Comparison between Optimized Geometries and ... - ACS Publications

Using one-scale-factor scaling, scale factors of 0.82, 0.89, 0.98, 0.93, 0.96, and ...... for furan22 and the d0, 2,5-d2, 3,4-d2, d4, 2-d1, 3-d1, 2,3-...
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15056

J. Phys. Chem. 1996, 100, 15056-15063

Comparison between Optimized Geometries and Vibrational Frequencies Calculated by the DFT Methods A. A. El-Azhary*,† Institut fu¨ r Theoretische Chemie, UniVersita¨ t Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany

H. U. Suter Institut fu¨ r Physikalische und Theoretische Chemie, UniVersita¨ t Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany ReceiVed: February 29, 1996; In Final Form: June 11, 1996X

Optimized geometries, vibrational frequencies, and scale factors were calculated for furan and thiophene with the HF, MP2, LDA, BVWN, BLYP, and B3LYP methods of theory using the 6-31G**, cc-pVDZ, and cc-pVTZ basis sets. The agreement between the optimized and experimental geometries was in the order B3LYP, MP2, LDA, BVWN, BLYP, then HF. The calculated frequencies by the unscaled BVWN force field had the smallest average error in the mid-IR region, but using one-scale-factor scaling, those calculated by the scaled B3LYP force field had the lowest average error. Using one-scale-factor scaling, scale factors of 0.82, 0.89, 0.98, 0.93, 0.96, and 0.96 were obtained by the HF, MP2, BLYP, B3LYP, LDA, and BVWN force fields, respectively, using the 6-31G** basis set. The effect of the basis set on the calculated bond angles, frequencies, and scale factors by the DFT methods was minor, but except with the LDA method, the agreement between the calculated and experimental bond lengths can be arranged in the order cc-pVDZ, 6-31G**, then cc-pVTZ basis set.

Introduction The study of vibrational spectra of small organic molecules is greatly enhanced by the help of ab initio calculations, which are in principle able to reproduce the harmonic force fields to any desired accuracy. Despite this theoretical possibility, it is often useful to work with scaled force fields, in the way defined by the pioneering work by Pulay and co-workers.1 This scaling procedure will take into account the inadequacy of the methods (missing correlation, basis set effects) as well as the anharmonicity of the potential. To be a reasonable starting point for a scaling procedure, the used method must show a systematic behavior. For the Hartree-Fock procedure, for example, it is known that the frequencies are uniformly overestimated by about 10%. The advance in computer hardware and the desire to get more accurate predictions of the vibrational spectra mobilized the use of the more time consuming methods, including electron correlation. The simplest of these methods is the MøllerPlesset (MP2) method,2 which gained interest especially after the development of the force fields and diople moment derivatives calculated by the analytical method.3 Typically, the MP2 frequencies are overestimated by about 5%.4 Although the MP2 force fields gained some interest for the vibrational analysis,5-12 MP2’s computational demand still limited its wide application. Recently, the DFT method was introduced for the vibrational analysis. The DFT method gained interest due to its excellent accuracy to CPU time ratio.13 Currently, three types of density functionals are in commen use. The first type uses only a function of the electron density and is sometimes called local density approximation (LDA). The VWN (Vosko-WilkNussair) functional14 belongs to that class. The second type * Author to whom correspondence should be addressed. † Permanent address: Department of Chemistry, Faculty of Science, Cairo University, Giza, Egypt. E-mail: [email protected]. X Abstract published in AdVance ACS Abstracts, August 15, 1996.

S0022-3654(96)00618-1 CCC: $12.00

also uses the gradient of the electron density in the functional; from those the Becke correction15 seems to be the most prominent. Therefore, the BVWN and BLYP functionals belong to that class. The development of mixed functionals16 defines a third class. In these functionals the different terms are mixed with empirically determined parameters. The so-called B3LYP functional13 will be used as an example of that class of functionals. It is hoped that the quality of the calculation is improved in the chain LDA < (BVWN, BLYP) < B3LYP. There are already several publications that compare these functionals.1-13,17-20 In addition, in a previous publication we compared the optimized geometries, frequencies, intensities, scale factors, and scaled force fields calculated by the HF, MP2, and B3LYP methods.21 But in these publications, the most promising B3LYP functional was not considered (since these publications were reported before the development of the B3LYP functional), the basis set effect was not considered (although this is expected to have minor effect,18,19) or the calculated vibrational intensities13 or the scale factors20 were the main interest. The aim of the present publication is to investigate in detail the differences between the optimized geometries, frequencies, and scale factors calculated by the BLYP, B3LYP, LDA, and BVWN methods in addition to the HF and MP2 methods. Further, to investigate the basis set effect, three basis sets were chosen. These are the 6-31G**, cc-pVDZ, and cc-pVTZ basis sets. The two molecules furan and thiophene were selected for this investigation, because on one hand they are rather well characterized experimentally, while on the other hand they are large enough not to be trivial cases. Both molecules have been the subject of detailed vibrational analysis22,23 and normal coordinate analysis using empirical force fields.24 Harmonic frequencies and IR absorption intensities at the MP2/DZP level were also reported for both molecules.5 © 1996 American Chemical Society

Optimized Geometries and Calculated Frequencies

J. Phys. Chem., Vol. 100, No. 37, 1996 15057 TABLE 2: Equilibrium Geometry for Thiophenea

TABLE 1: Equilibrium Geometry for Furan 6-31G** coordinate exptlb O1-C2 C2dC3 C3-C4 C2-H6 C3-H7 C5O1C2 O1C2C3 C2C3C4 O1C2H6 C3C2H6 C2C3H7 C4C3H7 µ

1.362 1.361 1.431 1.075 1.077 1.06.5 110.7 106.0 115.9 133.4 126.1 127.9 0.66

HF 1.343 1.339 1.441 1.069 1.070 1.07.2 110.8 105.6 116.2 132.9 126.8 127.6 0.77

6-31G**

MP2 MP2 DFT DFT DFT DFT full fc BLYP B3LYP LDA BVWN 1.364 1.365 1.426 1.074 1.075 106.6 110.5 106.2 115.7 133.8 126.2 127.6 0.64

