Theory of damped quantum rotation in nuclear magnetic resonance spectra. II. Numerical simulations for the benzene rotor

Abstract

In Part I of this series of papers, the damped quantum rotation (DQR) theory, formulated originally for hindered threefold molecular rotors in solids, was generalized to the N-fold case. The stochastic dynamics of such objects, evidenced in NMR line shapes, was shown to be more complicated than in the standard model of classical jumps between the wells of the N-fold torsional potential. Actually, it comprises certain quantum rate (i.e., coherence-damping) processes subject to the requirements of the Pauli principle. The jump picture is recovered only when the quantum rates fit specific patterns. In this work, one of the ways of approaching such a classical limit is identified for the benzene rotor. This is inferred from a quantum mechanical model whose validity was earlier confirmed for a methyl group. Based on that model, theoretical calculations for the benzene ring dynamics in a clathrate crystal, 1-(9-anthryloxy)anthraquinone/benzene-d6, confronted with the pertinent literature data, point to possible deviations from the classical limit. However, the predicted DQR effects are too small to be observed in solid echo 2H NMR spectra of the C6D6 isotopomer. The chances of detecting the effects are improved when Carr-Purcell echo 1H spectra of a single crystal of the isotopomer including C6H6 as a guest are considered. The substantial differences in the sensitivity to the DQR effects of the spectra of protonated and deuterated benzene are concerned with different magnitudes of the intramolecular dipolar spin couplings. The dynamic isotope effect (C6D6 vs C6H6), which is small in this case, is only of secondary importance. Legitimacy of the use of the jump model in 2H NMR line shape studies of benzene-d6 is fully confirmed by the present considerations. However, the physical significance of the dynamic parameters extracted from such studies is shown from a new perspective.

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

“…(2), in which rate constants k 01 and k 01 were given values of (α 2 + β 2 )τ and 4β 2 τ , respectively [see Eqs. (13) and (14)]. The spectra obtained using these two methods match each other rather well.…”

“…2,16 For the methyl group considered from the stochastic perspective, this takes place when α 2 τ a = 3β 2 τ b , see Eqs. (13) and (14). It may be of interest to notice that, in the classical limit, the same line shape function can be modeled by two completely different MC schemes.…”

“…16 However, in the studies of DQR effects for the systems such as the benzene protons, use of ) echo or other sophisticated NMR techniques would be indispensable. 13 Our previous experience with such experiments is that the assumption about ideal performance of the radio frequency pulses must be abandoned in the line shape analyses. This necessitates numerical evaluation of the evolution of the density matrix in the pulse duration.…”

“…All of this afforded evaluation of the quantities on the rhs of Eqs. (13) and (14), including their dependence on temperature. A detailed presentation of these results out of the wider context of the derivation of the effective equation of motion, Eq.…”

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

“…For inert molecular rotators like the benzene ring, the inclusion of coherences between different TR states appears to be of a critical significance for the calculated rate constants to have physical sense. 39 For the methyl groups it has minor significance unless, such as in the present case, the dynamics near the classical limit are of interest. These peculiar features of the methyl group, as compared to rotators with large moment of inertia, are concerned with substantial differences between the corresponding frequency spectra.…”