Model Studies of the Optical Rotation, and TheoreticalDetermination of its Sign for β-Pinene and trans-Pinane

ANGELIKA BARANOWSKA,1,2 KRZYSZTOF Z. ŁACZKOWSKI,3 ANDRZEJ J. SADLEJ1

1Department of Quantum Chemistry, Faculty of Chemistry, Nicolaus Copernicus University,7, Gagarin St., PL-87100 Torun, Poland

2Department of Physical Chemistry, Faculty of Chemistry, University of Santiago de Compostela,E-15782 Santiago de Compostela, Spain

3Department of Organic Chemistry, Faculty of Chemistry, Nicolaus Copernicus University,7, Gagarin St., PL-87100 Torun, Poland

Received 22 May 2009; Revised 12 July 2009; Accepted 2 August 2009DOI 10.1002/jcc.21403

Published online 23 September 2009 in Wiley InterScience (www.interscience.wiley.com).

Abstract: We have carried out extensive studies on the basis set dependence of the calculated specific optical rotation(OR) in molecules at the level of the time–dependent Hartree–Fock and density functional approximations. To reachthe limits of the basis set saturation, we have devised an artificial model, the asymmetrically deformed (chiral) methane(CM) molecule. This small system permits to use basis sets which are prohibitively large for real chiral molecules andyet shows all the important features of the basis set dependence of the OR values. The convergence of the OR has beenstudied with n-aug-cc-pVXZ basis sets of Dunning up to the 6–ζ . In a parallel series of calculations, we have used therecently developed large polarized (LPolX) basis sets. The relatively small LPolX sets have been shown to be competitiveto very large n-aug-cc-pVXZ basis sets. The conclusions reached in calculations of OR in CM concerning the usefulnessof LPolX basis sets have been further tested on (S)-methyloxirane and (S)-fluoro-oxirane. The smallest set of the LPolXfamily (LPol–ds) has been found to yield OR values of similar quality as those obtained with much larger Dunning’saug-cc-pVQZ basis set. These results have encouraged us to carry out the OR calculations with LPol–ds basis sets forsystems as large as β-pinene and trans-pinane. In both cases, our calculations have lead to the correct sign of the ORvalue in these molecules. This makes the relatively small LPol–ds basis sets likely to be useful in OR calculations forlarge molecules.

© 2009 Wiley Periodicals, Inc. J Comput Chem 31: 1176–1181, 2010

Key words: ab initio calculations; density functional calculations; optical rotation; basis set; terpenes

Introduction

Accurate theoretical calculations of the specific optical rotation(OR) in chiral molecules are the ultimate hope for predictions oftheir absolute configuration. However, this task is by no meanseasy and several examples of disagreement between theoreticaland well-established experimental OR signs are reported in liter-ature,1–4 nothing to say about the numerical OR values. To remedythe observed disagreement some of the authors invoke several,usually neglected, effects like solute–solvent interactions,5, 6 vibra-tional corrections,7–12 and the importance of high-order correlationeffects.4, 9, 13 One of the possible reasons for the failure of compu-tational methods is also the use of basis sets which are too smallor insufficiently flexible to describe the simultaneous effect of elec-tric and magnetic fields of the radiation. Recently, Pedersen et al.14

reported an extensive study of OR in (S)-fluoro-oxirane with a seriesof aug-cc-pVXZ basis sets of Dunning and coworker15–17 and found

a very strong and discouraging basis set dependence of the calculatedOR values. Similar conclusions follow also from earlier calcula-tions, in particular from the extensive study of (S)-methyloxiraneand (R,R)-dimethylthiirane by Cheeseman et al.18

The present article focuses on the basis set issue. The conver-gence of the calculated OR data with respect to the systematicextension of the basis set will be investigated for a small modelsystem, deformed methane molecule. The model which is hereafterreferred to as the chiral methane (CM) is sufficiently small to per-mit the use of basis sets whose size would be prohibitively largefor calculations on real chiral systems. The present OR calculationsfor CM will be carried out with augmented correlation consistent,

Additional Supporting Information may be found in the online version ofthis article.

Correspondence to: A. Baranowska; e-mail: teoang@chem.uni.torun.pl

© 2009 Wiley Periodicals, Inc.

Model Studies of the Optical Rotation 1177

Table 1. Geometrical Parameters of Chiral Methane Molecule.

