Approximate quantum chemical methods for modelling

carbohydrate conformation and aromatic interactions: β-

cyclodextrin and its adsorption on a single-layer graphene

sheet

Panichakorn Jaiyong and Richard A. Bryce*

Division of Pharmacy and Optometry, University of Manchester, Oxford Road, Manchester

M13 9PL, UK

*E-mail: r.a.bryce@manchester.ac.uk

1

Abstract

Noncovalent functionalization of graphene by carbohydrates such as -cyclodextrin (CD)

has the potential to improve graphene dispersibility and its use in biomedical applications.

Here we explore the ability of approximate quantum chemical methods to accurately model

CD conformation and its interaction with graphene. We find DFTB3, SCC-DFTB and

PM3CARB-1 methods to provide the best agreement with density functional theory (DFT) in

calculation of relative energetics of gas-phase CD conformers; however, the remaining

NDDO-based approaches we considered underestimate the stability of the trans,gauche

vicinal diol conformation. This diol orientation, corresponding to a clockwise hydrogen

bonding arrangement in the glucosyl residue of CD, is present in the lowest energy CD

conformer. Consequently, for adsorption on graphene of clockwise or counterclockwise

hydrogen bonded forms of CD, calculated with respect to this unbound conformer, the

method provides closer agreement with DFT values than PM7 and PM6-DH2 approaches.

These findings suggest approximate quantum chemical methods as potentially useful tools to

guide the design of carbohydrate-graphene interactions, but also highlights the specific

challenge to NDDO-based methods in capturing the relative energetics of carbohydrate

hydrogen bond networks.

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1. Introduction

Carbohydrates are fundamental components in biological systems with a range of key roles.

As mediators of recognition, they determine ABO blood type and are implicated in immunity

and inflammation.1 In metabolism, they are substrates for energy production. Carbohydrates

are also important structural compounds, conferring stability to cell walls across animals and

plants. Reflecting this diversity of role, there has been increasing efforts to employ

carbohydrates in a wide range of applications, from therapeutics, formulations and food

additives through to novel materials and biofuels.2

Correspondingly, these varied functions of carbohydrates arise from their underpinning

complexity in structure and interactions. In addition to a large degree of isomerism in linkage,

flexibility in conformation and subtle stereoelectronic effects, carbohydrates are also

frequently found to interact with aromatic residues in protein active sites;2b for example, in T.

cruzi trans-sialidase,3 a substrate lactosyl moiety stacks between tryptophan and tyrosine

residues; and a triad of tyrosines in the carbohydrate binding module of Trichoderma reesei

cellobiohydrolase I interacts with the glucosyl residues of cellulose.4 There has been recent

interest in developing polysacccharide-graphene nanomaterials.5 For graphene-

polysaccharide nanocomposites, graphene has the ability to increase the tensile strength of

some polysaccharide adsorbates from 20% up to 134%.6 Computational simulations of the

hydrated cellulose-graphene interface predicted significant affinity of glucosyl OH and CH

groups for the graphene system.7 The common occurrence of carbohydrate- interactions is

reflected by favourable computed water-benzene and methane-benzene interaction energetics,

of -3.3 and -1.5 kcal/mol respectively at the CCSD(T)/complete basis set level.8

3

Computational methods provide valuable insights into the atomistic detail of carbohydrate

structure and interactions which in many cases are currently inaccessible by experimental

means. However, due to the size and flexibility of carbohydrates, the application of high

level, computationally expensive quantum mechanical (QM) techniques, such as the

CCSD(T) method, are not routinely possible. More computationally efficient approaches

include density functional theory (DFT) methods; recent developments in this area have led

to DFT approaches which account for dispersion interactions, eg. the Minnesota suite of

functionals such as M06-2X,9 or DFT-D methods which use an empirical R-6 correction for

dispersion, such as B97X-D.10

However in order to study the large number of atoms frequently encountered in carbohydrate-

based systems, more approximate quantum chemical approaches are required. With a

comparable performance to DFT-D and three orders of magnitude faster,11 semiempirical QM

methods offer a route to modelling these larger systems. Recent semiempirical methods11a

often include specific dispersion and hydrogen bond correction terms, such as the PM6-DH

methods,12 PM613 and most recently PM7.14 Interestingly, the interaction of a range of

unsaturated molecules with single layer graphene15 was studied by the PM6-DH2 method,

providing good agreement in adsorption energies to DFT values and experiment. Other

approaches include the PM3-D* method,16 specifically parameterised for carbohydrate-

aromatic interactions and PM3CARB-1 method,17 fitted to reproduce aspects of carbohydrate

conformation. Alternative to NDDO-based approaches and comparable in computational

efficiency is the density functional tight-binding (DFTB) method.18 The most recent

implementation, DFTB3, is derived from a third-order expansion of the DFT total energy

with respect to charge density fluctuations around a reference charge density.19 DFTB3 has

4

been shown to perform well for hydrogen-bonded complexes, improving the description of

geometries that depend on nonbonded interactions.20,21

In this work, we evaluate the ability of a range of approximate quantum chemical methods to

model the conformation and aromatic interactions of β-cyclodextrin (βCD), a cyclic

oligosaccharide of seven -D-glucose units. The molecule forms a cylindrical shape, with a

nonpolar interior and polar exterior (Figure 1). βCD derivatives have been used as excipients

in formulations of therapeutics.22 Recently, βCD has been experimentally studied as a

noncovalent adsorbate on graphene surfaces23 and was shown to reduce graphene

hydrophobicity, confirmed experimentally by contact angle measurements.24 Improving

dispersibility in polar solvents is important for realising graphene’s potential in drug and gene

delivery applications. Unlike covalent attachment of organic functional groups, noncovalent

functionalisation does not disrupt the -system of graphene, preserving its intrinsic properties

while improving aqueous solubility.

