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Comparison between Optimized Geometries and Vibrational Frequencies Calculated by the

DFT Methods

A. A. El-Azhary*,

Institut fur Theoretische Chemie, UniVersitat Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany

H. U. Suter

Institut fur Physikalische und Theoretische Chemie, UniVersitat 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 thiophenewith the HF, MP2, LDA, BVWN, BLYP, and B3LYP methods of theory using the 6-31G**, cc-pVDZ, andcc-pVTZ basis sets. The agreement between the optimized and experimental geometries was in the orderB3LYP, MP2, LDA, BVWN, BLYP, then HF. The calculated frequencies by the unscaled BVWN forcefield had the smallest average error in the mid-IR region, but using one-scale-factor scaling, those calculatedby the scaled B3LYP force field had the lowest average error. Using one-scale-factor scaling, scale factorsof 0.82, 0.89, 0.98, 0.93, 0.96, and 0.96 were obtained by the HF, MP2, BLYP, B3LYP, LDA, and BVWNforce 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, theagreement 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 moleculesis greatly enhanced by the help of ab initio calculations, whichare in principle able to reproduce the harmonic force fields toany desired accuracy. Despite this theoretical possibility, it isoften useful to work with scaled force fields, in the way definedby the pioneering work by Pulay and co-workers.1 This scalingprocedure will take into account the inadequacy of the methods

(missing correlation, basis set effects) as well as the anharmo-nicity of the potential. To be a reasonable starting point for ascaling procedure, the used method must show a systematicbehavior. For the Hartree-Fock procedure, for example, it isknown that the frequencies are uniformly overestimated by about10%. The advance in computer hardware and the desire to getmore accurate predictions of the vibrational spectra mobilizedthe use of the more time consuming methods, including electroncorrelation. The simplest of these methods is the Mller-Plesset (MP2) method,2 which gained interest especially afterthe development of the force fields and diople momentderivatives calculated by the analytical method.3 Typically, theMP2 frequencies are overestimated by about 5%.4 Althoughthe MP2 force fields gained some interest for the vibrationalanalysis,5-12 MP2s computational demand still limited its wideapplication.

Recently, the DFT method was introduced for the vibrationalanalysis. The DFT method gained interest due to its excellentaccuracy to CPU time ratio.13 Currently, three types of densityfunctionals are in commen use. The first type uses only afunction of the electron density and is sometimes called localdensity approximation (LDA). The VWN (Vosko-Wilk-Nussair) functional14 belongs to that class. The second type

also uses the gradient of the electron density in the functional;from those the Becke correction15 seems to be the mostprominent. Therefore, the BVWN and BLYP functionals belongto that class. The development of mixed functionals16 definesa third class. In these functionals the different terms are mixedwith empirically determined parameters. The so-called B3LYPfunctional13 will be used as an example of that class offunctionals. 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 thesefunctionals.1-13,17-20 In addition, in a previous publication wecompared 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 mostpromising B3LYP functional was not considered (since thesepublications were reported before the development of theB3LYP functional), the basis set effect was not considered(although this is expected to have minor effect,18,19) or thecalculated vibrational intensities13 or the scale factors20 werethe 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, andBVWN methods in addition to the HF and MP2 methods.Further, to investigate the basis set effect, three basis sets werechosen. These are the 6-31G**, cc-pVDZ, and cc-pVTZ basissets. The two molecules furan and thiophene were selected forthis investigation, because on one hand they are rather wellcharacterized experimentally, while on the other hand they arelarge enough not to be trivial cases. Both molecules have beenthe subject of detailed vibrational analysis22,23 and normalcoordinate analysis using empirical force fields.24 Harmonicfrequencies and IR absorption intensities at the MP2/DZP levelwere also reported for both molecules.5

* Author to whom correspondence should be addressed. Permanent address: Department of Chemistry, Faculty of Science, Cairo

University, Giza, Egypt. E-mail: [email protected] Abstract published in AdVance ACS Abstracts,August 15, 1996.

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

S0022-3654(96)00618-1 CCC: $12.00 1996 American Chemical Society


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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 calculationswere performed with and without correlating the 1s orbitals ofthe heavy elements acronymed as fixed core (fc) and MP2-(full). The MP2 calculation with the cc-pVTZ basis set wasnot performed due to the large amount of CPU time required.The number of basis functions used for furan/thiophene withthe 6-31G**, cc-pVDZ, and cc-pVTZ basis sets are 95/99, 90/94, and 206/210, respectively. The geometry optimization andforce field calculations were done under the C2V symmetryconstraints. 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 withthe Gaussian program and at the HF/6-31G** level with the

CADPAC program, the force fields were obtained analyticallyat the corresponding optimized geometries. For those calculatedwith MP2/6-31G** for furan and thiophene and MP2/cc-pVDZfor furan, the force fields were calculated numerically, at the

corresponding optimized geometry, using the simplex algorithmwith a step size of 0.001 bohr. To minimize the translationaland rotational contamination of the force fields calculatednumerically, the optimized geometries were obtained with thelargest component of the Cartesian energy gradient of less than10-5 hartree/bohr. While for those calculated at the HF/6-31G** level the largest component of the Cartesian energygradient was less than 10-4 hartree/bohr. For those calculatedwith the Gaussian program, the default parameters were used.

