Introduction

Cyclodextrins (CDs) are homochiral macrocyclic oligosaccharides formed by linking D-(+)glucopyranose subunits together via α-(1→4) glycosidic bonds [1]. CDs are obtained through the partial degradation of the amylose component of starch by the enzyme cyclodextrin glucosyltransferase [2, 3]. The structural formulae of the most common naturally occurring CDs (α, β, and γ; with six, seven, and eight glucopyranose units, respectively) can be found in several works [1, 4]. α, β, and γ are also the CDs that have received the most attention from researchers in the literature [4]. Figure 1 shows structural representations of these three CDs (both top and side views of each).

Fig. 1
figure 1

Original X-ray diffraction structures of the CDs used in this work. Hydrogen atoms either did not appear in the original data or have been removed for clarity

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These carbohydrates are commonly described as being toroidal or shallow truncated conic in shape rather than cylindrical, due to the chair conformation of each glucopyranose building block. They have a hydrophilic exterior (shell), making them soluble in water; hydroxyl groups are present in both their outer and inner rims. Figure 2 shows a schematic of a single glucose unit from a CD.

Fig. 2
figure 2

Glucose unit from a CD, showing the atom labeling scheme applied

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In CDs, the inner rim contains primary hydroxyls (O6–H). The outer rim has secondary hydroxyls (O2–H and O3–H). CDs also have a cavity lined with hydrogen atoms and glycosidic oxygen bridges (O1), meaning that the cavity is weakly lipophilic (hydrophobic) [1] and can therefore capture hydrophobic molecules. This phenomenon is known as forming a molecular inclusion complex, also called a host–guest complex [8,9,10,11]. Various aspects of cyclodextrin chemistry have been discussed in several reviews [12,13,14,15].

CDs have been applied in many fields [4, 10, 16], including industry [17,18,19], pharmaceutical science [8, 20,21,22,23], environmental protection [24,25,26], food [27, 28], and analytical chemistry [29,30,31,32]. Applications of CDs usually make use of their ability to host guest molecules, and so the applications of a particular CD depend mainly on its diameter and the general conformation of its inner cavity [21].

Although CDs have a truncated cone structure, the glucopyranose units are very flexible as a consequence of the rotation and rapid proton exchange of the primary and secondary hydroxyl groups, and the C5–C6 bond shows a high degree of rotation. The glycosidic linkage is also able to rotate to some extent, meaning that each unit has a certain degree of mobility relative to the other units. This flexibility of the glucopyranose units of CDs makes it extremely difficult to describe a cyclodextrin using a single structure, as many conformers are possible for a particular CD at room temperature [33]. These conformers are expected to be in equilibrium in aqueous solution [33].

Several X-ray diffraction (XRD) structures have been determined for CDs and their derivatives [5,6,7, 34,35,36]. These structures vary in accordance with the conditions under which the crystals were synthesized and their level of hydration [6, 33], as varying these parameters can change the dominant conformer. The XRD structure observed in the solid phase may not be the most populated conformer in solution, meaning that XRD cannot unequivocally identify the structures and stabilities of CDs in solution [33]. Moreover, a complete conformational search of all possible conformers would lead to a huge amount of structures, even for the smallest CD, α.

Due to the size of these systems (CDs and host–guest complexes), the methods that have traditionally been used to determine their structures are molecular mechanics [37,38,39,40], semiempirical methods [41, 42], and low-level DFT and ab initio methods [42,43,44].

