International Conference on Gas Hydrates (ICGH) (6th : 2008)


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COMPUTATIONAL CHARACTERIZATION OF 13C NMR LINESHAPES OF CARBON DIOXIDE IN STRUCTURE I CLATHRATE HYDRATES  Saman Alavi,   Peter Dornan, and Tom K. Woo Department of Chemistry, University of Ottawa, Ottawa, ON, K1N 6N5, CANADA   ABSTRACT Nonspherical large cages in structure I (sI) clathrates impose non-uniform motion of nonspherical guest molecules and anisotropic lineshapes in NMR spectra of the guest. In this work, we calculate the lineshape anisotropy of the linear CO2 molecule in large sI clathrate cages based on molecular dynamics simulations of this inclusion compound. The methodology is general and does not depend on the temperature and type of inclusion compound or guest species studied. The nonspherical shape of the sI clathrate hydrate large cages leads to preferential alignment of linear CO2 molecules in directions parallel to the two hexagonal faces of the cages. The angular distribution of the CO2 guests in terms of a polar angle θ and azimuth angle  and small amplitude vibrational motions in the large cage are characterized by molecular dynamics simulations at different temperatures in the stability range of the CO2 sI clathrate. These distributions are used to calculate the NMR powder spectrum of CO2 at different temperatures. The experimental 13 C NMR lineshapes of CO2 guests in the large cages show a reversal of the skew between the low temperature (77 K) and the high temperature (238 K) limits of the stability of the clathrate. Good agreement between experimental lineshapes and calculated lineshapes is obtained. No assumptions regarding the nature of the guest motions in the cages are required.  Keywords: gas hydrates, chemical shift anisotropy, carbon dioxide clathrate    Corresponding author: Phone: +1 613 265 5335 Fax +1 613 998 7833 E-mail: NOMENCLATURE γ magnetogyric ratio σ chemical shielding tensor B0 external magnetic field strength σPA chemical shielding tensor in principle axis frame σLAB chemical shielding tensor in laboratory frame σii eigenvalues of chemical shielding tensor in PA frame (i = 1,2,3) σαα eigenvalues of chemical shielding tensor in LAB frame (α = x,y,z) R(α,β,γ) rotation matrix in terms of Euler angles R(θ,) rotation matrix for linear molecules σi,LAB(T;t) lab frame chemical shielding tensor for cage i at time t and temperature T. )(, Ti  time averaged eigenvalues of σi,LAB(T;t)  INTRODUCTION NMR chemical shifts and chemical shift anisotropy are widely used to characterize the structure and dynamics of guest species in clathrates. [1-11]  A difference between the chemical shift of the guest in the clathrate with the free guest indicates the encapsulation of the guest in the clathrate cages. Peak integration of the NMR spectra of the guests, along with knowledge of the structure and number of cage types in the clathrate allows for the assignment relative populations of the guests in different cages. The chemical shift anisotropy for the guest species at each temperature can be used to characterize the dynamics of guest in the clathrate.  The standard method for extracting dynamic information from NMR lineshapes [12]  of guest molecules is to assume a model for the guest motion in the cage and fit the adjustable parameters in the model to reproduce the experimental lineshape. Different models for guest dynamics are typically required to match the experimental data at different temperatures. The dynamics of CO2 molecules in structure I (sI) clathrates have been characterized using this approach. The sI clathrates have a cubic unit cell composed 46 water molecules arranged in six large 14-sided cages (5 12 6 2 ) with average cavity radii of 4.33 Å and two 12-sided cages (5 12 ) with average cavity radii of 3.95 Å. The large cages in Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008. the sI clathrate are flattened in the direction perpendicular to the two hexagonal faces of the cage and non-spherical guest molecules such as CO2 will be preferentially aligned with the long axis parallel to the hexagonal faces, as depicted in the inset of Figure 1. This anisotropic spatial distribution leads to incomplete averaging of the chemical shift tensor observed in the 13 C NMR spectra of the CO2 guests in the large cages of the sI clathrate and anisotropic lineshapes shown in Figure 1. In contrast, the more spherically symmetric small cages of the sI clathrate show isotropic NMR lineshapes for the guest molecules.  