Results and Discussion

Optimized Geometries. The theoretically optimized andexperimental geometries determined for furan and thiophene

TABLE 1: Equilibrium Geometry for Furan

6-31G**

coor-dinate exptlb HF

MP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

O1-C2 1.362 1.343 1.364 1.366 1.382 1.364 1.351 1.383C2dC3 1.361 1.339 1.365 1.366 1.371 1.361 1.361 1.369C3-C4 1.431 1.441 1.426 1.427 1.444 1.435 1.422 1.443C2-H6 1.075 1.069 1.074 1.075 1.086 1.079 1.088 1.080C3-H7 1.077 1.070 1.075 1.076 1.087 1.080 1.089 1.082C5O1C2 1.06.5 1.07.2 106.6 106.6 106.4 106.8 107.2 106.3

O1C2C3 110.7 110.8 110.5 110.5 110.5 110.5 110.4 110.5C2C3C4 106.0 105.6 106.2 106.2 106.3 106.0 106.0 106.3O1C2H6 115.9 116.2 115.7 115.7 115.4 115.8 115.9 115.4C3C2H6 133.4 132.9 133.8 133.9 134.1 133.7 133.8 134.1C2C3H7 126.1 126.8 126.2 126.2 126.6 126.5 126.5 126.6C4C3H7 127.9 127.6 127.6 127.5 127.1 127.5 127.4 127.1 0.66 0.77 0.64 0 .87 0.60 0.63 0.52 0.64

cc-pVDZ

HFMP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

1.343 1.361 1.363 1.381 1.364 1.351 1.3821.343 1.375 1.378 1.376 1.364 1.365 1.3731.444 1.434 1.436 1.445 1.438 1.425 1.4451.075 1.086 1.088 1.092 1.086 1.095 1.0871.077 1.087 1.089 1.094 1.087 1.096 1.088

107.2 106.7 106.8 106.5 106.9 107.2 106.4110.9 110.8 110.8 110.5 110.6 110.5 110.5105.5 105.8 105.9 106.2 106.0 105.9 106.3116.3 115.8 115.8 115.4 115.7 115.8 115.4132.8 133.4 133.5 134.1 133.8 133.8 134.1126.7 126.2 126.3 126.5 126.6 126.6 126.5127.8 127.9 127.9 127.3 127.4 127.5 127.20.66 0.64 0.74 0.47 0.51 0.40 0.52

cc-pVTZ

HFDFT

BLYPDFT

B3LYPDFTLDA

DFTBVWN

1.340 1.379 1.361 1.348 1.3801.336 1.365 1.354 1.355 1.3621.441 1.440 1.432 1.418 1.4391.067 1.081 1.075 1.085 1.075

1.068 1.082 1.076 1.086 1.077107.2 106.4 106.8 107.2 106.3110.9 110.4 110.4 110.4 110.4105.5 106.4 106.2 106.1 106.4116.4 115.6 115.9 116.1 115.6132.7 134.0 133.6 133.6 134.0126.7 126.5 126.5 126.5 126.5127.7 127.1 127.3 127.4 127.10.66 0.60 0.61 0.51 0.63

a Bond lengths in angstroms and angles in degrees; is the dipolemoment in debye. b References 31 and 32.

TABLE 2: Equilibrium Geometry for Thiophenea

6-31G**

coor-dinate exptlb HF

MP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

S1-C2 1.714 1.725 1.714 1.717 1.754 1.736 1.715 1.755C2dC3 1.369 1.345 1.375 1.376 1.378 1.367 1.368 1.376C3-C4 1.423 1.437 1.418 1.419 1.437 1.429 1.415 1.437C2-H6 1.078 1.071 1.077 1.078 1.087 1.081 1.090 1.082C3-H7 1.080 1.074 1.079 1.080 1.091 1.084 1.093 1.085C5S1C2 92.1 91.3 92.0 91.9 91.4 91.5 91.1 91.3S1C2C3 111.5 111.8 111.6 111.6 111.4 111.5 111.3 111.4

C2C3C4 112.5 112.5 112.4 112.4 112.9 112.8 112.6 112.9S1C2H6 119.8 120.4 120.3 120.2 119.8 120.0 120.1 119.9C3C2H6 128.7 127.8 128.1 128.2 128.7 128.5 128.6 128.7C2C3H7 123.2 123.6 123.1 123.2 123.2 123.3 123.2 123.2C4C3H7 124.3 123.8 124.5 124.4 123.9 124.0 124.2 123.9 (0.52 (

0.04)0.90 0.45 0.78 0.59 0.62 0.54 0.59

cc-pVDZ

HFMP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

1.729 1.725 1.726 1.756 1.738 1.718 1.7571.349 1.387 1.388 1.382 1.371 1.372 1.3791.437 1.426 1.427 1.438 1.431 1.417 1.4381.078 1.090 1.091 1.095 1.088 1.097 1.0891.080 1.092 1.093 1.097 1.090 1.099 1.092

