# A DFT study of vibrational frequencies and ^{13}C NMR chemical shifts of model cellulosic fragments as a function of size

## Abstract

Plane wave and molecular orbital density functional theory calculations with periodic models and oligomer fragments based on Iβ cellulose showed relationships among hydrogen bond (H-bond) lengths, angles, energies, and vibrational frequencies. Significantly, the ^{13}C NMR chemical shifts (δ^{13}C), glycosidic and hydroxymethyl torsion angles, H-bond vibrational frequencies, and H-bond geometries results all suggest the predominance of the crystallographic structure C rather than structure A of Iβ cellulose as reported by Nishiyama et al. (J Am Chem Soc 124(31):9074–9082, 2002). The results reported herein also clarified that the δ^{13}C and δ^{13}C′ data from Erata et al. (Cellul Commun 4:128–131, 1997) correspond to δ^{13}C from the origin and center chains of cellulose, respectively. Moreover, this work discusses the use of cellulose oligomer fragments for their potential use in understanding cellulose assembly.

## Keywords

Density functional theory (DFT) Nuclear magnetic resonance spectroscopy (NMR) Infrared spectroscopy Natural bond order analysis (NBO) Thermodynamics Cellulose Hydrogen bonding Vibrational modes Vibrational spectroscopy Cellulose structure## Introduction

Plant cell walls (PCW) are heterogeneous structures composed of polymers such as cellulose, hemicellulose, pectins, lignins, and glycoproteins (McQueen-Mason and Cosgrove 1994; Cosgrove 2005). Ongoing investigation of the formation, molecular structure, and the potential interactions among PCW polymers could provide valuable and necessary information about PCW architecture. Improved understanding of the H-bonds occurring among PCW polymers could guide the exploration of methods to break these bonds, and could lead to the development of methods to more efficiently decompose cellulose into smaller units during biofuel production (Shen and Gnanakaran 2009). Moreover, understanding the H-bonds that occur among PCW polymers is necessary for understanding how these polymers interact during PCW assembly. The work herein used molecular orbital, and projector-augmented planewave density functional theory (DFT) (Hohenberg and Kohn 1964; Kohn and Sham 1965; Kresse and Furthmüller 1996) calculations to study the “Network A” (Net A) hydrogen-bonding (H-bonding) network that has been suggested as the dominant form in cellulose (Nishiyama et al. 2002).

The goal of the present work was to develop a method that provided precise ^{13}C NMR chemical shifts (δ^{13}C), then to use the method to study the relationships among O–H vibrational frequencies and molecular structure (i.e., glycosidic and hydroxymethyl torsion angles, and H-bond lengths, strengths, and angles). These calculated relationships between spectroscopic parameters and structure could then be used to interpret spectra of disordered cellulose. Although our work focused on the interactions of oligomer models of cellulose and the comparison of our results with cellulose data, the methods developed herein could be applicable for the study of cellulose surfaces, disordered cellulose, cellulose assembly, and cellulose interactions with other PCW components.

Crystallographic and spectroscopic data can aid in the elucidation of cellulose structure; however, relying solely on crystallographic data could be inadequate for the elucidation. Both ^{13}C NMR data (Blackwell 1977; Erata et al. 1997; Atalla 1999; Nishiyama et al. 2002; Sternberg et al. 2003; Zhao et al. 2007), and infrared (IR) data (Blackwell et al. 1970) are available for cellulose. However, the OH-stretching region (3,300–3,800 cm^{1}) obtained from IR spectroscopy is difficult to interpret (Blackwell et al. 1970), because these frequencies depend on both H-bond lengths and strengths, and IR bands in the OH-stretching region are broad (Gallina et al. 2006).

Our previous work developed a method for precisely calculating NMR chemical shifts (δ^{13}C) for cellulose-proxy models (Kubicki et al. 2013), which were compared with NMR data (Erata et al. 1997; Sternberg et al. 2003). We explored the relationships among torsion angles, H-bonding and δ^{13}C.

The χ-torsion angles may be causing the observed upfield shift of δ^{13}C4 for cellulose chains on the surfaces and/or disordered cellulose (Wickholm et al. 1998; Newman and Davidson 2004; Malm et al. 2010; Fernandes et al. 2011; Harris et al. 2012). We report the effect of the hydroxymethyl torsion angles χ_{1} (O5–C5–C6–O6) and χ_{2} (C4–C5–C6–O6) on the calculated C4, C5, and C6 δ^{13}C. Moreover, this study reports the precision of the correlation between the calculated and experimental glycosidic torsion angles (Φ and Ψ), and the correlation precision between the calculated and experimental H-bond O–O lengths and O–H–O angles of the intramolecular O2-H–O6 and O3–H–O5, and intermolecular O6–H–O3 H-bonds (Nishiyama et al. 2002). In addition, we compared the calculated and ring-puckering parameter (θ) (Cremer and Pople, 1975) of our periodic Iβ cellulose model with the experimental θ values of Iβ cellulose crystal structures A and C that were reported by Nishiyama et al. (2002). Furthermore, this work compared the periodic model results with the crystallographic data (Nishiyama et al. 2002) for the intermolecular distances between the origin chain axial H atom on C1 and one of the two methylene H atoms on the hydroxymethyl group of the center chain C6 atom. The systematic study of torsion angles, bond distances, and θ, could lead to refinements in our knowledge of cellulose structure and ultimately to improved understanding of PCW chemistry.

