Journal of Molecular Structure 924–926 (2009) 493–503

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Journal of Molecular Structure

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An effective scaling frequency factor method for scaling of harmonic vibrationalfrequencies: Application to toluene, styrene and its 4-methyl derivative

Piotr Borowski a,*, Manuel Fernández-Gómez b, Maria-Paz Fernández-Liencres b,Tomás Peña Ruiz b, Manuel Quesada Rincón b

a Faculty of Chemistry, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 3, 20-031 Lublin, Polandb Department of Physical and Analytical Chemistry, University of Jaén, 23071 Jaén, Spain

a r t i c l e i n f o

Article history:Received 28 September 2008Received in revised form 26 November 2008Accepted 2 December 2008Available online 11 December 2008

Keywords:Vibrational spectraScaling proceduresSQMESFFTransferability

0022-2860/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.molstruc.2008.12.005

* Corresponding author. Tel.: +48 81 537 56 14.E-mail addresses: pibcio@vsop404.umcs.lublin.pl

(M. Fernández-Gómez), liencres@ujaen.es (M.-P.ujaen.es (T.P. Ruiz), mqrpeli@yahoo.es (M.Q. Rincón).

a b s t r a c t

The predictive capabilities of the newly proposed effective scaling frequency factor (ESFF) method andtransferability of scaling factors are checked. A set of three related but with different structural motifs mol-ecules, i.e. toluene, styrene and 4-methylstyrene, was used. Four sets of optimized local scaling factors weregenerated on the basis of different choices of bands from the experimental IR spectra. In all cases the spectralrange was 3000–400 cm�1. The best fitting was obtained with the so-called Set A (66 experimental bandsand 14 optimized local scaling factors). Only slightly worse results were obtained with the Set B containingmerely eight local scaling factors. Different statistical tests demonstrate good transferability of local scalingfactors as well as better performance of the ESFF approach than Pulay’s SQM method. In particular, the RMSdeviation calculated for 66 vibrational modes with ESFF is 28% lower than that obtained with SQM method.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

The usefulness and capabilities of vibrational spectroscopy as astructural tool has been remarkably enhanced since the advent ofaccurate theoretical prediction of vibrational spectra. Thus, it hasbecome almost routine to apply it to systems for which otherstructural methods are very bothersome and difficult to use. Fur-thermore, it can contribute in a noteworthy way to understandingfast processes to elucidate reaction paths from the prediction ofvibrational spectra of transient short-lived species.

The knowledge of a vibrational spectrum relies on finding thehessian matrix for equilibrium geometry of a molecule, i.e. the to-tal energy second derivatives, also known as force constants (po-tential parameters). For many years, such information have beencoming from an empirical origin. For most except a few small mol-ecules, the number of such potential parameters is far larger thanthat of experimental data, e.g. vibrational frequencies. Thus, get-ting numerical values for force constants was possible only afterthe solution of the so-called inverse vibrational problem in which,after a least-square procedure, we pursue a final matrix that pro-vides the vibrational frequencies least deviating from experimentalones.

ll rights reserved.

(P. Borowski), mfg@ujaen.esFernández-Liencres), truiz@

The direct vibrational problem, on the other hand, has becomeaffordable since the appearance of quantum mechanical methods,ab initio and DFT, and computation facilities powerful enough toobtain the theoretical force constants as the energy second deriva-tives at the equilibrium configuration of the molecule. However,the incomplete treatment of electron correlation in approxima-tions introduced when solving Schrödinger equation, finiteness ofthe basis sets and the presence of anharmonicity, make the calcu-lated vibrational frequencies to be usually somewhat larger thanthe observed ones in a rather systematic pattern.

Scaling of theoretical force constants or theoretical vibrationalfrequencies has become one of methods correcting for the short-comings of theoretical predictions with very low computationalcost. Different scaling methods have been proposed over the years(see for instance [1–5] and some others). Recently [6], a new meth-od called effective scaling frequency factor (ESFF) method ofscaling of harmonic vibrational frequencies was proposed. Thescaling factors are built up from the diagonal elements of the po-tential energy distribution (PED) matrix calculated on the basisof a non-redundant natural coordinates set and the so-called localscaling factors attributed to different types of coordinates. The pre-liminary test on toluene in the range of 3000–400 cm�1 led to a fit-ting tighter (according to the RMS deviations) than that obtainedwith the most-accepted Pulay’s SQM method [1,2], i.e. 2.48 vs.3.29 cm�1. The ESFF method was then tested [7] using the trainingset of 30 molecules proposed by Baker and coworkers [2]. The re-sults point towards somewhat better performance of the ESFF

494 P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503

method as compared to SQM, as well as somewhat better transfer-ability of scaling factors.

The above-mentioned training set includes a wide variety of or-ganic molecules. This diversity is apparently the main reason forsome disagreement between scaled and experimental frequencies,as the transferability of scaling factors between molecules is lessobeyed in the case of totally different molecular topologies. In or-der to further investigate the predictive capabilities of the ESFFmethod and transferability of local scaling factors, this time amongrelated molecules, a study of vibrational spectra of three systems:toluene, styrene and 4-methylstyrene was carried out. As in ourfirst work [6], the attention was focused on the modes in the rangeof 3000–400 cm�1 for all molecules. It is shown that ESFF-scaledfrequencies reproduce well the experimental spectra. The agree-

0

0.25

0.5

0.75

1

Tra

nsm

ittan

ce

3000 2500 2

Wavenu

0

0.25

0.5

0.75

1

Tra

nsm

ittan

ce

3000 2500 2

a

b

Fig. 1. IR spectra of (a) styre

ment is better than in the case of SQM-scaled frequencies. It is alsodemonstrated that the local scaling factors are very well transfer-able for these related compounds.

2. Experimental

Infrared spectra of toluene (1), styrene (2) and 4-methylstyrene(3) (Aldrich, +99%) were recorded in the liquid phase at room tem-perature using a Brucker Vector 22 FTIR spectrometer at 1 cm�1

resolution. Averaging of 100 scans was carried out. Spectra of (2)and (3) are depicted in Fig. 1; spectrum of (1) was reported before[6]. Actually, vibrational spectra of the title compounds have beenstudied before (see e.g. [8–10] for (1), [11,12] for (2), [13–15] for(3) and references therein). The consistency of conditions, under

000 1500 1000 500

mber [cm−1

]

000 1500 1000 500

ne, (b) 4-methylstyrene.

a

cb

P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503 495

which all three spectra were obtained, was the reason of redoingthe experiment.

3. Computational details

3.1. DFT calculations

Geometries of molecules 1–3 were optimized at the DFT (see,e.g. [16]) level of theory using B3LYP [17,18] exchange-correlationpotential and 6-311G** basis set [19]. The force constant matricesin cartesian coordinates, fðxÞ, were then obtained at the same level.Files with molecular geometries and force constants were storedfor subsequent ESFF and SQM calculations. This part of researchwas carried out with the parallel version of the ab initio quantumchemistry PQS package [20,21].

3.2. ESFF calculations

3.2.1. The choice of internal coordinatesThe primitive internal coordinates for the 4-methylstyrene mol-

ecule are depicted in Fig. 2. The force constant matrix in cartesiancoordinates fðxÞ was transformed into natural internal coordinaterepresentation (shown in Table 1) originally defined in [22]. Theonly exception is the definition of torsion of the methyl and vinylgroups with respect to the benzene ring as well as the torsion ofthe vinyl group itself – the Hilderbrandt definition [23,24] wasadopted.