91.3 92.0 92.0 91.3 91.5 92.0 91.2111.8 111.6 111.6 111.5 111.5 111.4 111.5112.6 112.4 112.4 112.9 112.7 112.6 112.9120.5 120.3 120.3 119.8 120.1 120.0 119.9127.7 128.1 128.1 128.7 128.4 128.7 128.7123.6 123.2 123.1 123.2 123.4 123.2 123.2123.9 124.5 124.5 123.9 123.9 124.2 123.90.79 0.67 0.67 0.39 0.46 0.37 0.43

cc-pVTZ

HFDFT

BLYPDFT

B3LYPDFTLDA

DFTBVWN

1.718 1.743 1.726 1.704 1.7441.344 1.374 1.363 1.363 1.3711.432 1.430 1.423 1.409 1.4301.069 1.082 1.077 1.087 1.0771.071 1.086 1.080 1.090 1.080

91.5 91.5 91.7 92.2 91.4111.7 111.4 111.4 111.3 111.4112.6 112.9 112.7 112.6 112.9120.6 119.9 120.2 120.1 120.0127.7 128.7 128.4 128.6 128.6123.6 123.3 123.4 123.2 123.3123.8 123.9 123.9 124.2 123.80.81 0.47 0.52 0.43 0.49

a See corresponding footnote in Table 1. b References 31, 32, and34.

Optimized Geometries and Calculated Frequencies J. Phys. Chem., Vol. 100, No. 37, 1996 15057


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by microwave spectroscopy31-34 are depicted in Tables 1 and2, respectively. The atom numbering employed is shown inFigure 1. By examination of the results in Tables 1 and 2,several observations can be made. With the exception of thebond lengths calculated by the LDA method, it is clear that allthe bond lengths determined by the DFT method using the6-31G** and cc-pVDZ basis sets are overestimated in ac-cordance with the previous observations.17-19 The only excep-tion is the CdC bond length determined with a B3LYP/6-31G**calculation for thiophene, which is predicted to be only 0.002

too short. Also the bond lengths determined at the MP2/cc-pVDZ level, fc and full, are overestimated,4 with the exceptionof the O-C bond length for furan, which is calculated to beonly 0.001 too short with the MP2(full) method. In going tothe larger cc-pVTZ basis set, the calculated bond lengths bythe BLYP and BVWN methods are still overestimated, but forthose calculated by the B3LYP method, as well as with the LDAmethod for all three basis sets, some are overestimated and someare underestimated.

For the LDA method, the O-C and C-C bond lengths arecalculated to be about 0.01 too short.17,19,35 The CdC bondlength is almost exactly predicted by the 6-31G** basis set17,35

but predicted to be about 0.004 too long by the cc-pVDZ

basis set and 0.006 too short by the cc-pVTZ basis set. Thebond angles calculated by the different DFT methods are closeto each other to within a few tenths of a degree. 20 The largestdifference is for the COC and CSC bond angles by the BVWNand LDA methods, where the difference is as high as 0.9. Thecalculated geometries by the MP2(full) and MP2(fc) methodsare close to each other to within 0.001 in most of the casesand 0.003 at most for bond lengths and 0.1at most for bondangles. The calculated bond lengths at the HF level, as well-known,17 are generally underestimated.

Among the seven methods used in this study, Tables 1 and2, the optimized geometry obtained by the B3LYP method hasthe best agreement with the experimental geometry, followedby the MP2 optimized geometry. The HF optimized geometry

is the worst. The BVWN and BLYP geometries are close toeach other, with the optimized geometry calculated by theBVWN method being slightly better, but both are worse thanthe B3LYP geometry. The optimized geometry calculated bythe LDA method behaves differently from those obtained bythe other methods, but it is generally better than that obtainedby the BVWN method and worse than the B3LYP geometry.The agreement between the calculated and experimental geom-etries can then be arranged in the order B3LYP, MP2, LDA,BVWN, BLYP, then HF.

Hertwig and Koch (HK)36 calculated the average error forthe difference between the experimental and optimized geom-etries 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 MP2levels. It is clear that the results in the current study for thetwo heterocyclic molecules do not support such a conclusion,although the basis sets used in our study are quite different fromthose used by HK. The reason for the difference from HK isthat in the chosen set of molecules there are molecules, forexample F2, for which a single reference treatment, such as MP2,is not expected to yield reasonable results. It is exactly, theadvantage of density functionals, that they are able to treat somecases for which an explicit multireference treatment wouldotherwise be mandatory. One such case is ozone,37 and anotheris the polyene radicals.38 A detailed discussion of the variouscorrelation effects in the density functional theory compared to

standard ab initio treatments may be found in the recent workof Handy and co-workers.39

Florian and Johnson11 concluded for formamide that the LDAand MP2 optimized geometries are comparable to each other.The results in Tables 1 and 2 for furan and thiophene do notsupport this conclusion.