## Methods

*tg*/Net A (Fig. 1) H-bond network. (See the Electronic Supplemental Information (ESI) for the Cartesian coordinates of the initial structures.) Hereafter, except for P, the models are referenced by the number of the glucose residues in their respective oligomeric molecules and the number of oligomers in the model. For example, the 4 × 3 model consists of three cellotetraose models arranged to give a

*tg*/Net A H-bond network between each cellotetraose. In addition, 4 × 3 × 3 and P models (Fig. 2) were used to determine if oligomer stacking affected the calculated results. The 4 × 3 × 3 and P models were more realistic than the smaller glucan chain models with respect to cellulose, so the 4 × 3 × 3 and P models could have provided results that were more precise than the smaller models. The 4 × 3 × 3 model consists of three stacked 4 × 3 models, and the P model unit cell consists of four stacked 4 × 3 models.

The shaded glucose residues in each model shown in Fig. 2 are those that were analyzed for their ^{13}C NMR chemical shifts, glycosidic and hydroxymethyl torsion angles, and their H-bond lengths, angles, energies, and vibrational frequencies. We chose a central glucose residue unit or units within each model for analysis, because those units would best represent the bulk structure of cellulose, as compared with the glucose residues on the reducing or non-reducing ends of the glucans. The single glucose unit models (i.e., 1 × 1, 1 × 2, and 1 × 3) are obvious exceptions to this procedure. Unless otherwise stated, the notation that we used for the torsion angles and atom numbers agrees with the IUPAC notation found at a www.chem.qmul.ac.uk/iupac.

Labeling schemes from the experimental NMR data include two sets of ^{13}C NMR chemical shifts (δ^{13}C) labeled as either C- or C′-nuclei (Erata et al. 1997; Sternberg et al. 2003). For the crystallographic data, the C-atoms were categorized based on whether they were from the center or origin chain of cellulose (Nishiyama et al. 2002). Based on our results, the C NMR data correlated with the crystallographic data for the origin chain, whereas the C′ NMR data more precisely correlated with the crystallographic data from the center chain. Therefore, throughout the current work, the nomenclature for the periodic model results for the origin (P[o]) and center (P[c]) chains will correspond to those conventions (i.e., the P[o] results correspond to the δ^{13}C data, and the P[c] results correspond to the (δ^{13}C′ data).

The glucose, cellobiose, cellotriose, and cellotetraose-based models and the 4 × 3 × 3 model (Fig. 2) were energy minimized using Gaussian 09 (G09) (Frisch et al. 2009). We used the meta hybrid density functional M05-2X coupled with the 6-31G(d,p) basis set (Hohenberg and Kohn 1964; Kohn and Sham 1965; Krishnan et al. 1980; Clark et al. 1983; Zhao et al. 2006; Papajak et al. 2011). We chose the M05-2X meta-hybrid density functional based on the utility of this method to calculate the properties of biomolecules (McNamara and Hillier, 2007). In addition, M05-2X energy minimization calculations of the 3 × 3 Net A model showed that the H-bond O6–O2 and O3–O5 distances and their respective O–H–O vibrational frequencies converged for the 6-31G(d,p); additional basis functions had negligible effect on the calculated bond lengths and frequencies (ESI Figure S1).

For the DFT calculations with G09, the initial geometry of each model was subjected to one of two distinct energy minimization methods in the gas phase. One optimization was performed without symmetry or atomic constraints for a fully relaxed minimization (FR). The other method, fixed coordinate or FC, was performed using the keyword *opt* = *modredundant* in G09 (Frisch et al. 2009). The FC method froze the C1–C6 and the O5 atoms of each glucose residue, while allowing the hydroxyl groups and H-atoms to minimize unconstrained. The FC method forced the models to maintain the lateral position of the glucan chains.

Subsequent frequency calculations in G09 using M05-2X/6-31G(d,p) followed each optimization to determine the vibrational frequencies of each model, and to determine if each structure resided at a potential energy minimum. The resulting frequencies were not scaled. Note that although the 4 × 3 × 3 model underwent FR minimization, the model did not converge to an energy minimum and did not undergo a frequency calculation; however, when the calculation was terminated, the energy of this model was changing by less than 0.01 kJ/mol; therefore, the structure was likely near a stationary point on its potential energy surface (PES).

The P model was energy minimized using the Vienna Ab-Initio Simulation Package (VASP) (Kresse and Furthmüller 1996). Projector-augmented planewave pseudopotentials were used with the PBE gradient-corrected exchange correlation function for the 3D periodic DFT calculations. The choice of electron density and atomic structure optimization parameters were based on literature values (Bućko and Hafner 2005; Li et al. 2011). An energy cut-off of 77,190 kJ/mol was used with an electronic energy convergence criterion of 9.6 × 10^{−6} kJ/mol. Atomic structures were relaxed until the energy gradient was less than 1.93 kJ/mol/Å. 2 × 2 × 2 k-point samplings were used. Atoms were first allowed to relax with the lattice parameters constrained to the experimental values, and then the atoms and lattice parameters were allowed to relax to obtain the structures; energies and spectroscopic properties reported herein. The dispersion-correction parameters were 40 Å for the cutoff distance (Bućko and Hafner 2005), and 0.75 for the scaling factor (s_{6}) and 20 for the exponential coefficient (d) in the damping function (Grimme 2006). Energy minimization was carried out in two stages. First, atoms were allowed to relax with the lattice parameters constrained to experimental values. Second, the atoms and lattice parameters were allowed to relax simultaneously. A subsequent frequency calculation provided the vibrational frequencies for the fully relaxed periodic model; the resulting frequencies were not scaled.

