J. Phys. Chem. 1993,97, 8387-8392
8387
Simulation of One-Dimensional *HNMR Line Shapes? R. J. Schadt, E. J. Cain, and A. D. English' Du Pont Central Research and Development, Experimental Station, Wilmington, Delaware 19880-0356 Received: April 18. 1993
The determination of the molecular mechanics involved in a bond reorientation process from the simulation of one-dimensional *HN M R spectra is considered for a variety of models. Anisotropic line shape distortions and the accompanying refocusing efficiency factors are essential in identifying the general characteristics of the motional process. The utility of employing a basis set of simulations for an initial analysis of experimental data where a distribution of correlation times is required to fit the experimental data is illustrated for the 120° jump or r-flip model.
Introduction Determination of the geometry of molecular bond motion with respect to a molecular framework is one objective of solid-state 2H NMR experiments. If the motion is a discrete jump, the vector angle which is determined from the NMR line shape or other data must be related via a model to a molecular mechanics geometry. Furthermore, due to the symmetry of the nuclear spin Hamiltonian, supplementaryjump angles produce identical NMR observables; however, supplementaryjump angles do nor in general produce supplementary molecular mechanics angles. When simulating experimentalline shapes, estimation of an initial value for the jump angle for exchange between sites related by molecular symmetry is straightforward; whereas, for jumps between sites not related by molecular symmetry the initial value@) are not obvious. Examples of jumps between sites related by molecular symmetry are abundant for and methy17J2-26groups. Examples of jumps between sites not related by molecular symmetry for C-DZ7-33 and N-D bond~29.30.3~.35 are also widely available. For each case, it is very useful to examine a basis set of line shapes to evaluate the applicability of proposed models. In order to determinewhich types of basis sets will find the greatest utility, it is useful to classify the general types of molecular bond jumps and their temporal regimes. In one-dimensional NMR experiments, observation of line shape distortions due to anisotropic relaxation can facilitate the identification of both the geometrical and temporal nature of two general classes of motion: (a) small-angle jumps/ restricted diffusion29 and (b) large-angle jumps. Quadrupole echo, onedimensional 2H NMR line shapes for small-angle jumps and restricted diffusion are virtually indi~tinguishable,2~ but in favorable cases they may be resolved by multidimensional NMR line shape analysis via characteristic intermediate exchange patterns (off-diagonal line shape i n t e n ~ i t y ) . ~ ~ Anisotropic relaxation of ZH NMR line shapes can be observed if the exchange rate can be adequately manipulated and the distribution of rates is not too broad. Such observations are not possible if the rate of motion is always fast at any accessible tempcratureZ9or a broad distribution of correlation times8J0~2934 due to underlying structural heterogeneity (distributionin packing arrangements) such as that found in disordered systems (glasses, mixed crystals, partially anisotropic fluids) causes dynamical heterogeneity. Additionally, an initial evaluation of the model of the molecular motion is facilitated by examination of the temporal and temperature dependence of the integrated line shape or quadrupolar echo refocusing effi~iency.8.10,22.24~29~~1~~~ These data not only are useful in selected cases for verifying whether Contribution No. 643 1.
* To whom correspondence should be addressed. 0022-3654/93/2091-8381$04.00/0
the jump angle is fairly small or fairly large but also are most useful in establishing the width of the distribution of correlation times present at any temperature. With this information, a basis set of line shapes can be constructed which facilitates the selection of parameters for quantitatively fitting the simulations to the experimental observations. L i e Shape Calculation Procedure Line shapes were calculated using a Silicon Graphics R-4D/ 280SX or Cray YMP system for various correlation times (10-10 s I T~ I lo2 s) and delay times (0.1, 20, 40, 80, and 160 ps) between quadrature pulses43(3.2 ps pulse length) for both 120° vector jumps (r-flip model) and for internal rotations of loo, 20°, 30°, 60°, looo, and 120° about a tetrahedral axis. All simulations considered exchange only between two equally populated sites with a quadrupolar coupling constant (e2qQ/h) of 170 kHz and r] = 0. Each calculated line shape was corrected for echo distortions" and finite pulse length4sand was convoluted with Gaussian and Lorentzian broadenings. Corrections due to molecular motion during the radio-frequency pulses were not attempted.46 All illustrated line shapes include a Gaussian broadening of 500 Hz and a Lorentzian broadening of 1 kHz (full width at half-maximum). For a specific simulation of a ZH NMR experimental line shape, Gaussian broadening (1-3 is typically determined from low-temperature spectra and is intended to account primarily for TZ(dipolar) interactions to both ZHand lH neighboringnuclei. The Lorentzian broadening is determined by the value used in processing of the experimental FID. When rC < l C 7 s or T~ > 10-l s, the quadrupolar echo refocusing efficiency approaches unity, and the line shape becomes independent of T,. Superposition of an inhomogeneous logGaussian distribution function, P(log(.r,)), with the calculated line shapes which are also each weighted by the echo refocusing efficiency factor R(1og ( T ~ ) produces ) a line shape characterized by a mean correlationtime, a standard deviation of a log-Gaussian distribution of correlation times, and a mean refocusing efficiency factor, R. Results and Discussion Geometry and Time Scale of Motion. The determination of the geometry of motion with respect to the molecular framework from NMR observables is dependent on a large number of variables. Even for the simplest case of a two-site, equal population,jump characterized by a single correlation time, this process can be ambiguous. NMR spectra of solids are described by second rank tensors; consequentially, equivalent one-dimensional NMR line shapes result from supplementaryjump angles. While the relationship between the vector jump angle and an equivalent internal rotation angle about a tetrahedral axis (Figure 0 1993 American Chemical Society
Schadt et al.