Several studies have reported the use of symmetrical conformers to describe the structures and stabilities of CDs and their derivatives in the gas phase and solution [40, 43,44,45,46,47,48,49]. In 1988, Koehler et al. [40] carried out a molecular dynamics study using the GROMOS force field to characterize the dynamical behavior of α-CD in aqueous solution. The mean geometry of α-CD over time was presented, and significant differences were reported between the conformation of this CD in aqueous solution and that of α-CD in crystal form. Pinjari et al. presented a study that explicitly addressed the symmetrical conformers of α-, β-, and γ-CD [45]. They used the PM3 and B3LYP methods to investigate intramolecular hydrogen bonding and molecular electrostatic potential. A subsequent study in 2007 explored symmetrical conformers using Hartree–Fock (HF) theory and the B3LYP density functional method [46]. That work examined intramolecular hydrogen interactions by means of the GIAO chemical shifts of hydroxyl protons, and electron density topology using the quantum theory of atoms in molecules (QTAIM). Also in 2007, Karpfen et al. [47] studied symmetrical β-CD using the B3LYP and HF methods, forcing C7 symmetry in all calculations. They found several stable hypothetical conformers and discussed cooperative hydrogen bonds. Moreover, Anconi et al. studied symmetrical α-CD conformers using HF and BLYP in the gas phase and in aqueous solution (applying the PCM model as an implicit solvation scheme). In that work, they identified the most favorable structures in the gas phase and aqueous solution by studying thermodynamic quantities. In the same year, Snor et al. reported a study of anhydrous β-CD using the B3LYP method in which C7 symmetry was imposed. In 2008, Jiménez et al. [48] presented a study of α-CD using HF, B3LYP, and X3LYP, in which geometry optimization was performed and the effects of PCM solvation were examined. More recently, Deshmukh et al. studied the cooperative hydrogen interactions in α-, β-, and γ-CD conformers using B3LYP and by imposing C n symmetry during optimization, where n is the number of glucose units [49]. Also in 2011, Stachowicz et al. [50] reported a study of the interactions of symmetrical conformers of β-CD with positive metal ions. A very recent study by Jaiyong et al. [51] depicted the interactions of symmetrical β-CD conformers using several methods, including density functionals with high-quality basis sets and other approximate quantum-chemical methods. They also used conformers that were previously studied by Snor et al. [44] and Stachowicz et al. [50] in order to compare values and trends. Those studies focused on specific characteristics or phenomena of CDs. However, there has not been an exhaustive study of all three CDs using state-of-the-art methodologies in the last few years.

In the last decade, modern computational methods such as the Minnesota series of density functionals [52] have gained prominence in the literature. Moreover, recent advances in high-performance computing and hardware setups make it possible to use more accurate methods to describe relatively large systems such as CDs. Thus, the aim of the work reported in the present paper was to update the data available from theoretical calculations of CDs using up-to-date methodologies, in order to lay the foundations for further studies on molecular inclusion complexes involving CDs.

Methods

System under study

The system under study consisted of symmetrical conformers built from the XRD structures of α-CD [5], β-CD [6], and γ-CD [7] (Fig. 1).

The conformers were generated by rotating the “free” bonds of the glucopyranose units: C2–O2 and C3–O3 for secondary hydroxyls, and C5–C6 and C6–O6 for primary hydroxyls (Fig. 2). Each dihedral was rotated by 10° while enforcing C n symmetry, where n is the number of glucopyranose units. The geometries obtained were first optimized using the semiempirical Hamiltonian PM6-D3H4 [53,54,55] as implemented in the MOPAC2016 calculation package [56]. As a result of the conformational search and optimization, all of the structures collapsed into a set of 24 stable symmetrical conformers (eight for each CD). These conformers differed from each other in the hydrogen interaction patterns of their inner and outer rims. Figure 3 shows the eight conformers for α-CD.

Fig. 3
figure 3

Symmetrical conformers of α-CD, generated using PM6-D3H4. One glucose unit is detailed in each conformer to aid clarity

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The obtained conformers were classified into three groups: A, B, and C, as also done by other authors in the literature [46, 49]. Conformers type A are characterized by O6–H···O6′ hydrogen interactions with the primary hydroxyls oriented to the cavity, forming a ring of hydrogen bonds between adjacent primary hydroxyls; B type conformers have O6–H···O5′ hydrogen interactions, and C type conformers have the primary hydroxyls pointing to the exterior of the CD with possible hydrogen interactions with the O5 of the same glucopyranose unit (O6–H···O5).

The conformers were also categorized according to the orientation of the hydrogen interaction pattern. Viewing from the top of the secondary (i.e., outer) rim, type 1 conformers present counterclockwise and clockwise orientations of their secondary and primary hydroxyls, respectively, while type 2 conformers exhibit the opposite orientation. Type 3 conformers present counterclockwise orientations of both their secondary and primary hydroxyls,whereas type 4 conformers show clockwise patterns for both secondary and primary hydroxyls. The reader is advised to be careful when comparing the geometries obtained here with those reported in the literature, as some differences may be observed, mainly regarding dihedral angles.