As mentioned above, the B3LYP optimized geometry hasgenerally the best agreement with the experimental geometry.The exceptions are the S-C bond length and the CSC bondangle for thiophene, which are generally poorly predicted byall the DFT methods and the HF method with the 6-31G** andcc-pVDZ basis sets. The S-C bond length is calculated to be0.02-0.04 too long by the BLYP, B3LYP, and BVWNmethods, but with the LDA method it is calculated to be onlya few thousandths of an angstrom too long. The CSC bondangle is underestimated by 0.6-1.0by 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 worstamong these molecules is 1,2,5-thiadiazole, where the N-Sbond length is calculated to be 0.033 too long and the NSN

bond angle is underestimated by 1.1. The reflection of thepoor prediction of the S-X bond length and the XSY bond

angle (where X and Y ) C or N) was such that the root-mean-square (rms) deviations of the calculated frequencies from theexperimental frequencies using the scaled force field for thesulfur-containing molecules were larger than their oxygenanalogs.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 too longand the CSC bond angle is underestimated by 0.7. On theother hand, at the MP2(full)/6-31G** level the S-C bond lengthis exactly predicted and the CSC bond angle is underestimatedby only 0.1. Also at the MP2(full)/cc-pVDZ level, the S-Cbond length is predicted to be 0.011 too long. Using thecc-pVTZ basis set, this error by the HF and DFT methods is

reduced.For furan using the BLYP and BVWN methods, with thethree basis sets, the O-C bond length is calculated to be about0.020 too long, while the COC bond angle is predicted by adifference of only 0.2 at most. With the LDA method, theO-C bond length is calculated to be about 0.011 too shortand the COC bond angle is overestimated by 0.7. But withthe B3LYP method the O-C bond length is predicted within0.002 and the COC bond angle is overestimated by about0.4. At the MP2 level, the O-C bond length is calculatedwithin 0.004 at most and the COC bond angle is overestimatedby 0.3 at most. It is concluded that, although the B3LYPmethod generally gives a better prediction of the experimentalgeometry than the other DFT methods used in our study, the

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

15058 J. Phys. Chem., Vol. 100, No. 37, 1996 El-Azhary and Suter


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MP2 optimized geometry, which is only slightly worse thanthe B3LYP optimized geometry, avoids large errors in predictingsome coordinates observed with the B3LYP method.21,40

The general trend of the bond distances could be comparedto the results of Neumann et al.39 The DFT methods, especiallythe LDA approximation, predict too long bond lengths, similarto the CASSCF calculations. This could be interpreted as theeffect of the static correlation. The HF method, on the otherhand, normally underestimates the bond lengths. Since B3LYPuses HF exchange potentials and may be seen as a hybrid HF/

DFT method, the geometries are expected to be a mixture ofHF 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 onthe optimized geometries. At the HF level the basis set changefrom 6-31G** to cc-pVDZ did not improve the optimizedgeometry even with the cc-pVTZ basis set. At the MP2 levelthe bond lengths are better predicted by the 6-31G** basis set,while the bond angles are better predicted by the cc-pVDZ basisset for furan, but they are similar for thiophene. At the DFTlevel, similar to that at the MP2 level, the bond lengths are betterpredicted with the 6-31G** basis set, while the agreementbetween the calculated and experimental bond angles by bothbasis sets is almost similar. With the cc-pVTZ basis set, the

calculated bond lengths are shorter than those calculated by theother two basis sets.18,19 Consequently the agreement betweenthe calculated and experimental bond lengths is significantlyimproved with the exception of those calculated by the LDAmethod and the CdC bond length with the B3LYP method,where the agreement became worse. The bond angles are almostunchanged.

Calculated Frequencies. The calculated frequencies forfuran and thiophene are depicted in Tables 3 and 4, respectively.For comparison, the experimental frequencies reported forfuran22 and thiophene23 are also included in Tables 3 and 4. Tofacilitate the comparison between the experimental and calcu-lated frequencies, the average error for the C-H stretchingmodes (

1,

2,

12, and

13in 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-Hstretching modes), and the total average error were calculated.These are included at the end of Tables 3 and 4. Since thestudied molecules belong to the C2Vsymmetry point group, their21 fundamental vibrations are classified as 8A1 +3A2 +7B1+3B2.

The calculated frequencies at the HF level are grosslyoverestimated, with an average error of about 150 cm-1, butthe overestimation decreases as the wavenumber decreases. Thisis due to the larger anharmonicity of the higher frequencymodes. The situation is similar with the MP2 calculation exceptthat the average error of the calculated frequencies from the

experimental frequencies is only about 80-45 cm-1

dependingon 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 forfuran. With the DFT calculations, the average error is signifi-cantly less than that for the MP2 or HF calculations11,17-19

mainly because the calculated frequencies for the C-H stretch-ing modes are closer to the corresponding experimental frequen-cies11 and most of the bond lengths calculated with DFTmethods are too long.

For the C-H stretching modes, the overestimation of thecalculated frequencies with DFT is inversely proportional tothe overestimation of the C-H bond lengths. The overestima-tion of the C-H bond lengths decreases in the order LDA,

BLYP, BVWN, then B3LYP. The overestimation of thecalculated 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-Hbond lengths are overestimated by about 0.012, 0.010, 0.005,and 0.003 , with the LDA, BLYP, BVWN, and B3LYPmethods, respectively. The corresponding average error of thecalculated frequencies is 67, 48, 86, and 130 cm-1.

