, and 1349 cm-1 on the basis of the absorption spectrum. We will return to this issue in examining the VCD spectrum. Six bands were reported in the range 14001500 cm-1. Those at 1438, 1458, and 1484 cm-1 are assigned to fundamentals 36, 37, and 38. Fundamental 35 can be assigned to either of the two bands at 1430 and 1435 cm-1 or both, if strong Fermi resonance is assumed. The DFT/B3LYP spectrum is qualitatively extremely similar to the MP2 spectrum. The most noticeable differences are the lesser splitting of fundamentals 18 and 19, and the somewhat different pattern of frequencies and intensities of fundamentals 30-32. The DFT/B3LYP calculations do not remove the ambiguity in assigning fundamentals 31 and 32. Calculated rotational strengths of 1 are also given in Table 1. VCD spectra obtained from calculated frequencies and rotational strengths are displayed in Figure 3. Experimental rotational strengths obtained from the experimental VCD spectrum are also given in Table 1. There is a large variation among the predicted VCD spectra, as was the case for the predicted absorption spectra. Again the MP2 and DFT/B3LYP spectra are very similar and in obvious one-to-one correspondence with the experimental spectrum for almost all fundamentals whose VCD is experimentally detectable. The SCF and DFT/LSDA spectra are very different from each other, from the MP2 and DFT/B3LYP spectra, and from experiment. The DFT/BLYP spectrum is more similar to the DFT/B3LYP spectrum than to the DFT/LSDA spectrum. Fundamentals 12-15, 17-21, 23, 24, 26, and 38 exhibit clearly detectable VCD, whose signs are in agreement with the calculated MP2 spectrum. The most obvious differences, quantitatively, are for fundamentals 14, 19, and 23. The VCD calculated for fundamentals 28-34 is also in qualitative agreement with the observed spectrum. Fundamentals 28, 29, and 30 exhibit the observed alternating (+/-/+) sign pattern. The alternating (-/+) sign pattern predicted for fundamentals 31 and 32 is in agreement with the VCD observed for the bands at 1338 and 1343 cm-1, supporting the assignment of these bands as fundamentals and, therefore, the 1349 cm-1 band as a nonfundamental. The predicted VCD of fundamentals 33 and 34 are both negative, that of fundamental 34 being the larger in intensity, in agreement with the negative VCD associated with the absorption bands at 1362/1365 cm-1. VCD is not clearly observed for the fundamentals 10, 11, 16, 22, 25, 27, and 35-37. Calculated VCD intensities are weak for all of these fundamentals. The predicted DFT/B3LYP VCD spectrum is very similar to the MP2 spectrum, overall. Both calculations predict the same sign for all fundamentals in the mid-IR spectral region except for fundamentals 4, 16, 21, 27, and 34. The VCD intensities of fundamentals 14 and 23 are somewhat greater, while for fundamentals 19 and 28 the opposite is the case. In the case of fundamentals 14, 19, and 23 the DFT/B3LYP VCD intensities are in better agreement with experiment; for fundamentals 16, 21, and 28 the agreement is worse. In the case of fundamentals 33 and 34, the larger VCD intensity is predicted to be associated with mode 33, not mode 34, as was the case with the MP2 calculation. Having assigned the fundamentals of 1 to the extent possible on the basis of the 6-31G* MP2 and DFT/B3LYP calculations, using both the absorption and VCD spectra, we can now

Spectra of 6,8-Dioxabicyclo[3.2.1]octane

J. Phys. Chem., Vol. 100, No. 22, 1996 9265

TABLE 1: Calculated and Experimental Frequencies, Dipole Strengths, and Rotational Strengths of 1a calculationsc experimentb νj

1484 1469 1458 1438 1435 1430 1365 1362 1349 1343 1338 1331 1315 1310 1276 1240 1211 1182 [1181] 1157 1124 1119 1089 1076 1033 1022 993 962 939 893 882 857 832 809 777 744 611 508 448 [373] [339] [252] [188]

