Theory of damped quantum rotation in NMR spectra. I. Fundamental aspects

Nonclassical dynamics of the methyl group in 1,1,1-triphenylethane. Evidence from powder 1H NMR spectra

According to the damped quantum rotation (DQR) theory, hindered rotation of methyl groups, evidenced in nuclear magnetic resonance (NMR) line shapes, is a nonclassical process. It comprises a number of quantum-rate processes measured by two different quantum-rate constants. The classical jump model employing only one rate constant is reproduced if these quantum constants happen to be equal. The values of their ratio, or the nonclassicallity coefficient, determined hitherto from NMR spectra of single crystals and solutions range from about 1.20 to 1.30 in the latter case to above 5.0 in the former, with the value of 1 corresponding to the jump model. Presently, first systematic investigations of the DQR effects in wide-line NMR spectra of a powder sample are reported. For 1,1,1-triphenylethane deuterated in the aromatic positions, the relevant line-shape effects were monitored in the range 99-121 K. The values of the nonclassicality coefficient dropping from 2.7 to 1.7 were evaluated in line shape fits to the experimental powder spectra from the range 99-108 K. At these temperatures, the fits with the conventional line-shape model are visibly inferior to the DQR fits. Using a theoretical model reported earlier, a semiquantitative interpretation of the DQR parameters evaluated from the spectra is given. It is shown that the DQR effects as such can be detected in wide-line NMR spectra of powdered samples, which are relatively facile to measure. However, a fully quantitative picture of these effects can only be obtained from the much more demanding experiments on single crystals.

A quantum mechanical alternative to the Arrhenius equation in the interpretation of proton spin-lattice relaxation data for the methyl groups in solids

The theory of nuclear spin-lattice relaxation in methyl groups in solids has been a recurring problem in nuclear magnetic resonance (NMR) spectroscopy. The current view is that, except for extreme cases of low torsional barriers where special quantum effects are at stake, the relaxation behaviour of the nuclear spins in methyl groups is controlled by thermally activated classical jumps of the methyl group between its three orientations. The temperature effects on the relaxation rates can be modelled by Arrhenius behaviour of the correlation time of the jump process. The entire variety of relaxation effects in protonated methyl groups have recently been given a consistent quantum mechanical explanation not invoking the jump model regardless of the temperature range. It exploits the damped quantum rotation (DQR) theory originally developed to describe NMR line shape effects for hindered methyl groups. In the DQR model, the incoherent dynamics of the methyl group include two quantum rate (i.e., coherence-damping) processes. For proton relaxation only one of these processes is relevant. In this paper, temperature-dependent proton spin-lattice relaxation data for the methyl groups in polycrystalline methyltriphenyl silane and methyltriphenyl germanium, both deuterated in aromatic positions, are reported and interpreted in terms of the DQR model. A comparison with the conventional approach exploiting the phenomenological Arrhenius equation is made. The present observations provide further indications that incoherent motions of molecular moieties in the condensed phase can retain quantum character over much broader temperature range than is commonly thought.

Unusual effects in variable temperature powder NMR spectra of the methyl group protons in 9,10-dimethyltriptycene-d₁₂

Variable temperature (1)H wide line NMR spectra of polycrystalline 9,10-dimethyltriptycene-d12 deuterated in the aromatic positions were studied. The spectra show different patterns in an unrepeatable dependence on the way of preparation of the powdered samples. Simultaneously, no anomalies were seen in the MAS and CPMAS proton-decoupled room-temperature (13)C spectra as well as in powder X-ray diffraction patterns. The effects observed in the (1)H spectra are tentatively explained in terms of a phenomenological model. For one of the examined samples it afforded a consistent interpretation of the entire series of temperature dependent spectra in terms of structural non uniformity of the solid material studied. Quantum character of the stochastic dynamics of the methyl groups in the investigated compound was confirmed, although these dynamics are close to the classical limit where the familiar random jump model applies.