Fig. 2. Primitive internal coordinates for the 4-methylstyrene molecule: (a) in-plane coordinates; symbols a, b and / refer to C–C–C, C–C–H and H–C–H valenceangles, respectively, (b) out-of-plane coordinates; (c) notation used in the definitionof the out-of-plane coordinates.

3.2.2. The choice of experimental bandsIn order to carry out the ESFF (as well as SQM) optimization of

scaling factors it is necessary to choose a set of bands from exper-imental spectra and to assign them to theoretical vibrationalmodes. Our attention was focused on the frequency range of3000–400 cm�1, i.e. the Ar–H and @C–H stretching modes as wellas very low energy modes were not taken into account. There aretwo reasons for this restriction. First, IR spectra are typically not re-corded below 400 cm�1. Second, spectra of aromatic compounds,in particular these containing the vinyl group, in the range above3000 cm�1 are complicated – the assignment of bands to vibra-tional modes is thus not unambiguous. It is not our intention toprovide in this paper detailed assignment of all bands on experi-mental spectra. We also do not plan to develop sets of scaling fac-tors for the future use in scaling procedures (maybe apart fromtheir applications to some aromatic hydrocarbons). For that pur-pose a more systematic study is needed, and we have alreadystarted to investigate these problems. Instead, we would like focusour attention on the predictive capabilities of the ESFF method, fac-tors transferability problem and further comparison of ESFF andextensively used over the past few decades SQM approaches. It istherefore desirable to use bands that are unambiguously assigned.Obviously such restriction will lower down the overall RMS valuefor the considered theoretical frequencies. However, we want toemphasize that our SQM calculations are based on exactly thesame set of experimental bands as ESFF. As it will be demonstratedinclusion of extra bands of imprecise locations deteriorates boththe ESFF and SQM results.

Tables 2–4 contain theoretical as well as selected experimentalfrequencies for molecules 1–3. The percentage contributions fromlocal modes are also reported unless the contribution is lower than15%. The initial SQM and ESFF calculations (not reported in thispaper) were carried out prior to making the assignment. In the lat-ter case, the local scaling factor for C@C stretching mode was esti-mated by dividing its experimental fundamental frequency inethylene (1623 cm�1) by the corresponding theoretical (DFT/B3LYP/6-311G**) value (1691 cm�1). The remaining factors were

taken from [6] and some of them were adopted for the descriptionof the vinyl group motions. All bands in the region of2000—1650 cm�1 were omitted. Besides, bands at 1523.0 and1106:5 cm�1 for toluene molecule were also omitted since thecomparison of the experimental and initially scaled (both SQMand ESFF) frequencies indicate that they do not have the theoreti-cal counterparts. Moreover, the band at 791:6 cm�1 for 4-methyl-styrene molecule was also omitted since the deviation from theclosest scaled frequency was above 20 cm�1. For styrene molecule,however, no additional band was omitted. As a consequence 23bands were chosen for the molecule 1, 24 – for 2 and 34 – for 3,which yields a set of 81 frequencies. In the last column of Tables2–4, entitled ‘‘Comments”, the following abbreviations were used:BB – broad (relative to the height) band, VBB – very broad band, IB– irregular-shaped band, VIB – very irregular-shaped band. Theywere assigned after the inspection of the shape of the enlargedbands. The spectrum of toluene requires additional comment.

496 P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503

The low-resolution spectrum obtained from SDBS database [25]exhibits a single, relatively broad band with a minimum on thetransmittance scale at about 1460 cm�1. It may be assigned as aris-ing from two theoretical modes: at 1504.60 and 1490:47 cm�1

(cf. Table 2). Increasing the resolution down to 1 cm�1 (the presentstudy) and enlarging the band reveals that this really is a superpo-sition of two modes with one mode showing up as a shoulder bandon the red wing of the other one. Due to very irregular shape of thisband(s) on our higher-resolution spectrum, the reported in Table 2experimental values of 1458.9 and 1455:0 cm�1 are probablyinaccurate.

3.2.3. Optimization of scaling factors for local modesThere are two main problems associated with the optimization

of scaling factors for local modes: (i) the choice of bands from theexperimental spectra and their assignment to theoretical modesand (ii) qualification of the internal coordinates according to theirtypes. The first of these items has already been discussed in thepreceding section. As for the second, four sets of local scaling fac-tors were generated; they are denoted A, B, B0 and C, respectively.Their (optimized) values are reported in Tables 5 and 6.

(1) Set A – the ESFF-A calculations. Out of the identified andassigned bands (cf. Tables 2–4) the shoulder bands (Sh) aswell as these, qualified as either VBB or VIB were not takeninto account in the optimization of local scaling factors. Thepossible high uncertainty in the determination of resonancefrequencies for these bands could cause significant errors inthe calculations. Sixty-six bands were thus considered: 19(as in [6]) for the molecule 1, 21 – for 2 and 26 – for 3. Four-teen local scaling factors were optimized (cf. Table 5). Fac-tors for a few local modes were preset to unity – it wasobserved that these local modes do not contribute much to

Table 1Natural internal coordinates for the 4-methylstyrene molecule.

Coord. no. Definition (cf. Fig. 2)

1–4 r1 � r4

5–7 r5 � r7

8–10 r8 � r10

11–16 R1 � R6

17 R7

18 R8

19 R9

20–22 ða1 � a2 þ a3 � a4 þ a5 � a6Þ=ffiffiffi

6p

ð2a1 � a2 � a3 þ 2a4 � a5 � a6Þ=ffiffiffiffiffiffi

12p

ða2 � a3 þ a5 � a6Þ=223–26 ðb1;2;3;4 � b01;2;3;4Þ=

ffiffiffi

2p

27 ða7 � a07Þ=ffiffiffi

2p

28 ða9 � a09Þ=ffiffiffi

2p

29,30 ðb5 � b6Þ=ffiffiffi

2p

ðb7 � b8Þ=ffiffiffi

2p

31,32 ð2/1 � b7 � b8Þ=ffiffiffi

6p

ð2a8 � b5 � b6Þ=ffiffiffi

6p

33–35 ð/2 þ /3 þ /4 � b9 � b10 � b11Þ=ffiffiffi

6p

ð2/2 � /3 � /4Þ=ffiffiffi

6p

ð/3 � /4Þ=ffiffiffi

2p

36,37 ð2b9 � b10 � b11Þ=ffiffiffi

6p

ðb10 � b11Þ=ffiffiffi

2p

38–40 ðs1 � s2 þ s3 � s4 þ s5 � s6Þ=ffiffiffi

6p

ðs1 � s3 þ s4 � s6Þ=2ð�s1 þ 2s2 � s3 � s4 þ 2s5 � s6Þ=

ffiffiffiffiffiffi

12p

41–44 c1 � c4

45 c5

46 c647 c748 c8

49 sHil1

50 sHil2

51 sHil3

normal modes considered in the present work. All 66 theo-retical frequencies were then scaled. The Set A was designedfor the test purpose only, for example in order to figure outwhich local modes can be put into one type.