- 1.
the crystallographic structure (N) was not minimized (i.e., the structure was as determined by Nishiyama et al. 2002);

- 2.
the H-atoms were allowed to relax but the C and O atoms were constrained (FC);

- 3.
all atoms were allowed to relax during the minimization (FR). These three models were used to determine if the crystallographic assignments were accurate relative to

^{13}C NMR data and to determine the crystallographic H-atom assignments; or - 4.
classical MD simulation (10 ns) of a Iβ 6 × 6 cellulose microfiber followed by energy minimization both using the CHARMM force field (Guvench et al. 2009; Raman et al. 2010). This model (MM) is a cluster that was extracted from the center of a 6 × 6 cellulose model (Zhao et al. 2013). We used the MM 3 × 3 × 4 model, taken from the center of the microfibril to approximate the structure of cellulose as predicted by CHARMM.

In order to evaluate the results from these methods for reproducing details of cellulose structure, we have subjected these periodic DFT-minimized models to the same ^{13}C NMR calculations and comparison against experimental ^{13}C NMR data as the G09-minimized models. We also used G09 to calculate the^{13}C NMR chemical shifts (δ^{13}C) for the MM model that was derived from an MD simulation for comparison with the NMR data.

Single-point GIAO NMR calculations (Wolinski et al. 1990; Schreckenbach and Ziegler 1995; Cheeseman et al. 1996; Adamo et al. 1998; Buhl et al. 1999; Karadakov 2006) using the mPW1PW91/6-31G(d) method and a multistandard approach using methanol as the internal standard determined the (δ^{13}C) of the C nuclei in each model (Sarotti and Pellegrinet 2009; Kubicki et al. 2013). The subsequent calculation of δ^{13}C with a method that differs from the method used for the energy minimization calculation is a standard procedure (Cheeseman et al. 1996; Frisch et al. 2009; Sarotti and Pellegrinet 2009). Furthermore, our recent paper showed that energy minimization calculations with projector-augmented planewave pseudopotential with the PBE gradient-corrected exchange correlation functional in VASP followed by single-point NMR calculations using the mPW1PW91/6-31G(d) level of theory in Gaussian 09 reproduced δ^{13}C within ±2 ppm for crystalline Iα and Iβ cellulose (Kubicki et al. 2013).

In water, the experimental NMR chemical shift of the C-atom of methanol (δ^{13}C) is 49.5 ppm (Gottlieb et al. 1997); this value was necessary for the multistandard calculations (Sarotti and Pellegrinet 2009). To improve the precision of the NMR results, these calculations included the self-consistent reaction field (SCRF) method integral equation formalism polarized continuum model (IEFPCM) with the permittivity of water (Cancès et al. 1997; Gogonea 1998). However, the NMR calculation for the extended clusters that were extracted from the periodic models were in the gas-phase rather than a SCRF due to computational costs; moreover, because the C-nuclei of interest in the periodic models are solvent inaccessible (Fig. 2), use of the SCRF was unnecessary for this model. Comparison of the calculated and experimental δ^{13}C shifts determined which models correlated best with the NMR data for cellulose Iβ (Erata et al. 1997; Sternberg et al. 2003).

In addition, following the energy minimization calculations with G09 or VASP, M05-2X/6-31G(d,p) NBO population analysis calculations determined the binding energies, E(2) (Panduranga et al. 2011) of the pertinent H-bonds for the oligomer models as well as extracted clusters from the periodic models (Glendening and Weinhold 1997, 1998). These analyses involved only the intra- and intermolecular H-bonded glucose residues (Fig. 1) shown in the shaded regions of Fig. 2.

The calculated NBO E(2) energies provide the energy in kJ/mol for the delocalization of electrons from an electron donor (e.g., O-atom) to an electron acceptor (e.g., hydroxyl-H atom). These calculated E(2) energies are not H-bond energies, but we related the two quantities using the following method. For example, the M05-2X/6-31G(d,p) NBO E(2) energy for the delocalization of electrons from the O-atom of one water model to a H-atom of a second H_{2}O model in a water dimer is 79 kJ/mol. The counterpoise corrected (Boys and Bernardi, 2002) M05-2X/6-31G(d,p)//M05-2X/6-31G(d,p) binding energy for the water dimer (i.e., ΔE = E_{water dimer} + E_{BSSE} − 2E_{water monomer}, where E_{BSSE} is the energy of the counterpoise correction for basis set superposition error) is 23 kJ/mol, which is comparable to the range of reported experimental data of 20.4–12.3 kJ/mol with reported uncertainties of approximately ±3.5 kJ/mol (Fiadzomor et al. 2008). Therefore, the NBO E(2) energies are 4–7× greater than the experimentally observed H-bond energy for a water dimer, so, we divided the NBO E(2) energies by a factor of 5.5 (i.e., the average of 4 and 7) for this work.

The calculated H-bond O–O lengths, O–H–O angles, and O–H vibrational frequency results in addition to the torsion angle results for Φ (O5–C4–O–C1′), Ψ (C4–O–C1′–C5′), χ_{1} (O5–C5–C6–O6), and χ_{2} (C4–C5–C6–O6) were obtained with GaussView version 5.0.9 (Frisch et al. 2009) from the energy minimized models. The ring-puckering parameter (θ) (Cremer and Pople 1975) for the periodic model was calculated using the online θ calculator of Shinya Fushinobu (http://www.ric.hi-ho.ne.jp/asfushi/) and was compared with the experimentally derived θ values of Nishiyama et al. (2002).

To evaluate the precision of the relationship between the δ^{13}C results and data, we used the mean unsigned error (MUE) and root mean-squared error (RMSE), as used previously (Watts et al. 2011).