D

13.1 3.9 20.7 11.9 5.8 6.7 11.1 14.9 8.3 21.8 11.4 37.6 5.8 28.1 0.8 11.5

DFT/B3LYPd R

fund

48 47 46 45 44 43 42 41 40 39 19.4 38

}

νj 3114 3105 3103 3093 3089 3082 3058 3052 3046 3044 1555

DFT/BLYP νj

D

R

νj

D

R

νj

3023 3017 3008 3004 2995 2993 2974 2965 2959 2952 1513

57.9 85.1 102.6 13.9 103.1 3.0 28.8 27.4 43.8 89.7 3.9

5.1 4.0 -10.9 -59.4 62.1 6.2 -5.0 5.7 -14.0 10.9 6.6

3050 3043 3041 3033 3015 3003 2991 2987 2985 2959 1479

32.5 7.0 54.8 23.0 79.2 33.8 13.3 22.0 23.9 75.4 11.0

24.3 -6.6 -55.4 35.8 -5.2 14.5 -0.5 7.4 -12.5 1.6 9.2

3174 3166 3163 3159 3155 3152 3117 3108 3104 3099 1582

1.3 1492 0.3 1475

8.9 4.2

0.7 1451 33.9 0.4 1432 15.1

1.9 -0.2 -0.2 1468

2.8

0.0 1423 15.7 -0.7 1534

R

D 48.4 73.5 95.8 16.3 69.4 21.0 26.4 27.1 63.7 51.2 4.1

58.8 101.7 112.3 22.3 86.2 28.3 36.1 35.3 70.4 59.1 3.8

0 31.9 -40.1 -26.5 5.4 33.6 -2.6 1.4 -1.1 2.1 6.7

37 1532 12.0 36 1513 5.4

8.1 3.5

1.0 0.4

35 1506

4.8

MP2d

DFT/LSDA

-0.5 38.6 -48.5 -34.2 6.2 43.4 -2.5 1.3 0.1 0.2 6.1

SCF

D 46.5 36.8 36.0 4.6 70.2 22.2 28.7 22.9 53.6 23.9 5.5

3.4 1565 14.7 1.7 1543 9.2 8.6

νj

R

D

R

99.0 1.2 3.0 3310 76.8 1.7 57.5 41.3 61.6 3296 90.0 -28.7 64.7 -96.3 -151.3 3282 60.2 7.7 9.7 19.7 33.2 3269 95.4 67.0 107.5 11.2 16.6 3255 38.2 -103.7 42.4 23.8 38.7 3244 39.4 46.4 54.2 1.2 2.2 3237 64.3 16.9 43.4 2.0 2.7 3226 38.2 10.4 72.6 5.0 6.6 3216 41.9 -8.0 45.2 -5.9 -8.9 3208 35.4 -6.6 4.1 7.8 6.6 1685 3.1 5.3 7.8 4.3

0.4 0.4

2.4

0.0

1.0 1657 10.3 0.1 1631 4.1 -0.2 1627

4.6

2.0 0.0 -0.9

34 1415 17.2 21.2 1.6 3.2 1369 21.4 -0.2 1361 5.3 -4.1 1439 11.6 18.1 -10.0 -12.3 1559 36.8 4.2 -18.8 33 1412 25.0 30.3 -15.5 -18.4 1366 19.3 -11.2 1359 27.2 -4.4 1435 11.7 24.1 -3.8 -0.8 1546 47.3 -17.0

} 24.9

32 31 30 29 28 27 -16.4 26 25 34.4 24 -29.8 52.3 -15.2 11.6

126.3 13.6 23 171.5 22 205.9 33.4 -9.6 21 17.0 19.3 20 77.4 20.3 19 190.8 -62.3 18 248.5 -80.0 17 32.6 6.4 16 120.2 88.0 15 120.1 63.3 14 216.0 -74.0 13 174.6 32.5 12 26.1 11 21.2 10 6.1 9 8.5 8 49.9 7 8.1 6 79.1 5 4 3 2 1

}

1401 1386 1376 1359 1349 1314 1278 1243 1214

14.6 11.8 39.4 0.8 30.9 1.7 12.8 4.2 25.9

21.0 15.0 47.6 0.7 27.4 2.3 15.4 8.7 31.1

12.6 -14.1 21.3 -2.8 0.4 -1.7 -8.1 -2.5 3.9

1187 112.6 144.0

5.5

15.3 -15.6 23.1 -1.7 0.1 -2.3 -8.4 -3.3 6.9

1355 1340 1327 1317 1299 1269 1239 1199 1170

18.0 8.1 29.9 0.2 34.9 1.6 10.7 1.3 18.1

12.5 -7.6 11.2 0.0 -1.2 -2.1 -5.7 -2.0 1.0

1340 1334 1326 1308 1296 1260 1228 1212 1177

26.8 4.5 46.2 18.8 4.3 2.3 18.5 11.3 109.3

20.4 18.6 78.0 149.9 180.7 25.0 117.3 64.5 103.2 145.3 8.9 16.7 6.3 3.9 32.5 4.5 58.0 35.2 11.7 0.5 86.0

22.1 18.7 78.6 194.5 238.0 32.6 144.2 70.8 137.4 191.8 13.7 23.8 10.6 3.8 53.6 7.5 71.2 49.8 16.5 1.4 127.1

0.1 19.8 32.5 -47.3 -34.9 -6.5 78.4 24.2 -30.1 26.5 -3.3 -3.0 -6.5 3.1 -4.4 -5.2 -3.3 0.5 7.1 0.8 -8.0

1.8 19.2 32.4 -49.1 -44.8 -7.2 92.4 24.7 -31.8 28.4 -4.0 -3.7 -9.9 4.1 -5.4 -4.8 -3.7 0.6 8.1 0.8 -9.0

1080 1056 1017 990 962 941 905 874 861 829 805 782 757 726 599 497 436 364 328 240 177

114.4 10.5 67.2 231.4 183.1 26.1 88.5 54.9 58.7 272.8 24.1 24.4 3.1 2.6 27.6 4.5 58.1 30.6 11.3 0.4 79.8

1419 1410 1403 1387 1373 1333 1302 1270 1233

12.5 17.6 44.2 3.3 18.4 3.5 12.0 2.7 26.2

24.9 24.4 54.1 3.2 16.2 4.5 13.7 5.5 31.4

6.2 1166 352.4 -8.0 1210 164.7 219.0

2.2 1146 38.8

1149 328.7 415.3 -9.6 -11.8 1092 191.0 -9.7 1124 50.1 1120 1095 1053 1047 1023 979 953 911 903 878 842 815 793 757 620 515 451 375 340 249 186