Here, we assess the ability of NDDO-based and DFTB-based quantum chemical methods to

evaluate βCD structure, specifically its hydrogen bonding conformers in open and closed

forms. From this analysis, we identify an underestimation of the stability of clockwise

hydrogen bonded conformers of CD via NDDO-based approaches. We then consider the

effect of this issue on modelling of adsorption of βCD on single layer graphene sheets,

comparing PM7, PM6-DH2 and DFTB3 methods with DFT calculations.

2. Computational methods

Conformations of βCD. β-cyclodextrin has a cylindrical shape with two hydrophilic rims of

hydroxyl groups located outside its molecular cavity, denoted the C2C3 and C6 rims

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according to the atom numbering of the exposed glucosyl hydroxyl groups (Figure 1a);

elsewhere these rims have sometimes been labelled the head and tail respectively.23 βCD can

form a closed or open structure: when βCD adopts a closed structure, its shape resembles a

truncated cone, as the primary 6-hydroxyl groups form strong interactions with each other

across glucose units, effectively closing access via the narrower C6 rim (Figure 1a-d).25 In

contrast, when βCD is in an open structure, it presents a hydrophobic cavity in a more

uniform cylindrical shape, suitable for inclusion of small or medium-sized guest molecules

(Figure 1e,f).

The in vacuo energetics of open and closed βCD conformers have been studied previously at

the B3LYP density functional level by Snor et al.25 and Stachowicz et al.26 In open and closed

forms, they evaluated clockwise (cw) and counter-clockwise (cc) orientations of

intramolecular hydrogen bonds lining the two rims; cw and cc are defined when viewing βCD

from above the smaller C6 rim. (We note alternative notation has also been used for

cyclodextrin cc and cw hydrogen bond networks, namely – and + respectively).27 The B3LYP

studies of βCD25-26 found that a closed conformation was lowest in energy and that a cc

hydrogen bond orientation along the C6 rim and cw along the C2C3 rim was preferred

(Figure 1a); this cccw conformer was lower in energy than three other low energy closed

geometries and two open structures (Figure 1b-f). Although defined according to a view

facing the C6 rim, these networks can more clearly be observed looking into the C2C3 rim

(Figure S1).

QM calculations. Here, we examine these four closed and two open low-energy

conformations of βCD, i.e. cccw, cccc, cwcw, cwcc, O-cwcw and O-cccc, at DFT and

approximate QM levels (Figure 1). As well as extending earlier B3LYP calculations, we

6

evaluate the six conformers using the B97X-D10 and M06-2X28 density functionals in

conjunction with the def2-TZVPP and def2-SVP basis sets. The M06-2X functional exhibits

good agreement with CCSD(T) calculations for monosaccharide conformational energetics29

and aromatic interactions:30 for example, M06-2X/def2-TZVPP calculations of the

conformational energies of 58 α-pyranose aldohexose isomers yielded a root-mean-square

(RMS) error in relative energy of 0.3 kcal/mol against CCSD(T)/complete basis set

calculations.29 Similarly, a good performance is found by the B97X-D hybrid density

functional for noncovalent interactions,10 including for aromatic hydrocarbon/graphene

complexes.31

The semiempirical NDDO-based Hamiltonians employed were PM6,13b PM7,14 PM6-DH2,12a

PM6-D3,12d PM3CARB-1,17 PM3-D*,16a PM332 and AM1.33 For the DFTB approaches, we

use the DFTB3 and SCC-DFTB methods,18,19,43 with parameter sets 3ob-2-1 and mio-1-1

respectively. For DFTB3, the third-order self-consistent charge (SCC) model was used with a

tolerance of 1108 e and the Slater-Kirkwood type dispersion model, with the Hubbard

derivatives in atomic units of 0.1575 for oxygen atoms, 0.1857 for hydrogen atoms and

0.1492 for carbon atoms, with their corresponding atomic covalent radii.34,35

Interaction energies of βCD/graphene complexes were computed using single layer graphene

flakes of two sizes, C96H24 and C1006H88. The C96H24 complexes were optimised at the M06-

2X/def2-SVP level of theory and their interaction energies (IE) with βCD computed at the

M06-2X/def2-TZVPP level with an ultrafine grid for numerical integration of two-electron

integrals. IEs were also computed for C96H24 and C1006H88 complexes using PM6-DH2, PM7

and DFTB3 methods. For the latter, the Broyden charge mixing method was used with a

third-order SCC tolerance of 1.0107 e.

