Quantum mechanical modeling of the structures, energetics and spectral properties of Iα and Iβ cellulose

As a test of the accuracy of the DFT-D2 computational methodology employed in the VASP calculations, comparisons of ωB97X-D and DFT-D2 results were made for the H₂O-H₂O and CH₃OH-CH₃OH dimers. The water dimerization energies (ΔE_dimer) are equal to −32, −26 and −26 kJ mol⁻¹ for the 6−311G(d,p), 6-311+G(d,p) and 6-311++G(d,p) basis sets, respectively. The DFT-D2 ΔE_dimer result was −25 kJ mol⁻¹. The calculated ΔE_dimer is approximately within the range of reported experimental data of −20.4 to −12.3 kJ mol⁻¹ with reported uncertainties of approximately ± 3.5 kJ mol⁻¹ (Fiadzomor et al. 2008; Rocher-Casterline et al. 2011). Addition of diffuse functions did improve the precision of the calculated results with respect to the water dimerization energy by −6 kJ mol⁻¹.

More relevant to the results on cellulose considered in this study are the H-bond distances and O-H stretching frequencies of the H-bonded OH group. The ωB97X-D/6-311G(d,p) and 6-311++G(d,p) basis sets resulted in O-O bond lengths of 2.811 and 2.835 Å and unscaled harmonic O-H symmetric stretching frequencies of 3,785 and 3,779 cm⁻¹, respectively. Thus, for H-bond distances and symmetric O-H stretching frequencies, the lack of diffuse functions in the basis set does not make a significant difference on the calculated values. In comparison, the DFT-D2 calculated values for ΔE_dimer, O-O H-bond distance and O-H stretching frequency associated with the H-bond in the H₂O-H₂O dimer are −25 kJ mol⁻¹, 2.856 Å, and 3,814 cm⁻¹, respectively. The experimental O-H frequency is 3,735 cm⁻¹ (Huisken et al. 1996), and O-O H-bond distance is 2.976 Å (Odutola and Dyke 1980). Therefore, DFT-D2 results agree well with the observed interaction energy, O-O distance and O-H stretching frequency. This is especially true if the DFT-D2 frequency is scaled by the 0.97 factor (3,814 × 0.97 = 3,670 cm⁻¹) derived by correlating DFT-D2 frequencies with SFG frequencies (see “Methods” section).

The CH₃OH-CH₃OH dimer is a test system that is even more analogous to the H-bonds found in cellulose. Although experimental data on the ΔE_dimer is not available for the methanol dimer, we suggest that the ωB97X-D/6-311++G(d,p) calculations on the methanol dimer are as accurate as they are for the water dimer. Hence, comparison of DFT-D2 and ωB97X-D/6-311++G(d,p) results can be used to evaluate the accuracy of the DFT-D2 energy results. The DFT-D2 ΔE_dimer is −29 kJ mol⁻¹ which compares favorably with −28 kJ mol⁻¹ calculated using ωB97X-D/6-311++G(d,p). Similarly, the O-H-O H-bond distance using DFT-D2 is 1.826 Å whereas the ωB97X-D/6-311++G(d,p) value is 1.875 Å. The experimental value for this bond length is 2.034 Å (Lovas et al. 1995), so the calculated results were within 0.2 Å of the observed H-bond length. The unscaled, calculated O-H stretching frequencies associated with this H-bond are 3,504 and 3,767 cm⁻¹ for DFT-D2 and ωB97X-D/6-311++G(d,p), respectively. In this case, the ωB97X-D/6-311++G(d,p) result overestimated the observed value of 3,527 cm⁻¹ (Han et al. 2011) by 126 cm⁻¹ (3 %, after scaling by 0.97); furthermore, when scaling is considered, and the DFT-D2 result underestimates the frequency by 128 cm⁻¹ (3 %) after scaling. We conclude that the DFT-D2 methodology used on cellulose in this study is accurate to approximately 10 and 3 % for the O-H distances and O-H stretching frequencies based on these comparisons with experimental and the ωB97X-D/6-311++G(d,p) results.

Table 1 contains the lattice parameters and glycosidic torsion angles from experiment (Nishiyama et al. 2002, 2003), 2-D periodic DFT calculations (Nishiyama et al. 2008), previous 3-D periodic DFT-D2 calculations (Bućko et al. 2011; Li et al. 2011) and the present study for Iα and Iβ cellulose. Although variations on the order of a few percent are present throughout when comparing calculated results versus experimental data and our calculated results against previous calculations, the model values are reasonably precise. The tg/NetA results agree best with experimental lattice parameters compared to tg/NetB, gt and gg, for both Iα and Iβ (Table 1). In general, the calculated lattice parameters for the tg/NetA model are less than observed and some of this discrepancy may be accounted for the temperature difference between experiment and theory (i.e., observations at 298 K and calculations at 0 K) causing thermal expansion in the former as compared with the latter. In addition, the tg/NetA models are predicted to have the lowest total electronic energies for both Iα and Iβ (Table 2), consistent with the interpretation of experimental data (Nishiyama et al. 2003). For Iα and Iβ, the energy differences between NetA and NetB conformations are −20 and −24 kJ/glucose residue, respectively.