-10.3 -2.6 28.1 -13.1 6.5 -1.9 -17.9 -1.9 11.6

20.3 8.8 13.4 -73.5 -32.2 -0.6 74.4 50.3 -17.0 20.4 -2.5 -4.6 -3.5 5.6 -5.8 -5.4 -3.0 -0.3 6.9 0.7 -7.4

1090 1084 1044 1038 1025 964 951 911 902 862 850 829 792 746 603 501 445 370 331 259 187

18.8 74.5 54.6 142.3 69.1 77.4 39.8 82.2 174.5 8.7 42.8 9.6 8.8 14.9 23.3 4.6 60.4 31.4 9.8 1.1 78.3

8.8 16.8 1530 9.1 2.7 -14.5 -17.0 1520 39.0 8.3 23.3 20.6 1509 34.0 6.3 -6.3 -3.6 1491 24.2 -6.6 3.4 0.2 1479 3.7 3.8 0.7 0.1 1442 3.5 -2.2 -7.9 -6.8 1386 16.9 -10.3 -3.8 -4.4 1357 24.7 -2.7 6.2 9.8 1331 62.9 15.4 0.0

-3.3 1309 429.8 -26.8

9.5 1174 254.0 326.5 -8.1 -14.1 1275 180.0 -12.0 -2.5 20.9 51.1 -84.0 -2.3 32.5 27.1 3.2 -3.6 1.8 6.2 1.8 -4.1 -1.3 -2.0 -6.1 -2.9 0.6 6.8 1.2 -9.4

1145 1130 1081 1071 1044 1010 986 932 923 895 872 845 803 767 628 516 458 382 344 268 190

16.0 27.4 83.0 87.9 192.8 19.5 95.2 73.4 103.0 118.6 8.6 13.3 20.4 4.5 34.6 4.9 59.5 43.4 14.4 0.5 106.6

14.4 29.1 88.3 114.1 268.7 24.6 118.1 85.1 140.6 164.1 13.1 18.8 28.4 4.1 54.2 7.8 69.1 56.3 17.3 1.0 136.4

-4.4 23.2 60.5 -59.6 -41.6 0.9 68.3 11.9 -25.0 31.4 -3.7 -0.2 -12.4 3.6 -3.4 -4.5 -3.5 -0.5 8.2 1.1 -9.1

-3.1 24.4 61.3 -58.3 -56.1 0.9 82.2 10.5 -23.5 33.2 -3.9 0.3 -17.6 6.1 -4.8 -4.2 -3.6 -0.5 8.5 1.1 -9.4

1220 1204 1156 1142 1128 1068 1046 1007 982 959 908 872 862 826 673 557 483 401 368 264 203

35.2 1.5 82.9 80.1 73.5 -19.9 116.0 -5.5 105.8 -30.3 50.7 -3.3 125.3 49.7 154.2 -8.0 22.7 10.4 45.3 11.7 9.5 -2.6 23.2 -10.4 6.2 -0.6 9.8 0.0 50.9 -3.5 5.7 -4.9 58.8 -3.9 51.0 2.3 11.9 7.0 1.1 0.8 104.7 -8.9

a νj in cm-1, D in 10-40 esu2cm2, R in 10-44 esu2cm2. Rotational strengths, R, are for the (+)-(1R,5S) enantiomer of 1. b From references 6 and 7. Obtained from absorption and VCD spectra in CCl4 and CS2 solution (see text). Frequencies in brackets were obtained from gas phase absorption spectra. Dipole and rotational strengths were obtained from absorption and VCD spectra assuming Lorentzian band shapes. VCD spectra were normalized to 100% enantiomeric excess. c All calculations use the 6-31G* basis set, the distributed origin gauge and GIAO-based distributed AATs. d Values of dipole and rotational strengths in italics are obtained using SCF APTs and AATs.

compare the accuracies of the various calculations in predicting the vibrational frequencies, dipole strengths, and rotational strengths. The percentage deviations from observed frequencies of the frequencies calculated using the SCF, MP2, DFT/LSDA, DFT/ BLYP, and DFT/B3LYP methods are plotted in Figure 4. Thirty-six frequencies were taken into account, fundamentals 22 and 35 being excluded from the analysis because of their uncertain assignment. Percentage deviations are uniformly positive for SCF and MP2, almost all positive for DFT/B3LYP, predominantly negative for DFT/BLYP, and evenly divided between positive and negative for DFT/LSDA. The ranges and

mean absolute values of the percentage deviations are as follows:

SCF

4.8/14.2%

11.5%

MP2

1.1/7.3%

4.6%

DFT/LSDA

-2.4/3.2%

1.0%

DFT/BLYP

-5.9/2.6%

2.0%

DFT/B3LYP

-1.2/5.2%

2.5%

The SCF frequencies are clearly the least accurate. It would also appear that the MP2 and DFT frequencies are ordered in accuracy: MP2 < DFT/B3LYP < DFT/BLYP < DFT/LSDA.