(2) Sets B and B0 – the ESFF-B and ESFF-B0 calculations. In order tocalculate the Set B of local scaling factors the same as beforeset of 66 experimental bands was used in the optimizationprocedure. On the other hand the Set B0 was generated usingbands that correspond to ‘‘pure” modes, i.e. modes, forwhich the contribution of one of the local modes is not lowerthan 60%. This yields 39 modes: 14 for the molecule 1, 12 –for 2 and 13 – for 3. This set of scaling factors will provide uswith additional information, whether effective scaling fac-tors for delocalized modes built as PED-weighted sums oflocal scaling factors obtained from localized modes preservetheir quality. In both B and B0 Sets, eight local scaling factorswere optimized (cf. Table 6). The analysis of local factorsreported in Table 5 reveals that some of them have similarvalues. In particular, this is the case of: (i) in plane C–C–C,C–C–H and H–C–H deformations; local modes correspond-ing to these deformations were qualified as a common type3 and (ii) out-of-plane Ar–H and @C–H deformations; localmodes corresponding to these deformations were qualifiedas a common type 5, which, in addition, includes out-of-plane Ar–C deformations. Types 1, 2 and 4 remain the sameas previously. Note, that due to the limited number of typesof scaling factors introduced so far (five types), we decidedto keep mC@C; sHil

C@C and cC@C as separate types (6, 7 and 8)in spite of having only one (or none in the case of toluene)of these coordinates per molecule. Bands associated primar-ily with these motions (modes 9, 24 and 28 for molecule 2and 11, 30 and 34 for molecule 3, cf. Tables 3 and 4) play acrucial role in the identification of vinyl derivatives of

Description Symbol

CAr–H stretching mAr—H

CVin–H stretching m@C—H

CMet–H stretching mMet

CAr—CAr stretching mAr

CAr—CVin stretching mAr—Vin

CVin—CVin stretching mC@C

CAr—CMet stretching mAr—Met

Benz. trig. deformation dAr

Benz. asym. deformation

Ar–H rocking qAr—H

Ar–Vin rocking qAr—Vin

Ar–Met rocking qAr—Met@C–H rocking q@C—H

Vin. deformation dVin

Met. sym. deformation dMet

Met. asym. deformation

Met. rocking qMet

Benz. puckering sAr

Benz. asym. torsion

Ar–H wagging cAr—H

@C–H wagging c@C—H

C@C wagging cC@CAr–Vin wagging cAr—VinAr–Met wagging cAr—Met

Ar–Vin torsion sHilAr—Vin

Vin. torsion sHilC@C

Ar–Met torsion sHilAr—Met

Table 2Toluene. Theoretical (DFT/B3LYP/6-311G**) frequencies (mðWDCÞ , cm�1) and intensities (I, km/mol) as well as the corresponding experimental frequencies (mðexptÞ , cm�1) andintensities (I) are given. Contributions (greater than 15%) of local modes (cf. Table 1) to all normal modes as well as the information regarding the shape of the experimental band(see text for details) are also reported.

No. mðWDCÞ I mðexptÞ I Contributions Comments

1 3186.21 16.818 99% mAr—H

2 3173.43 42.006 99% mAr—H

3 3165.13 8.400 99% mAr—H

4 3152.83 6.473 99% mAr—H

5 3150.52 9.260 99% mAr—H

6 3100.65 18.185 2949.6 M 99% mMet (asym.)7 3073.58 22.227 2920.2 M 100% mMet (asym.)8 3020.58 29.225 2872.9 W 100% mMet (sym.)9 1648.48 7.512 1604.5 M 65% mAr + 21% qAr—H

10 1626.44 0.421 68% mAr + 17% qAr—H11 1529.48 15.427 1495.5 S 36% mAr + 60% qAr—H12 1504.60 14.213 1458.9 M 20% qAr—H + 56% dMet (asym.) VIB13 1490.47 6.818 1455.0 Sh 90% dMet (asym.) VIB14 1470.44 0.036 24% mAr + 38% qAr—H + 34% dMet (asym.)15 1414.51 1.027 1378.9 W 89% dMet (sym.)16 1356.16 0.022 16% mAr + 80% qAr—H17 1327.79 0.179 82% mAr

18 1228.66 0.883 1210.1 W 22% mAr + 16% dAr + 17% qAr—H + 43% mAr—Met

19 1203.93 0.366 1178.3 W 21% mAr + 79% qAr—H20 1181.87 0.075 1156.1 W 20% mAr + 80% qAr—H BB, IB21 1111.84 5.696 1081.4 M 44% mAr + 43% qAr—H

22 1062.31 7.424 1040.9 W 63% qMet

23 1051.36 2.853 1029.8 M 66% mAr + 21% qAr—H

24 1017.91 0.048 1002.3 W 34% mAr + 63% dAr BB, VIB25 1000.35 0.505 23% mAr + 65% qMet26 998.99 0.068 15% sAr + 83% cAr—H27 975.57 0.003 92% cAr—H28 910.49 0.496 895.3 W 87% cAr—H

29 855.09 0.012 841.3 W 100% cAr—H VBB, VIB30 798.62 0.751 785.4 W 27% mAr + 36% dAr + 31% mAr—Met

31 744.60 44.985 728.5 VS 31% sAr + 58% cAr—H

32 713.43 26.973 694.3 VS 68% sAr + 26% cAr—H33 637.44 0.102 622.4 W 87% dAr IB34 529.08 0.583 521.7 W 73% dAr

35 476.49 9.074 464.3 S 54% sAr + 35% cAr—Met

36 415.01 0.010 84% sAr

37 343.91 0.359 81% qAr—Met

38 208.91 2.259 53% sAr + 18% cAr—H + 24% cAr—Met39 19.47 0.228 15% mAr + 56% sHil

Ar—Met

P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503 497

organic compounds and for that reason they deserve a spe-cial treatment. Again factors for a few local modes were pre-set to unity and all 66 theoretical frequencies were thenscaled.

(3) Set C – the ESFF-C calculations. When generating this set oflocal scaling factors the types of local modes were the sameas in the B or B0 Sets. This time all identified and assignedbands (81, cf. Tables 2–4) were used in the optimizationprocedure.

Recently, it is becoming a common practice to report statisticaluncertainties for scaling factors [26], at least in the case of sin-gle-parameter scaling approaches. Tables 5 and 6 include estimatesof statistical uncertainties for local scaling factors. They are, in fact,standard deviations of linear regression coefficients returned – inaddition to coefficients themselves – by microsoft excel statisticalfunction REGLINP. Their calculation is based on the assumptionthat the only appreciable error appears for scaled frequencies(the y-axis); all the remaining errors (e.g. inaccuracy of theoreticalfrequencies due to the inaccuracy of optimized geometries) areassumed to be negligible. It is worth noting that the reporteduncertainties are by an order (and sometimes by two) of magni-tude lower than in the case of single scaling parameter [26]. Theonly exceptions are scaling factors for sHil

C@C and c@C—H local modesof Set A. This is not surprising, however, since there are merely

two modes on the experimental spectra involving these localmotions.

We would also like to point out that the above-mentioned REG-LINP function carries out the optimization of regression coeffi-cients analogous to that implemented for local scaling factors inour home-made program. What has to be provided is a PED matrixas well as non-scaled theoretical and experimental frequencies.When calculating standard deviations we obtained exactly thesame set of local scaling factors, which confirmes the correctnessof our implementation. It should also be noted that the possibilityof using standard software (Excel) for obtaining local scalingfactors and scaled frequencies is another attractive feature of theESFF approach – no additional programming (like in the case ofthe SQM approach) has to be done as the PED coefficients are pro-vided by most standard quantum-chemical packages. However, webelieve that making the implementation of the ESFF algorithm atthe end of the frequency module of a given package would greatlyfacilitate ESFF calculations.

3.3. SQM calculations

In order to make a reliable comparison with the ESFF procedurethe eight parameter SQM calculations were carried out. All possiblescaling factors (i.e. eight factors, cf. [6]) that can be specified forhydrocarbons in the standard implementation of the SQM proce-

Table 3Styrene. See caption of Table 2.

