Chemical Physics Letters 599 (2014) 169–174

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Chemical Physics Letters

journal homepage: www.elsevier .com/ locate /cplet t

Fourier type potential energy function for conformational changeof selected organic functional groups

http://dx.doi.org/10.1016/j.cplett.2014.03.0290009-2614/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Fax: +45 8619 6199.E-mail address: kemskj@chem.au.dk (S.J. Knak Jensen).

Anita Rágyanszki a, Attila Surányi a, Imre G. Csizmadia a,b, András Kelemen c, Svend J. Knak Jensen d,⇑,Selma Yarligan Uysal e, Béla Viskolcz a

a Department of Chemical Informatics, Faculty of Education, University of Szeged, Boldogasszony sgt. 6, H-6725 Szeged, Hungaryb Department of Chemistry, University of Toronto, M5S 3H6 Toronto, Ontario, Canadac Department of Applied Informatics, Faculty of Education, University of Szeged, Boldogasszony sgt. 6, H-6725 Szeged, Hungaryd Department of Chemistry, Langelandsgade 140, Aarhus University, DK-8000 Aarhus C, Denmarke Department of Chemistry, Faculty of Arts and Science, Eskis�ehir Osmangazi University, TR-26480 Eskis�ehir, Turkey

a r t i c l e i n f o

Article history:Received 30 January 2014In final form 12 March 2014Available online 20 March 2014

a b s t r a c t

The energy changes associated with internal rotation of a functional group in a molecule depend on thetopology of the chemical environment. Energies obtained from electron structure and force-field calcula-tions have been analyzed by Fourier expansions. The findings show that rotation around bonds connect-ing atoms without lone pairs can be described with a one term Fourier-series. In contrast, two or threeterms are needed if the connected atoms have lone pairs. The analysis inspires adoption of a simplifiedFourier expansion that reproduces the data well, suggesting that Fourier-type-series with few termsare useful in describing any internal rotation analytically.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Perspective

Protein folding is a century old problem. With the aid of confor-mational analysis the biological problem became a chemical prob-lem. However, due to the complexity of the molecular system forcefield methods have been developed. Initially such molecularmechanics methods were based on spectroscopic data and laterfurther development utilized optimized potential energy functions.Yet the reliability of the analytic potentials used today may still bean open question. On the one hand very extensive expansion maybe accurate but impractical to use and truncated expansion may bepractical but not sufficiently accurate. Thus some compromise isneeded. The present Letter aims to analyse what functions arepractical to use yet may give accurate enough results for suchstudies.

1.2. Historic background

For several decades, during the second half of the 20th century,it was an unexplained phenomenon that the barrier to internalrotation (torsion) about a CAC single bond in ethane (CH3ACH3)

could be successfully calculated by just about any method,whereas the lower ‘anti’ barrier along the OAO single bond inhydrogen peroxide (HOAOH) was very difficult to compute. Theapparent discrepancy was clarified in 1978 when Cremer [1]studied hydrogen peroxide with a large polarized basis set whichproduced remarkably accurate results. Moreover, Peterson andCsizmadia [2] demonstrated that all critical points of thePotential Energy Hypersurface (PEHS) of three independencevariables of n-butane (H3CACH2ACH2ACH3) could be reproducedat a modest level of theory. Further studies at that time [3–5]indicated that the presence of lone pairs made the descriptionmore difficult than the modest distortion of CAH bondingelectron pair.

The mathematical representation of the rotation potential usinganalytic functions has its own story. The forecast of the internalrotation and its energy changes is essential because it reflect theextent of intramolecular interactions [6]. The potential energycurves and surfaces can be represented by suitably chosen analyt-ical functions. Such functions can describe the atomic motion with-in a molecule, or within complexes formed by the interaction of themolecules [7]. If this function is in a suitable form then it can beused as an approximate analytic solution of the Schrödinger equa-tion. Such equations can also be used for estimating rates of inter-nal rotation [8,9].

Several functional groups have a unique internal rotational de-gree of freedom around a symmetric axis [10]. In such cases fixed

170 A. Rágyanszki et al. / Chemical Physics Letters 599 (2014) 169–174

bond angles may be assumed for the internal torsional movements,thus only a single bond is engaged in the internal rotation [11] andthe potential energy depends on the dihedral angle only [12,13].

In the 1970’s the Fourier-series were investigated by Radomand Pople to determine the internal rotation around a single bond[9,14]. In most cases Fourier-series, with relative few terms wereused for the fitting of the curves.

In the late 1980’s Chung developed a method for calculatingtorsional energies with n-term Fourier-series. The full term expan-sion Fourier-series is recommended for higher accuracy in contrastto minimum number of terms [13]. Nevertheless the practicalquestion still remains; how many terms are needed to obtain reli-able results for r-bonds.