Synthetic infrared spectra were generated using the results from G09 and the program Molden (Schaftenaar and Noordik 2000).

## Results and discussion

### NMR results

#### NMR benchmarking

This section focuses on the structures of the models and the correlations between the calculated ^{13}C chemical shifts (δ^{13}C) with the cellulose NMR data. Developing a method that accurately reproduces experimental δ^{13}C and crystallographic data is important, because if the method can precisely reproduce those data, then we can have greater certainty in the accuracy of our study of cellulose H-bond chemistry. Furthermore, if the method can reproduce the spectroscopic and structural data precisely, then the method could be used for modeling other systems of interest that are not as well constrained by experimental data.

^{13}C for both cellulose Iα and Iβ when subjected to NMR calculations in Gaussian 09 (Frisch et al. 2009). However, Table 1 shows that the precision of the calculated δ

^{13}C with respect to the NMR data (Erata et al. 1997; Sternberg et al. 2003) depends on the geometry of the Iβ cellulose model. In comparison with observed interatomic distances in Iβ cellulose structure A (Nishiyama et al. 2002), we have determined the glycosidic O, the C1–C1 inter-chain, and C1–C1 inter-layer repeat distances for our models. The experimental values are 10.38, 8.20 and 7.78 Å, respectively. For the P FR model, the measured results are 10.40, 8.14, and 7.55 Å, respectively, which agree well with the results for structure A (Nishiyama et al. 2002). In general, the molecular cluster models have similar values, where the three distances for the 3 × 1 through 4 × 3 FR, and 4 × 3 × 3 models are 10.44(±0.07), 8.08(±0.19), and 7.71 (for the 4 × 3 × 3 model) Å, respectively.

δ^{13}C data of Sternberg et al. (2003) compared with mPW1PW91/6-31G(d)-calculated δ^{13}C results for the CHARMM (MM), fully constrained (N), the C-atom constrained periodic (FC), and the fully relaxed periodic (FR) Iβ cellulose models that all had initial structures based on the crystallographic data of Nishiyama et al. (2002)

C-nuclei (origin chain) | δ | δ | δ | δ | δ | δ |
---|---|---|---|---|---|---|

C1 | 105.6 | 93.4 | 95.7 | 105.6 | 105.9 | 105.1 |

C2 | 71.2 | 64.9 | 64.3 | 70 | 70.6 | 70.2 |

C3 | 74.8 | 64.4 | 63.4 | 72.3 | 72.9 | 70.6 |

C4 | 88.7 | 74.1 | 76.6 | 87 | 87.1 | 87.1 |

C5 | 72.2 | 68.6 | 63.7 | 71.9 | 72.4 | 71.6 |

C6 | 65.4 | 50.1 | 53.6 | 64.9 | 65.1 | 65.0 |

Slope/MUE (ppm) | 0.91/10.4 | 0.98/10.1 | 1.01/1.0 | 1.01/0.8 | 1.01/1.4 | |

y-intercept (ppm)/RMSE (ppm) | −3.19/11.2 | −8.57/10.3 | −2.20/1.4 | −1.41/1.1 | −2.22/1.9 | |

r | 0.902/15.3 | 0.981/12.1 | 0.996/2.5 | 0.996/1.9 | 0.991/4.2 | |

C′-nuclei (center chain) | δ | δ | δ | δ | δ | δ |

C1′ | 103.8 | 99.3 | 95.6 | 103.8 | 103.9 | 105.1 |

C2′ | 71.2 | 63.5 | 64.6 | 69.9 | 70.5 | 70.2 |

C3′ | 74 | 64.5 | 63.4 | 73 | 73.5 | 70.6 |

C4′ | 87.9 | 74.3 | 76.5 | 85.9 | 86.2 | 87.1 |

C5′ | 71.2 | 64.3 | 63.8 | 70.7 | 70.9 | 71.6 |

C6′ | 65.7 | 54.4 | 53.5 | 64.7 | 65.2 | 65.0 |

Slope/MUE (ppm) | 1.03/8.9 | 1.02/9.4 | 1.01/1.0 | 1.00/0.6 | 1.05/1.3 | |

y-intercept/RMSE (ppm) | −12.37/9.4 | −10.78/9.6 | −2.08/1.2 | −0.93/0.8 | −5.05/1.6 | |

r | 0.966/13.6 | 0.976/12.2 | 0.998/2.0 | 0.998/1.7 | 0.992/3.4 |

^{13}C results from the N model, the non-minimized model that Nishiyama et al. (2002) reported as the likely structure of cellulose Iβ, do not correlate precisely with the NMR data (Table 1). The results in Table 1 show that it was necessary to fully relax (model FR) model N during minimization in order to reproduce the experimental δ

^{13}C. The δ

^{13}C RMSEs for the FR, FC, and N periodic models were, 1.1, 1.4, and 10.3 ppm, respectively; therefore, N shows the poorest correlation of the three models with the NMR data of cellulose Iβ. In addition, model FC, which was minimized with relaxed H-atoms did not replicate the NMR data as precisely as the FR model did (Table 1 and Fig. 3). The results from model FC suggest that there was likely uncertainty in the assignment of the H-atom positions in model N by Nishiyama et al. (2002), and the results from model FR show that there was also uncertainty in the C- and O-atom positions in model N.

Notably, the model extracted from the MD simulation (MM) exhibited poorer precision with regard to the δ^{13}C data than the N, FC, or FR models (Table 1 and Fig. 3). Therefore, although the computational cost of obtaining the MM model was low, the δ^{13}C results obtained from that models suggest that it is inaccurately reproducing the structure of cellulose.