Geometry 1 Geometry 2

Atom X Y Z X Y Z

C 0.000000 0.000000 0.000000 0.000020 0.000015 −0.000118H 0.000000 0.000000 1.089000 0.234522 0.901137 −0.579257H 1.094877 0.000000 −0.313951 0.239115 0.239800 1.113529H −0.555014 −0.961312 −0.426099 −1.155736 −0.362055 −0.242393H −0.478202 0.828271 −0.405970 0.581977 −0.778972 −0.291167

Cartesian coordinates (Å).

n-aug-cc-pVXZ, basis sets of Dunning and coworker15–17 and n andX values up to q (quadruply augmented) and 6, respectively. Weshall also investigate the performance of recently developed largepolarized (LPolX) basis sets19 which are designed primarily for theaccurate representation of the first–order perturbed wave functions.

The analysis of the OR data for CM will be used to select rela-tively small basis sets which are likely to give reliable data for ORof real chiral molecules. This will be verified in calculations for (S)-methyloxirane and (S)-fluoro-oxirane. Finally, the basis sets selectedfor model calculations on CM and tested on these two moleculeswill be used to establish the sign of OR in (1S,5S)-β-pinene and(1S,2S,5S)-trans-pinane.

The article is organized as follows. In section following “Intro-duction” the details of the computational studies are briefly pre-sented. Section on “The Chiral Methane Model” reports and dis-cusses the results of the basis set studies performed for the CMmodel. Conclusions of this study are used for the prediction ofspecific rotation of four test chiral systems in section on “Calcu-lations for Real Chiral Molecules: Oxiranes, Pinene, and Pinane.”The article is summarized in the last section.

Computational Details

All OR studies have been carried out at the level of theTDDFT/B3LYP approximation with the London atomic orbitals(LAOs)25–27 used to account for the effect of the magnetic field per-turbation. Additionally, the TDHF calculations have been carriedout for CM. The conclusions on the basis set requirements follow-ing the model CM specific rotation study have been found to be verysimilar for the two methods. Hence, only the TDDFT/B3LYP valuesare reported in the article whereas the TDHF results are collected inthe Supporting Information. The software used in OR calculationsis dalton 2.0 set of programs.20 The OR results are reported in theunits of specific rotation, i.e., 10−1deg cm2g−1, and will be referedto as the OR-units throughout for simplicity.

The Chiral Methane Model

The CM molecule, which is the basic model used in the presentstudy, is created from the real nonchiral system by the distortion ofits tetrahedral geometry. Two specific structures, whose geometrydata are give in Table 1, have been selected for the presentation ofour investigations. They follow from a large number of pilot stud-ies and their choice for reporting the representative OR data gives

meaningfully large numerical values. Simultaneous deformationsof the CH bond length and HCH angles were found necessary toobtain large enough OR values.

To select the frequency value for calculations on the model CMmolecule, we have carried out a series of TDDFT calculations withtwo selected basis sets, aug-cc-pVDZ and d-aug-cc-pVTZ. The cor-responding results are presented in Figure 1 and show very similarpattern of ω–dependence of OR for both basis sets. On the basis ofthese numerical experiments, we have selected the value of ω forother basis sets as equal to 0.2 au (λ = 227.8 nm). This value of ω

leads to significantly large OR data and simultaneously is far awayfrom resonances.

The results of the OR calculations for CM in n-aug-cc-pVXZbasis sets are reported in Table 2. One should note that the totalnumber of basis functions (nL , Table 2) grows rapidly with theincrease of n and X, and for some multiply augmented basis setsthe corresponding calculations have been found too demandingfor the available computational resources. Moreover, the increaseof the number of diffuse functions leads to near-linear dependenciesamong functions of the molecular set which results in deletion ofsome of the molecular orbital (MO) components from the total setof functions. Whenever these dependencies led to the removal offive or more functions (numbers in italics), the further extensionof the basis set has been abandoned (empty entries in Table 2).

Figure 1. TDDFT/B3LYP specific rotation α(ω) frequency scan per-formed for chiral methane model in aug-cc-pVDZ (denoted aVDZ) andd-aug-cc-pVTZ (denoted daVTZ) basis sets. Symbols g1 and g2 denoteGeometry 1 and Geometry 2, respectively.