1.366 1.366 1.427 1.075 1.076 106.6 110.5 106.2 115.7 133.9 126.2 127.5 0.87

1.382 1.371 1.444 1.086 1.087 106.4 110.5 106.3 115.4 134.1 126.6 127.1 0.60

1.364 1.361 1.435 1.079 1.080 106.8 110.5 106.0 115.8 133.7 126.5 127.5 0.63

1.351 1.361 1.422 1.088 1.089 107.2 110.4 106.0 115.9 133.8 126.5 127.4 0.52

1.383 1.369 1.443 1.080 1.082 106.3 110.5 106.3 115.4 134.1 126.6 127.1 0.64

coordinate S1-C2 C2dC3 C3-C4 C2-H6 C3-H7 C5S1C2 S1C2C3 C2C3C4 S1C2H6 C3C2H6 C2C3H7 C4C3H7 µ

exptlb

HF

MP2 full

MP2 DFT DFT DFT DFT fc BLYP B3LYP LDA BVWN

1.714 1.369 1.423 1.078 1.080 92.1 111.5 112.5 119.8 128.7 123.2 124.3 (0.52 ( 0.04)

1.725 1.345 1.437 1.071 1.074 91.3 111.8 112.5 120.4 127.8 123.6 123.8 0.90

1.714 1.375 1.418 1.077 1.079 92.0 111.6 112.4 120.3 128.1 123.1 124.5 0.45

1.717 1.376 1.419 1.078 1.080 91.9 111.6 112.4 120.2 128.2 123.2 124.4 0.78

cc-pVDZ

1.754 1.378 1.437 1.087 1.091 91.4 111.4 112.9 119.8 128.7 123.2 123.9 0.59

1.736 1.367 1.429 1.081 1.084 91.5 111.5 112.8 120.0 128.5 123.3 124.0 0.62

1.715 1.368 1.415 1.090 1.093 91.1 111.3 112.6 120.1 128.6 123.2 124.2 0.54

1.755 1.376 1.437 1.082 1.085 91.3 111.4 112.9 119.9 128.7 123.2 123.9 0.59

cc-pVDZ

HF

MP2 full

MP2 fc

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

1.343 1.343 1.444 1.075 1.077 107.2 110.9 105.5 116.3 132.8 126.7 127.8 0.66

1.361 1.375 1.434 1.086 1.087 106.7 110.8 105.8 115.8 133.4 126.2 127.9 0.64

1.363 1.378 1.436 1.088 1.089 106.8 110.8 105.9 115.8 133.5 126.3 127.9 0.74

1.381 1.376 1.445 1.092 1.094 106.5 110.5 106.2 115.4 134.1 126.5 127.3 0.47

1.364 1.364 1.438 1.086 1.087 106.9 110.6 106.0 115.7 133.8 126.6 127.4 0.51

1.351 1.365 1.425 1.095 1.096 107.2 110.5 105.9 115.8 133.8 126.6 127.5 0.40

1.382 1.373 1.445 1.087 1.088 106.4 110.5 106.3 115.4 134.1 126.5 127.2 0.52

HF

MP2 full

MP2 fc

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

1.729 1.349 1.437 1.078 1.080 91.3 111.8 112.6 120.5 127.7 123.6 123.9 0.79

1.725 1.387 1.426 1.090 1.092 92.0 111.6 112.4 120.3 128.1 123.2 124.5 0.67

1.726 1.388 1.427 1.091 1.093 92.0 111.6 112.4 120.3 128.1 123.1 124.5 0.67

1.756 1.382 1.438 1.095 1.097 91.3 111.5 112.9 119.8 128.7 123.2 123.9 0.39

1.738 1.371 1.431 1.088 1.090 91.5 111.5 112.7 120.1 128.4 123.4 123.9 0.46

1.718 1.372 1.417 1.097 1.099 92.0 111.4 112.6 120.0 128.7 123.2 124.2 0.37

1.757 1.379 1.438 1.089 1.092 91.2 111.5 112.9 119.9 128.7 123.2 123.9 0.43

cc-pVTZ

cc-pVTZ HF

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

1.340 1.336 1.441 1.067 1.068 107.2 110.9 105.5 116.4 132.7 126.7 127.7 0.66

1.379 1.365 1.440 1.081 1.082 106.4 110.4 106.4 115.6 134.0 126.5 127.1 0.60

1.361 1.354 1.432 1.075 1.076 106.8 110.4 106.2 115.9 133.6 126.5 127.3 0.61

1.348 1.355 1.418 1.085 1.086 107.2 110.4 106.1 116.1 133.6 126.5 127.4 0.51

1.380 1.362 1.439 1.075 1.077 106.3 110.4 106.4 115.6 134.0 126.5 127.1 0.63

a

Bond lengths in angstroms and angles in degrees; µ is the dipole moment in debye. b References 31 and 32.

Computational Details The calculations were done with the HF,25 MP2,2 BLYP,26 B3LYP,13 LDA,14 and BVWN methods using the 6-31G**,27 cc-pVDZ,28 and cc-pVTZ28 basis sets. The MP2 calculations were performed with and without correlating the 1s orbitals of the heavy elements acronymed as fixed core (fc) and MP2(full). The MP2 calculation with the cc-pVTZ basis set was not performed due to the large amount of CPU time required. The number of basis functions used for furan/thiophene with the 6-31G**, cc-pVDZ, and cc-pVTZ basis sets are 95/99, 90/ 94, and 206/210, respectively. The geometry optimization and force field calculations were done under the C2V symmetry constraints. All calculations were done using the Gaussian9429 program except those at the HF/6-31G** and MP2/6-31G** levels for both molecules and the MP2/cc-pVDZ level for furan, where the CADPAC30 program was used. For calculations with the Gaussian program and at the HF/6-31G** level with the

HF

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

1.718 1.344 1.432 1.069 1.071 91.5 111.7 112.6 120.6 127.7 123.6 123.8 0.81

1.743 1.374 1.430 1.082 1.086 91.5 111.4 112.9 119.9 128.7 123.3 123.9 0.47

1.726 1.363 1.423 1.077 1.080 91.7 111.4 112.7 120.2 128.4 123.4 123.9 0.52

1.704 1.363 1.409 1.087 1.090 92.2 111.3 112.6 120.1 128.6 123.2 124.2 0.43

1.744 1.371 1.430 1.077 1.080 91.4 111.4 112.9 120.0 128.6 123.3 123.8 0.49

a

See corresponding footnote in Table 1. b References 31, 32, and

34.

