Electron Spin Resonance Volume 4

A Review of the Literature Published Between June 1975 and November 1976

By P. B. Ayscough

The Royal Society of Chemistry

Copyright © 1977 The Chemical Society
All rights reserved.
ISBN: 978-0-85186-781-6

Contents

Chapter 1 Relaxation, Lineshapes, and Polarization By P. W. Atkins, 1,
Chapter 2 EN DOR and ELDOR By K. Mobius, 16,
Chapter 3 Triplets and Biradicals By A. Hudson, 30,
Chapter 4 Transition-metal Ions By A. L. Porte, 40,
Chapter 5 Inorganic and Organometallic Radicals By M. C. R. Symons, 84,
Chapter 6 Organic Radicals: Structure By B. C. Gilbert, 111,
Chapter 7 Organic Radicals: Kinetics and Mechanisms of their Reactions By R. C. Sealy, 144,
Chapter 8 Organic Radicals in Solids By T. J. Kemp, 163,
Chapter 9 Radical Ions, Ion Pairs, and Dynamic Processes By R. F. Adams, 198,
Chapter 10 Biological Systems By P. F. Knowles and B. Peake, 212,

CHAPTER 1

Relaxation, Lineshapes, and Polarization

BY P. W. ATKINS

Relaxation has had a quiet episode, with few papers of major significance apart from a most helpful and detailed review of saturation transfer spectroscopy, a technique referred to in early Reports. The general impression is that there has been a general dispersion of relaxation theorists either into chemically induced spin polarizations or into more general techniques of studying molecular motion in liquids. The brevity of this chapter reflects this dispersal.

1 General Relaxation Theory

A fundamental assessment of relaxation and entropy production has been made by Lenk. The quantum-statistical expression for entropy production a has been used as the starting point of a description of spin-relaxation. This approach leads to a relation for a in terms of a quantum correlation function over a dissipation operator connected with the flux operator and a generalized force. The final expression for the phenomenological coefficient of response theory yields the conventional expression for Ti.

Previous Reports have underlined the significance of projection operator formulations, this importance stemming from their ability to isolate and to separate the fast and slow modes of motion, the former being the rapid molecular motions responsible for the latter, the relatively slow spin relaxations. Diestler ‘ has presented an extensive and detailed exposition of a projection operator technique for the treatment of a two-level system in contact with a thermal bath. He uses the Zwanzig-Mori projection operator formalism to derive exact equations of motion for the correlation function and population characterizing the two-level system. He shows that in the van Hove weak-coupling limit, the correlation function and the excess population decay exponentially, and he establishes the relation between the two characteristic time-constants. His approach resembles that reviewed by Kivelson and Ogan (and quoted with approval in an earlier Report) and he attempts to pinpoint the differences. Hills has extended an earlier projection operator calculation of analytical expressions for lineshapes under conditions of slow motion by incorporating a radiation Hamiltonian into the formalism. This has the advantage of treating both radiation and lattice interactions with equal emphasis, and of permitting the removal of the linear response approximation. The practical outcome is that one obtains an analytical description of the effects of saturation on the lineshapes of slowly tumbling molecules.

The evolution of a spin density-matrix is of central attention both for theories of spin relaxation and in the analysis of spin polarization. Evans has developed a projection operator technique to derive a master equation for the ensemble-averaged matrix that accounts for the initial correlation of the spin and lattice degrees of freedom. This equation is Redfield-like with a generalized averaged tetradic, identical with that of Deutch. If the initial conditions of the combined lattice-spin density matrix are replaced by a source with anisotropy, then the equation of motion for the ensemble-averaged spin density matrix is found to have an effective source term that is isotropic and another that arises from the cross-correlation of the spin-lattice interaction and the anisotropic component of the source. The results of the analysis are applied to a TM-CIDEP calculation, the significant development being that the relaxation rate is made dependent on the intersystem crossing rate. The effect is to diminish the effective triplet spin-lattice relaxation time beneath the true magnetic resonance value.

The effect on the linewidth of a chemical exchange between a radical ion and its precursor is familiar and well understood, but the effect on spin-lattice relaxation itself has not been studied in depth, and in cases where lineshapes have been observed in detail there have remained small discrepancies in the fit at intermediate exchange rates. The models on which these calculations are based envisage instantaneous electron transfer, with no flight time and no change in the electron spin state, and assume that the rate is independent of the nuclear spin state of the molecules involved. Cheng and Weissman have made careful measurements on KTCNE in DME and analysed the Bloch equations with a view to resolving these problems. In the slow-exchange limit T1, as measured by progressive saturation of individual lines, becomes shorter with increasing exchange rate, as does T2. In the fast-exchange limit T1 reverts to its value in the absence of exchange. This is a nice, elegant paper which has implications for theories of spin relaxation and models of electron transfer processes.