9266 J. Phys. Chem., Vol. 100, No. 22, 1996

Figure 4. Percentage deviations of calculated and experimental frequencies for 1. Fundamentals 22 and 35 are excluded.

However, it must be remembered that experimental frequencies are lowered relative to true harmonic frequencies by several percent as a result of anharmonicity.18 That is, anharmonicity corrections are comparable to the deviations of the calculated frequencies. Unfortunately, we do not yet know the harmonic frequencies of 1. However, comparisons of 6-31G* calculations for 11 small molecules whose harmonic frequencies have been determined found the following mean absolute percentage deviations with respect to harmonic/observed frequencies:19 SCF 9.0/12.7%; MP2 3.3/6.4%; DFT/LSDA 3.5/2.5%; DFT/BLYP 3.3/2.6%; DFT/B3LYP 2.0/4.5%. If observed frequencies are used the accuracies are SCF < MP2 < DFT/B3LYP < DFT/ BLYP < DFT/LSDA. However, if harmonic frequencies are used, the accuracies are SCF < DFT/LSDA < DFT/BLYP ∼ MP2 < DFT/B3LYP. We can reasonably expect that correcting the experimental frequencies of 1 for anharmonicity will similarly alter the ordering of accuracies. Predicted and experimental dipole strengths and rotational strengths are compared in Figures 5 and 6, respectively. Thirtyone calculated dipole strengths and 18 rotational strengths were plotted against experimental data. The dipole strengths of fundamentals 22 and 35 and the rotational strength of fundamental 32 were excluded because of the uncertainties in their assignments. The mean absolute deviations of calculated and experimental dipole and rotational strengths are as follows:

SCF MP2 DFT/LSDA DFT/BLYP DFT/B3LYP

D (10-40 esu2cm2) 39.5 20.4 38.2 25.8 15.6

R (10-44 esu2cm2) 33.7 18.2 27.4 18.8 16.5

The relative accuracies are SCF < DFT/LSDA < DFT/BLYP < MP2 < DFT/B3LYP for both dipole and rotational strengths, confirming the conclusions arrived at from the qualitative comparison of absorption and VCD spectra (Figures 1-3). Surprisingly, there appears to be somewhat greater scatter in the plot of calculated and experimental dipole strengths (Figure 5) than in the case of rotational strengths (Figure 6). This is

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Figure 5. Calculated and experimental dipole strengths for 1. Fundamentals 22 and 35 are excluded. Dipole strengths are obtained using DFT/B3LYP (9), DFT/BLYP (b), DFT/LSDA (2), MP2 ([), and SCF (1) HFFs and APTs. Dipole strengths obtained using the DFT/B3LYP HFFs and SCF APTs (0) and the MP2 HFFs and SCF APTs (]) are also shown.

unexpected for two reasons: the rotational strengths are intrinsically less accurate than the dipole strengths since correlation is not fully included in the AATs; and VCD spectra are less accurately measurable than absorption spectra. The MP2 and DFT calculations of dipole and rotational strengths include correlation in the HFF, APTs, and AATs. It is of interest to examine the relative importance of the correlation contributions to the HFFs, on the one hand, and to the APTs and AATs, on the other hand. Accordingly, dipole and rotational strengths have been obtained using MP2 and DFT/ B3LYP HFFs, coupled with SCF APTs and AATs. The results are given in Table 1 and Figures 5 and 6. Calculated dipole and rotational strengths are much more strongly affected by inclusion of correlation in the HFF than in the APTs and AATs. This is not unexpected. Electric and magnetic dipole transition moments are linearly dependent on the APTs and AATs and on the vibrational normal coordinates, described by the SλR,i matrix. The latter, however, are nonlinearly related to the HFF. Small changes in HFF elements can create very large changes in SλR,i matrix elements. Thus, the contribution of correlation to the HFF can lead to disproportionately larger contributions to dipole and rotational strengths. In contrast, contributions to the APTs and AATs are proportionately reflected in the dipole and rotational strengths. In view of the excellent performance of the MP2 and DFT/ B3LYP methods when implemented at the 6-31G* basis set level, we have also carried out calculations using the smaller 3-21G basis set. Absorption and VCD spectra obtained from 3-21G DFT/B3LYP, MP2, and SCF frequencies, dipole strengths, and rotational strengths are compared to the experimental spectra in Figures 7-9. Again, MP2 and DFT/B3LYP spectra are quite similar and both are very different from the SCF spectra. Again, MP2 and DFT/B3LYP spectra are closer to the experimental spectra. However, the agreement is much worse than for the

Spectra of 6,8-Dioxabicyclo[3.2.1]octane

Figure 6. Calculated and experimental rotational strengths of (+)(1R,5S)-1. Fundamental 32 is excluded. Rotational strengths are obtained using DFT/B3LYP HFF and APTs, and semi-DFT/B3LYP AATs (9); DFT/BLYP HFF and APTs, and semi-DFT/BLYP AATs (b); DFT/LSDA HFF and APTs, and semi-DFT/LSDA AATs (2); and MP2 HFF and APTs, semi-MP2 AATs ([); SCF HFF, APTs, and AATs (1). Rotational strengths obtained using the DFT/B3LYP HFFs and SCF APTs and AATs (0) and the MP2 HFFs and SCF APTs and AATs (]) are also shown.