One of the larger variations from observation is for the b-lattice parameter of Iα where our closest calculated value is 5 % shorter (Table 1) than found in Nishiyama et al. (2003). These discrepancies are not surprising because the b-lattice parameter is strongly influenced by van der Waals forces between sheets which are likely to be the least accurate component of the DFT-D2 methodology.

The structure of Iβ cellulose is predicted more accurately than Iα (Table 1). For the tg/NetA conformation, the largest discrepancy is the a-lattice vector with a 3 % error. The better agreement for Iβ versus Iα is likely due to the monoclinic symmetry of the Iβ unit cell serving to limit the relaxation of the simulation cell. This symmetry effect seems to be significant in spite of the fact that the simulation cell was doubled in size along the c-direction compared to the Iα simulation cell.

Glycosidic torsions (Scheme 1-Φ₁, Φ₂, Ψ₁, Ψ₂) are critical parameters in determining whether or not cellulose twists (French and Johnson 2009), and these calculated structural parameters are also compared with experiment and previous calculations in Table 1. The discrepancies of our calculated values with those determined by Nishiyama et al. (2002, 2003) are relatively small. For both Iα and Iβ, the calculated torsion angles differ from experiment by at most 6 °. The reported experimental standard deviation for the glycosidic torsion angles are approximately 3° for Iα and up to 20° for Iβ cellulose (Nishiyama et al. 2003). Hence, these model deviations from experiment of ≤6° are reasonably precise. Previous calculations resulted in similar differences from experiment of approximately 5 % (Bućko et al. 2011; Li et al. 2011; Nishiyama et al. 2008). Unfortunately, due to the limited size of the simulation cell, the possible twist along the cellulose chains cannot be directly investigated. In addition, the experimental uncertainty and computational error will not allow for evaluation of potential subtle, long-range twisting. Nonetheless, this model of cellulose that strictly excludes twisting will be shown to reproduce experimentally observed spectra accurately. Models including any twist must reproduce experimental observables more accurately in order for twisting to be considered a necessary component of the cellulose structure.

Nishiyama et al. (2002, 2003) determined that the torsions of the hydroxymethyl groups (χ) are in the tg conformation for both Iα and Iβ cellulose based on X-ray diffraction patterns and structural refinements. Consequently, previous studies using DFT-D2 methods to model these structures focused solely on the tg conformations of the hydroxymethyl groups (Bućko et al. 2011; Li et al. 2011). However, based on classical molecular dynamics simulations of cellulose, Matthews et al. (2012) have suggested that Iα undergoes a conformational change from tg to gg, and Iβ switches from tg to gt at temperatures of 500 K. Furthermore, χ of native cellulose surfaces may not be constrained to the values observed in the interior due to the interactions with water and other plant biopolymers (Fernandes et al. 2011; Harris et al. 2012; Newman and Davidson 2004; Viëtor et al. 2002; Štrucova et al. 2004). Hence, we have also investigated the gt and gg conformations in addition to tg of Iα and Iβ cellulose in order to predict energy and structural differences accompanying hydroxymethyl group rotations. Although the calculated lattice parameters, glycosidic torsions and energies of these other conformations do not suggest that they exist in large percentages in the particular samples analyzed (Tables 1, 2), consideration of these other forms is worthwhile.

Table 3 lists the experimental and calculated χ₁ and χ₂ torsion angles for the hydroxymethyl groups in Iα and Iβ. Our calculated values are similar to the Nishiyama et al. (2003) values for the tg/NetA conformations, especially considering experimental uncertainty. All values deviate from the ideal values of 180° and −60°. The gt and gg torsion angles also are predicted to deviate by only a few degrees from the ideal values of +60/+60° and −60°/−60°, respectively. These changes in the hydroxymethyl group torsion angles induce changes in the a and b lattice parameters of Iα cellulose that are outside of the range of computational accuracy, but the changes in lattice parameters for Iβ cellulose are smaller than for Iα (Table 1). This result suggests that mixtures of tg/gt/gg conformers may be more difficult to detect via X-ray and neutron diffraction in Iβ compared to Iα cellulose. Relative to the tg conformers, the calculated energy changes are +11 and +24 kJ mol⁻¹ for the gt and gg conformers in Iα, respectively, and +19 and +26 kJ/glucose residue for the gt and gg conformers in Iβ, respectively (Table 2). Hence, higher temperatures could induce transitions to other conformations when entropic effects are considered.