Molecular flexibility and structural instabilities in crystalline l-methionine

We have investigated the dynamics in polycrystalline samples of l-methionine related to the structural transition at about 307K by incoherent inelastic and quasielastic neutron scattering, X-ray powder diffraction as well as ab-initio calculations. l-Methionine is a sulfur amino acid which can be considered a derivative of alanine with the alanine R-group CH3 exchanged by CH3S(CH2)2. Using X-ray powder diffraction we have observed at ~190K an anomalous drop of the c-lattice parameter and an abrupt change of the β-monoclinic angle that could be correlated to the anomalies observed in previous specific heat measurements. Distinct changes in the quasielastic region of the neutron spectra are interpreted as being due to the onset and slowing-down of reorientational motions of the CH3-S group, are clearly distinguished above 130K in crystalline l-methionine. Large-amplitude motions observed at low frequencies are also activated above 275K, while other well-defined vibrations are damped. The ensemble of our results suggests that the crystalline structure of l-methionine is dynamically highly disordered above 275K, and such disorder can be linked to the flexibility of the molecular thiol-ether group.

Spin-lattice relaxation of the methyl group protons in solids revisited: damped quantum rotation approach

Proton spin-lattice relaxation of the methyl group in solids had been one of the most thoroughly addressed theoretical problems in nuclear magnetic resonance (NMR) spectroscopy, considered at different levels of sophistication. For systems with substantial quantum tunneling effects, several quantum mechanical treatments were reported, although in practical applications the quantum models were always augmented with or replaced by the classical jump model. However, the latter has recently proved invalid in the description of NMR line shape effects in variable-temperature spectra of hindered methyl groups, while the competing theory of damped quantum rotation (DQR) was shown to be adequate. In this work, the spin-lattice relaxation issue for the methyl protons is readdressed using the latter theory. The main outcome is that, while the existing formulas for the relaxation rates remain unchanged, the crucial parameter entering them, the correlation time of the relevant random process, need to be reinterpreted. It proves to be the inverse of one of the two quantum-rate constants entering the DQR model, neither of which, when taken separately, can be related to the jump process. It can be identified with one describing the life-time broadening of the tunnel peaks in inelastic neutron scattering (INS) spectra of the methyl groups. Such a relationship between the relaxation and INS effects was reported from another laboratory long ago, but only for the low-temperature limit where thermal population of the excited torsional levels of the methyl group can be neglected. The whole spectrum of cases encountered in practical relaxation studies on protonated methyl groups is addressed for the first time. Preliminary experimental confirmation of this novel approach is reported, based on already published NMR data for a single crystal of methylmalonic acid. The once extensively debated issues of quenching of the coherent tunneling and of the classical limit in the dynamics of the methyl groups are readdressed and presented in a consistent manner.

A heuristic model of damped quantum rotation effects in nuclear magnetic resonance spectra

The damped quantum rotation (DQR) theory describes temperature effects in NMR spectra of hindered molecular rotators composed of identical atoms arranged in regular N-gons. In the standard approach, the relevant coherent dynamics are described quantum mechanically and the stochastic, thermally activated motions classically. The DQR theory is consistent. In place of random jumps over one, two, etc., maxima of the hindering potential, here one has damping processes of certain long-lived coherences between spin-space correlated eigenstates of the rotator. The damping-rate constants outnumber the classical jump-rate constants. The jump picture is recovered when the former cluster appropriately around only as many values as the number of the latter. The DQR theory was confirmed experimentally for hindered methyl groups in solids and even in liquids above 170 K. In this paper it is shown that for three-, four-, and sixfold rotators, the Liouville space equations of NMR line shapes, derived previously with the use of the quantum mechanical reduced density matrix approach, can be be given a heuristic justification. It is based on an equation of motion for the effective spin density matrix, where the relevant spin hamiltonian contains randomly fluctuating terms. The occurrence of the latter can be rationalized in terms of fluctuations of the tunneling splittings between the torsional sublevels of the rotator, including momentary liftings of the Kramers degeneracies. The question whether such degeneracy liftings are physical or virtual is discussed. The random terms in the effective hamiltonian can be Monte Carlo modeled as piecewise constant in time, which affords the stochastic equation of motion to be solved numerically in the Hilbert spin space. For sixfold rotators, this way of calculating the spectra can be useful in the instances where the Liouville space formalism of the original DQR theory is numerically unstable.