No. mðWDCÞ I mðexptÞ I Contributions Comments

1 3221.38 15.026 99% m@C—H (asym.)2 3189.62 12.943 99% mAr—H

3 3181.40 32.138 99% mAr—H

4 3172.44 17.837 99% mAr—H

5 3162.43 0.109 99% mAr—H

6 3155.74 8.169 98% mAr—H

7 3142.17 3.468 98% m@C—H (sym.)8 3128.12 15.074 99% m@C—H

9 1689.94 11.983 1629.6 M 60% mC@C

10 1644.45 1.161 1600.6 W 65% mAr + 21% qAr—H

11 1617.59 3.307 1575.1 M 65% mAr + 20% qAr—H12 1528.64 11.047 1494.1 S 34% mAr + 60% qAr—H13 1483.00 4.782 1448.8 M 31% mAr + 47% qAr—H14 1451.27 6.082 1412.6 M 61% dVin (sym.)15 1362.66 1.856 1334.0 W 72% qAr—H

16 1345.92 0.908 1316.7 W 57% mAr + 22% q@C—H

17 1318.07 2.647 1289.2 W 50% mAr + 17% q@C—H

18 1226.28 2.384 1201.9 W 27% mAr + 17% qAr—H + 30% mAr—Vin

19 1205.40 0.128 1181.2 W 18% mAr + 74% qAr—H20 1183.41 0.199 1156.1 W 19% mAr + 80% qAr—H21 1111.52 4.894 1082.4 M 46% mAr + 39% qAr—H22 1056.00 0.170 52% mAr + 17% qAr—H + 27% q@C—H

23 1040.34 6.207 1020.2 M 27% mAr + 18% dAr + 46% q@C—H

24 1029.19 16.357 991.2 S 30% c@C—H + 66% sHilC@C

25 1014.57 0.160 39% mAr + 58% dAr

26 998.40 0.373 982.0 Sh 15% sAr + 79% cAr—H27 976.05 0.030 91% cAr—H28 928.81 43.835 908.3 S 17% cAr—H + 76% cC@C29 927.42 1.148 69% cAr—H30 849.96 0.405 839.9 W 100% cAr—H VBB31 798.45 32.886 775.7 VS 37% sAr + 33% cAr—H + 20% cAr—Vin

32 787.35 0.190 23% mAr + 37% dAr + 25% mAr—Vin

33 710.66 44.125 697.1 VS 37% sAr + 55% cAr—H

34 655.26 0.030 55% sAr + 18% cAr—H + 26% c@C—H35 635.04 0.027 620.5 W 87% dAr BB36 561.65 5.032 554.0 W 36% dAr + 32% dVin

37 449.07 1.316 443.0 Sh 23% dAr + 15% qAr—Vin + 16% dVin

38 446.47 8.006 436.3 M 52% sAr + 27% cAr—Vin

39 411.36 0.319 83% sAr

40 233.82 0.471 59% qAr—Vin + 32% dVin

41 203.94 2.151 45% sAr + 20% cAr—H + 27% cAr—Vin42 29.11 0.001 48% sHil

Ar—Vin + 19% sHilC@C

498 P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503

dure [27] were optimized. Two cases were considered: in the firstthe least-squares fit was made on the basis of 66 bands (what cor-responds to ESFF-B calculations), in the second – on the basis of all81 bands (what corresponds to ESFF-C calculations). They arecalled SQM-B and SQM-C calculations, respectively. In both SQMcalculations, the recommended values of weights, being 1000�inverse experimental frequency, were used. Weighting the fre-quencies in this way in the least-squares fit provides slightly betterresults (the lower RMS value) compared to the procedure, in whichequal weights are given to all frequencies. Note, that in all ESFF cal-culations the latter way of weighting was used. The SQM calcula-tions were optimized using SQM program developed by Bakerand coworkers [2,27], but other implementations are also available[28]. The SQM optimized scaling factors are summarized in Table 7.

4. Results and discussion

Tables 8–10 contain ESFF-scaled and SQM-scaled frequencies,the differences Dm between all scaled and experimental frequen-cies as well as the RMS values. The only frequencies included inthe Tables are these used in the factors’ optimization procedure.Note, that although the band positions reported in Tables 2–4 aregiven with a precision of 0:1 cm�1, the differences and RMS valueswere calculated, for convenience reason, using band positions that

follow from infrared band analysis of spectra handling program weused (two decimal figures).

4.1. Comparison of the ESFF-scaled and experimental frequencies

The analysis of the RMS values calculated for all (i.e. 66 or 81)scaled frequencies reveals, that the best results were obtained inthe ESFF-A calculations (RMS 3:11 cm�1). This is not a surprisesince the approach uses the largest number of scaling factors anda set of well-defined and resolved bands. Nearly comparableESFF-B (RMS 3:30 cm�1) and ESFF-B0 (RMS 3:35 cm�1) calculationsare only slightly worse in spite of reducing the number of opti-mized scaling parameters almost by a factor of 2. Inclusion of ‘‘sus-picious” bands in the ESFF-C calculations worsens the RMS valueby about 0:8 cm�1 (RMS 4:15 cm�1), which is probably due to theinaccurate determination of the resonance frequencies from theexperimental spectra for these bands. This is actually the case ofboth ESFF and SQM calculations (SQM-B: RMS 4:65 cm�1 andSQM-C: RMS 5:21 cm�1).

4.1.1. ESFF-A calculationsThe RMS values obtained using the Set A of local scaling factors

are very low for all molecules considered in this work. They are alllower than 4 cm�1 (the typical resolution of the FTIR spectrometer)

Table 44-Methylstyrene. See caption of Table 2.

No. mðWDCÞ I mðexptÞ I Contributions Comments

1 3220.68 15.836 99% m@C—H (asym.)2 3179.55 9.975 98% mAr—H

3 3168.19 27.018 99% mAr—H

4 3152.99 14.926 98% mAr—H

5 3150.79 16.107 98% mAr—H

6 3141.45 3.410 97% m@C—H (sym.)7 3126.94 16.327 98% m@C—H

8 3101.51 17.551 2944.8 S 100% mMet (asym.)9 3073.82 22.597 2921.2 S 100% mMet (asym.)

10 3019.74 34.174 2863.8 S 100% mMet (sym.)11 1689.59 14.957 1629.1 S 60% mC@C

12 1654.25 7.542 1608.3 M 63% mAr + 20% qAr—H VBB, IB13 1604.69 2.696 1570.7 M 66% mAr IB14 1547.19 21.719 1512.4 VS 35% mAr + 53% qAr—H

15 1494.97 9.186 1460.3 M 78% dMet (asym.) IB16 1489.82 6.769 1450.2 M 90% dMet (asym.) VBB, VIB17 1458.92 2.094 1417.9 M 60% dVin (sym.) VIB18 1436.96 5.009 1403.0 S 37% mAr + 35% qAr—H19 1413.80 0.167 1378.4 M 89% dMet (sym.)20 1353.79 2.607 1318.6 M 43% qAr—H + 31% q@C—H21 1335.00 0.112 1296.4 M 47% mAr + 28% qAr—H VBB22 1317.06 2.529 1278.6 M 58% mAr

23 1235.71 0.376 1214.5 M 22% dAr + 23% qAr—H + 29% mAr—Met

24 1229.70 2.515 1203.4 M 41% mAr + 16% qAr—H + 20% mAr—Vin

25 1207.02 3.201 1180.2 M 19% mAr + 65% qAr—H

26 1145.28 5.385 1115.6 S 33% mAr + 56% qAr—H27 1060.45 11.917 1040.0 Sh 64% qMet28 1051.94 4.617 1031.7 M 71% q@C—H29 1034.74 1.473 1018.2 S 29% mAr + 55% dAr