1.3. Scope

We are considering the dihedral torsion or internal rotation oftypical organic functional groups, like a methyl group (H3C–).The simplistic view-point is that the rotation is always the sameno matter wherever it is located in any molecule. A more sophisti-cated point of view is that the characteristic of the methyl rotationis influenced somewhat by its environment and most certainly bythe group it is attached to. The first question we wish to investigateis how large such a nearest neighbour interaction is in compoundsH3C-X, where X may be chosen horizontally or vertically along theperiodic table:

The second question we will explore is if it is possible to quantifythe changes in energy for the rotation about homonuclear diatomicunits such as (X–X) as the number of lone pairs of X increases, as inthe series:

Table 1Fits of the Fourier series (1) to the rigid- and relaxed torsional potential energy curves (PEC)X and X-Y systems, (X,Y = O, N, S). The data are arranged according to the number of term

DE0 a1 b1 a2

C-X family (n = 1)CAC Rigid 6.68 6.63 �1.12 � 10�2 –

Relax 5.86 5.99 �1.07 � 10�1 –CAN Rigid 4.05 4.04 �6.31 � 10�3 –

Relax 4.97 4.83 6.42 � 10�1 –CAO Rigid 1.96 1.97 �1.04 � 10�4 –

Relax 2.97 2.99 2.50 � 10�1 –CAS Rigid 1.34 1.35 �3.79 � 10�3 –

Relax 2.72 2.75 �4.30 � 10�5 –

X-Y family (n = 2)OAO Rigid 14.0 10.1 �7.34 � 10�1 13

Relax 12.3 17.1 �1.62 8.1SAS Rigid 11.5 3.16 �2.28 � 10�1 11

Relax 15.2 5.65 �5.80 � 10�1 14SAO Rigid 17.0 �1.82 3.92 � 10�2 17

Relax 13.5 7.02 �6.789 � 10�1 12

X-Y family (n = 3)NAO Rigid 27.0 2.07 �1.03 �1

Relax 28.7 3.33 �1.37 �1NAS Rigid 24.8 2.17 � 10�1 �3.46 � 10�1 8.8

Relax 19.6 4.31 �5.44 � 10�1 �1NAN Rigid 13.9 18.2 �1.34 7.3

Relax 12.1 12.2 �2.52 � 10�1 11

As a third question we wish to examine the rotation around asingle bond (CAO or OAO) in compounds containing more thanone r-bonds resulting in variable chemical environments. In thesecase studies we would examine the effect

of the presence of a methyl group in an OAO torsion(III) and in aH3CAO torsion(I) and the introduction of an OH group in aH3CAO torsion(II).

In this Letter we explore rotations around bonds in small mol-ecules. However, the perspective is to extend the study to side-chain rotations in amino acid residues. As a modest step in thatdirection we have included the simplest chiral amino acid residue(alanine) among the considered species.

2. Methods

Let DE be the energy change associated with rotation of a func-tional group relative to the lowest energy. DE can be expanded in aFourier-series with the general form:

DE ¼ DE0 þXn

j¼1

aj cosj2pk/

360þ bj sin

j2pk/360

� �ð1Þ

where DE0 is a constant, n is the number of terms in the expansionand k is a constant indicating the periodicity of the rotating group(i.e., k = 3 for a –CH3 group).

, DE (kJ/mol), obtained by DFT [B3LYP/6-31G(d)] calculations for fully hydrogenated C-s, n, in the Fourier series.

b2 a3 b3 R2

– – – 1– – – 0.9936

– – 1– – – 0.9963– – – 1– – – 0.9996– – – 0.9999– – – 0.9968

.4 �1.95 – – 0.99972 �1.55 – – 0.9994.3 �1.64 – – 0.9997.2 �2.95 – – 0.9972.1 �7.42 � 10�1 – – 0.9997.7 �2.48 – – 0.9987

4.9 4.24 – – 0.99921.5 2.06 – – 0.99941 � 10�1 1.82 �22.6 �2.54 0.99957.0 �8.16 � 10�1 �1.97 �3.47 � 10�1 0.99949 �1.09 3.41 �7.66 � 10–1 0.9977.7 �4.85 � 10�1 4.42 �2.74 � 10�1 0.9904

Figure 1. Fits of (1) with n = 1 to the torsional potential energy curves (PEC), DE (kJ/mol), for fully hydrogenated C-X families, (X = C, N, O, S). Open (blue) and filled(red) symbols indicate data for rigid- and relaxed rotation, respectively. Dataobtained at the B3LYP/6-31G(d) level of theory. (For interpretation of the referencesto colour in this figure legend, the reader is referred to the web version of thisarticle.)