CADPAC program, the force fields were obtained analytically at the corresponding optimized geometries. For those calculated with MP2/6-31G** for furan and thiophene and MP2/cc-pVDZ for furan, the force fields were calculated numerically, at the corresponding optimized geometry, using the simplex algorithm with a step size of 0.001 bohr. To minimize the translational and rotational contamination of the force fields calculated numerically, the optimized geometries were obtained with the largest component of the Cartesian energy gradient of less than 10-5 hartree/bohr. While for those calculated at the HF/631G** level the largest component of the Cartesian energy gradient was less than 10-4 hartree/bohr. For those calculated with the Gaussian program, the default parameters were used. Results and Discussion Optimized Geometries. The theoretically optimized and experimental geometries determined for furan and thiophene

15058 J. Phys. Chem., Vol. 100, No. 37, 1996 by microwave spectroscopy31-34 are depicted in Tables 1 and 2, respectively. The atom numbering employed is shown in Figure 1. By examination of the results in Tables 1 and 2, several observations can be made. With the exception of the bond lengths calculated by the LDA method, it is clear that all the bond lengths determined by the DFT method using the 6-31G** and cc-pVDZ basis sets are overestimated in accordance with the previous observations.17-19 The only exception is the CdC bond length determined with a B3LYP/6-31G** calculation for thiophene, which is predicted to be only 0.002 Å too short. Also the bond lengths determined at the MP2/ccpVDZ level, fc and full, are overestimated,4 with the exception of the O-C bond length for furan, which is calculated to be only 0.001 Å too short with the MP2(full) method. In going to the larger cc-pVTZ basis set, the calculated bond lengths by the BLYP and BVWN methods are still overestimated, but for those calculated by the B3LYP method, as well as with the LDA method for all three basis sets, some are overestimated and some are underestimated. For the LDA method, the O-C and C-C bond lengths are calculated to be about 0.01 Å too short.17,19,35 The CdC bond length is almost exactly predicted by the 6-31G** basis set17,35 but predicted to be about 0.004 Å too long by the cc-pVDZ basis set and 0.006 Å too short by the cc-pVTZ basis set. The bond angles calculated by the different DFT methods are close to each other to within a few tenths of a degree.20 The largest difference is for the COC and CSC bond angles by the BVWN and LDA methods, where the difference is as high as 0.9°. The calculated geometries by the MP2(full) and MP2(fc) methods are close to each other to within 0.001 Å in most of the cases and 0.003 Å at most for bond lengths and 0.1° at most for bond angles. The calculated bond lengths at the HF level, as wellknown,17 are generally underestimated. Among the seven methods used in this study, Tables 1 and 2, the optimized geometry obtained by the B3LYP method has the best agreement with the experimental geometry, followed by the MP2 optimized geometry. The HF optimized geometry is the worst. The BVWN and BLYP geometries are close to each other, with the optimized geometry calculated by the BVWN method being slightly better, but both are worse than the B3LYP geometry. The optimized geometry calculated by the LDA method behaves differently from those obtained by the other methods, but it is generally better than that obtained by the BVWN method and worse than the B3LYP geometry. The agreement between the calculated and experimental geometries can then be arranged in the order B3LYP, MP2, LDA, BVWN, BLYP, then HF. Hertwig and Koch (HK)36 calculated the average error for the difference between the experimental and optimized geometries using the 6-311G* basis set. It was concluded that the optimized geometries calculated at the DFT/BLYP and DFT/ BP levels are better than those calculated at the HF or MP2 levels. It is clear that the results in the current study for the two heterocyclic molecules do not support such a conclusion, although the basis sets used in our study are quite different from those used by HK. The reason for the difference from HK is that in the chosen set of molecules there are molecules, for example F2, for which a single reference treatment, such as MP2, is not expected to yield reasonable results. It is exactly, the advantage of density functionals, that they are able to treat some cases for which an explicit multireference treatment would otherwise be mandatory. One such case is ozone,37 and another is the polyene radicals.38 A detailed discussion of the various correlation effects in the density functional theory compared to

El-Azhary and Suter

Figure 1. Atom numbering and internal coordinates for furan (X ) O) and thiophene (X ) S).

standard ab initio treatments may be found in the recent work of Handy and co-workers.39 Floria´n and Johnson11 concluded for formamide that the LDA and MP2 optimized geometries are comparable to each other. The results in Tables 1 and 2 for furan and thiophene do not support this conclusion. As mentioned above, the B3LYP optimized geometry has generally the best agreement with the experimental geometry. The exceptions are the S-C bond length and the CSC bond angle for thiophene, which are generally poorly predicted by all the DFT methods and the HF method with the 6-31G** and cc-pVDZ basis sets. The S-C bond length is calculated to be 0.02-0.04 Å too long by the BLYP, B3LYP, and BVWN methods, but with the LDA method it is calculated to be only a few thousandths of an angstrom too long. The CSC bond angle is underestimated by 0.6-1.0° by all the DFT methods. Similar observations were found for thiazole,40 1,3,4-thiadiazole, and 1,2,5-thiadiazole21 at the B3LYP/6-31G** level. The worst among these molecules is 1,2,5-thiadiazole, where the N-S bond length is calculated to be 0.033 Å too long and the NSN bond angle is underestimated by 1.1°. The reflection of the poor prediction of the S-X bond length and the XSY bond angle (where X and Y ) C or N) was such that the root-meansquare (rms) deviations of the calculated frequencies from the experimental frequencies using the scaled force field for the sulfur-containing molecules were larger than their oxygen analogs.10,21,40 In the case of thiophene at the B3LYP/6-31G** level, the S-C bond length is calculated to be 0.022 Å too long and the CSC bond angle is underestimated by 0.7°. On the other hand, at the MP2(full)/6-31G** level the S-C bond length is exactly predicted and the CSC bond angle is underestimated by only 0.1°. Also at the MP2(full)/cc-pVDZ level, the S-C bond length is predicted to be 0.011 Å too long. Using the cc-pVTZ basis set, this error by the HF and DFT methods is reduced. For furan using the BLYP and BVWN methods, with the three basis sets, the O-C bond length is calculated to be about 0.020 Å too long, while the COC bond angle is predicted by a difference of only 0.2° at most. With the LDA method, the O-C bond length is calculated to be about 0.011 Å too short and the COC bond angle is overestimated by 0.7°. But with the B3LYP method the O-C bond length is predicted within 0.002 Å and the COC bond angle is overestimated by about 0.4°. At the MP2 level, the O-C bond length is calculated within 0.004 Å at most and the COC bond angle is overestimated by 0.3° at most. It is concluded that, although the B3LYP method generally gives a better prediction of the experimental geometry than the other DFT methods used in our study, the