Figure 1. A schematic representation of the anisotropic 13 C NMR lineshape for the CO2 guests in large sI cages at three temperatures. The CO2 guest in a large cage is shown in the inset.  Figure 1 shows that the 13 C NMR lineshapes of CO2 in the large cages change drastically with temperature.  At low temperatures (blue line) the lineshape of CO2 in the sI large cages has a skew similar to solid α-phase CO2 [12]  but with a smaller span. At intermediate temperatures (green line), the skew of the peak is almost zero and at higher temperatures near the decomposition temperature of the clathrate (red line), the direction of the skew of the lineshape is inverted compared to the low temperature line. The span of the peak also decreases at high temperatures. These lineshapes are interpreted in terms of the different spatial distributions and dynamics of CO2 guests in the large cages at different temperatures. [12,13]  Reasonable models describing the angular distribution of the guests in clathrate cages are developed and the adjustable parameters in the model are determined by fitting the predicted lineshapes from these models to experiments. For example, the high temperature motion of CO2 in the large cages has been modeled [14]  by a precessional motion with a maximum range of βmax from the equatorial plane, or equivalently with the polar angle θ ranging from θmin = 90° – βmax to θmax = 90° + βmax. Fitting the prediction of this model to the experimentally observed span, a value of βmax = 31° is obtained for the range of the distribution of the CO2 guests about the equatorial plane. Different models are obviously required to describe the dynamics of the guest motion at high and low temperature limits. [12]  In this work, we present a molecular dynamics calculation method to determine the spatial orientations of the molecules required for calculating NMR chemical shift anisotropy of guest molecules in clathrates. In this method, no assumptions about the nature of the guest motion is required and a uniform methodology is applicable to all temperatures.  Furthermore, the method developed is general and can be used to predict chemical shift anisotropies of guests in other inclusion compounds, pure solid phases with dynamically disordered components, or dynamically mobile side-chains of protein molecules. The Hamiltonian for a spin-1/2 nucleus in a static external magnetic field B0 is, 0Bσ1I  )(H    (1) where σ is the second-rank chemical shielding tensor, γ is the magnetogyric ratio, I is the nuclear spin angular momentum operator and 1 is the unit tensor. [15]  In the principal axes (PA) coordinate system, the chemical shielding tensor of a molecule is diagonal,            33 22 11 00 00 00    PAσ ,  (2) and the principle axis eigenvalues are ordered so that, 11 ≤ 22 ≤ 33. The chemical shielding tensor is characterized by the isotropic chemical shielding iso = (11 + 22 + 33)/3, span  = 33 – 11, skew  = 3(iso –  22)/, and anisotropy Δ = 33 – iso. [16]  Chemical shifts are defined as δii = ref – ii where ref is the isotropic chemical shielding Chemical Shift Low T High T of a reference material, tetramethylsilane (TMS) in the case of 13 C. In the laboratory frame (LAB), a molecule in a solid is characterized an orientation determined by the three Euler angles α, β, and γ. In the case of weak dipolar coupling between the guest and host cage, the chemical shielding tensor depends only on the molecular orientation and not on the position inside the cage. In the weak coupling limit, the chemical shielding tensor of the guest in the laboratory frame, σLAB, is related to σPA through the rotation matrix R(α,β,γ), ),,(),,(),,(  RσRσ PALAB 1  .         (3) For linear CO2, the second-rank chemical shielding tensor in its PA frame in Eq (2) reduces to,            33 11 11 00 00 00    PAσ .     (4) The two equal eigenvalues in the directions perpendicular to the molecular axis are shown as 11. Without loss of generality, the isotropic chemical shielding is chosen as the origin and the chemical shielding tensor is written in traceless form, [17, 18]              200 010 001 11 traceless PAσ . (5) In the laboratory frame, with a choice of an external reference z-axis, the orientation of a linear molecule is determined by spherical polar angle θ and azimuth angle  and the chemical shielding tensor σLAB, is related to σPA through the rotation matrix R(θ,), ),(),(),(  RσRσ PALAB 1  .   (6) Using the standard form of the rotation matrices [19]  with the principal axis chemical shielding given in Eq (5) we get,                         2 cos312sinsin 2 3 2sincos 2 3 2sinsin 2 32 sin 2 sin31 2 sin2sin 2 3 2sincos 2 32 sin2sin 2 32 sin 2 cos31  11),( traceless LABσ    (7) In a solid, the θ and  angles are fixed within the range of small lattice vibrations. In an clathrate, guest molecules move inside host cages and have different orientations in the cages at different times. A molecular dynamics simulation of CO2 guests in a sI clathrate at a temperature T gives the orientation of the CO2 guests in each large cage i at different times, t. Applying Eq (7), we can determine the chemical shielding tensor, σi,LAB(T;t), at each time t with corresponding eigenvalues σi,xx(T;t), σi,yy(T;t), and σi,zz(T;t). By averaging over all times in a simulation trajectory, averages of the three eigenvalues, )(, Ti   for each cage i can be determined. To simulate the powder spectra for the CO2 guests, the )(, Ti  for each cage are used to generate a lineshape intensities using the method described by Olivieri [20]  Briefly, the frequency for the chemical shift tensor for cage i, in the powder spectrum i(Θ,Φ) is given as,   )2cos2sin12cos3(3 ,),(   ii isoii       (8) where Θ and Φ are the polar angles which determine the orientations of all cages to the external magnetic field B0 in a powder sample and Δ  = 33  (22  11)/2 and η = (22 11)/Δ.  The peak intensity for each cage is obtained by binning the values of the chemical shielding, i(Θ,Φ) in a grid of 1000 points with respect to the values of cosΘ and 360 points corresponding to values of Φ. Further details are described in Ref. [20]. The lineshapes for all cages i in the simulation cell are added up as required to generate an overall powder pattern for the sample. In the simulation setup, the external magnetic field B0 is chosen to be parallel to the z-axis of the cubic simulation cell. At 400 fs intervals in the 1 ns trajectory of the MD simulation, the angles θi(t) and i(t) are determined for each guest i and σi,LAB(T;t) and the corresponding three eigenvalues i,αα(T; t) are calculated from the rotation matrix. For each guest, )(, Txxi , )(, Tyyi , and )(, Tzzi  values are calculated by averaging the i,αα(T; t) over time. At each temperature, if the average chemical shielding tensor eigenvalues have not converged to within 5 ppm, the simulation is extended a further 1 ns until this convergence criterion is met. The converged, time- averaged cage eigenvalues are then subjected to the powder averaging of Eq. 8 to generate a powder average spectral lineshape for that cage. All cage lineshapes are added to give the simulated sample lineshape at each temperature.  COMPUTATIONAL METHODOLOGY The chemical shielding tensor for the isolated CO2 guest molecules in the PA frame is calculated using the Gaussian 03 suite of programs [21]  with the gauge-invariant atomic orbital (GIAO) method [22-25]  at the MP2/6-311++G(d,p) level of theory. To determine the magnitude of the dipolar coupling between the guest and host cage, the chemical shielding tensor for CO2 guests in isolated sI clathrate small cages at the HF/6- 31+G(d) level were also performed. Calculations were performed for the guest placed at the center of the cage and displaced by 0.2 Å from the center of the cage with three mutually perpendicular orientations for the CO2 molecule. Molecular dynamics simulations are performed with force fields developed in our previous simulations on clathrate hydrates. [26]  Water and carbon dioxide molecule are considered rigid with intermolecular potentials taken as the sum of van der Waals (Lennard-Jones 12-6) and electrostatic point charge potentials centered on the carbon and oxygen atoms,                                        ji ij ji ij ij ij ij ij r qq rr V 0 612 4 4    .        (9) Point charges qi and qj located on the atomic nuclei i and j on different molecules are used to model electrostatic intermolecular interactions. Lennard-Jones potential parameters between unlike atom-type force centers are calculated using, ij = (ii jj) 1/2  and ij = (ii+jj)/2. The extended simple point charge (SPC/E) model is used for water [27]  and carbon dioxide Lennard- Jones parameters are taken from the model of Harris and Yung. [28]  Point charges on CO2 were chosen to reproduce the experimental gas phase quadrupole moment of –4.3 esu cm2.