In addition, we found that the δ^{13}C NMR results from the origin chain of the periodic structure (P[o]) (Fig. 2) correlated most precisely with the C-atom data (Erata et al. 1997; Sternberg et al. 2003), and the results from the center chain of the periodic structure (P[c]) correlated most precisely with the C′-atom data (Table 1 and Fig. 3). These assignments for the origin and center chains were ambiguous in Erata et al. (1997) and Sternberg et al. (2003); that is, it was uncertain whether the C-atom assignments were for particular glucose residues within a single glucan chain in cellulose, or whether the δ^{13}C represented C-atoms from different glucan chains within cellulose. Our results provided a clarified interpretation of the NMR data.

Although the oligomer fragment models (Fig. 2) used for this work do not represent crystalline cellulose, they were used to model structures that could occur during cellulose assembly or degradation. When compared with the NMR data, the results from models that were energy minimized with G09 using the FC models did not correlate as precisely as the fully relaxed (FR) model results did (Table ESI 1).

Among the G09-minimized models, the δ^{13}C results for FR 3 × 1 exhibited the most precise correlation (i.e., linearity and low errors) with the NMR data for cellulose Iβ (Table 1 and Fig. 3). This result suggests that once a glucose unit in a cellulose chain is H-bonded to other cellulose chains, it could exhibit δ^{13}C that are similar to those of the bulk crystalline cellulose. Although the periodic models (FC and FR) provided the most precise correlation with the NMR data, the 3 × 1 provided remarkable precision, considering its structural simplicity relative to cellulose. We conclude that the computational methodology used here to predict δ^{13}C values for cellulose is precise to within ± 2 ppm, based on the maximum errors (Table 1), provided that the model structure is realistic.

### Comparison to crystalline cellulose ^{13}C NMR

_{1}and χ

_{2}torsion angles for the 3 × 1, P, and MM models, and the data for the crystal structures A and C (Nishiyama et al. 2002). Table 2 shows that the χ

_{1}and χ

_{2}results from model FR P agree more precisely with experimental origin chain data from structure C but not structure A (Nishiyama et al. 2002), than do the results from model FR 3 × 1 and MM. These results for the origin chain further corroborate with the NMR results. Although neither of the χ

_{1}or χ

_{2}results agree precisely with the corresponding center chain data from structure A or C, the δ

^{13}C values of the center chain reproduce observed δ

^{13}C values as precisely as the origin chain. Thus, the discrepancy between the structure A and C χ

_{1}and χ

_{2}with the calculated P values could be due to error in the calculated structure, error in the experimentally-derived χ

_{1}and χ

_{2}values, or insensitivity of the δ

^{13}C to χ

_{1}and χ

_{2}.

Model | Origin chain | ||||
---|---|---|---|---|---|

Φ (°) | Ψ (°) | χ | χ | θ | |

P FR | −93 | −144 | 165 | −76 | 1.2 |

3 × 1 FR | −94 | −139 | 154 | −87 | 6.1 |

MM | −90 | −146 | 171 | −66 | 7.6, 0.7 |

A | −99 (2) | −142 (1.9) | 170 (0.3) | −70 (0.3) | 10.2 |

C | −94 (2.0) | −144 (1.1) | 164 (0.2) | −77 (0.2) | 3.2 |

Model | Center chain | ||||
---|---|---|---|---|---|

Φ (°) | Ψ (°) | χ | χ | θ | |

P FR | −94 | −145 | 168 | −74 | 2.8 |

3 × 1 FR | −93 | −140 | 154 | −87 | 6.1 |

MM | −98 | −144 | 180 | 75 | 2.2, 10.9 |

A | −89 (2.0) | −147 (1.6) | 158 (0.3) | −83 (0.3) | 6.7 |

C | −92 (2.0) | −146 (1.4) | 161 (0.2) | −80 (0.2) | 3.2 |

_{1}and χ

_{2}for periodic model FR fit the data from structure C better than they do for structure A. The initial structure of model P was that of the experimentally-based structure of structure A obtained from Nishiyama. Our results indicate that during the energy minimization calculation, structure A became more like structure C. Nishiyama et al. (2002) chose structure A over structure C because structure C had a close contact (2.138 Å) between the axial H on C1 of the origin chain (C1H

*o*) and the one of the methylene H atoms of the center chain (C6H

*c*) of cellulose Iβ. However, the DFT-D2 energy-minimized structure A of Nishiyama et al. (2002) with full atomic and lattice parameter relaxation (i.e., model P FR), the interaction distance between the H-atoms of C1H

*o*and C6H

*c*was 2.139 Å, which differs from the structure C distance by only 0.001 Å (Fig. 4). Thus, the DFT-D2 calculations are predicting that structure C is lower in energy than structure A and that the 2.138 Å C1H

*o*–C6H

*c*H-atom distance does not cause prohibitive steric interactions. This result suggests that structure C from Nishiyama et al. (2002) more likely describes the structure of cellulose Iβ than structure A does.

Significantly, further evidence that supports structure C as being a better approximation of cellulose Iβ structure than structure A from Nishiyama et al. (2002) is the ring puckering parameter (θ). Nishiyama et al. (2002) reported θ for the origin chain and center chains of 10.2 and 6.7 for structure A (Table 2), respectively, and θ of 3.2 for both the origin and center chains of structure C. We reported periodic DFT-calculated θ values (model P FR) of 1.2 and 2.8 for the origin and center chains, respectively (Kubicki et al. 2013). Therefore, the calculated θ results from P FR correlate better with those from structure C than they do with those from structure A of Nishiyama et al. (2002).