In the mid-IR region, the average errors obtained by the fourDFT methods are close to each other, but the error obtained

with the BVWN method is generally the lowest. This may beinterpreted as an advantage of the BVWN method over the morepopular BLYP method. On the other hand, the highest averageerror is obtained by the B3LYP method with the 6-31G** andcc-pVTZ basis sets, but with the cc-pVDZ basis set, the BLYPmethod has the highest average error. It is interesting to noticethat for the out-of-plane modes with the 6-31G** and cc-pVDZbasis 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 bythe MP2 method in the mid-IR region is about twice that bythe DFT methods, but with the cc-pVDZ basis set the averageerror is close to those obtained by the DFT methods.

All calculated frequencies by the B3LYP method are over-estimated, with the exception of only a few bands for thiophenewith the 6-31G** and cc-pVDZ basis sets, and also theoverestimation decreases gradually with the decrease of wavelength,41 a behavior similar to that observed at the HF and MP2levels. On the other hand, the calculated frequencies by theBLYP method are underestimated except for those correspond-ing to the C-H stretching modes, which are overestimated byabout 40 cm-1 and only a few bands for thiophene with the6-31G** and cc-pVDZ basis sets. As a result, scaling of theBLYP force fields using one-scale-factor scaling might notimprove the agreement between the calculated and experimentalfrequencies as a scale factor close to 1.0 is expected. On the

other hand, scaling of the B3LYP force fields is expected tosignificantly improve the agreement between the calculated andexperimental frequencies.10,21,40

Noting that the optimized geometries by the BVWN andBLYP methods are close to each other, with the one calculatedby the BVWN method being slightly better, the calculatedfrequencies by the BVWN and BLYP methods are similar toeach other except that in the mid-IR region the calculatedfrequencies by the BVWN method have a smaller average errorand some bands are overestimated. Also the C-H frequenciesare overestimated by about 80 cm-1 compared to about 40 cm-1

with the BLYP method. Those calculated by the LDA methodare similar to those calculated by the B3LYP method exceptfor the C-H frequencies, which are overestimated by about 60cm-1 compared to about 130 cm-1 by the B3LYP method, andin the mid-IR region some bands are underestimated. This isbecause, compared to the experimental bond lengths, some bondlengths are calculated to be too long and some are calculatedto be too short, while those calculated by the B3LYP methodare generally too long.

The calculated frequencies at the HF level showed littledependence on the basis sets used. Those calculated at the MP2level in the mid-IR region with the cc-pVDZ basis set have anaverage error about half that with the 6-31G** basis set, whichis comparable to those obtained with DFT calculations. Thisis because the bond lengths calculated with the MP2 and cc-pVDZ basis set are longer than those calculated with the



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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 thecalculated frequencies by the B3LYP method are the smallestwith the cc-pVDZ basis set, while those calculated by the LDAmethod are the smallest with the cc-pVTZ basis set.

Recently, Nonella et al.42 used the BP86 method with the6-31G** basis set to calculate frequencies for p-benzoquinone,1,4-naphthoquinone, and naphthalene. The corresponding root-mean-square (rms) deviations, excluding the C-H stretchingmodes, are 21, 19, and 16 cm-1, respectively. For comparisonwith our results, we calculated the average errors for the abovementioned molecules. These are 15, 14, and 12 cm-1, respec-tively. These average errors are close to those obtained in ourstudy by the BVWN/6-31G** force fields. In addition, mostof the calculated frequencies by the BP86/6-31G** force fields

for the above mentioned molecules are overestimated. This maysuggest that the BP86 and the BVWN methods behave similarly.

As was concluded before at the MP2/DZP level by Simandriset al.,5 the calculated frequencies for furan and thiophene inthis work support the experimental assignment for the d0isotopomer for both molecules.

It is worth noting that, although a comparison between thecalculated and experimental IR absorption intensities is notincluded in our study, Stephens et al.13 showed that thecalculated IR absorption and vibrational circular dichroismspectra for 4-methyl-2-oxetanone by the B3LYP method are inexcellent agreement with the experimental spectra and aresignificantly more accurate than those obtained by the LDA,BLYP, and HF methods. Also, those calculated at the MP2level are only slightly worse than those calculated by the B3LYPmethod.10,21,40 It was concluded also that the 6-31G* basis set

TABLE 3: Comparison between the Experimental and Calculated Frequencies for Furana

6-31G**

sym no exptlb HFMP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN sym no exptlb HF

MP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

A1 1 3167 3463 3379 3375 3217 3298 3233 3256 B1 15 1267 1412 1322 1319 1255 1296 1257 12742 3140 3430 3353 3349 3 186 3268 3208 3223 16 1180 1322 1268 1266 1159 1220 1229 11603 1491 1680 1545 1543 1 471 1529 1509 1479 17 1040 1165 1093 1091 1027 1071 1043 10304 1384 1549 1462 1459 1379 1430 1416 1387 18 873 961 891 889 862 890 866 8685 1140 1265 1187 1184 1135 1173 1162 1145 B2 19 838 994 810 804 790 839 796 8026 1066 1161 1128 1126 1056 1098 1106 1056 20 745 862 756 751 726 762 727 7347 995 1090 1047 1046 984 1022 1004 988 21 603 659 627 623 601 623 629 600