[29,30] The intermolecular potential parameters used in the simulations are given in Table 1. To equilibrate the initial configurations of the sI CO2 clathrate, isotropic NPT molecular dynamics simulations with the Nosé-Hoover barostat algorithm [31,32]  were performed on a periodic 333 (35.8 Å per side initial dimensions) replica of the cubic sI clathrate hydrate cage with the DL_POLY 2.17 molecular dynamics program. [33]  The simulation cell has a total of 162 large cages and 54 small cages with one CO2 guest per cage. Three temperatures 77, 150, and 238 K at ambient pressure, and 274 K at 35 bar, were simulated. Thermostat and barostat relaxation times of 0.5 and 2.0 ps, respectively, were used. The equations of motion were integrated with a time step of 1 fs using the Verlet leapfrog algorithm. [34,35]  Long-range electrostatic interactions were calculated using the Ewald summation method [34,35]  and all intermolecular interactions were calculated within a cutoff distance of Rcutoff  = 15.0 Å. A total simulation time of 100 ps is used with an initial temperature scaled equilibration period of 30 ps. To obtain the angular distributions of the CO2 molecules in the large cages at different temperatures, NVE simulations were performed for a minimum total simulation time of 1 ns, starting with configurations equilibrated by the previous NPT runs. If the ensemble averages of the eigenvalues of the chemical shielding tensor for each cage had not converged within this 1 ns time frame, further runs in 1 ns increments were performed until convergent eigenvalues were obtained.  Table 1. Electrostatic point charges and Lennard- Jones interaction parameters for SPC/E water and carbon dioxide.  RESULTS AND DISCUSSION The calculated eigenvalues of the NMR chemical shielding and shift tensors for gas-phase CO2 in the PA frame are given in Table 2. The calculated isotropic chemical shift of CO2 with respect to TMS is ~125 ppm, which is in excellent agreement with the experimental value of 124 ppm. [36]  To estimate the effect of dipolar coupling of the guest with the cage atoms, the chemical shielding tensor of CO2 was calculated at the center of a sI small clathrate cage in three mutually perpendicular orientations. The eigenvalues of the chemical shielding tensor ii from these three Atoms  q / e ii / Å ii / kJmol -1 O (water) H (water) C O -0.8476 +0.4238 +0.6645 -0.33445 3.166 0.000 2.785 3.064 0.6502 0.0000 0.2411 0.6901 configurations differed from the free CO2 molecule by a maximum of 3.3 ppm. The ii for three configurations with the CO2 guest displaced by 0.2 Å from the center of the cage (along the x- direction) were also calculated and the values differed from the free CO2 eigenvalues by a maximum of 3.5 ppm. The ii of the guests were not appreciably affected by encapsulation in the clathrate cages and the assumption of only retaining angular dependence in determining the calculated chemical shielding tensor is reasonable.  Table 2. Calculated eigenvalues of the 13 C NMR chemical shielding tensors for isolated CO2 at the MP2/6-311++G(d,p) and HF/6-31+G(d) levels. The chemical shielding of CO2 inside isolated sI clathrate small cages at the HF/6-31+G(d) level for several placements of the CO2 guest in the sI small cage are also given. The displacement of the guest from the center of the cage (in the x-direction), the θ, and  angles are given for each case.   The experimental 13 C NMR shifts for solid α- phase carbon dioxide [37]  and the CO2 clathrates [2, 3,7-10]  have been reported. At 77 K, the NMR spectrum of solid CO2 has an observed span  = 335 ppm. [12,37]  Our calculated value for the span for free CO2 is 326 ppm which is in good agreement with the experimental result. Superimposed snapshots at multiple time intervals of the rotational motion of a CO2 guest molecule in a large cage from MD simulations at three temperatures are shown in Figure 2. At 77 K, the range of the polar angle θ is relatively small and the distribution around , (the angle in the equatorial plane of the cage) is not homogenous. At low temperatures, the rotation of the CO2 guests in the large cages about the equatorial plane is hindered which suggests a hopping mechanism for the rotation of the CO2 guest about . At the high temperature limit (238 K), the distribution about the -angle is homogenous (free rotation about the equatorial plane) with a broader range of θ–angles. At 150 K, the distribution about the - angle is more uniform than the 77 K simulation, but not totally isotropic. Both hopping and hindered rotational motion mechanisms are operating at this temperature.  