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Ab initio QM and DFT calculations used the Gaussian 09 quantum chemistry package.36

PM3-D* calculations used the Gaussian 03 software.37 PM3CARB-1 and SCC-DFTB

calculations used the semiempirical QM module of AMBERTools13 in AMBER12.38,43 Other

NDDO-based semiempirical QM methods used the MOZYME module of MOPAC2012

version 14.083L.39 Calculations using DFTB3 employed the program DFTB+ version 1.2.2.40

3. Results

3.1 Gas-phase βCD conformation

Density functional theory methods. We first examine the stability of six hydrogen bond

conformers of βCD in the gas phase, two open and four closed (Figure 1). This follows the

work of Snor et al.25 and Stachowicz et al.26 at the B3LYP/6-31G* and B3LYP/6-31G**

levels of theory respectively. These studies find the most stable βCD structures to be cccw

and cccc conformers (Figure 1a,b), differing by only 0.1 kcal/mol in stability at both QM

levels (Table 1).25-26 Likewise, cwcc and cwcw conformers are similar in energy, and 2 - 3

kcal/mol less stable than cccw (Table 1). Finally, the open structures are 18 - 20 kcal/mol

higher in energy than cccw (Table 1); for O-cwcw, this reflects the weaker OH-ether

hydrogen bonds in its C6 rim compared to OH-OH bonds in closed βCD; and for O-cccc, the

absence of hydrogen bonds at the C6 rim.25-26

At the M06-2X/def2-TZVPP level of theory, we find that the six minimum energy structures

of βCD in vacuo agree well with the X-ray crystallographic structure (Table S1); the

crystallographic disorder in hydroxymethyl torsion angles and (Table S1) corresponds to

the presence of a range of hydroxymethyl rotamers. The most stable βCD structure at the

M06-2X/def2-TZVPP level was found to be cccw, in agreement with the earlier B3LYP

8

studies (Table 1).25-26 This cccw conformer has the shortest hydrogen bond distances of the

four closed structures, at both C2C3 and C6 rims, with corresponding O3-H…O2 and O6…

H-O6 distances averaged over the seven glucosyl rings of 2.11 and 1.83 Å respectively

(Tables S2 and S3). We note this O6…H-O6 hydrogen bond distance is short, close to that

observed from high level QM calculations for small water clusters.41,42

Interestingly, distinct from B3LYP predictions, both M06-2X/def2-TZVPP and

B97X-D/def2-TZVPP levels of theory find the cccw conformer to be 1 kcal/mol more

stable than both cccc and cwcw structures in the gas phase (Table 1). This indicates that

flipping the hydrogen bond network of cccw at the C2C3 rim as well as the C6 rim is

energetically unfavourable. The remaining closed conformer, cwcc, lies 2.6 kcal/mol above

cccw, in line with the B3LYP/6-31G* prediction (Table 1). The two open conformers, O-

cwcw and O-cccc, are 17.1 and 17.4 kcal/mol respectively less stable than cccw in the gas

phase at the M06-2X/def2-TZVPP level, similar to the corresponding 18.0 and 20.2 kcal/mol

values predicted via B3LYP/6-31G* (Table 1).

We note that this M06-2X/def2-TZVPP//M06-2X/def2-TZVPP energetic profile for the six

βCD conformers is reproduced to a mean unsigned error in energy of 0.3 kcal/mol using

geometries optimized at the M06-2X/def2-SVP geometry (Table 1). However, also

computing energies at the M06-2X/def2-SVP level leads to an increase in MUE to 2.9

kcal/mol (Table 1); this indicates the importance of the larger more polarized def2-TZVPP

basis set in accurately capturing the relative stability of the various cooperative hydrogen

bond networks. Using the def2-TZVPP basis set with the B97X-D functional for βCD

geometries and energies gives an overall MUE in energy of 1.4 kcal/mol, although the

maximum deviation from M06-2X/def2-TZVPP over closed conformations is only 0.3

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kcal/mol (Table 1). These results together provide a consensus that, in vacuo, cccw is the

most favoured of the six β-cyclodextrin structures, energetically distinct from the other closed

conformers, and that the open forms are considerably less stable than closed geometries.

Approximate QM methods. We now compare the ability of more approximate QM levels of

theory to reproduce the βCD energetic profile from DFT. Specifically, the NDDO-based

methods employed are PM6,13b PM7,14 PM6-DH2,12a PM6-D3,12d PM3CARB-1,17 PM3-D*,16a

PM332 and AM133; the two DFTB-based approaches are SCC-DFTB and DFTB3.18,19,43

All NDDO-based QM methods find the four closed conformations from the

M06-2X/def2TZVPP surface to be stable, with a maximum all-atom RMS deviation from the

DFT geometries of 0.28 Å. However, in contrast to the preceding DFT calculations,

PM3CARB-1 is the only NDDO-based method to find the cccw conformer to be lowest in

energy (Table 1). This reflects an AM1 analysis of CD by Köhler et al.,44 which predicted a

cccc conformer to be most stable in vacuo. Indeed, here we find that all but PM3CARB-1 and

AM1 of the NDDO-based methods predict the second most favoured structure as cwcc, and

cccw as only third most stable (Table 1). For open forms of CD, O-cccc is not predicted to

be a stable minimum except via PM3 and PM3-D* (Table 1). The AM1 method predicts an

intermediate O-cccc structure, with a partial network of intramolecular hydrogen bonds at the

C6 rim (Table S3). PM3CARB-1 only approximately captures the relative stabilities of the

closed and O-cwcw structures (Table 1).

The DFTB3 and SCC-DFTB methods predict all six βCD conformers as stable minima.

Notably both DFTB methods predict cccw as lowest in energy, although the detailed

energetic profile of closed structures is not quantitatively reproduced relative to

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M06-2X/def2-TZVPP and B97X-D/def2-TZVPP levels (Table 1). Nevertheless, DFTB3,

SCC-DFTB and PM3CARB-1 are the only methods to produce MUEs in energy below 7

kcal/mol over the six βCD conformers (Table 1).