### The effect of hydroxymethyl torsion angles (χ) on δ^{13}C4, δ^{13}C5, and δ^{13}C6

^{13}C4 (Wickholm et al. 1998; Newman and Davidson, 2004; Malm et al. 2010; Fernandes et al. 2011; Harris et al. 2012) values are upfield shifted from the bulk cellulose by 4 to 5 ppm. This observation has been interpreted as rotations of χ

_{1}and χ

_{2}. Previous calculations have shown that δ

^{13}C5 and δ

^{13}C6 are relatively insensitive to changes in χ

_{1}and χ

_{2}(Kirschner and Woods 2001; Gonzalez-Outeiriño et al. 2006). Therefore, using the energy minimization and NMR calculation methodologies that produced precise agreement between calculated and observed δ

^{13}C chemical shifts, we explored the relationship between those δ

^{13}C and the hydroxymethyl torsion angles (Fig. 5). For these comparisons, we used all of the models (i.e., FR, 3 × 1 and larger), rather than just those that correlated favorably with the NMR data, because the other models used for this work could be useful for elucidating the structure of other cellulose allomorphs, surface or internal glucans, and disordered cellulose. See ESI Table S2 for the numerical results plotted in Fig. 5.

Figure 5 shows the relationship between the torsion angles χ_{1} and χ_{2} with the calculated δ^{13}C4, δ^{13}C5, and δ^{13}C6 for the DFT minimized models. These results show that χ_{1} and χ_{2} range from 153° to 170° and -90° to -72°, respectively for the models ranging in size from 3 × 1 to 4 × 3, the 4 × 3 × 3 model, and the fully relaxed P model. However, δ^{13}C4, δ^{13}C5, and δ^{13}C6 remain relatively constant over the ranges of χ_{1} and χ_{2}: 86.2(±1.4), 71.0(±0.9), and 63.8(±1.3) ppm, respectively. These standard deviations approach the RMSE of the fully relaxed periodic model (Fig. 3), so ascertaining these shifts using DFT could be problematic. These results reaffirm that δ^{13}C5, and δ^{13}C6 are insensitive to χ_{1} and χ_{2} rotations for the *tg* conformation (Kirschner and Woods 2001; Gonzalez-Outeiriño et al. 2006) and do not support the hypothesis that χ_{1} and χ_{2} change are necessarily the cause the observed upfield shift for δ^{13}C4 (Wickholm et al. 1998; Davidson et al. 2004; Malm et al. 2010; Fernandes et al. 2011; Harris et al. 2012).

Prior NMR and molecular dynamics studies suggested that polar solvents such as water can disrupt H-bonding networks in oligosaccharides and cause changes in hydroxymethyl torsion angles (Kirschner and Woods 2001; Gonzalez-Outeiriño et al. 2006). The current work did not incorporate explicit solvent molecules and the initial structures were based on crystallographic data. Calculations on the effects of explicit solvation on χ_{1} and χ_{2} angles in periodic and oligomer proxies of cellulose would be worthwhile, but are outside the scope of this study.

### Glycosidic Torsion Angles (Φ,Ψ) effect on δ^{13}C1 and δ^{13}C4

The Φ and Ψ results from FR P origin and center chains agree more precisely with the data from structure C than with the data for structure A (Table 2) from Nishiyama et al. (2002). The Φ and Ψ results further suggest the full relaxation of the periodic model during its energy minimization produced torsion angles that agree better with the experimental data of structure C than with the data from structure A. The P model Φ and Ψ, χ_{1} and χ_{2}, and δ^{13}C results all suggest that structure C, rather than structure A, is the observed structure of cellulose Iβ. The FR 3 × 1 model results for the torsion angles did not agree with the data as well as the FR P model results did; this discrepancy could be due to the lack of intermolecular interactions between glucan chains in the FR 3 × 1 model. The MD simulation Φ and Ψ results from model MM also fit the data for structure C with good precision, except that Φ (−98°) for the center chain underestimates Φ (−92°) for structure C by 6°.

The differences between the non-minimized structure A of Nishiyama et al. (2002) and the DFT-minimized structure A that better correlated with structure C of Nishiyama et al. (2002) are subtle. When superimposed, the shaded regions of model P (Fig. 2) from structure A and the P FR model that better correlates with structure C showed minor differences in ring positions, hydroxymethyl torsion angles, and hydroxyl group torsion angles. However, based on the NMR results, torsion angles, θ, and the C1H*o*–C6H*c* H-atom distance, as discussed in the previous paragraphs, these minor structural differences culminated in structure C correlating more precisely to the structure of cellulose Iβ than structure A does.

The glycosidic torsion angles from model FR P agree with both the crystallographic data for the glycosidic torsion angles and NMR data for cellulose Iβ; however, although the results from FR 3 × 1 agree precisely with the NMR data, they do not agree as precisely with the crystallographic data for the glycosidic torsion angles, likely because the FR 3 × 1 model did not capture the chemistry of cellulose. However, the small glucan chain models that we used for this work could provide data that is useful for interpreting data obtained from fragments of cellulose that occur during cellulose degradation or to better understand cellulose assembly, because cellulose would be unlikely to have the properties of crystalline cellulose during formation and degradation.