Optimized Geometries and Calculated Frequencies MP2 optimized geometry, which is only slightly worse than the B3LYP optimized geometry, avoids large errors in predicting some coordinates observed with the B3LYP method.21,40 The general trend of the bond distances could be compared to the results of Neumann et al.39 The DFT methods, especially the LDA approximation, predict too long bond lengths, similar to the CASSCF calculations. This could be interpreted as the effect of the static correlation. The HF method, on the other hand, normally underestimates the bond lengths. Since B3LYP uses HF exchange potentials and may be seen as a hybrid HF/ DFT method, the geometries are expected to be a mixture of HF and pure DFT results. However, this is not necessarily true, as the example of the C-S bond in thiophene shows. Finally, we compare the effect of changing the basis set on the optimized geometries. At the HF level the basis set change from 6-31G** to cc-pVDZ did not improve the optimized geometry even with the cc-pVTZ basis set. At the MP2 level the bond lengths are better predicted by the 6-31G** basis set, while the bond angles are better predicted by the cc-pVDZ basis set for furan, but they are similar for thiophene. At the DFT level, similar to that at the MP2 level, the bond lengths are better predicted with the 6-31G** basis set, while the agreement between the calculated and experimental bond angles by both basis sets is almost similar. With the cc-pVTZ basis set, the calculated bond lengths are shorter than those calculated by the other two basis sets.18,19 Consequently the agreement between the calculated and experimental bond lengths is significantly improved with the exception of those calculated by the LDA method and the CdC bond length with the B3LYP method, where the agreement became worse. The bond angles are almost unchanged. Calculated Frequencies. The calculated frequencies for furan and thiophene are depicted in Tables 3 and 4, respectively. For comparison, the experimental frequencies reported for furan22 and thiophene23 are also included in Tables 3 and 4. To facilitate the comparison between the experimental and calculated frequencies, the average error for the C-H stretching modes (ν1, ν2, ν12, and ν13 in Tables 3 and 4), in-plane modes (excluding the C-H stretching modes), out-of-plane modes, mid-IR modes (all vibrational modes excluding the C-H stretching modes), and the total average error were calculated. These are included at the end of Tables 3 and 4. Since the studied molecules belong to the C2V symmetry point group, their 21 fundamental vibrations are classified as 8A1 + 3A2 + 7B1 + 3B2. The calculated frequencies at the HF level are grossly overestimated, with an average error of about 150 cm-1, but the overestimation decreases as the wavenumber decreases. This is due to the larger anharmonicity of the higher frequency modes. The situation is similar with the MP2 calculation except that the average error of the calculated frequencies from the experimental frequencies is only about 80-45 cm-1 depending on the basis set used and on the molecule. On the other hand, some of the out-of-plane modes, contrary to the HF calculations, are underestimated, especially with the 6-31G** basis set for furan. With the DFT calculations, the average error is significantly less than that for the MP2 or HF calculations11,17-19 mainly because the calculated frequencies for the C-H stretching modes are closer to the corresponding experimental frequencies11 and most of the bond lengths calculated with DFT methods are too long. For the C-H stretching modes, the overestimation of the calculated frequencies with DFT is inversely proportional to the overestimation of the C-H bond lengths. The overestimation of the C-H bond lengths decreases in the order LDA,

J. Phys. Chem., Vol. 100, No. 37, 1996 15059 BLYP, BVWN, then B3LYP. The overestimation of the calculated frequencies decreases in the order B3LYP, BVWN, LDA, and BLYP, with the order of LDA and BLYP reversed. As an example, for furan with the 6-31G** basis set, the C-H bond lengths are overestimated by about 0.012, 0.010, 0.005, and 0.003 Å, with the LDA, BLYP, BVWN, and B3LYP methods, respectively. The corresponding average error of the calculated frequencies is 67, 48, 86, and 130 cm-1. In the mid-IR region, the average errors obtained by the four DFT methods are close to each other, but the error obtained with the BVWN method is generally the lowest. This may be interpreted as an advantage of the BVWN method over the more popular BLYP method. On the other hand, the highest average error is obtained by the B3LYP method with the 6-31G** and cc-pVTZ basis sets, but with the cc-pVDZ basis set, the BLYP method has the highest average error. It is interesting to notice that for the out-of-plane modes with the 6-31G** and cc-pVDZ basis sets the B3LYP method has the lowest average error, around 10 cm-1, which is about half that with the other methods, but with the cc-pVTZ basis set it is the highest, about 20 cm-1. Also with the 6-31G** basis set, the average error obtained by the MP2 method in the mid-IR region is about twice that by the DFT methods, but with the cc-pVDZ basis set the average error is close to those obtained by the DFT methods. All calculated frequencies by the B3LYP method are overestimated, with the exception of only a few bands for thiophene with the 6-31G** and cc-pVDZ basis sets, and also the overestimation decreases gradually with the decrease of wave length,41 a behavior similar to that observed at the HF and MP2 levels. On the other hand, the calculated frequencies by the BLYP method are underestimated except for those corresponding to the C-H stretching modes, which are overestimated by about 40 cm-1 and only a few bands for thiophene with the 6-31G** and cc-pVDZ basis sets. As a result, scaling of the BLYP force fields using one-scale-factor scaling might not improve the agreement between the calculated and experimental frequencies as a scale factor close to 1.0 is expected. On the other hand, scaling of the B3LYP force fields is expected to significantly improve the agreement between the calculated and experimental frequencies.10,21,40 Noting that the optimized geometries by the BVWN and BLYP methods are close to each other, with the one calculated by the BVWN method being slightly better, the calculated frequencies by the BVWN and BLYP methods are similar to each other except that in the mid-IR region the calculated frequencies by the BVWN method have a smaller average error and some bands are overestimated. Also the C-H frequencies are overestimated by about 80 cm-1 compared to about 40 cm-1 with the BLYP method. Those calculated by the LDA method are similar to those calculated by the B3LYP method except for the C-H frequencies, which are overestimated by about 60 cm-1 compared to about 130 cm-1 by the B3LYP method, and in the mid-IR region some bands are underestimated. This is because, compared to the experimental bond lengths, some bond lengths are calculated to be too long and some are calculated to be too short, while those calculated by the B3LYP method are generally too long. The calculated frequencies at the HF level showed little dependence on the basis sets used. Those calculated at the MP2 level in the mid-IR region with the cc-pVDZ basis set have an average error about half that with the 6-31G** basis set, which is comparable to those obtained with DFT calculations. This is because the bond lengths calculated with the MP2 and ccpVDZ basis set are longer than those calculated with the

15060 J. Phys. Chem., Vol. 100, No. 37, 1996

El-Azhary and Suter

TABLE 3: Comparison between the Experimental and Calculated Frequencies for Furana 6-31G** sym no exptlb A1