Theory of damped quantum rotation in nuclear magnetic resonance spectra. III. Nuclear permutation symmetry of the line shape equation

The damped quantum rotation (DQR) theory describes manifestations in nuclear magnetic resonance spectra of the coherent and stochastic dynamics of N-fold molecular rotors composed of indistinguishable particles. The standard jump model is only a limiting case of the DQR approach; outside this limit, the stochastic motions of such rotors have no kinematic description. In this paper, completing the previous two of this series, consequences of nuclear permutation symmetry for the properties of the DQR line shape equation are considered. The systems addressed are planar rotors, such as aromatic hydrocarbons' rings, occurring inside of molecular crystals oriented in the magnetic field. Under such conditions, oddfold rotors can have nontrivial permutation symmetries only for peculiar orientations while evenfold ones always retain their intrinsic symmetry element, which is rotation by 180 degrees about the N-fold axis; in specific orientations the latter can gain two additional symmetry elements. It is shown that the symmetry selection rules applicable to the classical rate processes in fluids, once recognized as having two diverse aspects, macroscopic and microscopic, are also rigorously valid for the DQR processes in the solid state. However, formal justification of these rules is different because the DQR equation is based on the Pauli principle, which is ignored in the jump model. For objects like the benzene ring, exploitation of these rules in simulations of spectra using the DQR equation can be of critical significance for the feasibility of the calculations. Examples of such calculations for the proton system of the benzene ring in a general orientation are provided. It is also shown that, because of the intrinsic symmetries of the evenfold rotors, many of the DQR processes, which such rotors can undergo, are unobservable in NMR spectra.

A field-cycling NMR investigation of resonant spin-lattice relaxation features arising from tunnelling methyl groups

(1)H nuclear spin-lattice relaxation has been investigated in sodium acetate trihydrate and sorbic acid using field-cycling NMR in the solid state. The relaxation is dominated by the reorientation of the methyl groups. Resonant features arising from coherent tunnelling are observed in both the magnetic field dependence of the spin lattice relaxation rate, T(1)(-1)(B(z)) and in the inverse temperature dependence, T(1)(-1)(1/T). The two systems have different barrier heights and tunnelling frequencies, providing different perspectives on the tunnel resonance phenomena. The magnetic field dependence enables different spectral density components to be separately investigated and in the carboxylic acid, sorbic acid, concerted proton transfer in the hydrogen bonds is also identified at low field and low temperature. The methyl hindering barriers and the correlation times characterising the reorientational dynamics has been accurately determined in both materials.

Hindered rotation of the silyl group in liquid-phase NMR spectra of 9-silyltriptycene derivatives: a comparison with the methyl analogues

Three 9-silyltriptycene derivatives, 1,4-dichloro- (DCST), 1,4-dibromo- (DBST), and 1,4-dimethyl-9-silyltriptycene (DMST) were synthesized, and temperature-variable (1)H NMR spectra thereof were measured. Below 220 K for DMST, and below 250 K for DCST and DBST, rotation of the silyl group becomes practically frozen on the NMR time scale. Iterative line shape analysis of the silyl proton spectra reveals that the familiar Alexander-Binsch line shape equation, employing only one rate constant for the observed rate process, is adequate in these cases. This is at a striking variance with the behavior of the methyl group in the analogous compounds, investigated by us recently, where fingerprints of the damped quantum rotation effect, a phenomenon once predicted by us, are clearly visible in the experimental spectra of the methyl protons. In the damped rotation approach, the relevant dynamics are described in terms of two quantum rate processes characterized by two coherence-damping constants, and the Alexander-Binsch model is obtained as a limiting case where these two constants become equal. The possible reasons of the differences between the dynamics of the silyl and methyl groups in the same molecular environment are discussed.

Fingerprints of Damped Quantum Rotation Observed in Solid-State Proton NMR Spectra

(1)H NMR spectra of the methyl group in an oriented crystal sample of methylmalonic acid with all three non-methyl protons replaced by deuterons are interpreted in terms of the damped quantum rotation (DQR) theory of NMR line shapes. The DQR approach offers a perfect theoretical reproduction of the observed spectra while the conventional Alexander-Binsch line-shape model shows evident defects in the present case. The temperature trends of the quantities characterizing the coherent and incoherent dynamics of the methyl group in the DQR approach (the effective tunnelling frequency and two coherence-damping rates) derived from the spectra are fairly reproduced using a model reported previously. The present findings provide further evidence of limitations to the validity of the common belief that molecular rate processes in condensed phases are necessarily classical.