^{13}C1 and δ

^{13}C4 for the DFT minimized models. The calculated glycosidic torsion angles for all of the models used for our work range from −101º to −88º for Φ

_{calc}, and from −150º to −133º for Ψ

_{calc}, and the calculated δ

^{13}C1 and δ

^{13}C4 range from 83–87 to 101–106 ppm. (ESI Table S3 provides the chemical shift and glycosidic torsion angle results shown in Fig. 6.) Although δ

^{13}C1 and δ

^{13}C4 both vary with Φ and Ψ, the variation is scattered. These results suggest that it could be problematic to correlate

^{13}C NMR spectra with glycosidic torsion angles. Consequently, we did not pursue using them as a parameters for linking

^{13}C NMR and structure in disordered cellulose.

### H-bond lengths, angles, energies, and vibrational frequencies

H-bond geometries of the DFT minimized models (Fig. 2) could represent stages in cellulose crystallization or amorphization. ESI Table S4 shows the numerical results for the H-bond lengths, angles, energies, and vibrational frequencies discussed in this section.

O–O H-bond lengths and O–H–O H-bond angles for the intramolecular (O2–H–O6 and O3–H–O5) and intermolecular (O6–H–O3) H-bonds of the fully relaxed (FR) periodic (P) models, the FR 3 × 1 model, the molecular dynamics derived model (MM), and the data from Nishiyama et al. (2002) structure A

Model | O2–O6 (Å) | O2–H–O6 (°) | O3–O5 (Å) | O3–H–O5 (°) | O6–O3 (Å) | O6–H–O3 (°) | RMSE O–O (±Å) | RMSE O–H–O (± °) |
---|---|---|---|---|---|---|---|---|

N A[o] | 2.77 | 159 | 2.76 | 137 | 2.89 | 144 | ||

FR P[o] | 2.75 | 172 | 2.76 | 137 | 2.67 | 167 | 0.13 | 15 |

MM[o] | 2.77 | 166 | 2.66 | 152 | 2.96 | 155 | 0.07 | 11 |

N A[c] | 2.87 | 165 | 2.71 | 162 | 2.71 | 157 | ||

FR P[c] | 2.68 | 169 | 2.74 | 163 | 2.73 | 166 | 0.11 | 6 |

MM[c] | 2.78 | 165 | 2.84 | 126 | 2.95 | 146 | 0.17 | 22 |

FR 3 × 1 | 2.85 | 162 | 2.84 | 153 | n.a. | n.a. |

For the intramolecular H-bond O3–O5 distance and H-bond O3–H–O5 angle, Table 3 shows that there is precise agreement between the results (FR P) and data (N) structure A (A) for both the origin [o] and center [o] models. However, the corresponding results for the intramolecular H-bond O2–O6 and O6–O3 distances, and the intermolecular H-bond O2–H–O6 and O6–H–O3 angles do not agree as precisely as the H-bond O3–O5 distances and H-bond O3–H–O5 angles did. For instance, the H-bond O2–O6 distance (2.75 Å) result for the FR P[o] model correlates precisely with the datum from N A[o] (2.77 Å), but the H-bond O2–H–O6 angles from the FR P[o] model (172º) and N A[o] (159º) do not agree as precisely (Table 3). The MM and 3 × 1 results show similar discrepancies with the N data for the three H-bonds. If model FR P is reproducing structure C of Nishiyama et al. (2002), rather than structure A, then it could be that the H-bond O3–O5 distance and H-bond O3–H–O angle are similar for both structures A and C. The differences between the H-bond O3–O5 lengths and O3–H–O5 angles for structures A and C could be subtle, and could exhibit a degree of uncertainty, because H-bond positions are generated through modeling rather than measurements when using neutron fiber diffraction crystallography (Nishiyama et al. 2002).

Comparison of the calculated bond lengths and bond angles from the DFT-minimized models with the vibrational frequencies and the calculated energies of the H-bonds could be useful for studying cellulose assembly during biosynthesis and disordering. The calculated vibrational frequencies could provide insight into the interactions of cellulose polymers during aggregation or degradation and the energies of the H-bonds. Furthermore, because IR spectra are difficult to interpret because of broad peaks in the O–H stretching region for cellulose (Blackwell et al. 1970; Gallina et al. 2006), these calculated results could be applied to the interpretation of cellulose IR spectra. Significantly, the methods developed here could also be applied to determine the interactions that occur between other PCW polymers.

Figures 7–9 show the relationship between H-bond lengths, angles, NBO E(2) energies, and vibrational frequencies for the intramolecular O2–H–O6 (Fig. 7) and O3–H–O5 (Fig. 8) H-bonds, and the intermolecular O6–H–O3 (Fig. 9) H-bonds.

The NBO E(2) energies range from −57 to −102, −33 to −141, and −39 to −127 kJ/mol for the O2–H–O6, O3–H–O5, and O6–H–O3 H-bonds, respectively (Figs. 7a, 8a, 9a). As discussed in the Methods section, we divided the E(2) energies by 5.5 to obtain H-bond energies that compare with the available data for the energy of a water dimer H-bond. This division gives estimated O2–H–O6, O3–H–O5, and O6–H–O3 H-bond energies of −10 to −19, −6 to −26, and −7 to −23 kJ/mol, respectively. Therefore, each of these H-bonds is moderately strong, as expected for organic molecules (Jeffrey 1997). Because these E(2) energies were for a single, central glucose residue of each respective model, they provide approximate H-bond energies per glucose residue. Correlation of these H-bond energy results with the precise NMR data obtained for these models could provide information about the energy gained from cellulose assembly and the energy required to disassemble cellulose microfibrils (Zhao et al. 2013).