8388 The Journal of Physical Chemistry, Vol. 97, No. 32, I993 10" Internal Rotation
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1) is easily calculated, equivalent internal rotations about a tetrahedral axis, based upon the vector jump angle symmetry, are not trivially obvious (Figure 2). For example, Figures 1 and 2 indicate that a two-site 120° or 60° vector jump (*-flip of an aromaticring) is equivalent toa two-site 1 3 3 O or -64'internal rotation about a tetrahedral axis. As another example, a transgauchejump of a methylene group in a system such as polyethylene or polystyrene requires an internal rotation of 1 0 5 O about a tetrahedral axis (97O vector jump). These examples illustrate the type of angular relationships which must be examined to evaluate the angle relevant to molecular mechanics from a vector jump model. Fast exchange one-dimensional 2H NMR line shapes do not uniquely characterize the vector jump angle involved, unless the population of the two sites is known from other considerations. As an example, the classic q* = 1, UQ* = uQ/2 fast exchange line shape is produced by a locus (population, jump angle) of values, only one of which is the equal population tetrahedral jump case.47 Because of this uncertainty in the geometry of motion from examination of the fast exchange line shapes alone, various types of experiments must be employed which rely on anisotropic relaxation effects to distort the observed line shape when the jump frequency occurs at the appropriate "intermediate" exchange rate for the experiment being utilized. Although some experiments which rely upon anisotropic T1 or T ~ relaxation Q have been used to distinguish various motional models, the most extensively utilized method relies upon aniso-
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tropic TZrelaxation. When the rate of molecular motion u, is comparable to UQ, anisotropictransverse relaxation ( Tz)resulting from fluctuations of the quadrupolar interaction becomes observable in ZH NMR line shapes as a result of the orientational dependenceof TZwhich distorts the solid echo decay as compared tothe FID.4 Characteristiclineshapechangesoccurasa function of exchange rate with exchange between two orientations separated by a frequency difference A@, Whether the line shape shows characteristics of fast, intermediate, or slow exchange depends on whether AUT,is much less than, approximately equal to, or much greater than one, respectively. If the intermediate rate regime is encountered, then after the delay time T I in the quadrupolar echo pulse sequence, magnetization from those regions where TZ< 71 will not be completely refocused but may be characterized by the quadrupolar echo refocusing efficiency. Figures 3-8 show that when the NMR line shapes become independent of 71, in the slow ( T > ~ 10-l s) and fast ( T < ~ le7 s) motion limits, the refocusing efficiency is near unity. As a function of correlation time, the refocusing efficiency factor reaches a minimum near 1 V sZ9 for a wide variety of jump angles; the value at the minimum is dependent upon the jump angle. Moreover, the value in the intermediate rate regime is dependent upon the magnitude of the jump angle (Figures 3-8). Thus, the experimentallydeterminedvalue of the echo refocusing efficiency at its minimum is useful in differentiating small-angle (30°). The quadrupolar refocusing efficiency48for a small-angle, equal population jump not only can be diagnostic of the geometry of the motion when the jump angle is less than 30' but also is quite sensitive to the rate of motion. Figures 3,4, and 5 illustrate that the refocusing efficiency factor depends not only on the rate of motion but also on the jump angle over this range of angles. The utility of these calculations when compared to the experimental results is limited by the T2 (dipolar) contribution to refocusing of magnetization because the dipolar interaction, as well as the correlation time for the jump, is temperature dependent. This limitation is to some degree mitigated by the relatively small dipolar interactionspresent even in selectivelydeuterated systems. For jump angles larger than 30°,essentially identical behavior is observed for all angles,29 and this behavior is illustrated in Figure 6 for a 120° internal rotation (tetrahedral jump). Note that the refocusing efficiency is always greater than 5%, which
Simulation of One-Dimensional 2H NMR Line Shapes
The Journal of Physical Chemistry, Vol. 97, No. 32, 1993 8389