8 871 1090 885 882 856 886 869 860 errorc CH 293 213 209 48 130 67 86A2 9 863 956 812 805 835 880 844 846 in-planed 132 54 51 12 33 20 9

10 728 856 705 698 681 729 698 687 out-of-plane 110 29 31 27 10 24 2111 613 663 577 573 593 614 605 595 mid-IRe 124 45 44 17 25 21 13

B1 12 3161 3457 3373 3369 3211 3292 3226 3250 total 156 77 75 23 45 30 2713 3129 3417 3 343 3 339 3175 3258 3198 3212 SFf 0.819 0.891 0.894 0.985 0.928 0.963 0.96514 1556 1768 1620 1617 1552 1614 1588 1560 errorc 25 31 32 24 13 19 28

cc-pVDZ

no. HFMP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN no. HF

MP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

1 3457 3345 3334 3204 3285 3226 3244 15 1385 1285 1282 1229 1271 1241 12502 3423 3316 3308 3174 3256 3200 3212 16 1305 1250 1250 1143 1204 1207 11443 1671 1524 1520 1457 1518 1501 1466 17 1145 1062 1059 1011 1053 1022 10164 1534 1433 1430 1364 1415 1403 1374 18 957 880 878 859 887 863 8655 1249 1168 1164 1120 1159 1158 1130 19 985 829 823 799 846 803 812

6 1156 1119 1116 1050 1090 1091 1050 20 851 762 755 723 758 723 7337 1073 1022 1021 969 1005 985 976 21 658 630 628 602 622 628 6008 952 879 875 851 881 863 856 286 176 167 36 117 59 749 998 848 843 847 887 851 860 120 32 29 25 20 17 1710 847 729 719 688 734 702 695 101 13 16 23 12 21 1611 659 605 599 595 614 606 597 113 25 24 24 17 18 1712 3450 3337 3327 3198 3278 3218 3238 146 54 51 26 36 26 2813 3410 3304 3296 3163 3245 3190 3200 0.827 0.912 0.917 0.995 0.938 0.970 0.97414 1755 1590 1585 1536 1600 1576 1546 20 27 28 27 15 19 30

cc-pVTZ

no HFDFT

BLYPDFT

B3LYPDFTLDA

DFTBVWN no HF

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

1 3432 3203 3279 3211 3244 15 1408 1255 1295 1240 12762 3401 3174 3251 3187 3213 16 1306 1170 1199 1215 11463 1662 1455 1513 1490 1466 17 1156 1015 1062 1034 10174 1534 1365 1415 1399 1375 18 960 867 894 874 872

5 1256 1129 1167 1154 1142 19 1000 808 858 818 8226 1153 1045 1086 1094 1045 20 859 726 762 727 7367 1078 976 1015 995 980 21 659 604 625 630 6038 954 859 888 872 863 263 35 112 46 759 1012 848 893 861 861 122 20 24 13 1410 855 693 739 709 700 109 19 18 15 1111 661 599 618 611 601 117 20 22 14 1312 3425 3196 3272 3203 3238 145 23 39 20 2513 3390 3163 3241 3178 3203 0.833 0.993 0.938 0.976 0.97114 1743 1535 1595 1568 1547 25 24 11 15 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 CHstretching modes. f Scale factor.



6/8

is optimal to reproduce an accurate representation of the IRabsorption spectra with the B3LYP method.

Scale Factors. Table 5 shows the internal coordinate

definition used to convert the Cartesian coordinate force fieldsto internal coordinate force fields for furan and thiophene. Thescaling of the force fields was performed as described else-where.7,10 Although scaling was performed only for the d0isotopomers of furan and thiophene, in a second calculationscaling was done for all the isotopomers of furan and thiophenefor which the vibrational assignment was reported.22,23 Theseare the d0, 2,5-d2, 3,4-d2, d4, 2-d1, 3-d1, and 2,3,5-d3isotopomersfor furan22 and the d0, 2,5-d2, 3,4-d2, d4, 2-d1, 3-d1, 2,3-d2, 2,4-d2, 2,3,5-d3, and 2,3,4-d3 isotopomers for thiophene.23 Thedifference between the scale factors when scaling was done forthe d0isotopomers only and that when all the isotopomers wereincluded was in most of cases not more than 0.005. Forconsistency with the unscaled frequencies, the results of scaling

reported here are for the d0 isotopomers only. The obtainedscale factors and the corresponding average errors are in Tables3 and 4 for furan and thiophene, respectively.

It is clear that scaling the HF and MP2 force fieldssignificantly improved the average error, bringing them closeto or even smaller than some of those obtained by the DFTmethods. The average scale factors for the two studiedmolecules at the HF and MP2 levels are 0.82 and 0.89 with the6-31G** basis set and 0.83 and 0.91 with the cc-pVDZ basisset, respectively. These values are close to those obtained forsimilar five-membered heterocyclic molecules with the6-31G**,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 tothose previously reported,4,43 0.8929 and 0.9427, at the HF andMP2 levels, respectively.