A2 B1

1 2 3 4 5 6 7 8 9 10 11 12 13 14

HF

3167 3140 1491 1384 1140 1066 995 871 863 728 613 3161 3129 1556

3463 3430 1680 1549 1265 1161 1090 1090 956 856 663 3457 3417 1768

MP2 MP2 DFT DFT DFT DFT full fc BLYP B3LYP LDA BVWN 3379 3353 1545 1462 1187 1128 1047 885 812 705 577 3373 3343 1620

3375 3349 1543 1459 1184 1126 1046 882 805 698 573 3369 3339 1617

3217 3186 1471 1379 1135 1056 984 856 835 681 593 3211 3175 1552

3298 3268 1529 1430 1173 1098 1022 886 880 729 614 3292 3258 1614

3233 3208 1509 1416 1162 1106 1004 869 844 698 605 3226 3198 1588

3256 3223 1479 1387 1145 1056 988 860 846 687 595 3250 3212 1560

sym

no

B1

exptlb

15 1267 16 1180 17 1040 18 873 19 838 20 745 21 603 CH in-planed out-of-plane mid-IRe total

B2 errorc

SFf errorc

HF

MP2 full

MP2 fc

DFT DFT DFT DFT BLYP B3LYP LDA BVWN

1412 1322 1165 961 994 862 659 293 132 110 124 156 0.819 25

1322 1268 1093 891 810 756 627 213 54 29 45 77 0.891 31

1319 1266 1091 889 804 751 623 209 51 31 44 75 0.894 32

1255 1159 1027 862 790 726 601 48 12 27 17 23 0.985 24

1296 1220 1071 890 839 762 623 130 33 10 25 45 0.928 13

1257 1229 1043 866 796 727 629 67 20 24 21 30 0.963 19

1274 1160 1030 868 802 734 600 86 9 21 13 27 0.965 28

cc-pVDZ no.

HF

MP2 full

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3457 3423 1671 1534 1249 1156 1073 952 998 847 659 3450 3410 1755

3345 3316 1524 1433 1168 1119 1022 879 848 729 605 3337 3304 1590

MP2 fc

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

3334 3308 1520 1430 1164 1116 1021 875 843 719 599 3327 3296 1585

3204 3174 1457 1364 1120 1050 969 851 847 688 595 3198 3163 1536

3285 3256 1518 1415 1159 1090 1005 881 887 734 614 3278 3245 1600

3226 3200 1501 1403 1158 1091 985 863 851 702 606 3218 3190 1576

3244 3212 1466 1374 1130 1050 976 856 860 695 597 3238 3200 1546

no.

HF

MP2 full

MP2 fc

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

15 16 17 18 19 20 21

1385 1305 1145 957 985 851 658 286 120 101 113 146 0.827 20

1285 1250 1062 880 829 762 630 176 32 13 25 54 0.912 27

1282 1250 1059 878 823 755 628 167 29 16 24 51 0.917 28

1229 1143 1011 859 799 723 602 36 25 23 24 26 0.995 27

1271 1204 1053 887 846 758 622 117 20 12 17 36 0.938 15

1241 1207 1022 863 803 723 628 59 17 21 18 26 0.970 19

1250 1144 1016 865 812 733 600 74 17 16 17 28 0.974 30

cc-pVTZ no

HF

DFT BLYP

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3432 3401 1662 1534 1256 1153 1078 954 1012 855 661 3425 3390 1743

3203 3174 1455 1365 1129 1045 976 859 848 693 599 3196 3163 1535

DFT B3LYP

DFT LDA

DFT BVWN

3279 3251 1513 1415 1167 1086 1015 888 893 739 618 3272 3241 1595

3211 3187 1490 1399 1154 1094 995 872 861 709 611 3203 3178 1568

3244 3213 1466 1375 1142 1045 980 863 861 700 601 3238 3203 1547

no

HF

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

15 16 17 18 19 20 21

1408 1306 1156 960 1000 859 659 263 122 109 117 145 0.833 25

1255 1170 1015 867 808 726 604 35 20 19 20 23 0.993 24

1295 1199 1062 894 858 762 625 112 24 18 22 39 0.938 11

1240 1215 1034 874 818 727 630 46 13 15 14 20 0.976 15

1276 1146 1017 872 822 736 603 75 14 11 13 25 0.971 27

a Frequencies are in cm-1. b Reference 22. c Average error in cm-1. d Excluding CH stretching modes. e All vibrational modes excluding CH stretching modes. f Scale factor.

6-31G** basis set. With DFT, the calculated frequencies, similar to the HF ones, showed little dependence on the basis set used. In the mid-IR region, the average error of the calculated frequencies by the B3LYP method are the smallest with the cc-pVDZ basis set, while those calculated by the LDA method are the smallest with the cc-pVTZ basis set. Recently, Nonella et al.42 used the BP86 method with the 6-31G** basis set to calculate frequencies for p-benzoquinone, 1,4-naphthoquinone, and naphthalene. The corresponding rootmean-square (rms) deviations, excluding the C-H stretching modes, are 21, 19, and 16 cm-1, respectively. For comparison with our results, we calculated the average errors for the above mentioned molecules. These are 15, 14, and 12 cm-1, respectively. These average errors are close to those obtained in our study by the BVWN/6-31G** force fields. In addition, most of the calculated frequencies by the BP86/6-31G** force fields

for the above mentioned molecules are overestimated. This may suggest that the BP86 and the BVWN methods behave similarly. As was concluded before at the MP2/DZP level by Simandris et al.,5 the calculated frequencies for furan and thiophene in this work support the experimental assignment for the d0 isotopomer for both molecules. It is worth noting that, although a comparison between the calculated and experimental IR absorption intensities is not included in our study, Stephens et al.13 showed that the calculated IR absorption and vibrational circular dichroism spectra for 4-methyl-2-oxetanone by the B3LYP method are in excellent agreement with the experimental spectra and are significantly more accurate than those obtained by the LDA, BLYP, and HF methods. Also, those calculated at the MP2 level are only slightly worse than those calculated by the B3LYP method.10,21,40 It was concluded also that the 6-31G* basis set

Optimized Geometries and Calculated Frequencies

J. Phys. Chem., Vol. 100, No. 37, 1996 15061

TABLE 4: Comparison between the Experimental and Calculated Frequencies for Thiophenea 6-31G** sym no. exptlb A1

A2 B1

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3126 3098 1409 1360 1083 1036 839 608 898 683 565 3125 3098 1507

HF 3427 3390 1597 1530 1207 1106 890 658 1053 804 618 3425 3376 1740

MP2 MP2 DFT DFT DFT DFT full fc BLYP B3LYP LDA BVWN 3346 3312 1488 1436 1132 1094 867 630 867 673 550 3343 3299 1571

3341 3308 1483 1433 1131 1092 883 628 855 667 542 3338 3294 1568

3189 3144 1416 1360 1082 1030 803 591 877 647 555 3187 3131 1515

3271 3229 1470 1407 1112 1059 832 610 920 683 575 3269 3215 1578

3207 3168 1477 1374 1062 1054 856 609 883 654 570 3205 3155 1549

3226 3181 1419 1373 1099 1034 802 592 888 655 556 3224 3167 1524

exptlb

sym

no.