6-31G* calculations, and the 3-21G calculations alone would not permit a straightforward assignment of the experimental spectra. It is clear that this requires the utilization of a basis set of greater size than the 3-21G split-valence basis set. This result is not unexpected.11 The vibrational spectra of 1 were assigned previously by Wieser and co-workers.6,7 A scaled quantum mechanical (SQM) HFF was obtained from the SCF 3-21G HFF of 1, using 14 transferred scale factors. On the basis of the absorption and Raman spectra predicted using this SQM HFF, the fundamentals of 1 were assigned. The SQM HFF was subsequently refined, using this assignment, to optimize the fit of calculated to experimental frequencies. Our assignment, based on the 6-31G* MP2 and DFT/B3LYP calculations of the absorption and VCD spectra of 1, differs significantly from that of Wieser and coworkers, in which (i) the two bands at 1119 and 1124 cm-1 were both assigned as fundamentals (22 and 23); (ii) the band at 1211 cm-1 was assigned as nonfundamental; (iii) the band at 1349 cm-1 was assigned as fundamental 33; while (iv) the bands at 1362 and 1365 cm-1 were assigned (together) to the single fundamental, 34. Wieser and co-workers further predicted the VCD spectrum of 1 using their SQM HFF.6 Several methodologies were used to calculate VCD intensities: the atomic polar tensor (APT) model; the fixed partial charge (FPC) model; and the charge flow (CF) model. The APT model discards distributed AATs in the calculation of rotational strengths.4 The FPC model is obtained from the APT model by making the approximation λ PRβ ) qλδRβ

where qλ is the “fixed partial charge” of nucleus λ.4 This approximation modifies dipole strengths as well as rotational

J. Phys. Chem., Vol. 100, No. 22, 1996 9267

Figure 7. Unpolarized vibrational absorption spectra of 1: (a) experimental spectrum;6,7 (b-d) calculated spectra. HFFs and APTs are calculated using the (b) DFT/B3LYP, (c) MP2, and (d) SCF methodologies. The basis set is 3-21G. Predicted spectra use Lorentzian band shapes with γ ) 4 cm-1 for all bands. Fundamentals are numbered.

strengths. The CF model is a modification of the FPC model in which additional “charge flow” parameters are introduced. The predicted VCD spectra of Wieser and co-workers are limited in accuracy both by the errors in the SQM force field and by the severe approximations introduced by the APT, FPC, and CF models.20 We make no attempt to analyze further the origins of the differences in their calculated and experimental rotational strengths. Predicted absorption and VCD spectra for the C-H stretching spectral region obtained from the 6-31G* frequencies, dipole strengths and rotational strengths given in Table 1, are presented in Figure 10. As in the mid-IR spectral region, there are large variations in the predicted spectra. Experimental spectra have not been reported for this spectral region, however, and the agreement of the predicted and experimental spectra cannot yet be evaluated. In view of the congestion of this spectral region and the certainty of major perturbations due to Fermi resonance with overtone and combination bands, analysis of the C-H stretching spectra will undoubtedly prove more difficult than has been the case for the mid-IR spectra. Discussion DFT methods are increasingly widely used.21 The primary goal of this work has been to further explore the accuracy of vibrational spectra calculated using DFT. The relative accuracies of vibrational spectra predicted using different density functionals can be used to define the relative accuracies of the functionals. In addition, vibrational spectra predicted using SCF, MP2, and DFT methods permit comparison of the newer DFT method to the more established SCF and MP2 methods. 6,8Dioxabicyclo[3.2.1]octane is an ideal molecule for this purpose. It is chiral, permitting study by VCD in addition to unpolarized absorption spectroscopy. It has been thoroughly studied

9268 J. Phys. Chem., Vol. 100, No. 22, 1996

Figure 8. Unpolarized vibrational absorption spectra of 1: (a) experimental spectrum;7 (b-d) calculated spectra. HFFs and APTs are calculated using the (b) DFT/B3LYP, (c) MP2, and (d) SCF methodologies. The basis set is 3-21G. Predicted spectra use Lorentzian band shapes with γ ) 4 cm-1 for all bands. Fundamentals are numbered.

experimentally.6,7 Its size and asymmetry provide vibrational spectra of substantial but not overwhelming complexity. We have previously reported DFT calculations of the vibrational spectra of a number of smaller chiral molecules.11,22-24 In this work, we take advantage of the recent implementation of analytic derivative methods for the calculation of AATs using GIAOs.5 When using medium-sized basis sets such as 6-31G*, GIAOs provide substantially more accurate AATs than field-independent atomic orbitals (FIAOs),24 until very recently the only available option. Our results confirm the prior findings11,22,23 that the new hybrid functionals, such as B3LYP, are significantly superior in accuracy to the local and nonlocal functionals, such as LSDA and BLYP, widely used in DFT calculations. This is not immediately obvious from the vibrational frequencies alone, because the errors in the DFT harmonic frequencies are comparable in magnitude to the contributions of anharmonicity. The use of the vibrational intensities is an alternative, and superior, methodology. In addition, the DFT/B3LYP and MP2 methods are again found to be of similar accuracy. The MP2 method is, as expected, substantially superior to the uncorrelated SCF method. Vibrational intensities are a function of both the HFF, which determines vibrational frequencies and normal coordinates, and the APTs and AATs. In this work, we have again shown22 that it is the accuracy of the HFF which predominantly determines the accuracy of predicted spectra. The inclusion of correlation in the calculation of the HFF is a sine qua non in generating spectra resembling experimental spectra. The inclusion of correlation in the APTs and AATs has a much smaller impact on the quality of predicted spectra. In addition, correlation must be accurately included in the calculation of the HFF. This work

Ashvar et al.