Selected H-bond and O-O distances are listed in Table 4. In general, the DFT-D2 values are up to 0.2 Å less than the experimental values, whereas the CHARMM-based values from a 6 × 6 × 40 Iβ microfibril are typically 0.1 Å greater than the experimental values. The calculated distances are comparable to those obtained with DFT-D2 and wB97X-D/6-311++G(d,p) for the water and methanol dimers discussed above. These discrepancies can be considered small for computational methods, especially considering the uncertainties in the experimental values. However, the DFT-D2 model over-estimation of the H-bond strengths could have an effect on the calculated O-H stretching frequencies discussed below because O-H vibrations are highly sensitive to H-bond strengths. This will be particularly problematic for the Iα O3-H3-O5 and O6-H2-O2 O-H stretches and the Iβ O6-H6-O3 stretch where the frequencies may be 200 cm⁻¹ lower than observation because of the stronger calculated H-bonding (see “Vibrational frequencies” section below). In addition, the DFT-D2 calculations do not predict an O6-H6-O1 H-bond in Iα cellulose observed by Nishiyama et al. (2003) (Table 4).

To test the accuracy of our frequency calculations on crystalline solids, we compared calculated and observed vibrational frequencies of crystalline cellobiose. There is no controversy about the cellobiose crystal structure, so this model provides a firm link between structure and vibrational frequencies. Figure 2a, b illustrates that the correlation between the modeled and observed frequencies is excellent (IR: slope 1.00, intercept −2 cm⁻¹, R² 0.998; Raman: slope 1.00, intercept 2 cm⁻¹, R² 0.998). A perfect 1:1 correlation would result in a slope of 1.0, intercept of 0.0 and R² value of 1.0. Hence, we conclude that our computational methodology is accurate for carbohydrate vibrational frequencies.

The correlations of calculated vibrational frequencies with observed IR and Raman spectra are excellent for both Iα and Iβ cellulose (Fig. 2c-f). The slopes and R² values of the tg/NetA conformations deviate from their ideal values by <2 %. The maximum error for the intercept is 20 cm⁻¹ for the Iα Raman spectrum (Fig. 2d; Table 5). Paradoxically, the maximum error is found for the tg/NetA conformation that best matches the crystal structure (Table 1) and ¹³C NMR chemical shifts (see below). This result makes it difficult to distinguish the correct model of cellulose structure based on IR and Raman spectra because all the models are reasonably consistent with observed frequencies. We think the reason for the discrepancy between observation and the model tg/NetA is that the higher frequency O-H stretches that appear in the calculated tg/NetB conformation and observed spectra. Natural cellulose samples are probably a mixture of different conformations (Nishiyama et al. 2008); hence, the observed spectra may pick up frequencies that are not due to the most stable and predominant conformation.

Vibrational modes visualized are generally consistent with previous spectral assignments of IR and Raman bands, but detailed analysis of vibrational modes is beyond the scope of this paper. Based on these statistics it could be impossible to distinguish among the four conformations (tg/NetA, tg/NetB, gt and gg). Many vibrational modes such as C-C stretches and CH₂ angle bends should not be sensitive to the long-range order because short-range covalent forces dominate them. However, the O-H stretching region between 3,000 and 3,500 cm⁻¹ should be affected by longer-range structure because H-bonding distances would be affected by changes in lattice parameters. H-bond distances are known to have a readily observable correlation with O-H stretching frequencies.

Consequently, we performed correlations between calculated and observed O-H stretching modes separately. The correlation statistics for all models were poorer compared to the overall correlations as expected, but the gt and gg models resulted in better fits compared to the Iα IR and Raman frequencies, respectively. The tg/NetB model better matched observed Iβ IR and Raman O-H stretching frequencies.

There could be a number of reasons for these discrepancies. For example, we note that the IR spectrum for Iα cellulose of Blackwell (1977) exhibits a peak near 3,408 cm⁻¹. The model that best produces this higher frequency (i.e., less H-bonded) O-H stretch is the Iα/tg/NetB conformation. We have concluded above based on comparisons of structures to experiment and calculated energetics, that the Iα/tg/NetA conformation should be dominant, but mixtures of networks A and B are possible in cellulose (Nishiyama et al. 2008). The fact that our calculations reproduce other observed frequencies so accurately, while this one vibrational mode can deviate from observation by over 100 cm⁻¹, leads us to conclude that it is not inaccuracy of the model calculations but rather structural heterogeneity that leads to the presence of this 3,408 cm⁻¹ peak in the observed spectrum. A similar discrepancy exists between the calculated and observed IR spectrum of Iβ cellulose; but in this case, the network A conformation is approximately 60 cm⁻¹ under-estimated compared to experiment (Table 4). Another potential reason why the preferred Iα/tg/NetA conformation does not reproduce some of the observed higher frequency O-H stretches such as that at 3,408 cm⁻¹ could be that the observed spectra are detecting O-H groups at the surface of the cellulose. The H-bonding of surface OH groups could be less strong than internal OH groups and lead to higher frequency O-H stretches. Lastly, the scaling factor for the O-H stretches could be different from that obtained based on the CH₂ stretches.