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axis. allows observation of these highly distorted and informative line shapes for favorable (S/N) cases. This result is essentially identical to that found for a *-flip motion of an aromatic ring (see Figure 7). Hence, for jump angles larger than about 30°, the line shape distortions due to anisotropic T2 relaxation can not be used to identify both the geometry and rate of motion; however, by employing the anisotropy of the spin-lattice relaxation ( T I and T ~ Q librations ), (stochastic fluctuations) and various largeangle jumps may also be differentiated by ZH N M R line shapes4' Figure 7 illustrates line shape distortions due to anisotropic relaxation for the ubiquitous ?r-flip model of These line shapes and their associated refocusing efficiency factors can uniquely identify this type of motion where the geometry is an exact *-flip and the distribution of flipping rates is a singularity. Figure 8 illustrates that when the geometry of motion for a pure *-flip is compared to that of a slightly perturbed ?r-flip (coupled
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N M R calculated line shapes ($qQ/h = 170 kHz) and refocusing efficiency factors for a s-flip model as a function of the correlation time (TE) and refocusing time ( T i ) . to a 20° internal rotation of the 1,4-axis of the phenylene ring about a tetrahedral axis), the line shapes (71 = 20 FS; e2qQ/h = 170 kHz) are dramatically altered in the intermediate exchange regime and less so in the rapid motion limit. This illustrates that the line shapes are more sensitive to this minor change in the geometry of the motion than are therefocusing efficiency factors. If the experimental line shape shows no Tz distortion, then the relaxation is entirely controlled by the nonselective static dipolar interactions whose transverse relaxation time T2d < 500 ps.48 Considering only a distribution of correlation times and one trajectory of motion, this case (T2 = T2d) will be observed only if the motion is either very slow or very fast or if the distribution Figure 7.
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Schadt et al.
8390 The Journal of Physical Chemistry, Vo1. 97, No. 32, 1993 crc ( H r l ) ,
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Figure 8. Comparison of ZH NMR calculated line shapes (71 = 20 ps and GqQ/h = 170 kHz) and refocusing efficiency factors at various values of correlationtimes ( r c )for a pure r-flip model (left column) and a model with *-flip simultaneously coupled to an internal rotation (20') of the 1,4-axisof the phenyl ring about a tetrahedral axis (right column).
of correlation times is so large that intermediate exchange line shapes are unobservable due to both a minimal population and fast T2relaxation. These three cases produce respectively a slow motion limit line shape,a fast motion limit line shape,or a weighted sum of the slow limit and fast limit line shapes. In all three cases, well-defined singularities in the resultant line shapes must be observed. When the experimental spectra do not show evidence of very well-defined singularities, the motion in the systemcannot be described by a single trajectory of motion. In this case, an inhomogeneous distribution of amplitudes of motion must be considered. Distributions of jump angles or librations are introduced by population-weightedsummations of the line shapes according to an inhomogeneous distribution function. For example, simulations of selectively deuterated nylon (C-D and N-D bond reorientation)29-3' and poly(pphenyleneterephtha1amide) ( d i p s of terephthalamide rings19920 and diamine ring@) experimental line shapes employed this procedure using inhomogeneous Gaussian distributions of librations. In another example, simulations of large-angle jumps of amide bonds in N-deuterated p~ly@-phenyleneterephthalamide)~~have included both inhomogeneous Gaussian distributions of librations and a small distribution (f2O) of the angle between the N-D bond and the axis about which the N-D bond jumps. In each of these cases, the angular distribution broadened the "singularities" in the line shapes and increased the intensity in the central portion of the line shape. In these examples, the rates of motion used to model thedynamical processes reflected in the experimental line shapes were indicated to be in either the slow or fast exchange regimes by the experimental results (see above). Distributions of Correlation Times. When the rate of motion reflected in the experimental line shape cannot be simulated with a single correlation time, then the simplest modification of the model is to introducean inhomogeneousdistribution of correlation times which is simulatedby a weighted superposition of line shapes for individual correlation t i m e s . * J o ~ ~For ~ . ~example, ~ *-flip