Figure 9. VCD spectra of (+)-(1R,5S)-1: (a) experimental spectrum;6 (b-d) calculated spectra. HFFs and APTs are calculated as for Figure 7; AATs are (b) semi-DFT/B3LYP, (c) semi-MP2, and (d) SCF. All calculations use SCF GIAO-based DAATs. The basis set is 3-21G. Predicted spectra use Lorentzian band shapes with γ ) 4 cm-1 for all bands. Fundamentals are numbered.

demonstrates that the MP2 and DFT/B3LYP methods are comparably successful in incorporating correlation in the case of 1 and more so than the DFT/BLYP and, especially, the DFT/ LSDA methods. Our results do not support the continued use of SCF HFFs in predicting VCD spectra.25 The calculations reported here incorporate correlation in the APTs and AATs to the maximum extent currently practicable: the APTs are correlated at the same level as the HFFs; the AATs are calculated using the distributed origin gauge and are “semicorrelated”, i.e., the APT contribution is correlated while the distributed AAT term is not. While, as discussed above, the inclusion of correlation in the APTs and AATs is less important relative to the HFF, it nevertheless does enhance the accuracy of calculated intensities. In any case, when correlation is included in the calculation of the HFF, there is no point in not including correlation in the APTs and AATs. Our calculations of AATs use mainstream modern quantum chemical methods and techniques. Most importantly, we use analytical derivative techniques26 whose development has been largely responsible for the increased application of quantum chemistry in recent years. Not only are analytical derivative methods highly efficient, but in addition they permit the use of perturbation-dependent basis sets, which vastly improve the accuracy of properties calculated using basis sets of modest size. In the case of nuclear position derivatives, such basis sets have been de rigueur for some time. In the case of magnetic field derivatives, the use of GIAOssthe standard magnetic fielddependent basis functionsshas grown less rapidly. However, it is by now standard in calculations of NMR shielding tensors27 and obviously should become so in the calculation of AATs. Despite the impressive accuracy of the bestsMP2 and DFT/ B3LYPscalculations in accounting for the vibrational frequencies and intensities of 1, there remain differences between theory

Spectra of 6,8-Dioxabicyclo[3.2.1]octane

J. Phys. Chem., Vol. 100, No. 22, 1996 9269 mental spectrum provides a direct confirmation of the prior work. There has been no direct determination of the conformation of 1. Wieser and co-workers reported that the “boat” conformation of 1 is predicted at significantly higher energy by 3-21G SCF calculations.6,7,9b Derivatives of 1 whose X-ray structures have been reported uniformly exhibit the “chair” conformation.30 The unambiguous agreement of our best ab initio calculations of the absorption and VCD spectra of 1 with the experimental spectra directly confirms that the conformation of 1 in solution is indeed the “chair” conformation. We have confirmed the finding of Wieser and co-workers that SCF calculations of the vibrational spectra of 1 are insufficiently accurate to permit the assignment of the fundamentals of 1. Wieser and co-workers achieved an assignment by empirical scaling of the 3-21G SCF HFF, using the protocol of Pulay and co-workers31 to provide an optimized fit to the experimental absorption frequencies and intensities. Our 6-31G* MP2 and DFT/B3LYP calculations have led to some revision of the assignment of Wieser and co-workers, illustrating the difficulty of arriving at accurate assignments by scaling SCF HFFs, even when carried out with great care using a substantial number of transferred scale factors. Conclusion

Figure 10. Calculated unpolarized vibrational absorption and VCD spectra of (+)-(1R,5S)-1 in the C-H stretching spectral region. HFFs and APTs are calculated using the (a) DFT/B3LYP, (b) DFT/BLYP, (c) DFT/LSDA, (d) MP2, and (e) SCF methodologies. AATs are (a) semi-DFT/B3LYP, (b) semi-DFT/BLYP, (c) semi-DFT/LSDA, (d) semi-MP2, and (e) SCF. All VCD calculations use SCF GIAO-based DAATs. The basis set is 6-31G*. Predicted spectra use Lorentzian band shapes with γ ) 10 cm-1 for all bands. Fundamentals are numbered. The upper frequency scale applies to spectra e.