Hydrogen bonding in NDDO-based approaches. Consequently, with the exception of

PMCARB-1, it appears that the NDDO-based methods favour cccc and cwcc βCD

conformers, corresponding to a counterclockwise […H-O2…H-O3…H-O2…] hydrogen

bond network at the C2C3 rim; by contrast, higher level QM calculations and DFTB-based

methods prefer cccw and cwcw conformers, with a clockwise […O2-H…O3-H…O2-H…]

orientation. This discrepancy is somewhat unexpected, given that PM7 and the PM6-DH

class of methods have been specifically parametrized to reproduce hydrogen bonding

energetics and geometries. Indeed hydrogen bond distances obtained via the various

approximate QM methods appear in reasonable agreement with DFT (Tables S2 and S3).

To understand the origin of this discrepancy in hydrogen bond energetics, we focus

specifically on the contribution of the intraresidue interaction between hydroxyl groups.

Therefore, we consider the 1,2-ethanediol model system (Figure 2a), a simple molecule

representing the vicinal O2…O3 interaction of the glucosyl residue. The ten possible

conformers of 1,2-ethanediol are defined according to its three intramolecular dihedral

angles, HaOaCaCb, OaCaCbOb and HbObCbCa, labelled 1, and 2 respectively; these each can

adopt t, g or g values. In work by Guvench and MacKerell45 using an extrapolated

CCSD(T)/cc-pVQZ level of theory, denoted HM-IE, the lowest energy conformation of 1,2-

ethanediol was found to be tGg, with gGg and gGg conformers lying 0.35 and 0.92

kcal/mol higher in energy respectively.

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Of these low energy conformers, tGg is representative of the cw intraresidue hydrogen bond

in βCD (Figure 2a), and gGg is the cc equivalent (Figure 2b). For clarity, from this point we

label these structures cw/tGg and cc/gGg. Both conformers share a gauche angle, which

allows the vicinal hydroxyl groups to hydrogen bond more favourably than in other 1,2-

ethanediol conformers. For tTt, no intramolecular hydrogen bond is possible. Previous

theoretical work has shown that the repulsive hydrogen-to-hydrogen distance between the

OH groups correlates with the stability of the three conformers; for example, at the HF/cc-

pVTZ geometry, the HaHb distance in gGg is 0.3 Å closer than in the favoured tGg

structure.45

Therefore, as models of cw and cc interactions within glucosyl residues of βCD, we compute

the energetic difference between cc/gGg and cw/tGg diol conformers, Ecc–cw. At the

M06-2X/def2-TZVPP and B97X-D/def2-TZVPP levels, cw/tGg is predicted as lower in

energy than cc/gGg, with Ecc–cw values of 0.3 and 0.2 kcal/mol respectively (Figure 3, Table 2).

These values agree well with the HM-IE estimate Ecc–cw of 0.4 kcal/mol45 and more closely than

MP2/def2-TZVPP and MP2/aug-cc-pVTZ calculations performed here, which both predict Ecc–

cw as 0.7 kcal/mol (Table S4). This energetic preference is preserved by the PM3CARB-1,

SCC-DFTB and DFTB3 methods, with Ecc–cw energy values of 1.1, 0.4 and 0.6 kcal/mol

respectively (Figure 3). The correct trend in Ecc–cw for PM3CARB-1 is unsurprising as the

Hamiltonian was parametrised using 1,2-ethanediol in its training set.17

However, all other NDDO methods uniformly predict the opposite preference, ie. that the

cc/gGg conformer is lower in energy than cw/tGg; Ecc–cw values range from 0.9 kcal/mol for

AM1 to 1.4 kcal/mol for both PM3 and PM3-D* (Figure 3). For example, the PM6-DH2

method has a Ecc–cw value of 1.3 kcal/mol, deviating from the HM-IE value by 1.7 kcal/mol.

12

Considering the cw intramolecular interaction is understabilised relative to cc in each of the

seven glucosyl residues of βCD, the resulting error of 11.9 kcal/mol appears commensurate

with the erroneous 14.2 kcal/mol overstabilisation of cccc relative to cccw via PM6-DH2

(Table 1). We can compute the energy difference per glucosyl residue between cc and cw

βCD conformers, cccc and cccw. This energy difference appears of a similar magnitude to the

Ecc–cw values obtained for the ethanol conformers (Figure 3). This observed discrepancy

suggests that a major issue in computing βCD stability using NDDO-based models could lie

in modelling of the vicinal O2…O3 interaction.

To further investigate the origin of this discrepancy, we compare cc/gGg and cw/tGg

geometries of 1,2-ethanediol obtained via PM6-DH2 and M06-2X/def2-TZVPP (Table 2).

For the cc/gGg conformer, there is good correspondence in overall conformation between

these two levels of theory: for example, the key internal angle of the OCCO torsion differs

by only 4° (Figure 2d; Table 2). The difference for 2 is 7° and zero for 1. Intramolecular

Oa…Hb and Ob

…Ha distances also agree well, with values of 3.24 and 2.31 Å respectively at

the M06-2X/def2-TZVPP level, and 3.19 and 2.22 Å via PM6-DH2 (Table 2).

By contrast, the PM6-DH2 geometry of the cw/tGg diol conformer is distinctly twisted

around the central CC bond, away from the M06-2X/def2-TZVPP conformation; the value

is 88° at PM6-DH2 and 60° at M06-2X/def2-TZVPP levels (Figure 2c, Table 2). Deviations

in 1 and 2 of 5° and 12° respectively in cw/tGg are also observed; these are 5° larger

differences than for cc/gGg in both 1 and 2 angles. The intramolecular Ob…Ha hydrogen

bond in cw/tGg appears strained, with a predicted Ob…Ha distance of 2.81 Å at PM6-DH2.