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DE was calculated in two ways, (i): by quantum mechanicsusing the B3LYP/6-31G(d) implementation of the density func-tional theory [15] in the GAUSSIAN09 software package [16] – amethod extensively used for small and medium sized moleculesand (ii) by force field methods, Amber/GAFF and Amber99SB[17–19] – tools routinely used for big molecules like proteins.The side chain methyl rotation in N-acetyl Alanine-methylamidewas also studied at the B3LYP/6-31G(d) level of theory and bythe force field method, Amber99SB. Using force-field methods forthe molecules of the sizes studied here may seem like overkillbut is it useful for estimation of the associated energy changeswhen proteins are studied. DE was calculated for a range of dihe-dral angles, /, in interval [0,2p].

The calculation of DE was done as a Potential Energy Curve (PEC) ofrigid rotation, DErigid, as well as a PEC of relaxed rotation, DErelax.The former is considered the less useful because in rigid rotationthe functional groups are rotated like mechanically rigid wheelswithout geometrical adjustment. In contrast, in relaxed rotationthe groups involved in the internal rotation would have the oppor-tunity to adjust their geometries during rotation. The PEC curve islower for the relaxed rotation than for the rigid rotation. However,the energy lowering needs not be the same for the transition statesas for the potential minima. DErigid and DErelax were both calculatedat the B3LYP/6-31G(d) level. DErelax was also calculated using forcefield methods to assess the quality of the analysis of data obtainedfrom the two levels of description.

3. Results

A Fourier series (1) was fitted to the DErigid and DErelax data forthe two families of compounds (H3C-X and X-Y where X, Y = O, N,S). The results are shown in Table 1. It appears that the fit is verysimple in the case of the H3C-X family as a single term (n = 1) willpresent the data very well. We note that the b1 coefficient is muchsmaller than the a1 coefficient, suggesting the PEC is close to beingan even function. It is also worth noting that a1 is very close to DE0.The variation of DE with / is shown in Figure 1. The quality of thefit is estimated by calculating the deviation between the fittedfunction and the raw data as a function of the dihedral angle. Wefind the maximum deviation to be smaller than 0.1 kJ/mol.

For the second family (X-Y) two or three terms are required.Again we note from Table 1 that the b-coefficients are much smal-ler than the a-coefficients. The DE plots for the X-Y family areshown in Figure 2.

Influences of substituents on DE were studied using the com-pounds I, II and III. The variations of DE in these cases are shownin Figure 3. This figure shows, that H3CAO rotations (left hand sideof Figure 3) can be represented fairly accurately by single term(n = 1) while the rotation about a OAO bond requires a two term(n = 2) expansion in the Fourier-series.

It appears from Figure 2 that hydrazine has a very shallow min-imum at the anti conformation [20], which increases the demandfor more terms in the Fourier expansion. Figure 4 illustrates howthe quality of the fit increases for hydrazine as the number of termsincreases. The case of n = 3 does a perfect job indeed.

The observation that the b-coefficients in (1) are much smallerthan the a-coefficients in most cases along with the closeness of E0

and a1 for n = 1 inspired us to adopt a simple even fitting functionfor DE as

DE ¼ DEmax

nþ 1þXn

j¼1

DEmax

2jcos

j2pku360

� �ð2Þ

where DEmax is determined from the fit. For example, the H3C-Xfamily can be represented by an expansion with one term as

DE ¼ DEmax

2þ DEmax

2cosð0:052uÞ ð3Þ

For the X-Y family -excluding the NAN and NAS members- thefunction is:

DE ¼ DEmax

3þ DEmax

2cosð0:0175uÞ þ DEmax

22 cosð0:0175 � 2uÞ ð4Þ

In the case of NAN and NAS we obtain the function:

Figure 2. Fits of (1) with n = 2, 3 to the torsional potential energy curves (PEC), DE (kJ/mol), for fully hydrogenated X-Y families, (X, Y = N, O, S). Open (blue) and filled (red)symbols indicate data for rigid- and relaxed rotation, respectively. Data obtained at the B3LYP/6-31G(d) level of theory. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

172 A. Rágyanszki et al. / Chemical Physics Letters 599 (2014) 169–174

DE ¼ DEmax

4þ DEmax

2cosð0:0175uÞ þ DEmax

22 cosð0:0175 � 2uÞ

þ DEmax

23 cosð0:0175 � 3uÞ ð5Þ

In Table 2 we compare the results obtained from fitting (2) to theDE data derived from the relaxed dft calculations to the force fieldresults.