B1

15 1256 16 1085 17 872 18 751 19 867 20 712 21 452 CH in-planed out-of-plane mid-IRe total

B2 errorc

SFf errorc

HF

MP2 full

MP2 DFT DFT DFT DFT fc BLYP B3LYP LDA BVWN

1407 1210 951 808 1029 813 484 293 118 104 113 147 0.819 29

1314 1134 912 781 857 732 457 213 50 15 38 71 0.888 24

1311 1133 910 778 847 726 453 209 49 20 39 71 0.894 26

1240 1084 846 713 833 701 437 48 14 21 16 23 0.980 23

1282 1115 876 742 880 728 453 130 28 10 22 43 0.925 16

1226 1068 866 754 833 700 458 67 22 17 20 30 0.962 21

1259 1099 853 711 845 710 438 88 17 14 16 30 0.964 26

cc-pVDZ no.

HF

MP2 full

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3416 3382 1587 1509 1181 1098 882 653 1035 789 613 3413 3367 1724

3299 3272 1466 1400 1088 1073 872 617 890 675 564 3296 3258 1537

MP2 fc

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

3293 3266 1463 1399 1087 1071 870 615 884 670 561 3290 3252 1535

3171 3133 1410 1342 1053 1021 799 590 882 647 556 3168 3120 1502

3253 3218 1462 1390 1085 1051 835 611 921 683 576 3250 3204 1563

3194 3162 1473 1358 1044 1033 852 606 882 651 569 3191 3149 1538

3208 3170 1412 1355 1073 1026 799 591 895 656 558 3205 3157 1511

no.

HF

MP2 full

MP2 fc

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

15 16 17 18 19 20 21

1379 1186 936 804 1012 800 481 283 103 92 99 134 0.829 23

1269 1093 892 770 857 728 454 170 25 8 19 47 0.918 27

1268 1092 890 769 852 724 452 164 23 10 18 46 0.914 26

1217 1057 838 712 834 695 439 36 24 15 21 24 0.992 26

1259 1098 869 748 878 722 454 120 17 10 15 34 0.936 19

1203 1041 858 753 830 692 457 62 24 19 22 30 0.972 26

1237 1075 845 711 848 705 439 73 17 13 16 26 0.970 28

cc-pVTZ no

HF

DFT BLYP

1 2 3 4 5 6 7 8 9 10 11 12 13 14

3396 3359 1560 1518 1195 1098 885 654 1051 800 617 3393 3344 1709

3174 3132 1395 1350 1077 1026 804 594 884 656 562 3172 3119 1496

a,c,d,e,f

DFT B3LYP

DFT LDA

DFT BVWN

3250 3211 1444 1397 1106 1055 839 615 929 694 581 3247 3197 1555

3183 3146 1452 1362 1051 1049 858 612 890 662 576 3180 3134 1526

3213 3171 1402 1363 1096 1031 803 595 898 666 564 3211 3157 1508

no

HF

DFT BLYP

DFT B3LYP

DFT LDA

DFT BVWN

15 16 17 18 19 20 21

1400 1199 944 806 1027 808 488 261 110 102 107 136 0.834 27

1242 1077 850 717 842 698 446 38 16 15 16 20 0.988 21

1284 1108 880 752 889 727 462 115 21 18 20 38 0.935 12

1224 1056 868 762 842 695 465 49 19 16 18 24 0.975 18

1262 1094 856 715 857 709 447 76 10 6 10 21 0.966 23

See corresponding footnote in Table 3. b Reference 23.

is optimal to reproduce an accurate representation of the IR absorption spectra with the B3LYP method. Scale Factors. Table 5 shows the internal coordinate definition used to convert the Cartesian coordinate force fields to internal coordinate force fields for furan and thiophene. The scaling of the force fields was performed as described elsewhere.7,10 Although scaling was performed only for the d0 isotopomers of furan and thiophene, in a second calculation scaling was done for all the isotopomers of furan and thiophene for which the vibrational assignment was reported.22,23 These are the d0, 2,5-d2, 3,4-d2, d4, 2-d1, 3-d1, and 2,3,5-d3 isotopomers for furan22 and the d0, 2,5-d2, 3,4-d2, d4, 2-d1, 3-d1, 2,3-d2, 2,4d2, 2,3,5-d3, and 2,3,4-d3 isotopomers for thiophene.23 The difference between the scale factors when scaling was done for the d0 isotopomers only and that when all the isotopomers were included was in most of cases not more than 0.005. For consistency with the unscaled frequencies, the results of scaling

reported here are for the d0 isotopomers only. The obtained scale factors and the corresponding average errors are in Tables 3 and 4 for furan and thiophene, respectively. It is clear that scaling the HF and MP2 force fields significantly improved the average error, bringing them close to or even smaller than some of those obtained by the DFT methods. The average scale factors for the two studied molecules at the HF and MP2 levels are 0.82 and 0.89 with the 6-31G** basis set and 0.83 and 0.91 with the cc-pVDZ basis set, respectively. These values are close to those obtained for similar five-membered heterocyclic molecules with the 6-31G**,10,21,40 0.80 and 0.90 at the HF and MP2 levels, respectively. The values of the scale factors with the 6-31G** basis set correspond to 0.91 and 0.94 at the HF and MP2 levels, respectively, for frequency scaling. These values are close to those previously reported,4,43 0.8929 and 0.9427, at the HF and MP2 levels, respectively.