This compares to a value of 2.31 Å via M06-2X/def2-TZVPP; and 2.22 Å in cc/gGg via

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PM6-DH2 (Figure 2, Table 2). The repulsive Hb…Ha distance is 0.32 Å longer for cw/tGg

relative to cc/gGg at the M06-2X/def2-TZVPP level; however, it is 0.86 Å longer via PM6-

DH2 (Figure 2c,d). Similar geometric discrepancies are found for the cw/tGg conformer via

other NDDO methods (Tables 2 and S5), except for PM3CARB-1; the latter method predicts

a value of 66°, closer to the M06-2X/def2-TZVPP estimate of 60° (Table 2). Similarly, less

deviation from DFT is found for DFTB3 and SCC-DFTB geometries of cw/tGg , with

differences in of 10° and 3° respectively (Table 2).

The underestimated stability of the distorted cw/tGg conformer via these semiempirical QM

methods, in which the trans,gauche vicinal OH interaction occurs, points to an

underpredicted strength in the clockwise O3-H…O2 hydrogen bond within glucosyl residues

of βCD. We note that the twisted angle of the diol is unable to manifest in the glucosyl

residues of βCD due the covalent constraints of the cyclic system; this suggests an even less

stable vicinal diol trans,gauche interaction is predicted in βCD via NDDO methods.

3.2 βCD/graphene complex

βCD/C96H24 model. We now turn to consider the ability of approximate QM methods to

accurately compute βCD interactions with single-layer graphene in the gas phase. We first

compare their performance in modelling βCD-C96H24 adsorption energies with DFT

calculations, at the M06-2X/def2-TZVPP//M06-2X/def2-SVP level of theory. As suggested

by transmission electron microscopy of βCD/graphene composite,23 we orient the βCD cavity

channel perpendicular to the graphene surface. For the βCD/C96H24 system, we place the

larger C2C3 rim of βCD at the graphene interface, and compare the interaction of the two

lowest energy hydrogen bond conformers of βCD, ie. cccw and cccc.

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The cccw conformer of βCD binds with an IE of -29.7 kcal/mol at the M06-2X/def2-TZVPP

level; this “instantaneous” IE is calculated with respect to the monomer geometries found in

the bound βCD/C96H24 complex. We report IEs subsequently as IE per glucosyl residue, ie. in

this case, a value of -4.2 kcal/mol (Table 3). The cccc conformer binds essentially

isoenergetically to C96H24, with an IE per glucosyl of 4.1 kcal/mol (Table 3). With the

inclusion of basis set superposition error, the corresponding IEs lower to 3.7 and 3.6

kcal/mol respectively; this modest level of BSSE reflects the sizable basis set employed here

and is commensurate with the BSSE found in other work on modelling graphene adsorption.46

The parity in interaction of cccw and cccc with the graphene flake is not unexpected, given

that, while the direction of the hydrogen bond network differs, the total number and

approximate orientation of CH and OH groups with respect to the C96H24 flake are the same

for cccw and cccc conformers. When an overall IE is computed, ie. with respect to binding

of the gas phase global minimum cccw conformer, the IE is reduced in magnitude for cccw

and cccc, with values of -3.6 and -3.2 kcal/mol respectively (Table 3); thus the difference in

IE between conformers, IEcccc-cccw, favours binding of cccw by 0.4 kcal/mol at the

M06-2X/def2-TZVPP level (Table 3).

For comparison, we apply the DFTB3, PM6-DH2 and PM7 methods to computation of

βCD/C96H24 interaction energies, each of which have been applied previously to modelling

graphene-based systems.15a,46-47 We find that the DFTB3 method predicts IEs for the βCD-

C96H24 interaction which are ~2 kcal/mol per glucosyl higher than the M06-2X values (Table

3). However the trends of the DFT calculations are reproduced by DFTB3, with

instantaneous IEs which are isoenergetic for cccw and cccc forms (Table 3); and a preference

in overall IE favouring cccw, with a IEcccc-cccw value of 0.9 kcal/mol (Table 3). Both PM6-

DH2 and PM7 methods also predict isoenergetic instantaneous IEs of βCD with C96H24, with

15

values of -7 kcal/mol and -10 kcal/mol respectively (Table 3). This high PM7 estimate

correlates with the shortest average βCD rim oxygen-graphene distances of the three

approximate QM methods, with values of 2.9 and 2.8 Å for cccw and cccc conformers,

compared to 3.0 – 3.1 Å for DFTB3 and PM6-DH2 (Table S6). When the overall IE is

computed, for binding of the global minimum cccw βCD conformer, the preference is

reversed to favour binding of the cccc conformer, by 1.8 and 1.3 kcal/mol for PM6-DH2 and

PM7 respectively (Table 3). The discrepancy from the M06-2X/def2-TZVPP value is

therefore 2.2 and 1.7 kcal/mol respectively, a consequence of the understabilisation of the

unbound cccw conformer by these two Hamiltonians.