Figures 7a, c and 8a, c also show that intramolecular H-bond NBO E(2) energies for the 3 × 2, 3 × 3, 4 × 2, 4 × 3, and periodic structure are stronger than those for the 3 × 1 and 4 × 1 models. These results suggest that the intermolecular H-bonds in the former structures could be adding stability to the intramolecular H-bond of those models. Therefore, the intermolecular H-bonds present in cellulose microfibrils could add thermodynamic stability of the intramolecular H-bonds in the microfibril, if these trends are correct. If experimentalists are able to discern greater intramolecular H-bond energies in cellulose fragments due to intermolecular H-bonds, then it could be possible to determine if cellulose assembles (disassembles) through the addition (loss) of single chains of cellulose (e.g., 3 × 1, 4 × 1, etc.), or through the addition (loss) of fragments (e.g., 3 × 2, 3 × 3, etc.) that exhibit intermolecular H-bonds.

Generally, as H-bond O–O lengths shorten (Figs. 7a, 8a, 9a) and O–H–O bond angles become more linear (Figs. 7b, 8b, 9b), the NBO E(2) energies increase for the intramolecular O2–H–O6 and O3–H–O5 H-bonds, which agrees with general chemistry principles. However, there is no discernible relationship among the model sizes, and their respective H-bond O–O lengths (or O–H–O bond angles) and NBO E(2) energies. Therefore, the relationship between H-bond lengths, angles, and energies might be ambiguous when studying cellulose assembly and disassembly.

Figures 7b, 8b, and 9b show the relationship between the H-bond O–O lengths and the unscaled vibrational frequencies, and Figs. 7d, 8d, and 9d show analogous trends between the H-bond O–H–O angles and unscaled vibrational frequencies. The frequencies for the FR P model were calculated using VASP, while those for the 3 × 1 through 4 × 3 models were calculated with G09. In general, the 3 × 1 and 4 × 1 models, which exhibit only intramolecular H-bonds, showed higher vibrational frequencies coupled with longer H-bond O–O distances and less linear O–H–O bond angles than did the 3 × 2, 3 × 3, 4 × 2, and 4 × 3 models, which contain intermolecular H-bonds. The trends follow previously reported trends between H-bond lengths and angles with vibrational frequencies. These results suggest that it could be possible to determine if cellulose assembles (degrades) through the addition (loss) of single chains, or multiple chains that exhibit intermolecular H-bonds; higher frequency/longer H-bonds on the fragments would suggest the presence of single chains, and lower frequency/shorter H-bonds would suggest the presence of multiple chains.

The O–H–O H-bond angle results could be useful for interpreting experiment IR data, because the results from the cellulose fragment models show that the intermolecular O–H–O H-bond angles are less linear than the intramolecular H-bond angles are for the 3 × 2, 3 × 3, 4 × 2, and 4 × 3 models. A similar but less distinct trend is present for the H-bond types and their respective H-bond O–O lengths. Therefore, combining crystallographic data, IR spectroscopic data, and NMR results and data could aid in distinguishing the types of H-bonds observed with IR spectroscopy.

^{−1}range appear, and the frequencies at approximately 3,700 and 3,800 cm

^{−1}become more intense. In general, each IR frequency is attributable to multiple stretching modes, rather than a single mode. For example, the calculated frequencies for the 3 × 2 model (Fig. 10a) from approximately 3,500–3,650 cm

^{−1}all correspond to O2–H, O3–H, and O6–H stretches. However, the trends seen in Fig. 10 could be useful, if they are experimentally observable, and this method could be useful for studying the formation of intermolecular interactions among other PCW polymers.

## Conclusion

The results with the Iβ cellulose *tg*/Net A H-bond network suggest that further DFT studies using periodic structures and small models for cellulose proxies could be useful for distinguishing other H-bond networks that could be present in cellulose during assembly and/or degradation. Those studies could lead to the differentiation of ordered (crystalline), disordered (amorphous), and surface cellulose samples. In addition, the calculated and correlated relationships among H-bond lengths, strengths, angles, and vibrational frequencies, coupled with precise NMR results could aid in the differentiation of cellulose Iβ from other allomorphs of cellulose. These results also provided information about relative H-bond strengths in Iβ cellulose that could be useful for determining the mechanisms of cellulose formation and degradation. Finally, the method developed by this work could also be useful for interpreting the interactions among other plant cell wall polymers.

Based on the results of this work, we argue that Nishiyama et al. (2002) should have reported their structure C as the structure of cellulose Iβ rather than their structure A because the NMR results for FR fit the NMR data better than those data from N. In addition, the χ_{1} and χ_{2} torsion angle results for FR fit the data for structure C better than those for structure A. Moreover, the Φ and Ψ torsion angle results for FR fit the data for structure C better than those for structure A. Significantly, the presence of the 2.139 Å bond distance between the H-atoms of C1H*o* and C6H*c* for periodic model FR, which Nishiyama et al. (2002) reported for their structure C but not for their structure A. Also, the precise correlation between the calculated ring-puckering parameters (θ) with structure C but not structure A. Finally, the H-bond lengths and angles for N structure A do not agree precisely with the results from the periodic model.

Additional results from this work showed that the effect of hydroxymethyl torsion angles (χ_{1}, χ_{2}) rotation by as much as 15º had minimal effect on the δ^{13}C4, δ^{13}C5 and δ^{13}C6. However, the effect of hydroxymethyl torsion angle (Φ, Ψ) rotation had discernable effects on δ^{13}C1 and δ^{13}C4, which could be useful for data interpretation for crystalline and non-crystalline cellulose, cellulose assembly constituents, and cellulose degradation products.

## Supplementary material

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