15062 J. Phys. Chem., Vol. 100, No. 37, 1996

El-Azhary and Suter

TABLE 5: Internal Coordinates for Furan and Thiophenea no. q1, q5 q2, q4 q3 q6, q7, q8, q9 q10 q11 q12, q13, q14, q15 q16, q17, q18, q19 q20 q21

modeb

description

r1, r5 r2, r4 r3 r6, r7, r8, r9 R1 + a(R2 + R5) + b(R3 + R4) (a - b)(R2 - R5) + (1 - a)(R3 - R4) β1 - β2, β3 - β4, β5 - β6, β7 - β8 τ6, τ7, τ8, τ9 b(τ1 + τ5) + a(τ2 + τ4) + τ3 (a - b)(τ4 - τ2) + (1 - a)(τ5 - τ1)

C-X stretchc CdC stretch C-C stretch C-H stretch ring deformation ring deformation CH rocking CH wagging ring torsion ring torsion

a See Figure 1 for definition of r, R, β, and τ internal coordinates. a ) cos 144° and b ) cos 72°. Values of normalization constants are not given. c X ) O for furan and X ) S for thiophene. b

At the DFT level, the average error of the B3LYP method is the lowest, with a significant improvement over that obtained with the unscaled force field. The average values of the 6-31G** and cc-pVDZ scale factors are 0.93 and 0.94, respectively. The value of the 6-31G** scale factor is the same as that reported for the above mentioned molecules,10,21,40 0.93. It is then concluded, for this set of five-membered heterocyclic molecules, that scaling the B3LYP force fields produces more consistent scale factors than those obtained with the MP2 or HF force fields. With the BLYP method scale factors of 0.98 and 1.00 are obtained with the 6-31G** and cc-pVDZ basis sets, respectively, but the average error is similar to or larger by 1 or 2 cm-1 than that with the unscaled force fields. This in fact was expected for reasons that were mentioned above. Practically, upon scaling, the average error for the C-H stretching modes improved, but at the expense of the average error for mid-IR modes, which became worse. In a separate calculation, the BLYP/6-31G** force field for furan was scaled using two scale factors, one for the C-H stretching modes and the other for the mid-IR modes. The obtained average error for the C-H stretching modes was only 2 cm-1, for the mid-IR modes it was 10 cm-1, and the total average error was 9 cm-1. The corresponding scale factors are 0.97 and 1.02 for the C-H stretching modes and the mid-IR modes, respectively. For comparison, the same calculation was repeated for the B3LYP/6-31G** force field for furan, and the obtained average errors were 1, 8, and 6 cm-1 for the C-H stretching modes, mid-IR modes, and all the modes, respectively. The corresponding scale factors are 0.92 and 0.95, respectively. It is clear that the use of a unique scale factor for the C-H stretching modes improved the average error of the BLYP force field to be close to that obtained by the B3LYP force field. This might suggest that a further increase of the number of scale factors might overcome the difference between the BLYP and B3LYP force fields. However, Rauhut and Pulay (RP)20 calculated one-scale-factor and multi-scale-factor scaling by the BLYP and B3LYP methods using the 6-31G* basis set for 20 molecules. They concluded that even with multi-scale-factor scaling the B3LYP performs uniformly better than the BLYP. Notice that the above scale factors for the C-H stretching modes by the BLYP and B3LYP methods are close to those obtained by RP,20 0.977 and 0.920, respectively. The average scale factors for furan and thiophene by the B3LYP and BLYP force fields using the 6-31G** basis set are 0.93 and 0.98, respectively. These values are close to those reported by RP20 with the B3LYP, 0.928, and BLYP, 0.990, force fields using the 6-31G* basis set. Recently, Scott et al.44 reported scale factors of 0.9614 and 0.9945 for frequency scaling

by the B3LYP and BLYP methods, respectively. These correspond to 0.9890 and 0.9243 for force field scaling. These values are close to those obtained by us and to those obtained by RP.20 On the other hand, there is no scale factor we know for the force fields calculated by the LDA and BVWN force fields. Scaling the BVWN force field is similar to scaling the BLYP force field with values of scale factors of 0.96 and 0.97, obtained with the 6-31G** and cc-pVDZ basis sets, respectively. Scaling of the LDA and B3LYP force fields is also similar to scale factors of 0.96 and 0.97 obtained with the 6-31G** and ccpVDZ basis sets, respectively, but the average error is larger than with the B3LYP force field (but smaller than that with the BLYP and BVWN force fields) due to the fact that in the midIR region some bands are underestimated and some are overestimated. The effect of basis set change on the scale factors was small, with the scale factors determined by the cc-pVDZ and cc-pVTZ basis sets close to each other and slightly larger than those obtained with the 6-31G** basis set. Conclusion In the present paper the optimized geometries, frequencies, and scale factors obtained for furan and thiophene with HF, LDA, BVWN, BLYP, B3LYP, and MP2 methods using the 6-31G**, cc-pVDZ, and cc-pVTZ basis sets were compared. The agreement between the calculated and experimental geometries can be arranged in the order B3LYP, MP2, LDA, BVWN, BLYP, and HF. Although the B3LYP optimized geometry is slightly better than the MP2 optimized geometry, the MP2 optimized geometry avoids large errors in predicting some coordinates observed by the B3LYP optimized geometry. The effect of changing the basis set on the bond angles was minor, but with the exception of the LDA method, the bond lengths are better predicted by the cc-pVTZ, followed by the 6-31G**, then the cc-pVDZ basis sets. Among the methods used in our study, the calculated frequencies in the mid-IR region by the BVWN force field have the lowest average error. Among the four DFT methods, those frequencies calculated by the B3LYP force field have the highest average error. This is with the exception of the cc-pVDZ basis set, where the average error is close to that obtained by the BVWN method, and the calculated frequencies by the BLYP force field have the highest average error. The calculated frequencies by the B3LYP force field are generally overestimated, but those calculated by the BLYP force field are underestimated, except for those corresponding to the C-H stretching modes. This is reflected such that scaling by the B3LYP force field produced the lowest average error, and the average error obtained by scaling the BLYP force field remained almost unchanged. The calculated frequencies by the BVWN force field are similar to those calculated by the BLYP force field, and those calculated by the LDA force field are similar to those calculated by the B3LYP force field. The effect of changing the basis set on the calculated frequencies and scale factors was minor. In summary, the present study indicates that scaled B3LYP force fields are preferred for vibrational analysis, where better agreement between the calculated and experimental geometries, frequencies, and intensities is expected. In the case where the optimized geometry is the main interest, the MP2 method is more reliable, but at the expense of more computational time. If unscaled force fields are to be used for the vibrational analysis, the BVWN force field is the most accurate force field among the DFT methods used in this study.