βCD/C1006H88 model. The DFTB3, PM6-DH2 and PM7 methods were then applied to

examine the gas phase interaction of all six βCD conformers with the substantially larger

C1006H88 graphene flake, binding via either its C2C3 or C6 rim (Figure 4). For interaction of

cccw and cccc conformers at the C2C3 rim, as for the smaller graphene flake, we find that the

DFTB3 estimate of interaction energy predicts that cccw is favoured, with an IEcccc-cccw of 0.6

kcal/mol (Table 4); PM6-DH2 and PM7 methods favour binding of cccc, by -2.3 and -1.2

kcal/mol per glucosyl. The preference of PM6-DH2 and PM7 methods for cc conformers is

clearly exhibited by overall βCD-C1006H88 interaction energies for cccc and cwcc complexes at

the C2C3 rim which are 1-2 kcal/mol higher than for cccw and cwcw (Figure 5a; Table S7).

The extent of understabilisation of the cccw conformation leads to a slightly positive IE for

O-cwcw in the gas phase.

It is possible to introduce a correction for this relative understabilisation of the cccw structure

for each of the three approximate QM methods, to reproduce the M06-2X/def2-TZVPP

energy difference for a given βCD conformer relative to cccw. For example, to correct the

16

overall IE of cccc via PM6-DH2, a total correction for βCD of 14.0 kcal/mol is applied.

When this correction is applied, the resulting IE per glucosyl at the C2C3 rim is rather similar

across the four closed conformers, varying only by 0.4 and 0.3 kcal/mol for PM6-DH2 and

PM7 respectively (Figure 5b; Table S8); for DFTB3, this range is 0.3 kcal/mol, with the

exception of the cwcc conformer, which is 2.7 kcal/mol less favoured than cccw (Figure 5b).

All three Hamiltonians predict that graphene prefers to interact with closed βCD conformers

relative to those open forms that remained open on optimization; corrected estimates of

IEclosed-open range from -1.0 to -2.6 kcal/mol (Table 4). Similarly, the methods agree in

predicting βCD to adsorb on graphene via its C2C3 rim, rather than the C6 rim, by -7.5, -10.0

and -16.1 kcal/mol in total IE at PM6-DH2, PM7 and DFTB3 levels respectively (IEC2C3-C6,

Table 4). This preference can be expected given the C2C3 rim is the larger of the two,

contacting graphene via two hydroxyl groups per glucosyl (O2 and O3). For the C6 rim, the

interface involves the CH2 and OH groups of the C6 exocyclic hydroxymethyl moiety in each

glucosyl, which, in closed or open form, are more obliquely oriented with respect to the

graphene surface than the O2-H and O3-H groups.

4. Discussion and conclusions

In this work, we have assessed computational methods for predicting the conformational

energetics of βCD and its interaction with graphene. For βCD in the gas phase, DFT methods

M06-2X and B97X-D provide a consensus profile of the relative stability of low lying

closed and open conformers. Specifically the closed cccw conformer is predicted to be most

favourable geometry, as found by earlier ab initio QM studies, with open structures

significantly higher in energy. In contrast to these previous studies, we find a preference for a

clockwise direction of the C2C3 rim hydrogen bond network.

17

On comparison with the DFT energetics for βCD, we find that the more computationally

expedient DFTB3 and SCC-DFTB methods give reasonable agreement, with the PM3CARB-

1 Hamiltonian reproducing some aspects. However the remaining NDDO methods appear to

underestimate the stability of the cw intramolecular hydrogen bond relative to the cc form.

This understabilisation of the cw energy minimum is manifest when considering overall

adsorption energetics of βCD conformers on graphene surfaces in vacuo.

This imbalance in interactions between O/H atom pairs in close proximity does not seem to

stem from the dispersion correction term, as PM3 and PM6 methods perform similarly to

PM6-D3 and PM6-DH2; rather it appears to arise from inadequacies in parameterisation of

the underlying QM Hamiltonian around trans orientations of the vicinal OH interaction. It is

possible that the NDDO methods underpredict stereoelectronic delocalisation into the CO

antibonding orbital associated with gauche stabilisation. Natural bond orbital deletion

analysis has been applied previously to 1,2-ethanediol conformers to explore the extent of

CO* and lone pair* delocalisation in the molecule.45 At the B3LYP/6-311++G** level

of theory, there are significant hyperconjugation contributions: the CO* interaction

favoured cw/tGg by 0.5 kcal/mol; and the lone pair* interaction favoured cc/gGg by 2.1

kcal/mol. Potentially, imbalance of these contributions could contribute to a destabilisation of

the tGg conformer. Atoms-in-molecules analysis suggests both these conformers are on the

cusp of possessing an intramolecular hydrogen bond.48,49 Whether electrostatic or

stereoelectronic in source, this inability to capture the subtle balance in H…H repulsion with

attractive O…H nonbonding interactions suggests further work is required to improve

parameterisation of the NDDO Hamiltonians.

18

Nevertheless, for noncovalent intermolecular interactions, methods such as PM6-DH2 and

PM7 have been shown to reproduce high level QM approaches with good accuracy; for

example, for the S66x8 benchmark set, PM6-DH2 and PM7 methods displayed RMS errors

in interaction energy of 0.8 and 1.0 kcal/mol.12e In another work, PM6-DH2 estimated the

adsorption energy of hydrocarbons on graphene as 1.8 kcal/mol per carbon atom, which

compares well with an experimental value of 1.7 ± 0.3 kcal/mol.46 Here we find βCD-

graphene interaction energies are overestimated by PM6-DH2 (3 kcal/mol per glucosyl

residue), DFTB3 (2 kcal/mol) and in particular PM7 methods (6 kcal/mol), relative to M06-

2X/def2-TZVPP calculations. However, all three approaches reproduce the essentially

isoenergetic instantaneous IE of cccc and cccw with the C96H24 flake. On application to the

larger βCD/C1006H88 system, all three semiempirical QM methods agree on a preference for

interaction with closed forms of βCD via its C2C3 rim. This concurs with a study of βCD-

graphene interactions using the COMPASS force field where Zhou et al.23 similarly found the

C2C3 rim to bind more favourably than the C6 rim, by 2.7 kcal/mol per glucosyl.