Figure 2. Snapshots of the orientations of a CO2 molecule in a large sI clathrate cage at 77, 150, and 238 K from MD simulations with the carbon atoms superimposed.  To determine the 13 C NMR lineshape for the sI clathrate sample, the z-direction of the simulation cell is chosen as the direction of the magnetic field with which the θ-angle is measured. Additionally, the -angle is referenced with respect to the x-axis of the simulation cell. These are the reference directions for calculating the NMR powder lineshape. The cage powder lineshape intensities are superimposed to get the overall sample lineshapes at different temperatures which are shown in Figure 3. No additional line broadening was applied when summing the individual cage spectra to get the sample spectrum. The calculated high temperature (274 K and 35 bar) spectrum is compared with the experimental results [3]  in Figure 3(a). At this temperature, a long simulation time of 5 ns was needed for the convergence of the cage eigenvalues. Excellent agreement between the experimental and calculated results is observed. The temperature dependence of the lineshapes at 77, 150, and 238 K shown in Figure 3(b) agrees very well with the experimental temperature dependence given in Ref. [12] and shown schematically in Figure 1. The span of the spectra decreases as the temperature increases and the skew is reversed in going from the low to the high temperature limit. Atoms  11 33 iso  MP2 HF -39.0 -24.0 285.6 284.2 68.6 78.7 0 Å, 90°, 0° 0 Å, 90°, 90° 0 Å, 0°, 0° 0.2 Å, 90°, 0° 0.2 Å, 90°, 90° 0.2 Å, 0°, 0° -23.4, -22.4 -23.2, -22.8 -23.3, -22.6 -23.5, -22.5 -22.4, -22.1 -21.2, -20.7 280.9 281.6 281.1 280.7 282.9 282.4 78.3 78.6 78.4 78.2 79.4 80.2 77 K 150 K 238 K Y Y X Y Z X Y  Figure 3. (Top panel) The simulated and experimental 13 C powder NMR lineshape at 274 K and 35 bar for CO2 in the small sI cages. Simulated lineshapes at ambient pressure are given in the bottom panel.  The high temperature CO2 motion was previously been modeled [14]  by the precession of the CO2 guest with a maximum range of 31° about the equatorial plane. This range is in good agreement with the width spatial distribution of the CO2 guests obtained in the simulation. In the MD simulations, the range of guest motion was determined without the need to input the experimental lineshape or make any assumption regarding the nature of the motion at this temperature. As seen in Figure 2, at low temperatures the rotational motion of the guests is hindered and furthermore, the cage water molecules are rotationally frozen. This leads to greater inhomogeniety in the large cage environments and broader NMR powder lineshapes. At high temperatures (238 K), CO2 guests are rotationally free with respect to rotation in the -angle and the water hydrogen distributions between the clathrate water oxygens is truly disordered due to rotation of water molecules in the clathrate sites. As a result, the guest peaks are narrower and the skew is reversed compared to the low temperature spectrum. In simulations performed for this study, the longest trajectory time considered is 6 ns. This simulation time is short compared to the time scale of the NMR experiment and the water proton rearrangements. The simulations will therefore not show the total extent of the peak sharpening as the temperature is increased.  SUMMARY AND CONCLUSIONS A general method for calculating the chemical shift lineshape anisotropy of guest molecules in clathrate hydrate compounds from molecular dynamics simulations has been developed for the case of weak host – guest dipolar coupling. The orientational distributions from molecular dynamics simulation along with time and powder angle averaging are used to calculate the cage chemical shielding tensors and the NMR lineshape produced by each guest molecule. The total predicted lineshape anisotropy is calculated from the superposition of the lineshapes of all guests. The approach is applied to calculate the temperature dependent 13 C NMR lineshape anisotropy of carbon dioxide in sI clathrates. The resulting lineshapes agree well with the experimental 13 C NMR spectrum at each temperature.  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