A crystallinity index for cellulose I based on the intensities of the 380 and 1,096 cm⁻¹ Raman peaks is available (Agarwal et al. 2010). Since these two vibrational modes may be related to the order/disorder of cellulose I, we examined the model vibrational modes of frequencies in these regions. Calculated frequencies were scaled by 0.97 based on matching observed and calculated sum frequency generation (SFG) modes (Lee et al. 2012), and we found that the observed 380 cm⁻¹ vibration is due to C1-C2-C3, C1-O1-C4 and C1-O5-C5 bending modes in our calculations. The 1,096 cm⁻¹ Raman peak consists of C1-O1, C1-C2 and C4-C5 stretches, C1-O1-C4 angle bends, and C5-C6H₂ twists. Note that Raman intensities were not calculated, so we cannot definitively assign these modes to Raman bands nor examine the intensity ratios, but these motions do provide a hypothesis for amorphization of cellulose I. The fact that the internal ring modes of the 380 cm⁻¹ peak lose Raman intensity suggests that the glucose residues may be distorting from the chair conformation. The predicted motions of the 1,096 cm⁻¹ peak involve both the glycosidic and hydroxymethyl torsions, which will affect the order both along and between the glucose residue chains.

The C1 and C4 atoms involved in the glycosidic torsions are in agreement with observed values well within the accuracy of our methodology. This could have three explanations. One, the δ¹³C signals from these atoms could be relatively insensitive to the 10 % error in Φ and Ψ mentioned above in our model. Two, there could be compensating errors in computing the δ¹³C values, allowing an inaccurate structure to provide fortuitous agreement with experiment. Three, model Φ and Ψ values could be more accurate than Table 1 would suggest.

The first explanation was tested by varying the Φ and Ψ values of a cellulose dimer model through (−86°, 151°), (−87°, 156°) and (−88°, 161°) with other atomic coordinates held constant. The C1 and C4 δ¹³C values varied by 2.6 and 1.1 ppm over this range, respectively. In contrast, ranges of approximately 15 and 12 ppm for the C1 and C4 δ¹³C values have been reported (Suzuki et al. 2009), but the range of Φ and Ψ values was from 0 to 360° in that study. The Suzuki et al. (2009) results within the window of Φ and Ψ values we investigated show small δ¹³C dependence, so the current results are actually consistent with this previous study. Hence, one can conclude that the C1 and C4 chemical shifts are relatively insensitive to changes in the glycosidic torsion angles within our computational accuracy. The second argument can be discounted because similar methodologies applied to simple molecules with known structures (e.g., cellobiose) predict ¹³C chemical shifts to within approximately 2 ppm. The third explanation cannot be tested with our methods because of the insensitivity of the C1 and C4 δ¹³C values mentioned above. We conclude that uncertainty in measuring Φ and Ψ angles and the actual variation in these cellulose structural parameters forces one to accept this level of uncertainty.

In a similar vein, the C4, C5 and C6 δ¹³C values should be sensitive to the χ₁ and χ₂ torsion angles (Suzuki et al. 2009). The fact that the calculated δ¹³C values for C4, C5 and C6 match experiment closely (Fig. 4) is another indication that the tg conformation predominates and that the model structures predicted are consistent with experiment. For example, the Iα/tg/NetA C5 and C6 chemical shifts differ from experiment by <1 ppm, whereas as those for the Iα/tg/NetB, gt, and gg conformations are in error by up to 4.2, 6.2 and 4.8 ppm, respectively. The δ¹³C4 value changes from 87.1 ppm in the tg conformation to 80.6 ppm in the gg conformation compared to the 87.9 and 88.7 ppm values of Sternberg et al. (2003). Consequently, our results strongly support the tg conformation.

The sensitivity of the C5 and C6 chemical shifts to hydroxymethyl group rotations was also tested by varying χ₁ from 160 to 170°. NMR calculations were performed without any further energy relaxation. The 10° difference between the 160 and 170° conformations resulted in changes of 1.2 and 0.4 ppm for the C5 and C6 atoms, respectively. These results are consistent with Suzuki et al. (2009) who predicted that the C5 atom should be most sensitive to changes in χ angle in this range. This magnitude of change should be observable in the ¹³C NMR spectra but is within the uncertainty of our computational methodology, so we cannot use the calculated δ¹³C values to further refine the expected χ₁ and χ₂ torsion angles.