motions commonly are not characterized by a single correlation time, but rather an inhomogeneous distribution of correlation times.1d.sJ0.2' Figures 7 and 9-12 illustrate the dependence of the anisotropic line shape distortions on the standard deviation of an inhomogeneous log-Gaussian distribution of correlation times centered at various mean correlation times for a *-flip model. Comparisonof the mean refocusing efficiency calculated for various distributions of correlation times (a = 0, 1,1.5,2,3) illustrates that this relationship is somewhat unexpected. When the correlation time distribution is sufficiently narrow, the refocusing efficiency will show a marked decrease in the intermediate rate regime. For single correlation time models this relationship of refocusing efficiency to the correlation time is illustrated in Figures 3-8. As shown in Figure 7, for a *-flip model with a single correlation time, the refocusing efficiency factor (compared to the slow or fast exchange value of unity) is 26% for ( r c )= 10-4s and = 20 ps and is 7% for ( r c )= 10-4 s and r1= 160 ps. (This may be compared with previous results for single intermediate rate models for various two-site jumps41v42 and three-site jumps22.41which found less than 20% refocusing efficiency.) These results for the refocusing efficiency for a single intermediate d i p p i n g rate (Figure 7)may be compared to those found for a log-Gaussian distribution of correlation times with a standard deviation (u) of 1 order of magnitude (Figure 9), where the variation of the mean refocusing efficiency factor as a function of T~ is attenuated by a factor of 2-3 by the imposition of a distribution. Near the slow and fast motion limits, the mean refocusing efficiency is reduced as the breadth of the distribution increases. In the intermediate rate regime, the refocusing efficiency is increased as the breadth of the distribution increases. This is a manifestation of the relative contributions to each line shape, as a function of the breadth of the distribution of correlation times, from (a) those line shapes with a refocusing efficiency near 100% (slow and fast motion limit) and (b) those line shapes with a minimal refocusing efficiency (intermediate rate regime). Furthermore, examination of these figures demonstrates that as the distribution of correlation times becomes larger, not only does the decrease in refocusing efficiency in the intermediate rate regime become much less pronounced but also the characteristic intermediate exchange regime line shape distortions become progressively less evident. For reasonably broad distributions (u 1 3 decades), the line shapes appear to be a simple combination of slow and fast motion limit line shapes, and the variability of the refocusing efficiency with correlation time is minimal. Figures 7 and 9-12 are very useful for obtaining an initial estimate of a mean correlation time and breadth of a correlation time distribution for a a-flip motion. This approach has found great utility in our work on ring dynamics in poly@-phenyleneterephthalamide)Is2l and enables one to greatly reduce the amount of calculations required for a quantitative line shape fit. Furthermore, these results are useful for estimating the temperature at which the proper observation window for a multidimensional experiment may be feasible. conclllsions A limitation of one-dimensional and multidinknsional NMR line shapes in defining a molecular dynamics model is the accessibility of the appropriate intermediate exchange rate window. When intermediate rate line shape distortions or offdiagonal line shape intensities are difficult to observe, as in disordered systems (glasses, mixed crystals, partially anisotropic fluids) with large structural heterogeneity or in the presence of phase transitions, the dynamical information which can be extracted is limited. When the intermediateexchangerate regime is accessible for onedimensional 2H NMR experiments,two basic kinds of motion can be distinguished on the basis of anisotropic line shape distortions and the quadrupolar echo refocusing
Simulation of One-Dimensional 2HNMR Line Shapes
The Journal of Physical Chemistry, Vol. 97, No. 32, 1993 8391
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8392 The Journal of Physical Chemistry, Vol. 97, No. 32, 1993 responsible for the vector jump defined by the NMR experiment is not uniquely defined. The *-flip basis set illustrated here provides an example of the utility of a basis set as a starting point for the initial interpretation of one-dimensional experimental ZH NMR line shapes. A quantitativefit of experimental2H NMR line shapes for selectively deuterated phenyl rings, or other 2H-substituted sites,undergoing a motion which to a good approximation is a A-flip is greatly facilitated by definingan initialvalue for both the mean correlation time and the standard deviation of an inhomogeneouslog-Gaussian distribution of correlation times. These initialvalues are available from the results given here via inspection. This approach, as opposed to an unguided search of simulationparameters, is vastly superior in efficiency.
Schadt et al.
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