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1178 Baranowska, Łaczkowski, and Sadlej • Vol. 31, No. 6 • Journal of Computational Chemistry

Table 2. Specific Rotation of Asymmetric Methane Obtained at the TDDFT/B3LYP/n-aug-cc-pVXZ (L) andTDDFT/B3LYP/n-aug-cc-pVXZ(cc-pVDZ) (S) Level of Theory (n = 1, d, t, q; X = D, T, Q, 5, 6) forλ= 227.8 nm.

Geometry 1 Geometry 2 Geometry 1 Geometry 2

Basis set nL nS [α]L [α]S [α]L [α]S Basic set nL nS [α]L [α]S [α]L [α]S

aVDZ 59 43 17.63 15.07 27.74 25.72 taVDZ 109 61 17.97 16.54 29.98 29.11aVTZ 138 66 17.23 15.56 28.94 27.25 taVTZ 242 98 17.43 17.21 29.23 28.73aVQZ 264 100 17.29 16.71 29.17 29.48 taVQZ 442 150 17.41 17.40 29.36 29.10aV5Z 447 147 17.36 17.03 29.10 29.59 taV5Z 219 17.40 29.17aV6Z 697 209 17.42 17.37 29.17 29.23 taV6Z 307 17.37 29.17daVDZ 84 52 17.81 16.46 29.88 29.00 qaVDZ 134 70 17.73 16.58 30.61 29.17daVTZ 190 82 17.53 17.16 29.31 28.73 qaVTZ 294 114 17.46 17.23 29.21 28.73daVQZ 353 125 17.41 17.30 29.29 29.15 qaVQZ 175 17.41 29.08daV5Z 583 183 17.39 17.47 29.21 29.19 qaV5Z 255 17.39 29.13daV6Z 258 17.36 29.21 qaV6Z 356 17.37 29.18

The symbol naVXZ stands for n-aug-cc-pVXZ basis set, nL and nS denote total number of basis functions in n-aug-cc-pVXZ and n-aug-cc-pVXZ(cc-pVDZ) sets, respectively. Specific rotation values reported in OR-units. Numbers in italicsdenote results obtained in the basis sets reduced by five or more functions due to the near-linear dependencies.

To cope with the dimensionality problem and near-linear depen-dence among basis set functions, we have tried to use smaller basissets on hydrogen atoms when systematically extending the set offunctions on the carbon atom, i.e., the molecular basis sets were ofthe form n-aug-cc-pVXZ(cc-VDZ), where the entry in parenthesesrefers to the basis set for hydrogen. This replacement of the hydro-gen basis set significantly reduces the total number (nS in Table 2)of basis functions. Yet, it saves the highly flexible description of thechiral center. Indeed, for large enough basis sets (aVQZ, daVQZ,taVQZ, and qaVQZ, see Table 2) on C the use of the small cc-pVDZbasis set on hydrogens does not lead to any significant deteriorationof the OR values.

For the two selected geometries of CM, the data of Table 2 showa stable convergence pattern. The OR values for Geometries 1 and2 converge to about 17.4 and 29.2 OR-units, respectively. Thesetwo results will be used as the corresponding reference values incalculations with LPolX basis sets. For Geometry 1 and large basissets on carbon and hydrogens, the calculated OR values reach thesaturation limits for aV5Z (nL = 447), daVQZ (nL = 353), taVTZ(nL = 242), and qaVTZ (nL = 294) sets. For the mixed (large onC, small on H) basis sets, the saturation limit is obtained for aV6Z(nS = 209), daV5Z (nS = 183), taVQZ (nS = 150), and qaVQZ(nS = 175) sets on carbon. The data for Geometry 2 show essentiallythe same pattern.

There is obviously certain interplay between the presence ofbasis functions with high values of the angular momentum quantumnumber and the basis set diffuseness. Once the diffuse functionsare added, the importance of high angular momentum functions isdiminished and one can use this observation to select the smallestsuitable basis set. Particularly interesting are the results with thereduced basis set on H. The reduction may require the use of slightlylarger basis sets on carbon (see Table 2). However, the total size ofthe molecular basis set is reduced at least more than twice and insome cases by more than a factor of 3.