30 1028.84 17.032 989.3 VS 31% c@C—H + 67% sHilC@C

31 1003.05 0.036 21% mAr + 66% qMet

32 970.15 0.046 965.0 Sh 91% cAr—H

33 961.83 0.477 940.6 M 17% sAr + 76% cAr—H BB34 924.87 41.473 903.5 VS 93% cC@C35 847.46 24.610 823.5 VS 76% cAr—H36 840.78 14.944 77% cAr—H37 833.87 0.917 40% mAr + 24% dAr

38 749.65 2.577 729.4 S 53% sAr + 16% cAr—H

39 718.97 0.699 712.1 M 23% dAr + 22% mAr—Vin + 23% mAr—Met

40 657.33 0.139 652.8 Sh 82% dAr IB41 654.66 0.342 642.2 M 49% sAr + 27% c@C—H42 534.72 3.002 527.0 M 23% dAr + 15% qAr—Vin + 38% dVin IB43 473.82 19.885 464.3 S 44% sAr + 20% cAr—Vin + 27% cAr—Met44 421.05 0.158 416.1 M 47% dAr VIB

45 413.25 0.302 82% sAr + 17% cAr—H

46 344.67 0.512 65% qAr—Met

47 299.03 0.049 35% cAr—Vin + 29% cAr—Met

48 206.68 0.450 55% qAr—Vin + 24% dVin

49 136.47 1.646 45% sAr + 24% cAr—H

50 35.52 0.007 15% sAr + 51% sHilAr—Vin + 16% sHil

C@C

51 21.72 0.242 20% mAr + 56% sHilAr—Met

P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503 499

and, in the case of toluene, even lower than 3 cm�1. The analysis ofindividual frequencies reported in Tables 8–10 reveals that thehighest deviation (defined as jDmj) between theoretical and exper-imental frequencies only slightly exceeds 7 cm�1 (in the case ofmolecules 2 and 3) and for only 13 out of 66 scaled frequencies thisdeviation is higher than 4 cm�1. Such a good performance is par-tially due to the large number of scaling factors. Reduction of thisnumber leads to Sets B and C.

4.1.2. ESFF-B calculationsThe Set B consists of eight local scaling factors. As it was already

discussed three of them are designed to account for three bandswhich are very characteristic for compounds containing vinylgroup. For that reason the deviations observed for these bands(molecules 2 and 3) should be very low. Indeed, they are: the high-est deviation is equal to 0:63 cm�1. Defining three ‘‘specific” typesof scaling factors also means that ‘‘the rest of each molecule” is de-

scribed by merely five scaling factors. Again the RMS values arevery low – they are all lower than 4 cm�1 and even lower than3 cm�1 for toluene. The highest deviation for individual frequenciesis again observed for molecule 3 – still it does not exceed 8 cm�1.Moreover, for only 18 out of 66 scaled frequencies this deviationis higher than 4 cm�1. It should be emphasized that such good re-sults, only slightly worse than in the case of ESFF-A calculations,were obtained with a very limited set of scaling factors. The discus-sion of the ESFF-B0 results will be postponed until Section 4.2.

4.1.3. ESFF-C calculationsThe Set C also consists of eight local scaling factors. This time

shoulder bands as well as bands qualified as VBB and VIB (cf. Sec-tion 3.2.2) were also used in the optimization procedure. Obviouslythe scaled frequencies will, on average, exhibit larger deviationsfrom the experimental values. This can be seen as an increase ofthe RMS value for all molecules: minor, about 0:1 cm�1, in the case

Table 5Types of internal coordinates and their numbers in molecules 1–3, ‘‘status”, valuesand standard deviations (SD) for local scaling factors for the Set A.

Type Local modes No. of coord. Status Set A

1 2 3 Value SD

1 mAr—H; m@C—H; mMet 8 8 10 Optimized 0.9500 0.00052 mAr 6 6 6 Optimized 0.9720 0.00153 dAr 3 3 3 Optimized 0.9820 0.00304 sAr 3 3 3 Optimized 0.9643 0.00545 qAr—H 5 5 4 Optimized 0.9781 0.00146 cAr—H 5 5 4 Optimized 0.9799 0.00297 mC@C 0 1 1 Optimized 0.9522 0.00278 sHil

C@C 0 1 1 Optimized 0.9510 0.01079 cC@C 0 1 1 Optimized 0.9769 0.0033

10 c@C—H 0 1 1 Optimized 0.9848 0.021711 dVin; q@C—H 0 4 4 Optimized 0.9775 0.002312 dMet (asym.) 2 0 2 Optimized 0.9767 0.003113 dMet (sym.) 1 0 1 Optimized 0.9796 0.005314 qMet 2 0 2 Optimized 0.9726 0.0020

15 Others 4 4 8 Fixed 1.0000

Table 7SQM optimized scaling factors.

No. Factor for SQM-B SQM-C

1 C–C stretching 0.9402 0.93982 C–H stretching 0.9020 0.90203 C–C–C in-plane deformation 0.9887 0.99354 C–C–H in-plane deformation 0.9602 0.95855 H–C–H in-plane deformation 0.9373 0.93436 C–C–C–C torsion 0.9298 0.92637 C–C–C–H torsion 0.9658 0.96958 H–C–C–H torsion 0.9240 0.9390

500 P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503

of styrene and quite significant, somewhat above 1 cm�1, in thecase of 4-methylstyrene, for which as many as eight bands ofuncertain location were added as compared to ESFF-A and ESFF-Bcalculations. The analysis of the data included in Tables 8–10 leadsto the conclusion that this increase is mostly due to a few out of alladded frequencies. The frequencies included already in the ESFF-Aand ESFF-B calculations almost always preserve their quality. Itshould be emphasized that worsening of the results when goingfrom B-type- to C-type-scaled frequencies is not an inherent fea-ture of the ESFF approach. It is also observed in the case of theSQM-scaled frequencies, for which RMS values increase whengoing from SQM-B to SQM-C calculations.

4.2. Transferability of local scaling factors

We should, in fact, consider two aspects of transferability oflocal scaling factors: (i) transferability ‘‘within the molecule” and(ii) transferability ‘‘among molecules”.

The first of these items can be formulated as follows: the localscaling factors are transferable within the molecule if, after calcu-lating them from rather localized modes (group vibrations), theyare capable of reproducing frequencies of delocalized modes, i.e.normal modes, involving a number of local vibrations. In order tocheck for this kind of transferability the ESFF-B0 calculations werecarried out. Set B0 of local scaling factors was obtained on the basisof ‘‘pure” modes (cf. Section 3.2.3). The modes that were not ‘‘pure”and therefore not taken into account in the optimization procedureare denoted ‘‘*” in Tables 8–10. However, they were scaled by theireffective scaling factors constructed from the Set B0 of local scalingfactors. It is immediately recognized that the quality of thesefrequencies (the ‘‘* frequencies”) is basically the same as in theESFF-B calculation: all but one show the same sign of the deviation

Table 6Types of internal coordinates and their numbers in molecules 1–3, ‘‘status”, values and st

Type Local modes No. of

1

1 mAr—H; m@C—H; mMet 82 mAr 63 dAr; qAr—H; dVin; q@C—H,dMet (asym.),dMet (sym.),qMet,qAr—Met; qAr—Vin 144 sAr 35 cAr—H; c@C—H; cAr—Met; cAr—Vin 66 mC@C 07 sHil

C@C 08 cC@C 0

9 Others 2

from the experimental values and in only 5 out of 27 cases the Dmvalues obtained in ESFF-B and ESFF-B0 calculations differ bysomewhat more than 1 cm�1. This is apparently due to the similar-ity of the values for B and B0 local scaling factors (cf. Table 6): allbut one (sAr, type 4) factors differ by less or even much less than10�3.