Brown has continued to examine phase-memory loss and has investigated the effects of molecular oxygen on the phase-memory time TM and the electron spin T1 of spin-labelled polystyrene in the temperature range 77 — 295 K. He identified two mechanisms by which molecular motion can influence TM. In the first, neighbouring spins contribute to the locally fluctuating field by dipolar interaction; in the second, which is operative at fast motional frequencies close to the Larmor frequency, the molecular motions influence T1 and indirectly determine TM by spin-lattice induced spin flips on the neighbour. When oxygen is present, TM and T1 have a minimum at the same temperature. The minimum in T1 is attributed to a time-dependent dipolar interaction between O2 and the radical spin, and the minimum in TM is ascribed to a Type 2 mechanism. The values of TM and T1 suggest correlation times of the order of 10-8 s at 77 K and 2 x 10-11 s for temperatures above the T1 minimum. These correlation times appear to be determined either by localized motion of O2 or by the O2 spin relaxation time. When the samples are in vacuo, T1 decreases monotonically as the temperature is raised, while TM shows a minimum at 140 K. The latter is traced to the onset of δ-relaxation.

In conventional treatments the pair-correlation function is used to describe some initial spatial distribution of molecules, and the subsequent time-development is treated in terms of a Green’s function or conditional probability: in the limit of large times these yield uniform distributions which lack pair-correlation effects. If the pair-correlation between spin-bearing molecules is related to a potential of average force then one obtains an effective force between them. This has to be inserted into a generalized diffusion equation. In their paper, Hwang and Freed set out to demonstrate that finite-difference techniques can be applied to this kind of problem to give explicit solutions for spin-correlation functions and spectral densities, even when the pair-correlation functions are so complex that they can be expressed only numerically. They concentrate on dipole-dipole relaxation, and give explicit examples based on hard-sphere Percus-Yevick equations (for ethane) and Debye-Hilckel theory (for ionic solutions). Analytic solutions are given which are appropriate for the proper boundary-value problem for the relative diffusion of molecules, which hitherto has largely been neglected in spin relaxation studies.

2. Motion

A pleasing review of the physical chemistry of the liquid state has been given by Kohler, Wilhelm, and Posch.13 Apart from structural information, they also review some dynamical aspects and treat orientational correlation and memory functions. The review is helpful as a summary of present work in the field. An even more complete account has been published by Rowlinson and Evans, who give detailed references.

Hydrodynamic theories have had notable success in describing modes of molecular motion underlying relaxation processes even though they were derived and, in most cases, apparently limited to the description of macroscopic objects in continuous fluids. Kowert and Kivelson have presented some new data on vanadyl acetylacetonate in several two-component solvents of various compositions. They found that a modified Debye equation gives reasonable results for the rotational correlation time τR for a number of solvent systems, the modification being the introduction of the generalized dimensionless K parameter through K = KA XA + KB XB. This parameter is independent of temperature, pressure, and viscosity in ideal mixtures, but the description fails for non-ideal solutions (such as those involving butanol near room temperature). The authors interpret the results by ascribing the rotational relaxation to long-wavelength (hydrodynamic) perturbations coupled to the probe molecule by nearest-neighbour interactions with a strength represented by K. This parameter can be interpreted in terms of hydrodynamic slip-stick conditions and related to the magnitudes of intermolecular torques.

The analysis of high-frequency molecular reorientation in liquids in terms of angular velocity correlation times and dynamical molecular models is often subject to uncertainties arising from slight differences in the definitions of correlation functions and correlation times. Kivelson has examined some of these problems and has presented an elucidation in a brief paper. Evans has looked at the cumulant expansion of a Fokker-Planck equation with the aim of demonstrating that, starting from the coupled rotational translational Fokker-Planck equation, the application of a projection operator generates an equation of motion for the orientational and positional correlation function in which diffusion tensors can be calculated directly on the basis of a one-parameter theory. The orientational correlation functions for the lth-rank spherical harmonics are in agreement with the high-density Fixman-Rider solutions. In an alternative approach to a similar problem, McMahon has described, in a brief note, how to apply a new Volterra-type integral equation to obtain rotational correlation functions, and employs a memory kernel in contrast to a memory function. This arises more naturally from the extended rotation models.

The Hubbard relation (essentially τR = φ/τJ connecting reorientational and angular momentum correlation times has played an important role in the quantitative theory of the spin-rotation interaction and has been investigated for linear molecules on the basis of the extended diffusion model and the Ivanov jump model. Whereas the M-diffusion model leads to an unphysical result (suggesting that the model is fundamentally unsound) the J-diffusion model leads to an acceptable form of the relation. When τJ2 [??] I/kT both the extended diffusion model and the Ivanov model give a relation similar to Hubbard’s original frictional model. When τJ2 > I/kT the various versions of the Hubbard relation differ enough to be distinguishable experimentally.