Computational details

In order to describe the geometries and wavefunctions of the obtained structures in an accurate way, the DFT hybrid functional M06-2X [52] and with Pople’s split valence double-ζ basis set 6-31G(d,p) were applied to all conformers. DFT-D3 corrections [54] were also implemented with zero damping [57]. To deal with solvation effects, the SMD method [58] was used as an implicit solvation model (SMD/M06-2X/6-31G(d,p)). In almost every DFT calculation we performed, water was used as the solvent, except for the 1H NMR studies, where dimethylsulfoxide (DMSO) was used. All DFT calculations were performed using the Gaussian09 package [59]. For all quantum calculations performed using the four methods described above, full eigenvector following geometry optimizations were carried out without any symmetry constraints.

In order to validate the methods used, the experimental XRD structures and the optimized conformers were compared in terms of bond distances and bond angles. The criterion used to compare these structures was the root mean square deviation (RMSD), as presented below:

$$ \mathrm{RMSD}=\sqrt{\frac{1}{N}\sum_{i=1}^N{\left(\frac{X_{\mathrm{theo},i}-{X}_{\mathrm{XRD},i}}{X_{\mathrm{XRD},i}}\right)}^2}\times 100\%. $$
(1)

In Eq. 1, N represents the number of determined values and X theo is the value of the parameter (bond length or bond angle) as calculated by the computational method. X XRD is the value of the same parameter as assessed experimentally using XRD.

For DFT calculations, frequencies were calculated to prove the existence of a real minimum for each conformer. Thermodynamic parameters were extracted from the results of gas-phase (M06-2X/6-31G(d,p)) and aqueous (SMD/M06-2X/6-31G(d,p)) Gaussian09 calculations in order to evaluate the solvation process based on the total energy of electrons plus nuclear repulsion (ΔE), enthalpy (ΔH aq), and free energy (ΔG aq) for calculations in aqueous solution, as well as the variation in the free energy during the solvation process (δΔG) as presented in the following equation:

$$ \delta \Delta G=\Delta {G}_{\mathrm{aq}}-\varDelta {G}_{\mathrm{gas}}. $$
(2)

Here, the subscripts “gas” and “aq” refer to calculations performed with the CDs in the gas phase and in aqueous solution, respectively.

ΔH aq and ΔG aq values were reported with respect to a selected reference value according to the following equation:

$$ \Delta Y=\Delta {Y}_i-\Delta {Y}_{\mathrm{ref}}, $$
(3)

where ΔY i is the value of the parameter (i.e., ΔH or ΔG) calculated for a given conformer and ΔY ref is a reference value for the same parameter (in this work, the reference value is the value obtained for one of the calculated conformers). This approach allows the reader to compare the results in a more comfortable way.

With the same aim, a new parameter was constructed from ΔE. The parameter ΔE/n is the total energy divided by the number of glucopyranose units. The results are presented using the appropriate minimum value (the values for the A3 conformer of α-CD in the gas phase and the B2 conformer of γ-CD in aqueous solution) as a reference in accordance with the equation

$$ \Delta E/n=\frac{\Delta {E}_i}{n_i}-\frac{\Delta {E}_{\mathrm{ref}}}{n_{\mathrm{ref}}}, $$
(4)

where the subscript i refers to the conformer of interest and “ref” indicates the reference conformer. This new parameter allowed us to compare CDs in terms of ΔE even when the CDs were different sizes. Also, from hereon, we use the term “total energy” to refer to the “total energy of electrons plus nuclear repulsion” for the sake of brevity.

Dipole moments were calculated at the SMD/M06-2X/6-31G(d,p) level of theory using water as an implicit solvent, as many computational studies have predicted high dipole moments along the z-axis for all three CDs [33].

Geometries and wavefunctions obtained via M06-2X were employed to calculate proton and 13C NMR spectra using the gauge including atomic orbitals (GIAO) method [60, 61], which has proven to be an adequate approximation for correlating theoretical chemical shifts with experimental NMR spectra [62, 63]. Chemical shifts are reported in ppm using the IUPAC convention. In all cases, tetramethylsilane (TMS) was used as a reference with a chemical shift of 0.0 ppm. The solvent used when calculating the theoretical 1H NMR spectra was dimethylsulfoxide (DMSO), in order to simulate experimental DMSO-d 6 solvation effects. For the calculations of 13C spectra, water was used as an implicit solvent to simulate a deuterium oxide (D2O) environment.