The transferability among related molecules naturally means,that the factors computed for one molecule can be successfullyapplied to obtain accurate spectrum of another molecule, or, alter-natively, factors computed for different molecules have similar val-ues. The problem of transferability of scaling factors among totallydifferent molecules has already been dealt with in detail [7]. Thistime, transferability among related molecules will be checked. Inour first paper on the ESFF procedure [6], we actually computedfive scaling factors for toluene alone using the same as in the pres-ent work set of 19 bands from its experimental spectrum. Thetypes of internal coordinates were exactly the same as types 1–5considered here (cf. Table 6). Thus, the procedure adopted therecorresponds to ESFF-B calculations of this work; the differenceconsists in including additional bands of two more related mole-cules in the present calculations. Comparing the values of scalingfactors from Table 2 of [6] with the first five factors of Table 6for Set B (this paper) leads to the conclusion that perfect agree-ment is obtained for types 1–3 of internal coordinates – these fac-tors differ by less or even much less than 10�3. The difference fortypes 4 and 5 is somewhat larger and amounts to �0.005. Sincethese types of internal coordinates describe rather low-frequencyvibrations (usually lower than 1000 cm�1), such a change in thescaling factor may cause an error in the predicted frequency smal-ler than 2:5 cm�1 assuming that one of these types of local motionscontributes 50% to the normal mode. In passing note, that f4 of a SetB is greater, while f5 is lower than the corresponding values of [6].Since normal modes are often superpositions of both sAr and cAr—H

local modes such errors are fortuitiously cancelled out – most ofthe scaled low-frequency vibrations for toluene molecule calcu-lated here and in [6] are basically the same, and the overall RMSfor toluene of 2:85 cm�1 (cf. Table 8) is only slightly higher thatof [6] (2:48 cm�1). We also want to point out that initial calcula-tions carried out for much larger set of related molecules, whichincludes benzene, toluene, all xylenes, all 3- and all 4-methyl

andard deviations (SD) for local scaling factors for Sets B, B0 and C.

coord. Status Set B Set B0 Set C

2 3 Value SD Value Value SD

8 10 Optimized 0.9500 0.0005 0.9500 0.9500 0.00066 6 Optimized 0.9740 0.0014 0.9748 0.9738 0.0016

13 18 Optimized 0.9773 0.0009 0.9768 0.9766 0.00093 3 Optimized 0.9724 0.0050 0.9685 0.9667 0.00547 7 Optimized 0.9795 0.0028 0.9792 0.9858 0.00221 1 Optimized 0.9531 0.0025 0.9533 0.9535 0.00311 1 Optimized 0.9536 0.0039 0.9538 0.9505 0.00471 1 Optimized 0.9772 0.0032 0.9773 0.9764 0.0039

2 4 Fixed 1.0000 1.0000 1.0000

Table 8Toluene. ESFF (A, B, B0 and C, see text) and SQM-scaled frequencies (cm�1) as well as differences Dm between scaled and experimental (cf. Tables 2–4) frequencies. The number ofscaled frequency is given in the first column. Symbol ‘‘*” next to the mðESFF-B0 Þ frequency is used in order to indicate that the corresponding band was not taken into account in theoptimization of local scaling factors. In the last row the root-mean-square (RMS) deviations (cm�1) for all scaled frequencies are reported.

No. mðESFF-AÞ Dm mðESFF-BÞ Dm mðESFF-B0 Þ Dm mðSQM-BÞ Dm mðESFF-CÞ Dm mðSQM-CÞ Dm

6 2945.99 �3.60 2945.94 �3.65 2945.94 �3.65 2945.89 �3.70 2945.94 �3.65 2945.89 �3.707 2920.18 0.00 2920.19 0.01 2920.19 0.01 2920.16 �0.02 2920.19 0.01 2920.16 �0.028 2869.92 �3.01 2869.96 �2.97 2869.96 �2.97 2870.00 �2.93 2869.96 �2.97 2870.00 �2.939 1607.46 2.98 1608.50 4.02 1609.18 4.70 1605.72 1.24 1607.99 3.51 1605.53 1.05

11 1494.11 �1.42 1494.44 �1.09 1494.47 �1.06 1494.76 �0.77 1493.71 �1.82 1494.04 �1.4912 1468.69 9.80 1467.41 8.5213 1455.48 0.45 1451.76 �3.2715 1378.90 0.05 1384.81 5.96 1384.18 5.33 1377.77 �1.08 1383.87 5.02 1376.10 �2.7518 1212.22 2.11 1211.73 1.62 1211.76 1.65* 1198.14 �11.97 1211.38 1.27 1198.14 �11.9719 1176.06 �2.23 1175.78 �2.51 1175.53 �2.76 1176.91 �1.38 1175.06 �3.23 1176.09 �2.2020 1154.54 �1.58 1154.26 �1.86 1154.00 �2.12 1155.98 �0.14 1153.54 �2.58 1155.15 �0.9721 1085.46 4.07 1084.94 3.55 1085.06 3.67* 1086.65 5.26 1084.43 3.04 1086.37 4.9822 1040.99 0.10 1038.11 �2.78 1037.28 �3.61 1042.03 1.14 1038.34 �2.55 1042.28 1.3923 1024.78 �5.02 1025.37 �4.43 1025.80 �4.00 1026.63 �3.17 1025.04 �4.76 1026.75 �3.0524 993.07 �9.25 999.50 �2.8228 891.50 �3.79 891.29 �4.00 890.84 �4.45 887.52 �7.77 896.04 0.75 890.83 �4.4629 842.97 1.68 840.10 �1.1930 786.46 1.09 785.49 0.12 785.52 0.15* 779.99 �5.38 785.25 �0.12 780.25 �5.1231 727.25 �1.23 727.67 �0.81 726.63 �1.85* 729.55 1.07 729.46 0.98 730.71 2.2332 692.27 �1.98 695.35 1.10 693.36 �0.89 694.81 0.56 693.98 �0.27 694.79 0.5433 625.63 3.22 622.83 0.42 622.58 0.17 629.59 7.18 622.39 �0.02 630.35 7.9434 520.46 �1.19 518.67 �2.98 518.49 �3.16 521.46 �0.19 518.37 �3.28 522.12 0.4735 466.25 1.97 464.87 0.59 463.80 �0.48* 464.87 0.59 464.70 0.42 465.07 0.79

RMS 2.55 2.85 2.95 4.31 3.71 4.30

P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503 501

derivatives of benzene confirm very good transferability of localscaling factors.

4.3. Comparison of the ESFF and SQM methods

In this section, the comparison of the results discussed in thepreceding section with these obtained with the SQM approach willbe given. The SQM-scaled frequencies along with deviations fromthe experimental values and the RMS values are also reported inTables 8–10. In the following we will concentrate on the compari-son between ESFF-B and SQM-B calculations giving a brief com-ment to C-type calculations at the end of this section.

Table 9Styrene. See caption of Table 8.