4. Discussion

It appears from Table 1 that not all r-bonded functional groupsbehave in an identical way during internal rotation (torsion). Someof them like H3C-X exhibit rather high symmetry with three iden-tical transition states [13]. These can be represented with a single(n = 1) cosine function. Rotation about other r-bonds, where thethree TS are not identical or where only a pair of non-identicalTS is in the potential, require a longer Fourier expansion (n = 2 or3). All of the potentials however were fitted very accurately withR2 values ranging from 0.9904 to 1 (Table 1).

The TS obtained by molecular mechanics (MM) simulationsagreed reasonably well (Table 2) for the C-X family. In the caseof the X-Y family noticeable deviations were observed in the casesof the SAO, SAS, NAO and NAN linkages, 33.9 kJ/mol, -18.6 kJ/mol,13.9 kJ/mol and 26.3 kJ/mol, respectively. This is understandablebecause traditional force field software were written to simulateprotein structure and protein folding. Thus discrepancies may be

expected in cases of bonds that are not common in proteins. SinceMM simulations, in general, aim to reproduce geometries and rel-ative stabilities of minima, the achieved TS energy approximationare really very good.

5. Conclusion

Fourier series, comprised by sin and cos terms are capable to fitany function. However, the potential energy functions consideredhere are close to being even functions which makes a pure cosexpansion satisfactory. Depending on the structural complexityof a given molecule, a 1-term or 2-term or 3-term expansionturned out to be adequate, using only cos expansion. For CAC,CAN, CAO and CAS bonds the 1-term expansion was accurate en-ough. For OAO, SAS and SAO bonds the 2-term expansion was sat-isfactory. In contrast, 3-term expansion was necessary to fit thepotential energy function associated with the internal rotationabout the NAO, NAS and NAN bonds. The CAC rotation potential,fitted by Molecular Mechanics (MM), agreed very well with thepresent Fourier series fit for the methyl rotation in ethane as wellas in alanine-diamide.

The ultimate purpose is to build a very accurate multi variablemathematical function for the Potential Energy Hypersurface offlexible molecules, such as peptides, of several internal bond-rota-tions. The present study made for a single dihedral rotation sug-gests that such an ultimate goal is achievable.

Figure 3. Fits of (1) to the potential energy curves for case studies of the environmental effects. Open (blue) and filled (red) symbols indicate data for rigid- and relaxedrotation, respectively. Data obtained at the B3LYP/6-31G(d) level of theory. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

Figure 4. Fits of (1) to torsional potential energy curves (PEC), DE (kJ/mol), for hydrazine with n = 1, 2, and 3 terms. Open and filled symbols indicate data for rigid- andrelaxed rotation, respectively. Data obtained at the B3LYP/6-31G(d) level of theory.

A. Rágyanszki et al. / Chemical Physics Letters 599 (2014) 169–174 173

Table 2Comparison of the maxima (or transition states) obtained by fitting (2) to the relaxedDE data (kJ/mol) from DFT [B3LYP/6-31G(d)] calculations to the force field data forfully hydrogenated C-X and X-Y families, (X,Y = O, N, S). Data for N-acetyl Alanine-methylamide are also presented.

A: Higher TS B: Lower TS

Method DFT Force field DFT Force field

C-X family (n = 1)CAC 11.78 12.05 – –CAN 10.11 9.35 – –CAO 6.01 4.82 – –CAS 5.51 5.17 – –H3C-Ala 15.62 14.91 – –

X-Y family (n = 2)SAO 47.86 13.96 18.77 11.42NAO 43.29 41.86 42.93 37.67SAS 34.42 53.02 23.08 47.86OAO 37.12 35.7 2.92 4.41

X-Y family (n = 3)NAS 36.66 22.74 36.66 21.81NAN 40.56 14.28 7.64 5.30

174 A. Rágyanszki et al. / Chemical Physics Letters 599 (2014) 169–174

Acknowledgement

The authors would like to thank Balázs Jójárt for helpful contri-butions and Miklós Krész for helpful discussions.

The authors acknowledge the financial support within TÁMOP-4.2.2.A-11/1/KONV-2012-0047 ‘New functional material and theirbiological and environmental answers’, TÁMOP-4.2.2.C-11/1/

KONV-2012-0010 ‘Supercomputer — the national virtual labora-tory’, HUSRB/1002/214/193 ‘Bile Acid Nanosystems as MoleculeCarriers in Pharmaceutical Applications’ and TÁMOP 4.2.4. A/2-11-1-2012-0001, ‘National Excellence Program Elaborating andoperating an inland student and researcher personal support sys-tem’, subsidized by the European Union and co-financed by theEuropean Social Fund.

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