Using the DFT results for the CDs in aqueous solution, further analysis of possible hydrogen interactions was then carried out. The first set of criteria used to compare hydrogen interactions were geometrical, i.e., interaction distances and angles, as several studies have reported correlations between these parameters and the strength and stability of the interactions [64,65,66,67]. However, these are not the definitive parameters for classifying interactions [67].

A second set of criteria were employed when exploring quantum chemical topology using the quantum theory of atoms in molecules (QTAIM) pioneered by Bader [68]. The first criterion to be established was the presence of a bond critical point (BCP) and a bond path between the interacting atoms. Further description and classification of these interactions was achieved using several descriptors of the electron density at the BCP, such as the electron density (ρ c), the Laplacian of the electron density (∇2 ρ c), the total energy density (H c), and the ellipticity of the electron density (ε c). The obtained parameters were used to classify the hydrogen interactions according to Nakanishi’s classification criteria [69, 70].

Results and discussion

Validating the geometries

A comparison of the geometries of the conformers obtained in this work with their corresponding XRD structures is shown in Fig. 4. This graphic shows the agreement of the XRD experimental bond lengths and angles with the corresponding quantities obtained using the SMD/M06-2X/6-31G(d,p) methodology. The results shown in the figure are averages across the eight conformers for each of α-, β-, and γ-CD, respectively.

Fig. 4
figure 4

Root mean square deviations (RMSDs) in bond lengths and angles between X-ray diffraction geometries and the geometries obtained using the SMD/M06-2X/6-31G(d,p) methodology. Water was used as solvent in the SMD implicit solvation scheme

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From Fig. 4, it is clear that the SMD/M06-2X/6-31G(d,p) methodology gives adequate results when compared to the XRD geometries, as the RMSDs in bond lengths and angles are always less than 2%. Comparisons of all the conformers regarding their geometrical parameters are presented in Tables S1–S6 and Fig. S1–S6 of the “Electronic supplementary material” (ESM). We could not detect any significant differences in the RMSDs in bond lengths and angles when comparing the conformers. For this reason, these geometrical parameters (bond lengths and bond angles) could not be used in this work to select the most probable conformers in aqueous solution. This result indicates that comparisons of solid-state structures with modeled gas-phase or solvated structures may not be adequate [33].

Thermodynamics

For DFT calculations, an analysis of some thermodynamic parameters of the conformers was performed, comparing the results calculated with and without implicit solvation. Figure 5 shows the results attained using the scheme M06-2X/6-31G(d,p) for the parameter ΔE/n, while Fig. 6 shows the results obtained using SMD with water as an implicit solvent.

Fig. 5
figure 5

Total energy of electrons plus nuclear repulsions per glucose unit (ΔE/n) of each optimized CD in the gas phase, calculated at the M06-2X/6-31G(d,p) level of theory. All values are presented relative to the minimum value (obtained for the A3 conformer of α-CD)

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Fig. 6
figure 6

Total energy of electrons plus nuclear repulsions per glucose unit (ΔE/n) of each optimized CD in aqueous solution, as calculated at the SMD/M06-2X/6-31G(d,p) level of theory. All values are presented relative to the minimum value (obtained for the B2 conformer of γ-CD)

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As can be observed from Fig. 5, the lowest energy (i.e., indicating the greatest stability) in the gas phase was obtained for conformer A3 of α-CD. In general, type A conformers are more stable than type B and C conformers, with a difference of over 12 kJ mol−1 per glucopyranose unit observed between the A3 and B2 conformers of α-CD. On the other hand, when using the SMD scheme (Figure 6), the behavior of this parameter shifts, especially for β- and γ-CD, for which the B and C conformers are more stable in aqueous solution. For α-CD, the difference observed in Fig. 6 is smaller than that seen in Fig. 5. This analysis suggests that B and C conformers may be more stable in aqueous solution than A conformers, and thus that B and C are the main conformers in aqueous solution. However, the results indicate that type A conformers are more likely to be observed in the gas phase than type B and C conformers, an inference supported by calculations in other works [46, 47]. However, the total energy may not be the best criterion for predicting this behavior.

Table 1 shows a comparison of the values of other thermodynamic parameters that can be used to decide whether a conformer is stable in aqueous solution. The absolute values obtained in all thermodynamic calculations are reported in Tables S7 and S8 of the ESM.