TABLE 4: Comparison between the Experimental and Calculated Frequencies for Thiophenea

6-31G**

sym no. exptlb HFMP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN sym no. exptlb HF

MP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

A1 1 3126 3427 3346 3341 3189 3271 3207 3226 B1 15 1256 1407 1314 1311 1240 1282 1226 12592 3098 3390 3312 3308 3144 3229 3168 3181 16 1085 1210 1 134 1 133 1084 1115 1068 10993 1409 1597 1488 1483 1416 1470 1477 1419 17 872 951 912 910 846 876 866 8534 1360 1530 1436 1433 1360 1407 1374 1373 18 751 808 781 778 713 742 754 7115 1083 1207 1132 1131 1082 1112 1062 1099 B2 19 867 1029 857 847 833 880 833 8456 1036 1106 1094 1092 1030 1059 1054 1034 20 712 813 732 726 701 728 700 7107 839 890 867 883 803 832 856 802 21 452 484 457 453 437 453 458 438

8 608 658 630 628 591 610 609 592 errorc CH 293 213 209 48 130 67 88A2 9 898 1053 867 855 877 920 883 888 in-planed 118 50 49 14 28 22 17

10 683 804 673 667 647 683 654 655 out-of-plane 104 15 20 21 10 17 1411 565 618 550 542 555 575 570 556 mid-IRe 113 38 39 16 22 20 16

B1 12 3125 3425 3343 3338 3187 3269 3205 3224 total 147 71 71 23 43 30 3013 3098 3376 3299 3294 3131 3215 3155 3167 SFf 0.819 0.888 0.894 0.980 0.925 0.962 0.96414 1507 1740 1571 1568 1515 1578 1549 1524 errorc 29 24 26 23 16 21 26

cc-pVDZ

no. HFMP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN no. HF

MP2full

MP2fc

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

1 3416 3299 3293 3171 3253 3194 3208 15 1379 1269 1268 1217 1259 1203 12372 3382 3272 3266 3133 3218 3162 3170 16 1186 1093 1092 1057 1098 1041 10753 1587 1466 1463 1410 1462 1473 1412 17 936 892 890 838 869 858 8454 1509 1400 1399 1342 1390 1358 1355 18 804 770 769 712 748 753 7115 1181 1088 1087 1053 1085 1044 1073 19 1012 857 852 834 878 830 848

6 1098 1073 1071 1021 1051 1033 1026 20 800 728 724 695 722 692 7057 882 872 870 799 835 852 799 21 481 454 452 439 454 457 4398 653 617 615 590 611 606 591 283 170 164 36 120 62 739 1035 890 884 882 921 882 895 103 25 23 24 17 24 1710 789 675 670 647 683 651 656 92 8 10 15 10 19 1311 613 564 561 556 576 569 558 99 19 18 21 15 22 1612 3413 3296 3290 3168 3250 3191 3205 134 47 46 24 34 30 2613 3367 3258 3252 3120 3204 3149 3157 0.829 0.918 0.914 0.992 0.936 0.972 0.97014 1724 1537 1535 1502 1563 1538 1511 23 27 26 26 19 26 28

cc-pVTZ

no HFDFT

BLYPDFT

B3LYPDFTLDA

DFTBVWN no HF

DFTBLYP

DFTB3LYP

DFTLDA

DFTBVWN

1 3396 3174 3250 3183 3213 15 1400 1242 1284 1224 12622 3359 3132 3211 3146 3171 16 1199 1077 1108 1056 10943 1560 1395 1444 1452 1402 17 944 850 880 868 8564 1518 1350 1397 1362 1363 18 806 717 752 762 715

5 1195 1077 1106 1051 1096 19 1027 842 889 842 8576 1098 1026 1055 1049 1031 20 808 698 727 695 7097 885 804 839 858 803 21 488 446 462 465 4478 654 594 615 612 595 261 38 115 49 769 1051 884 929 890 898 110 16 21 19 1010 800 656 694 662 666 102 15 18 16 611 617 562 581 576 564 107 16 20 18 1012 3393 3172 3247 3180 3211 136 20 38 24 2113 3344 3119 3197 3134 3157 0.834 0.988 0.935 0.975 0.96614 1709 1496 1555 1526 1508 27 21 12 18 23

a,c,d,e,f See corresponding footnote in Table 3. b Reference 23.



7/8

At the DFT level, the average error of the B3LYP method isthe lowest, with a significant improvement over that obtainedwith the unscaled force field. The average values of the6-31G** and cc-pVDZ scale factors are 0.93 and 0.94,respectively. The value of the 6-31G** scale factor is the sameas that reported for the above mentioned molecules,10,21,40 0.93.

It is then concluded, for this set of five-membered heterocyclicmolecules, that scaling the B3LYP force fields produces moreconsistent scale factors than those obtained with the MP2 orHF force fields. With the BLYP method scale factors of 0.98and 1.00 are obtained with the 6-31G** and cc-pVDZ basissets, respectively, but the average error is similar to or largerby 1 or 2 cm-1 than that with the unscaled force fields. Thisin fact was expected for reasons that were mentioned above.Practically, upon scaling, the average error for the C-Hstretching modes improved, but at the expense of the averageerror for mid-IR modes, which became worse.

