Article

Brownian motion description of activation energies from NMR-relaxation times for rotating molecular-groups

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Abstract

A general formula is derived (containing eigenvalues and eigenfunctions of some diffusion equation) for activation energies from NMR-relaxation times for rotating molecular groups. Numerical calculations for a trigonometric three well potential in the strong damping (Smoluchowski) limit show that there may be a considerable difference between activation energy and barrier height. In the low damping limit there is fairly good agreement.

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... (1) There is an NMR activation energy closely related to a barrier to CF 3 (or CH 3 ) reorientation. [31][32][33][34][35] In the Discussion section (Sec. IV), we divide this into an intramolecular and an intermolecular component and compare the two components in seven compounds similar to 1-3 (1-3 and four others). ...
... Seven representative compounds are listed in Table II of which 1-3 are shown in Fig. 1 (3) intramolecular and intermolecular steric interactions, and (4) intermolecular interactions somewhat arbitrarily divided into several types that the International Union of Pure and Applied Chemistry (IUPAC) bundles under the umbrella term of van der Waals interactions. 108 The segue between all these atomic and molecular interactions and the spin-spin dipolar interactions that are very well understood 27 [31][32][33][34][35] The relation τ = τ ∞ exp(E NMR /kT) presented in Eq. (9) provides the mean time between reorientations for a methyl (CH 3 ) or fluoromethyl (CF 3 ) group reorienting in a three-fold or six-fold potential [2][3][4][5]7,8,10,12,112 in a random (Poisson 113 ) process. The physical origin of τ −1 = τ −1 ∞ exp(−E NMR /kT) is the canonical ensemble. ...
... E NMR can be related to the barrier V for CF 3 or CH 3 reorientation. [31][32][33][34][35] The lower energy for the energy difference that enters into E NMR will not be zero as the naive model presented in the previous paragraph suggests, but will be the ground reorientational state which will be above the bottom of the reorientational barrier V . 31, 35 Flygare, 112 page 129, shows a reorientational energy level diagram for a CH 3 group with V = 1158 cm −1 = 13.8 kJ mol −1 . ...
Article
The dynamics of methyl (CH3) and fluoromethyl (CF3) groups in organic molecular (van der Waals) solids can be exploited to survey their local environments. We report solid state 1H and 19F spin-lattice relaxationexperiments in polycrystalline 3-trifluoromethoxycinnamic acid, along with an X-ray diffraction determination of the molecular and crystal structure, to investigate the intramolecular and intermolecular interactions that determine the properties that characterize the CF3 reorientation. The molecule is of no particular interest; it simply provides a motionless backbone (on the nuclear magnetic resonance(NMR) time scale) to investigate CF3 reorientation occurring on the NMR time scale. The effects of 19F–19F and 19F–1H spin-spin dipolar interactions on the complicated nonexponential NMRrelaxation provide independent inputs into determining a model for CF3 reorientation. As such, these experiments provide much more information than when only one spin species (usually 1H) is present. In Sec. IV, which can be read immediately after the Introduction without reading the rest of the paper, we compare the barrier to CH3 and CF3 reorientation in seven organic solids and separate this barrier into intramolecular and intermolecular components.
... This value is in reasonable agreement with the value of V crystal = 10.3 kJ mol À1 determined here from the electronic structure calculations, even though E NMR and V crystal are not the same parameter and relating them is a complicated problem in and of itself. [49,50] The parameter determined by the electronic structure calculation is very clear; it is a barrier height. V crystal f(q), where f(q) is an appropriately normalized angular function (approximately cos 3q for methyl group rotation in the present case) for some rotation angle q, is the potential function one would use in Schrçdinger's equation. ...
... Many factors come into play. The theoretical studies relating E NMR and V crystal [49,50] only involve methyl group rotation not methyl group rotation plus methoxy group libration. ...
... This value is in reasonable agreement with the value of V crystal = 10.3 kJ mol À1 determined here from the electronic structure calculations, even though E NMR and V crystal are not the same parameter and relating them is a complicated problem in and of itself. [49,50] The parameter determined by the electronic structure calculation is very clear; it is a barrier height. V crystal f(q), where f(q) is an appropriately normalized angular function (approximately cos 3q for methyl group rotation in the present case) for some rotation angle q, is the potential function one would use in Schrçdinger's equation. ...
... Many factors come into play. The theoretical studies relating E NMR and V crystal [49,50] only involve methyl group rotation not methyl group rotation plus methoxy group libration. ...
Article
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We bring together solid state (1)H spin-lattice relaxation rate measurements, scanning electron microscopy, single crystal X-ray diffraction, and electronic structure calculations for two methyl substituted organic compounds to investigate methyl group (CH3) rotational dynamics in the solid state. Methyl group rotational barrier heights are computed using electronic structure calculations, both in isolated molecules and in molecular clusters mimicking a perfect single crystal environment. The calculations are performed on suitable clusters built from the X-ray diffraction studies. These calculations allow for an estimate of the intramolecular and the intermolecular contributions to the barrier heights. The (1)H relaxation measurements, on the other hand, are performed with polycrystalline samples which have been investigated with scanning electron microscopy. The (1)H relaxation measurements are best fitted with a distribution of activation energies for methyl group rotation and we propose, based on the scanning electron microscopy images, that this distribution arises from molecules near crystallite surfaces or near other crystal imperfections (vacancies, dislocations, etc.). An activation energy characterizing this distribution is compared with a barrier height determined from the electronic structure calculations and a consistent model for methyl group rotation is developed. The compounds are 1,6-dimethylphenanthrene and 1,8-dimethylphenanthrene and the methyl group barriers being discussed and compared are in the 2-12 kJ mol(-1) range.