Table 1 Calculated values of the relative total energy of electrons plus nuclear repulsion (ΔE), the aqueous enthalpy (ΔH aq), the free energy (ΔG aq), and the solvation free energy (δΔG) for the conformers in aqueous solution; calculations were performed at the M06-2X/6-31G(d,p) level of theory with water included as an implicit solvent using SMD
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The values for all of the parameters presented in Table 1 indicate that the B and C conformers are the most stable, especially in β- and γ-CD, where solvation effects should be more pronounced due to an increase in the number of the hydroxyl groups in the molecule with respect to α-CD. Moreover, values of the parameter δΔG, which dictates whether the solvation process is spontaneous, indicate that the stabilities of B and C conformers are higher in solution than in the gas phase. These results agree with those obtained by Anconi et al. for α-CD using the BLYP hybrid DFT functional [43]; they observed similar solvation behavior for their type 2 and 3 conformers, which were similar to the type B and C conformers in this work. On the other hand, the high values of δΔG for the B and C conformers of β-CD contradict the experimental results, as β-CD is the least water soluble of all the natural CDs [21]. This constitutes a limitation of our model, as it cannot explain this experimental behavior, which has been the subject of several debates [33]. Moreover, no corrections were made to frequencies used to calculate the free energy at room temperature, so these data should only be compared with other values obtained in this work. However, this result is important in the context of selecting a set of structures that could be used as building blocks for future research into molecular inclusion complexes of CDs.

For the reasons given above, the implicit solvation formalism SMD is very useful for describing the thermodynamics and the order of stability of the studied conformers. It can also provide insight into the average conformations of these CDs in aqueous solution. Such results indicate the conformers that should be used when conducting computational studies of host–guest complexes.

Dipole moments

The dipole moments of CDs have also been investigated in other works. Several studies have suggested high dipole moments for the CDs presented in this work [12, 33]. The complementarity of the dipoles between CDs and guest molecules has also been discussed [71]. Table 2 displays the dipole moments for all 24 conformers treated in the present work.

Table 2 Dipole moments along the z-axis (in debyes), as calculated at the SMD/M06-2X/6-31G(d,p) level of theory
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Given that the structures studied here have symmetries close to C n , only the z-components of the dipole moments are displayed in Table 2; the other components are neglected. Twenty-three of the 24 structures have their dipoles oriented toward the secondary hydroxyls, with only conformer A1 of γ-CD having the opposite orientation. The highest dipoles were calculated for the B2 and B3 conformers of β- and γ-CD. The obtained results do not permit us to draw conclusions about the main conformations of the CDs in aqueous solution, even if we assume that CDs have high dipole moments, which is an assumption based only on theoretical calculations, not experiment [71,72,73]. Moreover, the A2 and A3 conformers have particularly high dipole moments, and the C-type conformers have very low ones when compared with B2 and B3, which makes us question the use of this parameter to select the conformers that best describe the CDs in solution. According to Pinjari et al. [46], their conformers of type A, which are almost identical to the A-type conformers proposed in this work, present low dipole moments when compared with the B and C types. That work also reported two negative values for the A4 conformers of β-CD and γ-CD, whereas only one negative value was detected (for the A1 conformer of γ-CD) in the present work. It should be noted that Pinjari et al. [46] calculated the dipole moments at the B3LYP/6-31G(d,p) level of theory. Additionally, there are some geometrical differences between the B- and C-type conformers in that work and those in the present work, meaning that not all of the conformers can be compared between works.

NMR studies

NMR spectroscopy is a very useful technique for determining chemical information. The obtained spectra are very sensitive to the geometrical structures and electron densities of the molecules studied. However, this sensitivity of NMR also has drawbacks in computational chemistry, as simulated NMR results are very sensitive to the particular method and basis set applied [61]. DFT hybrid functionals have been shown to yield good results for medium to large basis sets, with 6-31G(d,p) being close to the smallest basis set that can provide accurate results [61].

Typically, NMR is a “slow” spectroscopic technique when compared to the timescale for conformational changes of CDs at room temperature [74]. It should also be noted that magnetic pulses are applied to an enormous number of molecules, which means that NMR measurements are averaged over time and space. These limitations imply that signals from atoms located at the same position on different glucopyranose units are not distinguished in a NMR spectrum. For this reason, NMR spectra will always lead to symmetrical CDs, which makes this technique very interesting and useful for comparing experimental NMR spectra with the theoretical results obtained by GIAO.

Tables 3 and 4 show comparisons between theoretical (GIAO) and experimental results [36, 74] for the 1H and 13C chemical shifts, respectively, associated with the CDs studied in the present work.