Computation of the interaction of βCD with a carbon nanotube (CNT) using the same force

field similarly found a preference for the C2C3 rim, of 1.5 kcal/mol per glucosyl.50

PM6-DH2 and DFTB-D calculations have been successfully employed in rationalising the

preferential binding of small unsaturated molecules on graphene surfaces.15a,51 However, we

would suggest that caution is required in the application of NDDO-based approaches to

model carbohydrate binding to graphene surfaces which involve changes in the

carbohydrate’s intramolecular hydrogen bonding network. This is due to the underestimation

of the stability of the trans,gauche vicinal diol conformation. The DFTB3 method did not

appear to suffer from this issue and was able to reproduce the trends of higher level QM

calculations. As such, DFTB3 provided the best computational approach of the approximate

19

QM methods considered here for evaluation of overall IEs in large graphene-carbohydrate

systems. Accurate computational tools for prediction of absorption energies are essential for

directing the design of future carbohydrate-based adsorbants; such compounds have the

potential to improve the aqueous dispersibility and biocompatibility of graphene in the

development of graphene-based biomedical applications.

Acknowledgements

We thank Ian Hillier and Mark Vincent for helpful discussions.

Electronic Supplementary Information (ESI)

Geometries of optimised structures of βCD and 1,2-ethanediol; conformational energetics of

1,2-ethanediol; rim-plane distances of βCD/C96H24 complexes; interaction energies of

βCD/C1006H88 complexes.

20

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List of figures

Figure 1 CD geometries optimised at M06-2X/def2-TZVPP (a) cccw (b) cccc (c) cwcw (d)

cwcc (e) O-cwcw (f) O-cccc

Figure 2 Structures and relative energies of 1,2 ethanediols (a) cw/tGg (b) cc/gGg (c)

cw/tGg optimised at M06-2X/def2-TZVPP (in grey) and PM6-DH2 (in cyan) and (d) cc/gGg

optimised at M06-2X/def2-TZVPP (in grey) and PM6-DH2 (in cyan).

Figure 3 Relative energy (in kcal/mol), Ecc–cw, of conformations cc/gGg compared to cw/tGg

for 1,2-ethanediol, at a range of levels of theory (solid line). Relative energy per glucosyl of

cccc compared to cccw conformations of CD at same levels of theory (hatched lines).

Figure 4 Optimal structure of cwcc CD/C1006H88 complex at the PM6-DH2 level of theory,

where C2C3 rim faces the graphene surface.

Figure 5 (a) Overall interaction energies per glucosyl between βCD and C1006H88 computed

at PM6-DH2 (black), PM7 (hatched) and DFTB3 (stipled) levels of theory (b) Corrected

overall interaction energies per glucosyl residue. Energies in kcal/mol. Note: open geometries

collapsed to cccc marked with asterisk.

27

Table 1 Relative gas-phase conformational energetics of βCD (kcal/mol). Negative energies

in bold. Collapsed open structures in italics.

conformer

Methodcccw

()

cccc

()

cwcw

()

cwcc

()

O-cwcw

()

O-cccc

()

MSEe MUEf RMSEg

M06-2X/def2-TZVPP 0.0 1.2 0.9 2.6 17.1 17.4 -

M06-2X/def2-TZVPPa 0.0 1.1 0.9 2.5 17.8 18.1 0.2 0.3 0.4

M06-2X/def2-SVP 0.0 2.3 2.7 5.5 19.9 26.4 2.9 2.9 4.1

B97X-D/def2-TZVPP 0.0 1.0 1.2 2.3 19.5 22.8 1.3 1.4 2.4

B3LYP/6-31G**b 0.0 0.1 2.2 2.1 18.3d 19.6d 0.8 1.2 1.5

B3LYP/6-31G*c 0.0 0.1 2.8 2.6 18.0 20.2 0.7 1.1 1.5

DFTB3 0.0 6.2 4.7 10.9 15.4 26.9 4.1 4.7 5.8

SCC-DFTB 0.0 5.0 3.3 8.4 14.7 23.2 2.6 3.4 3.9

PM7 0.0 8.3 1.7 6.7 19.9 3.2 6.0 7.2 10.1

PM6-DH2 0.0 12.8 1.5 11.3 34.4 14.9 7.1 13.0 17.0

PM6-D3 0.0 11.3 4.7 7.0 29.4 11.6 5.8 11.2 14.4

PM6 0.0 12.5 4.7 6.4 29.0 12.2 6.1 11.3 14.7

PM3CARB-1 0.0 3.7 2.7 6.5 25.0 0.1 0.2 5.6 8.0

PM3-D* 0.0 5.7 6.0 1.1 0.9 12.8 9.0 10.7 14.9

PM3 0.0 5.6 3.4 1.3 0.4 10.2 8.9 9.7 13.8

AM1 0.0 5.9 7.6 0.1 6.1 0.7 5.1 7.3 9.1

aat the M06-2X/def2-SVP geometry bSnor et al.25 cStachowicz et al.26 dcalculations performed in this work

eMean signed error, fmean unsigned error and groot mean square error in relative energy with respect to M06-2X/def2-

TZVPP level of theory

28

Table 2 Relative energies (Ecc-cw) and optimal geometries of cw/tGg and cc/gGg conformers

of 1,2-ethanediols at a range of levels of theory (see Figure 2 for atom labelling). Energies in

kcal/mol, distances in Å and angles in deg.