Optimized Geometries and Calculated Frequencies Acknowledgment. A.A.E.-A. is grateful to Prof. T. A. Keiderling of the University of Illinois at Chicago for the use of the CADPAC program installed on the Department of Chemistry Titan minisupercomputer. A generous grant of computer time at the Centro Svizzero di Calcolo Scientifico (CSCS) in Manno is gratefully acknowledged. References and Notes (1) Pulay, P.; Fogarasi, G.; Pongar, G.; Boggs, J. E.; Vorgha, A. J. Am. Chem. Soc. 1983, 105, 7037. Pulay, P.; Fogarasi, G.; Pang, F.; Boggs, J. E. J. Am. Chem. Soc. 1979, 101, 2550. (2) Møller, C.; Plesset, M. S. Phys. ReV. 1934, 46, 618. (3) Simandiras, E. D.; Amos, R. D.; Handy, N. C. Chem. Phys. 1987, 114, 9, and references therein. (4) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (5) Simandiras, E. D.; Handy, N. C.; Amos, R. D. J. Phys. Chem. 1988, 92, 1739. (6) Tornkvist, C.; Bergman, J.; Liedberg, B. J. Phys. Chem. 1991, 95, 3119. Murphy, W. F.; Zerbetto, F.; Duncan, J. L.; Mckean; D. C. J. Phys. Chem. 1993, 97, 581. Tang, W.; Bally, T. J. Phys. Chem. 1993, 97, 4365. Gejji, S. P.; Hermansoon, K.; Lindgren, J. J. Phys. Chem. 1993, 97, 6986. Cadioli, B.; Gallinella, E.; Coulombean, C.; Jobic, H.; Berthier, G. J. Phys. Chem. 1993, 97, 7844. (7) El-Azhary, A. A. Spectrochim. Acta 1995, 51A, 995. (8) El-Azhary, A. A. Acta Chem. Scand. 1995, 49, 11. (9) El-Azhary, A. A. J. Chem. Res. 1995, 174, 1149. (10) El-Azhary, A. A.; Suter, H. U. J. Phys. Chem. 1995, 99, 12751. (11) Floria´n, J.; Johnson, B. G. J. Phys. Chem. 1994, 98, 3681. (12) Floria´n, J.; Johnson, B. G. J. Phys. Chem. 1995, 99, 5899. (13) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. (14) Vosko, S. H.; Wilk, L.; Nussair, M. Can. J. Phys. 1980, 58, 1200. (15) Becke, A. D. Phys. ReV. 1988, A38, 3098. (16) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (17) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J. Chem. Phys. 1993, 98, 5612. (18) Murray, C. W.; Laming, G. J.; Handy, N. C.; Amos, R. D. Chem. Phys. Lett. 1992, 199, 551. (19) Handy, N. C.; Murray, C. W.; Amos, R. D. J. Phys. Chem. 1993, 97, 4392. (20) Rauhut, G.; Pulay, P. J. Phys. Chem. 1995, 99, 3093. (21) El-Azhary, A. A. Spectrochim. Acta 1996, 52A, 33. (22) Rico, M.; Barranchina, M.; Orza, J. M. J. Mol. Spectrosc. 1967, 24, 133. (23) Rico, M.; Orza, J. M.; Morcillo, J. Spectrochim. Acta 1965, 21, 289. (24) Orza, J. M.; Rico, M.; Biarge, F. J. Mol. Spectrosc. 1966, 19, 1988. Scott, D. W. J. Mol. Spectrosc. 1969, 31, 451. Scott, D. W. J. Mol.

J. Phys. Chem., Vol. 100, No. 37, 1996 15063 Spectrosc. 1971, 37, 77. Ba´nki, J.; Billes, F.; Grofcsik, A. Acta Chim. Hungarica 1984, 116, 283. (25) Roothaan, C. C. J. ReV. Mod. Phys. 1951, 23, 69. (26) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. 1988, B37, 785. Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Chem. Phys. Lett. 1989, 157, 200. (27) Ditchfield, R.; Hehre, W. J.; Pople, J. A. J. Chem. Phys. 1971, 54, 724. Hehre, W. J.; Ditchfield, R.; Pople, J. A. J. Chem. Phys. 1972, 56, 2257. (28) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007. (29) Gaussian 94, Revision B.3; 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., Raghavachari, K., Al-Laham, M. A., Zakrzewski, V. G., Ortiz, J. V., Foresman, J. B., Peng, C. Y., Ayala, P. Y., Chen, 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., Head-Gordon, M., Gonzalez, C., Pople, J. A., Eds.; Gaussian, Inc.: Pittsburgh, PA, 1995. (30) Amos, R. D. CADPAC-The Cambridge Analytical Derivatives Package; SERC: Darebury, U. K., 1984. (31) Harmony, M. D.; Lawrie, V. W.; Kuczkowski, R. L.; Schwendeman, R. H.; Ramsey, D. A.; Lovas, F. J.; Lafferty, W. J.; Maki, A. G. J. Phys. Chem. Ref. Data 1979, 8, 619. (32) Bak, B.; Christensen, D.; Dixon, W. B.; Hansen-Nygaard, L.; Rastrup-Andersen, J.; Schottla¨nder, M. J. Mol. Spectrosc. 1962, 9, 124. Mata, F.; Martin, M. C.; Sørensen, G. O. J. Mol. Struct. 1978, 48, 157. (33) Bak, B.; Christensen, D.; Hansen-Nygaard, L.; Rastrup-Andersen, J. J. Mol. Struct. 1961, 7, 58. (34) Harris, B.; Le Fe´vre, R. J. W.; Sullivan, E. P. A. J. Chem. Soc. 1953, 1622. (35) Andzelm, J.; Wimmer, E. J. Chem. Phys. 1992, 96, 1280. (36) Hertwig, R. H.; Koch, W. J. Comput. Chem. 1995, 16, 576. (37) Murray, C. W.; Handy, N. C.; Amos, R. D. J. Chem. Phys. 1993, 98, 7145. (38) Sim, F.; Salahub, D. R.; Chin, S.; Dupuis, M. J. Chem. Phys. 1991, 95, 4317. (39) Neumann, R.; Nobes, R. H.; Handy, N. C. Mol. Phys. 1996, 87, 1. Mok, D. K. W.; Neumann, R.; Handy, N. C. J. Phys. Chem. 1996, 100, 6225. (40) El-Azhary, A. A.; Ghoneim, A. A.; El-Shakr, M. J. Chem. Res. 1995, (S) 354; (M) 2123. (41) See note added in proof in ref 10. (42) Nonella, M.; Tavan, P. Chem. Phys. 1995, 199, 17. Nonella, M. J. Mol. Struct. (THEOCHEM) 1996, 362, 7. (43) Pople, J. A.; Scott, A. P.; Wong, M. W.; Radom, L. Isr. J. Chem. 1993, 33, 345. (44) Scott, A. P.; Radom, L. J. Phys. Chem., submitted for publication.

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