No. mðESFF-AÞ Dm mðESFF-BÞ Dm mðESFF-B0 Þ Dm

9 1629.47 �0.08 1629.51 �0.04 1629.47 �0.0810 1603.62 2.99 1604.63 4.00 1605.30 4.6711 1576.02 0.94 1576.64 1.56 1577.30 2.2212 1493.48 �0.60 1493.72 �0.36 1493.72 �0.3613 1448.67 �0.09 1447.63 �1.13 1447.54 �1.22*

14 1419.27 6.67 1419.11 6.51 1418.62 6.0215 1330.76 �3.26 1329.96 �4.06 1329.52 �4.5016 1309.52 �7.14 1310.88 �5.78 1311.31 �5.35*

17 1286.61 �2.57 1286.19 �2.99 1286.48 �2.70*

18 1205.20 3.28 1204.94 3.02 1204.99 3.07*

19 1178.69 �2.50 1178.24 �2.95 1177.97 �3.2220 1156.11 �0.01 1155.81 �0.31 1155.54 �0.5821 1084.92 2.57 1084.53 2.18 1084.69 2.34*

23 1016.17 �3.99 1015.67 �4.49 1015.55 �4.61*

24 990.59 �0.64 990.60 �0.63 990.58 �0.652628 907.74 �0.57 907.82 �0.49 907.77 �0.543031 781.26 5.53 779.92 4.19 778.63 2.9033 692.92 �4.22 694.20 �2.94 693.04 �4.10*

35 623.22 2.74 620.47 �0.01 620.22 �0.2636 551.40 �2.55 549.21 �4.74 548.98 �4.97*

3738 436.58 0.26 435.64 �0.68 434.66 �1.66*

RMS 3.28 3.19 3.24

First, it should be noted that SQM also reproduces the experi-mental IR spectra with very high accuracy. The RMS value for all66 frequencies is 4:65 cm�1. The deviations for most of individualfrequencies are well below 8 cm�1 (this is the value for the averageerror in band positions reported in the manual to the SQM programwe used [27]), and even well below 6 cm�1. The highest deviations,close to 13 cm�1, are observed in the case of vinyl group stretchingmC@C modes of 2 and 3. In our ESFF-B calculations, we decided tokeep the mC@C stretching coordinate as a separate type. Note, thatin ESFF-A calculations factors for mAr and mC@C local modes differby as much as about 0.01. Such a procedure obviously significantlylowers the deviation for that mode. Thus, an additional, 9th type

mðSQM-BÞ Dm mðESFF-CÞ Dm mðSQM-CÞ Dm

1642.40 12.85 1629.57 0.02 1641.84 12.291601.91 1.28 1604.12 3.49 1601.71 1.081575.33 0.25 1576.14 1.06 1575.06 �0.021494.09 0.01 1492.98 �1.10 1493.36 �0.721449.61 0.85 1446.84 �1.92 1448.93 0.171413.74 1.14 1418.26 5.66 1412.25 �0.351333.83 �0.19 1329.13 �4.89 1332.84 �1.181312.10 �4.56 1310.45 �6.21 1311.31 �5.351286.31 �2.87 1285.70 �3.48 1286.04 �3.141195.99 �5.93 1204.55 2.63 1195.84 �6.081178.70 �2.49 1177.53 �3.66 1177.92 �3.271157.52 1.40 1155.09 �1.03 1156.69 0.571086.31 3.96 1084.05 1.70 1086.07 3.721019.85 �0.31 1015.10 �5.06 1020.04 �0.12

995.60 4.37 990.64 �0.59 1001.74 10.51979.95 �2.05 970.96 �11.04

904.18 �4.13 908.32 0.01 907.80 �0.51837.88 �1.97 834.76 �5.09

779.84 4.11 781.38 5.65 781.01 5.28695.92 �1.22 695.50 �1.64 696.76 �0.38627.18 6.70 620.04 �0.44 627.93 7.45554.77 0.82 548.83 �5.12 555.45 1.50

439.09 �3.91 442.51 �0.49435.35 �0.97 435.44 �0.88 435.48 �0.84

4.13 3.29 5.00

Table 104-Methylstyrene. See caption of Table 8.

No. mðESFF-AÞ Dm mðESFF-BÞ Dm mðESFF-B0 Þ Dm mðSQM-BÞ Dm mðESFF-CÞ Dm mðSQM-CÞ Dm

8 2946.80 2.03 2946.76 1.99 2946.76 1.99 2946.72 1.95 2946.75 1.98 2946.72 1.959 2920.42 �0.73 2920.42 �0.73 2920.42 �0.73 2920.41 �0.74 2920.42 �0.73 2920.41 �0.74

10 2869.12 5.35 2869.15 5.38 2869.16 5.39 2869.19 5.42 2869.15 5.38 2869.20 5.4311 1629.20 0.13 1629.26 0.19 1629.23 0.16 1642.07 13.00 1629.32 0.25 1641.51 12.4412 1614.42 6.08 1611.57 3.2313 1564.53 �6.21 1564.05 �6.69 1564.73 �6.01 1562.90 �7.84 1563.56 �7.18 1562.77 �7.9714 1513.20 0.80 1513.53 1.13 1513.59 1.19* 1511.72 �0.68 1512.85 0.45 1511.13 �1.2715 1460.35 0.02 1460.61 0.28 1460.00 �0.33 1457.26 �3.07 1459.58 �0.75 1456.57 �3.7616 1454.86 4.65 1451.10 0.8917 1425.51 7.60 1421.30 3.3918 1403.45 0.49 1402.68 �0.28 1402.75 �0.21* 1402.94 �0.02 1402.04 �0.92 1402.36 �0.6019 1378.23 �0.14 1384.13 5.76 1383.51 5.14 1377.08 �1.29 1383.20 4.83 1375.41 �2.9620 1320.09 1.50 1319.51 0.92 1319.10 0.51* 1324.22 5.63 1318.76 0.17 1323.27 4.6821 1301.15 4.74 1303.42 7.0122 1285.26 6.69 1284.98 6.41 1285.42 6.85* 1284.08 5.51 1284.55 5.98 1283.98 5.4123 1217.91 3.46 1216.82 2.37 1216.68 2.23* 1208.47 �5.98 1216.36 1.91 1208.58 �5.8724 1207.47 4.11 1208.19 4.83 1208.48 5.12* 1195.95 �7.41 1207.91 4.55 1195.53 �7.8325 1182.04 1.82 1181.71 1.49 1181.49 1.27 1178.70 �1.52 1181.06 0.84 1177.99 �2.2326 1119.12 3.50 1118.24 2.62 1118.21 2.59* 1119.91 4.29 1117.66 2.04 1119.51 3.8927 1036.57 �3.43 1040.45 0.4528 1027.63 �4.10 1027.50 �4.23 1027.19 �4.54 1030.45 �1.28 1026.83 �4.90 1029.94 �1.7929 1012.65 �5.58 1010.40 �7.83 1010.31 �7.92* 1016.69 �1.54 1009.86 �8.37 1017.30 �0.9330 989.94 0.64 989.93 0.63 989.93 0.63 995.24 5.94 989.87 0.57 1001.40 12.1032 954.82 �10.18 947.71 �17.2933 939.80 �0.81 940.36 �0.25 939.52 �1.09 934.91 �5.70 944.15 3.54 938.81 �1.8034 903.93 0.44 903.87 0.38 903.92 0.43 899.32 �4.17 903.48 �0.01 903.71 0.2235 830.76 7.30 829.40 5.94 828.91 5.45 831.71 8.25 833.65 10.19 833.21 9.7538 731.59 2.15 731.45 2.01 729.79 0.35* 731.78 2.34 731.25 1.81 732.24 2.8039 710.28 �1.81 709.53 �2.56 709.45 �2.64* 702.62 �9.47 709.34 �2.75 702.90 �9.1940 641.75 �11.04 649.82 �2.9741 638.64 �3.54 638.84 �3.34 637.49 �4.69* 639.25 �2.93 639.05 �3.13 639.69 �2.4942 525.40 �1.55 522.75 �4.20 522.53 �4.42* 528.14 1.19 522.40 �4.55 528.81 1.8643 465.59 1.31 462.58 �1.70 461.68 �2.60* 462.69 �1.59 463.00 �1.28 462.97 �1.3144 412.29 �3.78 415.47 �0.60