Energy/

Coordinate

MP2/

def2-TZVPP

M06-2X/

def2-TZVPP

B97X-D/

def2-TZVPPPM6-DH2

cw/tGg cc/gGg cw/tGg cc/gGg cw/tGg cc/gGg cw/tGg cc/gGg

Ecc-cw 0.71 0.32 0.17 -1.30

OaHb 3.57 3.25 3.58 3.24 3.60 3.26 3.87 3.19

ObHa 2.30 2.30 2.31 2.31 2.38 2.36 2.81 2.22

OaOb 2.78 2.82 2.77 2.81 2.81 2.84 3.07 2.75

HaHb 3.07 2.76 3.08 2.76 3.14 2.79 3.55 2.69

OaCaCb 111.2 111.0 111.1 110.9 111.9 111.6 110.7 110.0

ObCbCa 106.0 110.5 106.3 110.4 107.0 111.1 104.5 108.6

HbObCbCa (1) 167.6 73.5 168.3 73.4 166.8 72.1 173.2 73.1

HaOaCaCb (2) 50.5 43.2 50.6 42.6 52.0 43.3 62.4 35.3

OaCaCbOb () 61.0 57.0 60.5 56.7 62.1 57.6 88.5 52.4

Energy/

Coordinate

SCC-DFTB PM7 DFTB3 PM3CARB-1

cw/tGg cc/gGg cw/tGg cc/gGg cw/tGg cc/gGg cw/tGg cc/gGg

Ecc-cw 0.42 -0.93 0.59 1.10

OaHb 3.65 3.03 3.77 3.30 3.59 3.03 3.87 3.57

ObHa 2.25 2.18 2.62 2.44 2.15 2.14 2.64 2.65

OaOb 2.77 2.75 2.94 2.88 2.70 2.72 2.90 2.98

HaHb 3.10 2.53 3.40 2.91 3.01 2.53 3.59 3.25

OaCaCb 109.6 109.2 110.2 110.5 109.5 109.1 114.6 115.0

ObCbCa 108.0 109.3 104.2 109.6 108.0 109.5 108.3 112.8

HbObCbCa (1) 176.9 60.6 175.6 66.9 178.4 65.0 179.3 79.7

HaOaCaCb (2) 44.7 -30.9 57.8 42.5 38.9 25.4 -63.1 49.4

OaCaCbOb () 57.7 53.7 80.2 65.4 50.5 49.5 65.9 65.1

29

Table 3 Overall interaction energy (IE) per glucosyl residue of C96H24 with C2C3 rim of βCD

complexes with respect to global energy minimum cccw conformer (the hydrogen bond

network of C2C3 rim is underlined). Corresponding instantaneous IE values, with respect to

monomer in bound complex geometry, given in parentheses. IEcccc-cccw is IE of cccc relative

to cccw per glucosyl residue. Energies in kcal/mol.

MethodIE IEcccc-cccw

cccw/C96H24 cccc/C96H24

M06-2X/def2-TZVPPa -3.6 (-4.2) -3.2 (-4.1) 0.4 (0.1)

DFTB3 -6.1 (-6.2) -5.2 (-6.2) 0.9 (0.0)

PM6-DH2 -7.1 (-7.1) -8.9 (-7.1) -1.8 (0.0)

PM7 -10.1 (-10.2) -11.3 (-10.5) -1.3 (-0.3)

a at M06-2X/def2-SVP geometry

30

Table 4 Overall interaction energy (IE) per glucosyl residue of βCD/C1006H88 complexes

calculated via PM6-DH2, PM7 and DFTB3 methods at C2C3 and C6 rims with respect to

global energy minimum cccw conformer: average IE of closed conformations, <IEclosed>;

average difference in IE between closed and open conformers, IEclosed-open; and difference in

IE at C2C3 and C6 rims, IEC2C3-C6. IEs corrected for understabilisation of cccw conformer in

parentheses. Energies in kcal/mol.

method/rim

PM6-DH2 PM7 DFTB3

energy C2C3 C6 C2C3 C6 C2C3 C6

IEcccc-cccw -2.3 (-0.3) - -1.2 (0.2) - 0.6 (-0.1) -

<IEclosed> -5.9 (-4.7) -4.8 (-3.6) -9.3 (-8.5) -7.9 (-7.1) -7.2 (-7.3) -4.5 (-4.5)

IEclosed-open -6.5 (-2.6) -6.3 (-2.4) -3.7 (-1.1) -3.9 (-1.4) -3.0 (-2.4) -1.6 (-1.0)

IEC2C3-C6 -7.5 -10.0 -16.1

31

(a)

(b)

(c)

(d)

(e)

(f)

32

C2C3 rim

C6 rim

(a) cw/tGg (b) cc/gGg

CbCa

ObOa

Ha2

Ha

Ha1 Hb

Hb1

Hb2

CbCa

ObOa

Ha2

Ha

Ha1

Hb1

Hb2

Hb

(c) optimised cw/tGg (d) optimised cc/gGg

0.00 kcal/mol 0.32 kcal/mol

0.00 kcal/mol 1.30 kcal/mol

33

34

35

36

*

*

*

*