The observation of the interplay between highly diffuse and highangular momentum functions suggests that quite reliable OR datacan be obtained with very diffuse basis sets of relatively low value of

the angular momentum quantum number. Basis sets of this charac-teristics have been recently developed19 for the calculation of linearand nonlinear electrooptical properties of molecules. Four classesof these LPolX basis sets have been designed and will be referred toby symbols LPol-ds ([14s9p6d/6s5p3d] for C and [11s6p/4s3p] forH), LPol-fs ([14s9p6d4f /6s5p3d2f ] for C and [11s6p4d/4s3p2d]for H), LPol-dl ([14s9p6d/7s6p4d] for C and [11s6p/5s4p] for H),and LPol-fl ([14s9p6d4f /7s6p4d3f ] for C and [11s6p4d/5s4p3d] forH). If combined with the LAO formalism, these basis sets shouldalso be suitable for OR calculations. This is demonstrated by thedata of Table 3. One should note that for LPolX basis sets on allatoms the size of the total molecular (nL) basis set is much smallerthan the sizes of Dunning’s basis sets which give comparable accu-racy. Already for the smallest LPol-ds (nL = 88) set the calculatedOR data are close to the reference values established earlier.

To reduce the total basis set size we have used the same method-ology as that employed with Dunning’s sets, i.e., we have replacedLPolX sets on hydrogens with the small polarized ZPolX sets21, 22

([6s3p/3s1p] for H). The corresponding results are shown in Table 3.

Table 3. Specific Rotation of Asymmetric Methane Obtained at theTDDFT/B3LYP Level of Theory for λ = 227.8 nm Using LPolX (L) andLPolX(ZPolX) (LZ) Basis Sets.

Geometry 1 Geometry 2

Basis set nL nLZ [α]L [α]LZ [α]L [α]LZ

LPol-ds 88 60 17.36 17.37 29.26 30.30LPol-fs 142 74 17.20 16.98 29.14 29.63LPol-dl 113 69 17.22 17.43 29.40 28.36LPol-fl 194 90 17.43 17.45 29.32 28.56Reference value 17.4 29.2

Symbols nL and nLZ denote total number of basis functions in LPolX andLPolX(ZPolX) sets, respectively. Specific rotation values reported in OR-units. Numbers in italics denote results obtained in the basis sets reduced byfive or more functions due to the near-linear dependencies.

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Model Studies of the Optical Rotation 1179

It can be seen that using the combination LPolX(ZPolX) of largebasis set on C and small basis set on hydrogens works reasonablywell, though certain deterioration of the OR values occurs.

To conclude these model studies of OR for the CM molecule letus note that the use of Dunning’s basis sets of appropriate quality(at least aug-cc-pVQZ) is not feasible for most but the smallestmolecules. Essentially the same results can be obtained with LPol-dsset. Some further reduction (at the cost of certain deterioration of theOR results) of the total basis set size can be obtained with small ZPolsets on hydrogens. Hence, we choose LPol-ds and LPol-ds(ZPol)basis sets for OR calculations on real molecules.

Calculations for Real Chiral Molecules: Oxiranes,Pinene, and Pinane

The first case study is the calculation of OR for two stan-dard test molecules (S)-methyloxirane and (S)-fluoro-oxirane (seeFig. 2). The results of TDDFT/B3LYP calculations with LPolX andLPolX(ZPolX) basis sets are summarized in Table 4 and comparedwith the data obtained for Dunning’s basis sets. Also the total size(n) of each basis set is given.

The data of Table 4 are very encouraging in view of the possibilityof reliable calculations of OR for larger molecules. The LPol-dssets turn out to be essentially as good as the aVQZ sets of Dunning.However, the gain in the total basis set size is quite spectacular.Also smaller LPol-ds(ZPolX) sets are not much worse than the fullLPol-ds sets.