RMS 3.35 3.67 3.69 5.24 4.91 5.88

Table 11Comparison of ESFF-B and SQM-B methods. Number of scaled frequencies that exhibitthe deviation larger than 4 cm�1 (Test I), 6 cm�1 (Test II) and 8 cm�1 (Test III) from theexperimental values are reported for each of the molecules 1–3 and jointly for all 4molecules. In Test IV the number of scaled with a given method frequencies that are‘‘worse” (see text) than these scaled with the other, is given. The root-mean-square(RMS) deviations (cm�1) for all 66 scaled frequencies are reported in the last row.

Test 1 2 3 All molecules

ESFF SQM ESFF SQM ESFF SQM ESFF SQM

I 3 5 6 7 9 13 18 25II 0 3 1 2 3 5 4 10III 0 1 0 1 0 3 0 5IV 7 5 8 7 4 12 19 24

RMS 3.30 4.65

502 P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503

for vinyl group stretching mode, was added to SQM calculations.Significantly lower value of the RMS of 3:77 cm�1 was obtained thistime. However, it still remains by almost 0:5 cm�1 higher than in 8-parameter ESFF-B calculations in spite of extending the set of SQMscaling factors by one.

Additional information, named Test I–IV, comparing the twomethods are reported in Table 11. The information include thenumber of scaled frequencies that exhibit the deviation fromthe experimental values larger than 4, 6 and 8 cm�1 (Tests I–III)and, in Test IV, the number of scaled frequencies that are ‘‘worse”when scaled with one of the methods as compared to the other.Under the phrase ‘‘. . .the frequency is worse . . .” the followingis understood: if jDmj for the frequency scaled with method Xdoes not differ from jDmj for this frequency scaled with methodY by more than 1 cm�1 (i.e. by more than the resolution of thepresent experiment), the frequencies are said to be of the samequality; if that difference is greater than 1 cm�1, the scaled fre-quency exhibiting larger deviation from the experimental valueis said to be worse. For example, in the case of toluene (cf. Table8) the frequency 31 is of the same quality in both ESFF-B andSQM-B calculations (Dm values are �0:81 and 1:07 cm�1, respec-tively) and the frequency 9 is worse in ESFF-B as compared toSQM-B calculations (Dm values are 4.02 and 1:24 cm�1, respec-tively). In most of tests the ESFF is superior to SQM. Only intwo cases: Test IV for molecules 1 and 2 SQM is somewhat better;however, it does not change the final conclusion regarding thelower RMS values in the case of ESFF-scaled frequencies for allmolecules. The last column of Table 11 is of particular impor-tance, since the information concerns collectively all 66 frequen-cies. In particular, the number of frequencies that exhibit

deviation from the experimental values larger than 6 cm�1 ismore than twice as low in ESFF calculations as compared toSQM and no ESFF-scaled frequency exhibits deviation larger than8 cm�1. It should also be noted, that the overall RMS obtainedwith the ESFF method is 28% lower than that obtained with SQM.

Finally, we decided to carry out the so-called significance F-testfor the comparison of standard deviations. This test assumes thatthe populations from which the samples are taken are normal. Thisassumption may not be fulfilled in the present case, but the testmay still provide some useful information. The standard deviations(variances, to be precise), the ratio of which is to be calculated, areclosely related to the reported in Table 11 values of the root-mean-square (RMS) deviations. The sum of squares of deviations Dm hasto be divided by the number of degrees of freedom, i.e. number of

P. Borowski et al. / Journal of Molecular Structure 924–926 (2009) 493–503 503

frequencies (66 in this case) minus the number of scaling factors(i.e. 8). The statistic F (the ratio of variances) turned out to be1.986 which is higher than the critical value F58;58 of 1.856 ob-tained for 58 degrees of freedom at the significance level of 1%.Thus, the variance of the SQM method is significantly greater thanthat of the ESFF method at the 1% probability level, at least for therelated compounds considered in the present work.

Comparison of the results obtained in ESFF-C and SQM-C calcu-lations, in which larger (on average) differences between scaledand experimental frequencies are observed, leads to the conclusionthat frequencies inaccurately predicted by SQM approach are often‘‘repaired” by ESFF. The opposite statement is less frequentlyobeyed. Note that, for example, if the decision that ‘‘. . .the fre-quency is worse . . .” is arrived at on the basis the adopted differ-ences of, say, 6 and 8 cm�1, in the first case the Test IV for allfrequencies reads 3� 10 and in the second – 1� 6 in favor of ESFF.

5. Conclusions

In this work we report new results obtained with the recentlyproposed ESFF method for scaling of harmonic vibrational frequen-cies. The calculations involve three related molecules: toluene, sty-rene and 4-methylstyrene. The presented theoretical vibrationalspectra are of very high quality, higher than these obtained withthe well-known SQM method. A number of ad hoc designed setsof statistical treatments clearly show that ESFF method improvesSQM results as far as frequencies are concerned. The best resultswere obtained with the Set A of 14 local scaling factors optimizedon the basis of 66 IR well-defined fundamental bands of these mol-ecules. However, this set was selected for test purpose – 14 scalingfactors are definitely too many to carry out frequency scaling ofsimple hydrocarbons. The reduction of the number of local scalingfactors down to 8 (the Set B) only slightly deteriorates the ESFF-scaled frequencies. In addition, the overall RMS deviation is 28%lower than in the case of the corresponding 8-parameter SQMcalculations.

Transferability of ESFF local scaling factors within a moleculeand between related molecules was also checked. The Set B0 ofeight local scaling factors optimized on a basis of 39 bands cor-responding to ‘‘pure” modes scales all 66 (the same Set as B)harmonic frequencies nearly equally well as the Set B (optimizedfor all 66 bands). The RMS values calculated with these two setsare equal to 3.30 and 3:35 cm�1 for the Sets B and B0, respec-tively. The average RMS deviation attained at using PulaysSQM method turns out to be 4:65 cm�1. This clearly demon-strates the validity of the main assumption of the ESFF method,which is building effective scaling factors from local scaling fac-tors. The transferability of local scaling factors among relatedmolecules is also very satisfactory. Apart from high quality of

theoretical spectra (low values of RMS deviations) it should benoted that local scaling factors obtained on the basis of funda-mental bands of three molecules (toluene, styrene and 4-methyl-styrene) are capable of giving the scaled frequencies for toluenewhich are of comparable quality to these obtained with the aidof factors optimized for toluene itself.

Acknowledgements

P.B. thank the University of Jaén for the financial support thatenabled him to continue the work on this project. Also P.B. wishesto acknowledge Dr. Karol Pilorz for valuable discussions. Theauthors thank the Spanish Consejería de Innovación, Ciencia yEmpresa, Junta de Andalucía (PAI FQM 337 grant and FQM-P06-01864 project) for the financial support.

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