In a separate calculation, the BLYP/6-31G** force field forfuran was scaled using two scale factors, one for the C-Hstretching modes and the other for the mid-IR modes. The

obtained average error for the C-H stretching modes was only2 cm-1, for the mid-IR modes it was 10 cm-1, and the totalaverage error was 9 cm-1. The corresponding scale factors are0.97 and 1.02 for the C-H stretching modes and the mid-IRmodes, respectively. For comparison, the same calculation wasrepeated for the B3LYP/6-31G** force field for furan, and theobtained average errors were 1, 8, and 6 cm-1 for the C-Hstretching 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-Hstretching modes improved the average error of the BLYP forcefield to be close to that obtained by the B3LYP force field.This might suggest that a further increase of the number of scalefactors might overcome the difference between the BLYP and

B3LYP force fields. However, Rauhut and Pulay (RP)20calculated one-scale-factor and multi-scale-factor scaling by theBLYP and B3LYP methods using the 6-31G* basis set for 20molecules. They concluded that even with multi-scale-factorscaling the B3LYP performs uniformly better than the BLYP.Notice that the above scale factors for the C-H stretching modesby the BLYP and B3LYP methods are close to those obtainedby RP,20 0.977 and 0.920, respectively.

The average scale factors for furan and thiophene by theB3LYP and BLYP force fields using the 6-31G** basis set are0.93 and 0.98, respectively. These values are close to thosereported 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. Thesecorrespond to 0.9890 and 0.9243 for force field scaling. Thesevalues are close to those obtained by us and to those obtainedby RP.20

On the other hand, there is no scale factor we know for theforce fields calculated by the LDA and BVWN force fields.Scaling the BVWN force field is similar to scaling the BLYPforce field with values of scale factors of 0.96 and 0.97, obtainedwith the 6-31G** and cc-pVDZ basis sets, respectively. Scalingof the LDA and B3LYP force fields is also similar to scale

factors of 0.96 and 0.97 obtained with the 6-31G** and cc-pVDZ basis sets, respectively, but the average error is largerthan with the B3LYP force field (but smaller than that with theBLYP and BVWN force fields) due to the fact that in the mid-IR region some bands are underestimated and some areoverestimated.

The effect of basis set change on the scale factors was small,with the scale factors determined by the cc-pVDZ and cc-pVTZbasis sets close to each other and slightly larger than thoseobtained 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 the6-31G**, cc-pVDZ, and cc-pVTZ basis sets were compared.The agreement between the calculated and experimental geom-etries can be arranged in the order B3LYP, MP2, LDA, BVWN,BLYP, and HF. Although the B3LYP optimized geometry isslightly better than the MP2 optimized geometry, the MP2optimized geometry avoids large errors in predicting somecoordinates observed by the B3LYP optimized geometry. Theeffect of changing the basis set on the bond angles was minor,but with the exception of the LDA method, the bond lengthsare 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 havethe lowest average error. Among the four DFT methods, thosefrequencies calculated by the B3LYP force field have the highestaverage error. This is with the exception of the cc-pVDZ basisset, where the average error is close to that obtained by theBVWN method, and the calculated frequencies by the BLYPforce field have the highest average error. The calculatedfrequencies by the B3LYP force field are generally overesti-mated, but those calculated by the BLYP force field areunderestimated, except for those corresponding to the C-Hstretching modes. This is reflected such that scaling by theB3LYP force field produced the lowest average error, and theaverage error obtained by scaling the BLYP force field remainedalmost unchanged.

The calculated frequencies by the BVWN force field aresimilar to those calculated by the BLYP force field, and thosecalculated by the LDA force field are similar to those calculatedby the B3LYP force field. The effect of changing the basis seton the calculated frequencies and scale factors was minor.

In summary, the present study indicates that scaled B3LYPforce fields are preferred for vibrational analysis, where betteragreement between the calculated and experimental geometries,frequencies, and intensities is expected. In the case where theoptimized geometry is the main interest, the MP2 method ismore 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 amongthe DFT methods used in this study.

TABLE 5: Internal Coordinates for Furan and Thiophenea

no. modeb description

q1, q5 r1,r5 C-X stretchcq2, q4 r2,r4 CdC stretchq3 r3 C-C stretchq6, q7, q8, q9 r6,r7,r8,r9 C-H stretchq10 R1 + a(R2 + R5) + b(R3 + R4) ring deformationq11 (a - b)(R2 - R5) +

(1 - a)(R3 - R4)ring deformation

q12, q13, q14, q15 1 -2,3 -4,5 -6,7 -8

CH rocking

q16, q17, q18, q19 6,7,8,9 CH waggingq20 b(1 + 5) + a(2 + 4) + 3 ring torsionq21 (a - b)(4 - 2) +

(1 - a)(5 - 1)ring torsion

a See Figure 1 for definition ofr, R, , and internal coordinates.b a )cos 144and b )cos 72. Values of normalization constantsare not given. c X )O for furan and X )S for thiophene.



8/8

Acknowledgment. A.A.E.-A. is grateful to Prof. T. A.Keiderling of the University of Illinois at Chicago for the useof the CADPAC program installed on the Department ofChemistry Titan minisupercomputer. A generous grant ofcomputer time at the Centro Svizzero di Calcolo Scientifico(CSCS) in Manno is gratefully acknowledged.

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