and experiment. These can be attributed to a variety of causes. Undoubtedly, basis set error in our 6-31G* calculations is significant and the use of much larger basis sets should reduce the deviations. Methodological error remains, which will be reduced when more accurate methods than MP2 and DFT/ B3LYP become practicable. More accurate correlated methods than MP2 are well-known, but not yet practicable for molecules of the size of 1. More accurate functionals than B3LYP are not yet available but this situation will probably change in the near future. Anharmonicity and solvent effects, which are omitted from our calculations, are probably significant. We have commented already on the contribution of anharmonicity to frequencies; it is less clear how large is the contribution to intensities. (Note that recent studies of absorption and VCD intensities in trans-oxirane-2,3-d2 found relatively small contributions from anharmonicity.28) Lastly, some fraction of the deviations is likely to be due to experimental error, especially in the case of rotational strengths. VCD spectra are much less reliable than absorption spectra: signal-to-noise ratios are worse, polarization artifacts are present, and absolute calibration is less reliable. Rotational strengths are inevitably less accurate than dipole strengths. Future advances in the measurement of VCD should reduce the errors in rotational strengths. Our work confirms the absolute configuration and conformation of 16,7 and refines the assignment of its vibrational spectra. The absolute configuration of 1 was initially deduced indirectly, using chemical transformations from a precursor of known stereochemistry.29 The unambiguous agreement of our best ab initio calculations of the VCD spectrum of 1 with the experi-

In predicting the vibrational absorption and circular dichroism spectra of molecules comparable in size to 1, today’s ab initio quantum chemistry provides a number of options. HFFs, APTs, and AATs can be calculated at the uncorrelated, SCF level of approximation, the simplest and least accurate methodology. Alternatively, correlation can be included. Here, there are two options: the MP2 methodology and density functional theory (DFT). If DFT is used, a density functional must be selected. All of the above options can be executed using efficient analytical derivative methods. Finally, in addition to the methodological options, a basis set must be selected. We have assessed the relative accuracies of the various options when applied to 1. Our conclusions are as follows: 1. 6-31G* MP2 and DFT/B3LYP calculations are of similar accuracy and are in excellent agreement with experiment. 2. 6-31G* SCF, DFT/LSDA, and DFT/BLYP calculations are of significantly lower accuracy. 3. 3-21G MP2 and DFT/B3LYP calculations are of significantly lower accuracy. Thus, calculations of the spectra of molecules comparable in size to 1 should use the 6-31G* basis set and include correlation via either MP2 or DFT. If DFT is used, a hybrid functional such as B3LYP should be employed. With respect to the choice of correlated method, DFT is computationally less demanding and is therefore more cost-effective than MP2. We can expect a number of developments in the near future which will further enhance the accuracy and efficiency of calculations of absorption and VCD spectra. In particular, (1) the development of MP2 and DFT methods for calculating GIAO-based AATs, (2) the application of direct methods to the calculation of AATs; and (3) the development of density functionals of accuracy superior to B3LYP are all clearly foreseeable. Given the equally predictable enhancement of computing power, it is clear that calculations of better accuracy than any reported here for 1 will soon become practicable for molecules much larger than 1. The consequences for the practical utilization of VCD spectroscopy in the elucidation of the stereochemistry of chiral molecules will be dramatic. Acknowledgment. We are grateful to the San Diego Supercomputer Center for grants of computer time (to P.J.S.

9270 J. Phys. Chem., Vol. 100, No. 22, 1996 and P.R.T.). P.R.T. was supported by the National Science Foundation under Cooperative Agreement DASC-8902825. References and Notes (1) Trucks, G. W.; Frisch, M. J.; Andres, J. L.; Schlegel, H. B. J. Chem. Phys., submitted for publication. (2) Johnson, B. G.; Frisch, M. J. Chem. Phys. Lett. 1993, 216, 133. Johnson, B. G.; Frisch, M. J. J. Chem. Phys. 1994, 100, 7429. Komornicki, A.; Fitzgerald, G. J. Chem. Phys. 1993, 98, 1398. (3) Becke, A. D. J. Chem. Phys. 1993, 98, 1372, 5648. (4) Stephens, P. J. J. Phys. Chem. 1987, 91, 1712. (5) Bak, K. L.; Jørgensen, P.; Helgaker, T.; Ruud, K.; Jensen, H. J. Aa. J. Chem. Phys. 1993, 98, 8873. Bak, K. L.; Jørgensen, P.; Helgaker, T.; Ruud, K.; Jensen, H. J. Aa. J. Chem. Phys. 1994, 100, 6620. Bak, K. L.; Jørgensen, P.; Helgaker, T.; Ruud, K. Faraday Discuss. 1994, 99, 121. (6) Eggimann, T.; Shaw, R. A.; Wieser, H. J. Phys. Chem. 1991, 95, 591. (7) Eggimann, T.; Ibrahim, N.; Shaw, R. A.; Wieser, H. Can. J. Chem. 1993, 71, 578. (8) See, for example: Borden, J. H. In Pheromones; Birch, M., Ed.; North-Holland: Amsterdam, 1974; Chapter 8. Mori, K. The Synthesis of Insect Pheromones, 1979-89; Total Synth. Nat. Prod., 9; ApSimon, J., Ed.; Wiley: New York, 1992; Chapter 20. (9) (a) Shaw, R. A.; Ibrahim, N.; Wieser, H. Tetrahedron Lett. 1988, 29, 745. (b) Eggimann, T. Ph.D. Thesis, University of Calgary, 1991. (10) GAUSSIAN 92, Frisch, M. J. et al., 1992. GAUSSIAN 92/DFT, Frisch, M. J. et al., 1993. (11) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. (12) Stephens, P. J.; Jalkanen, K. J.; Amos, R. D.; Lazzeretti, P.; Zanasi, R. J. Phys. Chem. 1990, 94, 1811. (13) Stephens, P. J. J. Phys. Chem. 1985, 89, 748. (14) SIRIUS; Jensen, H. J. Aa.; Aagren, H.; Olsen, J. ABACUS; Helgaker, T.; Bak, K. L.; Jensen, H. J. Aa.; Jørgensen, P.; Kobayashi, R.; Koch, H.; Mikkelsen, K.; Olsen, J.; Ruud, K.; Taylor, P. R.; Vahtras, O. (15) Kawiecki, R. W.; Devlin, F. J.; Stephens, P. J.; Amos, R. D.; Handy, N. C. Chem. Phys. Lett. 1988, 145, 411. Kawiecki, R. W.; Devlin, F. J.; Stephens, P. J.; Amos, R. D. J. Phys. Chem. 1991, 95, 9817. (16) Hehre, W. J.; Radom, L.; Schleyer, P. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (17) Ibrahim, N.; Eggimann, T.; Dixon, E. A.; Wieser, H. Tetrahedron 1990, 46, 1503.