Table 3 Values of proton chemical shifts for the CDs of interest, as calculated at the SMD/M06-2X-D3/6-31G(d,p) level of theory with DMSO as the solvent using GIAO
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Table 4 Values of 13C chemical shifts for the CDs of interest, as calculated at the SMD/M06-2X-D3/6-31G(d,p) level of theory with H2O as the solvent using GIAO
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All chemical shifts were calculated using TMS as reference and are reported in parts per million (ppm). The results of theoretical NMR calculations are reported relative to experimental chemical shifts. All absolute shift values for 1H and 13C NMR spectra (Tables S9 and S10), along with correlation graphics for the 1H NMR data (Fig. S7) are presented in the ESM. According to the RMSD values and the correlation coefficients (R) between the theoretical and experimental NMR data, the theoretical results for the type B and C conformers are consistent with the corresponding experimentally determined proton and 13C chemical shifts.

Regarding the 1H NMR spectra, the theoretical results for the type B and C conformers correlate better with the corresponding experimental results based on the R values—especially B3 and C1, for which the R values are over 0.98. This indicates that linear scaling of the obtained chemical shifts may improve the theoretical NMR results and reduce RMSDs, as has been already reported [62, 63, 75]. However, the signal corresponding to the anomeric hydrogen (H1) is located at a significantly higher value of δ when compared with the other proton signals, which are grouped together. This behavior could lead to mistaken judgments about how well the experimental and theoretical data agree, due to the statistical limitations of the correlation analyses. For this reason, we also examined the RMSDs, which were lower for the B- and C-type conformers, in agreement with the correlation coefficients (R). Pinjari et al. [46] reported a GIAO study that used B3LYP to calculate proton chemical shifts. They reported linear correlations of the chemical shifts of active (O–H) protons with geometry- or electron-density-related parameters. Those results support the validity of the GIAO results obtained in this work, although only C–H protons were analyzed in the present work. Another critical marker is the average signal from CH2 protons (H6a,b), which is in better agreement with the experimental results for B and C conformers than for A conformers. It was found in previous experimental NMR studies that these methylene hydrogen atoms, which have near-isochronous shifts, are likely to be both positioned gauche with respect to H5 for CDs in solution [74], as they are in the B and C conformers.

Correlation analysis of the 13C spectra gives little information, since the R values are >0.99 for all conformers of the three CDs. However, the RMSD values corroborate the results from proton analysis, as smaller values were obtained for the B and C conformers.

The calculated NMR data are in good agreement with thermodynamic data obtained at the SMD/M06-2X/6-31G(d,p) level of theory.

Although experimental and theoretical proton shifts were obtained using DMSO as the solvent, this solvent is also able to form hydrogen bonds with CDs, meaning that CDs in DMSO show similar conformational behavior to CDs in aqueous solution. Thus, these results again point to a prevalence of B and C conformers of CDs in water.

Hydrogen interactions

Based on the XRD crystal structures of the CDs, it can be stated that intramolecular O–H···O interactions favor macrocyclic CD conformations, and may even govern the interactions of the CDs with molecules in their cavities (guests) [13, 76]. Therefore, studying the hydrogen interactions in CDs is a crucial step when attempting to accurately describe the structures and properties of the CDs.

Table 5 shows the hydrogen-bond distances along with their standard deviations (in parentheses) for the studied cyclodextrins. Table 6 presents similar information regarding their hydrogen interaction angles. It has been stated that these parameters are often correlated with other electron-density-based, thermodynamic, and spectroscopic parameters [64,65,66,67]. It has also been stated that, for the same type of interaction, the smaller the distance, the greater the strength. Also, due to the directional character of these interactions, O–H···O angles of close to 180° are favored.