According to the data of Table 3 one can hope that LPolX andLPolX(ZPolX) sets will also be useful to theoretically predict theOR sign in more subtle cases. To show this we have attemptedto predict theoretically the sign of OR in two notoriously diffi-cult cases, (1S,5S)-β-pinene (3) and (1S,2S,5S)-trans-pinane (4)(Fig. 2). Despite several studies the TDDFT/B3LYP/LAO calcu-lations did not so far succeed in correct prediction of the signof OR for these systems except for λ = 355 nm for 3, where,however, a considerable error in the order of magnitude has beenreported.4 Results obtained in the present study for 3 are comparedwith other theoretical and gas phase results in Table 5. The sign ofTDDFT/B3LYP/LPol-ds(ZPolX) results and experimental OR val-ues agree for all wavelengths. Moreover, large improvement of theagreement of the order of magnitude between theoretical and exper-imental results is achieved, as compared with aug-cc-pVDZ values.Because of the shortcomings of the TDDFT/B3LYP approximation,the excellent agreement of the data of Table 5 may be of course tosome extent fortuitous; however, we believe that the reported valuesare far closer to the basis set limit than other previously reported

Figure 2. Molecules 1–4 for which TDDFT/B3LYP specific rotationcalculations have been carried out.

Table 4. Specific Rotation of (S)–methyloxirane (1)a and (S)-fluoro-oxirane(2)b Obtained at TDDFT/B3LYP Level of Theory.

λ aVDZ aVTZ aVQZ L-ds LZ-ds

(S)–methyloxirane (1)355.0 11.9 30.5 33.0 33.0 33.1589.3 −14.8 −7.4 −6.5 −7.1 −6.4nc 146 322 596 222 180

(S)-fluoro-oxirane (2)355.0 31.9d 58.1d 61.7d 62.6 55.4nc 119 253 458 183 162

Symbol aVXZ stands for aug-cc-pVXZ basis set. Symbols L-ds and LZ-dsdenote LPol-ds and LPol-ds(ZPolX) sets, respectively. Specific rotation val-ues reported in OR-units, wavelengths in nm.aOptimized geometrical parameters and corresponding vibrational frequen-cies were obtained at the DFT/B3LYP/aug-cc-pVDZ level using gaus-sian 03 package23 (see the Supporting Information for details).bDFT/B3LYP/aug-cc-pVTZ geometry reported by Pedersen et al.14

cTotal basis set size.dLiterature result.14

TDDFT/B3LYP values for this system. From the point of view ofcomputational effort, particular attention should be given to the val-ues of the total molecular basis sets size. Also in this sense the LPolXbasis sets appear to win the competition with aug-cc-pVXZ.

In Table 6 we report TDDFT/B3LYP/LPolX(ZPolX) and aug-cc-pVDZ OR results calculated for 3 and 4 and compare them withavailable experimental data obtained for neat liquids.4 Althoughin this case the agreement of calculated results and experimentalvalues can be more fortuitous to our knowledge these are the firstλ = 589.3 nm theoretical TDDFT/LAO OR results whose signagrees with experimental OR data for these two systems. It should bestressed that the inclusion of the vibrational effects (and in the caseof the comparison with the experimental results obtained from themeasurements carried out in the neat liquids, also the solvent effects)may change the results substantially, as it was reported previously byother authors. In particular, vibrational averaging of the OR in (+)-β-pinene was shown to strongly influence the total specific optical

Table 5. Specific Rotation of (1S,5S)-β-pinene (3)a Obtained atTDDFT/B3LYP Level of Theory.

λ aVDZb aVDZ, this work LZ-ds Exp.

633 20.0 18.7 −5.3 −4.66 ± 0.6c

589 26.1 24.6 −3.6 −2.8d

355 257.1 256.8 149.7 69.7 ± 3.2c

Symbols aVDZ and LZ-ds denote aug-cc-pVDZ and LPol-ds(ZPolX) basisset, respectively. Specific rotation values reported in OR-units and comparedwith experimental gas phase results. Wavelengths in nm.aOptimized geometrical parameters and corresponding vibrational frequen-cies were obtained at the DFT/B3LYP/aug-cc-pVDZ level using gaus-sian 03 package23 (see the Supporting Information for details).bRef. 4. Reference values correspond to the calculations performed for theopposite enantiomer. The literature theoretical values reported in the presenttable are thus given with the opposite sign.cRef. 24.dRef. 4. Interpolated gas-phase specific rotation.