Ashvar et al. (18) Reference 16, Tables 6.39-6.41, pp 232-245. (19) Finley, J. W.; Stephens, P. J. J. Mol. Struct. (THEOCHEM) 1995, 357, 225. (20) Stephens, P. J.; Jalkanen, K. J.; Kawiecki, R. W. J. Am. Chem. Soc. 1990, 112, 6518. (21) Ziegler, T. Chem. ReV. 1991, 91, 651. (22) Stephens, P. J.; Devlin, F. J.; Ashvar, C. S.; Chabalowski, C. F.; Frisch, M. J. Faraday Discuss. 1994, 99, 103. (23) Devlin, F. J.; Finley, J. W.; Stephens, P. J.; Frisch, M. J. J. Phys. Chem. 1995, 99, 16883. (24) Bak, K. L.; Devlin, F. J.; Ashvar, C. S.; Taylor, P. R.; Frisch, M. J.; Stephens, P. J. J. Phys. Chem. 1995, 99, 14918. (25) For recent calculations of VCD spectra based on SCF HFFs see, e.g.: (a) Faglioni, F.; Lazzeretti, P.; Malagoli, M.; Zanasi, R.; Prosperi, T. J. Phys. Chem. 1993, 97, 2535. (b) Polavarapu, P. L.; Cholli, A. L.; Vernice, G. J. Am. Chem. Soc. 1992, 114, 10953. (c) Freedman, T. B.; Ragunathan, N.; Alexander, S. Faraday Discuss. 1994, 99, 131. (d) Nieman, J. A.; Keay, B. A.; Kubicki, M.; Yang, D.; Rauk, A.; Tsankov, D.; Wieser, H. J. Org. Chem. 1995, 60, 1918. (26) Amos, R. D. AdV. Chem. Phys. 1987, 67, 99. Pulay, P. AdV. Chem. Phys 1987, 67, 214. Yamaguchi, Y.; Osamura, Y.; Goddard, J. D.; Schaefer, H. F. A New Dimension to Quantum Chemistry: Analytic DeriVatiVe Methods in Ab Initio Molecular Electronic Structure Theory; Oxford University Press: New York, 1994. (27) Ditchfield, R. J. Chem. Phys. 1972, 56, 5688; Mol. Phys. 1974, 27, 789. Wolinski, K.; Hinton, J. F.; Pulay, P. J. Am. Chem. Soc. 1990, 112, 8251. Gauss, J. Chem. Phys. Lett. 1992, 191, 614. Schreckenbach, G.; Ziegler, T. J. Phys. Chem. 1995, 99, 606. (28) Bludsky, O.; Bak, K. L.; Jørgensen, P.; Spirko, V. J. Chem. Phys. 1995, 103, 10110. Bak, K. L.; Bludsky, O.; Jørgensen, P. J. Chem. Phys. 1995, 103, 10548. (29) Pecka, J.; Cerny´, M. Collect. Czech. Commun. 1973, 38, 132. (30) Brown, G. M.; Thiessen, W. E. Acta Crystallogr. 1969, A25, S195. Park, Y. J.; Kim, H. S.; Jeffrey, G. A. Acta Crystallogr. 1969, A25, S197. Noordik, J. H.; Jeffrey, G. A. Acta Crystallogr. 1977, B33, 403. Mundy, B. P.; Dirks, G. W.; Larsen, R. D.; Caughlan, C. N. J. Org. Chem. 1978, 43, 2347. Bartelt, K. E.; Fitzgerald, A.; Larsen, R. D.; Rees, M. S.; Mundy, B. P.; Emerson, K. J. Org. Chem. 1991, 56, 1958. (31) Fogarasi, G.; Pulay, P. Annu. ReV. Phys. Chem. 1984, 35, 191. Fogarasi, G.; Pulay, P. In Vibrational Spectra and Structure; Durig, J. R., Ed., Elsevier: New York, 1985; Vol. 14, p 125.

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