Table 5 Values of the mean hydrogen interaction distance and the standard deviations in these values (all in Å) for the CDs, as calculated at the SMD/M06-2X-D3/6-31G(d,p) level of theory
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Table 6 Values of the mean hydrogen interaction angle and the standard deviations in these values (all in degrees) for the CDs, as calculated at the SMD/M06-2X-D3/6-31G(d,p) level of theory
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From Table 5, it can be seen that the smallest distances are obtained for the O6–H···O6′ interactions of type A conformers; these also have angles of close to 180° (Table 6). These interactions are likely to be the strongest of all those considered, as also reported by Deshmukh et al. [49] and Pinjari et al. [46], contributing some extra rigidity to the CDs and almost closing the primary face, which may hinder possible complex formation with a guest molecule both thermodynamically and kinetically. Also, the planar configuration of the chain of hydrogen bonds prevents strong interactions of surrounding water molecules with the primary hydroxyls. For the secondary hydroxyls, the studied interactions are most favorable for the C1 and C4 conformers, as they have the smallest interaction distances for O3–H···O2′ and O2–H···O3, respectively. These interactions are very important for maintaining the CDs in a bucket-like conformation while retaining sufficient flexibility to accommodate an incoming guest and to interact with the solvent. For B-type conformers, the interaction distances for O6–H···O5′ are similar to those of secondary hydroxyls, but the nature (i.e., hydroxyl–ether) of these interactions, and the low angles of interaction, mean that they are likely to be weaker than the hydroxyl–hydroxyl interactions. Additionally, for C-type conformers, the distances and angles of the O6–H···O5 interactions imply that they may be of a dispersive nature.

Based on these results, it can be stated that the B- and C-type conformers are likely to be more flexible and to interact more strongly with the surrounding water molecules when in solution, in agreement with results of the thermodynamic and NMR analyses. However, geometrical parameters do not provide enough information to allow us to classify and sort these interactions. Therefore, to classify the interactions, a QTAIM analysis was performed. Bond critical points were calculated for all possible interactions in all 24 conformers. Table 7 presents the results of this analysis for conformers B3 and C1, as they presented the most accurate NMR results. Tables S11–S16 of the ESM show the results of QTAIM analysis for all the conformers.

Table 7 Mean values (and standard deviations) of the electron density (ρ c), the Laplacian of the electron density (∇2 ρ c), the total energy density (H c), and the ellipticity of the electron density (ε) at BCPs for hydrogen interactions in the conformers, as calculated at the SMD/M06-2X-D3/6-31G(d,p) level of theory with water as the solvent using QTAIM a
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According to Nakanishi’s classification of interactions [69, 70], based on the ρ c values, all the hydrogen interactions are typical hydrogen bonds. Since ∇2 ρ c > 0 for all the interactions, they can be classified as closed shell with a certain degree of covalency (given that H c < 0). As ε ≈ 0 for these interactions, the electron density distributions appear to be cylindrical, implying directional interactions (as opposed to other nondirectional weak interactions such as van der Waals) [77].

Noting the ρ c and ∇2 ρ c values for the interactions involving secondary hydroxyls (O3–H···O2′), it can be argued that the C1 conformers have the highest values of ρ c, suggesting that they exhibit the strongest interactions. However, the values of ∇2 ρ c suggest that the B3 conformers have the strongest charge-transfer character, although differences among the conformers are rather small. It is the opinion of the authors that these small differences do not favor one conformer over the other in the stability or strength of their interactions. In terms of interactions involving the primary hydroxyl groups, only B3 presents a hydroxyl–ether interaction (O6–H···O5′), as no BCP involving this group was found for the C1 conformers, meaning that the primary hydroxyls are free to interact with the surrounding medium. The hydroxyl–ether interactions observed in B3 are relatively stable compared to the O3–H···O2′ interactions in both B3 and C1. However, the groups involved in the O6–H···O5′ interactions are still able to interact with their surroundings due to the nonlinear geometries of the interactions.

Summary

We have presented the results of the computational modeling of several symmetrical conformers of α-, β-, and γ-cyclodextrin. The method SMD/M06-2X/6-31G(d,p) was found to accurately describe the conformers in terms of their bond distances and angles, as shown by a comparison with experimental X-ray diffraction geometries. The calculations showed that all of the studied conformers are stable in the gas phase and in aqueous solution and may coexist in a real system. Theoretical thermodynamical and NMR analyses indicated that the B and C conformers are the most populated in aqueous solution, while the A conformers dominate in the gas phase. The use of theoretically calculated dipole moments remains a questionable criterion for deciding which of the conformers most accurately describes a particular CD in solution. The main noncovalent interactions present in these conformers were found to be typical hydrogen bonds between the hydroxyl groups of the cyclodextrins. For the B-type conformers, hydroxyl–ether hydrogen bonds with O6–H groups were identified, while the C-type conformers do not show interactions involving O6–H. The results of this work will be used to construct a cyclodextrin model that can be applied when studying host–guest complexes involving cyclodextrins.