... This value is in reasonable agreement with the value of V crystal = 10.3 kJ mol À1 determined here from the electronic structure calculations, even though E NMR and V crystal are not the same parameter and relating them is a complicated problem in and of itself. [49,50] The parameter determined by the electronic structure calculation is very clear; it is a barrier height. V crystal f(q), where f(q) is an appropriately normalized angular function (approximately cos 3q for methyl group rotation in the present case) for some rotation angle q, is the potential function one would use in Schrçdinger's equation. ...
... Many factors come into play. The theoretical studies relating E NMR and V crystal [49,50] only involve methyl group rotation not methyl group rotation plus methoxy group libration. ...
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This paper brings together field emission scanning electron microscopy, single-crystal X-ray diffraction, and density functional electronic structure calculations in both an isolated molecule and a cluster of seven whole and fourteen half molecules of 4,4'-dimethoxybiphenyl to investigate coupled methyl-group rotation (over a barrier) and methoxy-group libration (meaning a rotation from the ground state not all the way to the transition state and back again). The structure of the isolated molecule, determined by the electronic structure calculations, is compared with the structure of the molecule found in the crystal. As the methyl group rotates from its ground state to its transition state, the methoxy group rotates 30° in the isolated molecule and 16° in the cluster. The calculated barriers for this coupled methyl-group rotation and methoxy group libration in the isolated molecule and in the crystal are 12.8 kJ mol(-1) and 10.3 kJ mol(-1) respectively, suggesting that intermolecular interactions in the crystal lower the barrier. These barriers are compared with the value of 11.5±0.5 kJ mol(-1) obtained from solid-state (1)H spin-lattice relaxation measurements [P. A. Beckmann and E. Schneider, J. Chem. Phys. 2012, 136, 054508.].
... We note for completeness, that the relationship between NMR activation energies and rotational potentials is complicated but for methyl group rotation, calculations indicate that the activation energies are 0-20 % less than the rotational barriers in the ranges discussed here. [37,38] As such, the NMR activation energies are a reasonably good measure of the rotational barriers. ...
Article
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The spin‐lattice relaxation time T1 and the spin‐spin relaxation time T2 for two identical spins I=☒ have been calculated for anisotropic reorientation in which the spin pair reorients randomly about an axis which, in turn, tumbles randomly. The results are applicable to liquids and solids provided that the correlation time for tumbling of the axis is small compared to T2. Although the two types of motion are independent, their contributions to relaxation are not. For nonviscous liquids, T1=T2. The results are generalized to multispin systems.
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A dynamic method for determining carbon‐13 spin—lattice relaxation times and nuclear Overhauser enhancements from spectra of samples with natural isotopic abundance is presented. The method, along with adiabatic rapid passage and relative intensity measurements, is used to determine the relaxation times and important relaxation mechanisms of all carbons in neat samples of o‐xylene and mesitylene. The results are interpreted in terms of over‐all molecular rotational diffusion and internal methyl rotation. The relaxation times in mesitylene indicate that internal methyl rotation is very rapid with respect to over‐all molecular diffusion. There is also evidence that the relatively free methyl rotation leads to relaxation of these carbons through spin—internal‐rotation interaction. In the case of o‐xylene the internal rotation is not well separated from the over‐all rotation, but approximate calculations show that the results are consistent with a rate of hindered rotation which does not greatly exceed the over‐all diffusion rate, with a barrier to rotation of 1 or 2 kcal/mole. Relatively large Overhauser enhancements and dipolar relaxation rates for the substituted ring carbons indicate that intermolecular dipole—dipole interactions are significant in the relaxation of these carbons. These intermolecular interactions are shown to be relatively unimportant, however, for carbons with directly bonded protons.
Article
Carbon-13 spin-lattice relaxation times and nuclear Overhauser enhancement factors are reported for borneol, camphor, isopinocampheol, and α-pinene in deuterochloroform solution at several temperatures. The random jump rates characterizing the internal motion of methyl groups are evaluated and the corresponding Arrhenius activation energies are determined. The error sources involved in such a procedure are discussed and compared to the conventional treatment, using single-temperature data. The activation energies for the nonassociated systems are found to agree reasonably well with the potential barriers obtained from the empirical force-field calculations.
Article
Spectroscopic techniques as NMR give information about correlation functions of the motion. For polymer chains, this motion is the result of internal rotations about several bonds. These rotations are supposed to be independent of each other and are described by one-dimensional rotational diffusion in a suitable potential. The general case is described and calculations are made for a trigonometric potential with three minima. The free rotational diffusion and the trans-gauche model are limiting cases of this model. The correlation functions decay on two time scales, for short times due to motion within the wells of the potential, which however not causes a decay to zero. On a longer time scale the functions decay further to zero by passage over the potential barriers. The model is applied to NMR-relaxation in hydro-carbon chains of lipid bilayers, where it gives another dependence upon chain position than previous models.
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A particle which is caught in a potential hole and which, through the shuttling action of Brownian motion, can escape over a potential barrier yields a suitable model for elucidating the applicability of the transition state method for calculating the rate of chemical reactions.
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DOI:https://doi.org/10.1103/RevModPhys.15.1