3 Slow Motion and No Motion

The technique of saturation transfer e.s.r. has been commented on in earlier Reports. In this technique the dispersion signal is detected under moderate microwave saturation and with rapid magnetic field modulation. The signal is detected at the modulation frequency, but 90° out of phase. The spectrum of immobilized radicals obtained in this way normally resembles the undifferentiated absorption, but if the radicals change their orientations significantly during the modulation period, then the spectrum changes. This makes the technique sensitive to motion on the time-scale 10-5 — 10-7 S (with 100 kHz modulation).

Mailer and Hoffman have reported observation of the temperature variation of e.s.r. absorption and rapid adiabatic passage dispersion signals for di-t-butyl nitroxide dissolved in s-butyl benzene and absorbed on silica. They observed adsorbate motion over a wide temperature range, and the combined techniques show that surface nitroxides show a broad distribution of correlation times and activation energies.

One of the principal applications of this work on saturation transfer e.s.r. is likely to be to the study of the molecular dynamics of biological macromolecules and supramolecular complexes. Dalton and co-workers have now published an extensive and comprehensive account of the instrumentation and theoretical background to this form of spectroscopy. Two saturation transfer e.s.r. detection schemes are discussed in detail: these are dispersion (detected φ/2 out of phase with respect to the 100 kHz modulation and adsorption detected φ/2 out of phase with respect to the second harmonic of the 50 kHz field modulation). Two theoretical approaches are also described. One uses coupled Bloch equations; the other is based on the stochastic Liouville equation for the density matrix with the orientation variables treated by transition rate matrix or orthogonal function expansion techniques. Both approaches agree with each other and with a variety of experimental spectra. The authors emphasize that the spectra depend on a number of relaxation processes other than rotational diffusion, and so it is important to ensure the accuracy of measured correlation times.

Polnaszek and Freed have examined anisotropic ordering, spin relaxation, and slow tumbling in liquid crystalline solvents for the 2,2,6,6-tetramethyl- 4-piperidone N-oxide radical. They examined the linewidths on the basis of the theory proposed earlier but modified for anisotropic ordering both in the motional narrowing and slow tumbling region. In a very detailed analysis they find that the motional narrowing r esults are normally consistent with isotropic rotational diffusion, but under a weak asymmetric ordering potential such that [FORMULA NOT REPRODUCIBLE IN ASCII] ~ -0.1; activation energies are characteristic of the twist viscous properties of the liquid crystals. They observed anomalous lineshape behaviour in the incipient slow-tumbling region, and this is not explained by the extrapolation of the appropriate parameters obtained from the motional narrowing region. This anomaly is discussed in terms of the anisotropic viscosity and the director fluctuations. The latter is predicted to be of negligible importance for the weakly ordered spin probe and qualitatively of the wrong behaviour. The anisotropic viscosity also leads to physically untenable conclusions. They therefore go on to discuss the anomaly in terms of slowly fluctuating intermolecular torques, leading to a frequency-dependent diffusion coefficient. While this accounts partially for the observations, the slowness of the fluctuations suggests a new model based on a local solvent structure around the spin probe, and which may persist over periods longer than the orientational time of the probe. It is shown that such a model could have the same formal spectral effects as anisotropic rotational diffusion, and could yield non-Debye like spectra densities of the type that might be able to explain the anomaly.

The computation of e.s.r. spectra would be simple and quick if the magnetic field were held constant (constant eigenvalues) and the frequency swept. In practice the frequency is held constant and the field is swept, so that the matrix has to be diagonalized at each field setting. Conventional perturbation theory cannot be applied unless nuclear Zeeman energies are negligible, and various other complications prolong conventional calculations. Harriman has presented a method based on the partitioning of the spin Hamiltonian, and found it suitable for application to organic radicals. Coffey et al. have described a fast algorithm for a computer for the simulation of the effects of rotational diffusion, electron and nuclear relaxation, microwave power, and modulation frequency on saturation transfer e.s.r. spectra. Comparison of the theoretical and experimental spectra for nitroxide labels suggests that, although the electron T1 has only a weak dependence on the rotational correlation time, the variation of the ratio of the electron to nuclear spin-lattice relaxation times is marked. The authors also show that it is necessary to consider strong nuclear relaxation when spectra are being simulated when the correlation times are close to the reciprocal of the nitrogen nuclear resonance frequency.

(Continues…)Excerpted from Electron Spin Resonance Volume 4 by P. B. Ayscough. Copyright © 1977 The Chemical Society. Excerpted by permission of The Royal Society of Chemistry.
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