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1180 Baranowska, Łaczkowski, and Sadlej • Vol. 31, No. 6 • Journal of Computational Chemistry

Table 6. Specific Rotation of (1S,5S)-β-pinene (3)a and(1S,2S,5S)-trans-pinane (4)a Obtained at TDDFT/B3LYP level of theoryfor λ = 589.3 nm.

aVDZb aVDZ, this work LZ-ds Exp.b

(1S,5S)-β-pinene (3)26.1 24.6 −3.7 −23.1

(1S,2S,5S)-trans-pinane (4)3.6 2.9 −69.0 −15.9

Symbol aVDZ and LZ-ds denote aug-cc-pVDZ and LPol-ds(ZPolX) basisset, respectively. Specific rotation values reported in OR-units and comparedto experimental results corresponding to neat liquids.aOptimized geometrical parameters and corresponding vibrational frequen-cies were obtained at the DFT/B3LYP/aug-cc-pVDZ level using gaus-sian 03 package23 (see the Supporting Information for details).bRef. 4. Reference theoretical values for β-pinene correspond to the calcula-tions performed for the opposite enantiomer. The literature theoretical valuesreported in the present table for 3 are thus given with the opposite sign.

rotation value changing it from −23.73 to −15.20 and −10.95 OR-units at the zero–point and at the 273 K.11

The basis set study performed for the model asymmetric methanemolecule shows that TDDFT/B3LYP OR results converge to stablevalue in the basis set of at least aug-cc-pVQZ quality, in agreementwith the latest reports by Pedersen et al.14 This leads to the conclu-sion that using basis set of aug-cc-pVDZ quality in OR calculationsmay lead to significant errors and in some cases to incorrect predic-tions of the OR sign. Because of large size of augmented correlationconsistent Dunning’s basis sets it is, however, very difficult to per-form routine OR calculations for a real system using aug-cc-pVTZset and practically impossible with aug-cc-pVQZ sets. Lately devel-oped LPol-ds basis set, combined with the ZPol set for hydrogenatoms, shorten the computational time and allow to avoid or stronglydiminish the near-linear dependence in the total set of functions.

Test calculations performed for (1S,5S)-β-pinene yielded resultsin a good agreement with experimental gas phase specific rota-tion and seem to be the best theoretical results available for thissystem. To our knowledge this has not been achieved in earlierTDDFT/B3LYP/LAO calculations. Comparison of λ = 589.3 nmtheoretical OR of (1S,5S)-β-pinene and (1S,2S,5S)-trans-pinanewith experimental results obtained from measurements in neat liq-uids also shows the agreement of theoretical and experimental sign,although in this case the agreement may be more fortuitous.

Summary

In this article, we have documented that the recently developedLPolX basis sets provide attractive computational tool for calcu-lations of optical properties of chiral systems. These basis sets havebeen generated initially for calculations of linear and nonlinear elec-tric properties upon the condition of the accurate representation ofthe first-order perturbed orbitals for the electric field perturbations.When combined with the LAO approach, the LPolX basis sets per-form well in calculations of the optical rotation. As shown by thestudy of the CM model, the LPolX sets are competitive to usuallymuch larger multiply augmented sets of Dunning with high values

of the cardinal number X. The conclusions reached in calculationsfor CM have been confirmed by the OR data obtained for the twooxiranes. Finally, the LPol-ds(ZPolX) basis sets gave the right signof OR in (1S,5S)-β-pinene and (1S,2S,5S)-trans-pinane.

Obviously, further studies, including wide range of real systems,including vibrational and solvent effects, are needed to definitelyjudge the performance of LPol-ds(ZPolX) set in optical rotationcalculations. It can be, however, concluded from our study that LPol-ds basis set combined with ZPolX set for hydrogen atoms gives muchbetter accuracy than the commonly employed aug-cc-pVDZ setsand can be used in OR calculations aiming at the comparison withexperimental gas phase results. This methodology will be followedin our future studies. Calculations for a wide series of “difficult”chiral molecules are in progress and will be reported elsewhere.

Concluding the present article it is worth to recall that theoxiranes as well as the bicyclic molecules being the subject ofour investigation represent a group of particularly demandingmolecules. For many of the other systems studied in the litera-ture, the TDDFT/B3LYP/aug-cc-pVDZ level of theory was shownto yield correct signs of specific rotation in many (rigid as wellas flexible) systems and the average errors are small enough forpractical applications.2

Acknowledgments

Calculations reported in this article have been carried out on thecomputer cluster at the Information & Communication TechnologyCentre of the Nicolaus Copernicus University, Torun, Poland.

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Journal of Computational Chemistry DOI 10.1002/jcc