ESR theory

The origin of ESR spectroscopy lies in the spin angular momentum possessed by the unpaired electron. Associated with this spin is a magnetic moment, g, which is proportional in size to the spin vector (S). 

It is clear from eqns. (2) and (3) that, the absence of the applied magnetic field, the two spin states Ms = are degenerate but that, in the presence of the applied field, this degeneracy is removed. The energy level separation between the two spin states has the value gpB and increases linearly with the magnetic field as shown in Fig. 1. Transitions between the two spin states are induced when electromagnetic radiation of the appropriate frequency v is applied and where hv matches the energy level separation between the two spin states 

The frequency necessary to induce the transition from one spin state to the 

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L.J. Berliner, ed., In vivo EPR (ESR) Theory and Applications, Kluwer Academic/Plenum Publishers, New York, 2004. 

Wertz, J., and J. R. Bolton, Electron Spin Resonance, McGraw-Hill, New York, 1972. A good treatment of ESR theory and applications. 

NO2 is a stable paramagnetic gaseous molecule at normal temperatures. The ESR parameters of NO2 trapped in a solid matrix have been well established from single-crystal ESR measurements and have been related to the electronic structure by molecular orbital studies [39]. Thus, the NO2 molecule has potential as a spin probe for the study of molecular dynamics at the gas-solid interface by ESR. More than two decade ago temperature-dependent ESR spectra of NO2 adsorbed on porous Vycor quartz glass were observed [40] Vycor is the registered trademark of Coming, Inc. and more information is available at their website. The ESR spectral line-shapes were simulated using the slow-motional ESR theory for various rotational diffusion models developed by Freed and his collaborators [41]. The results show that the NO2 adsorbed on Vycor displays predominantly an axial symmetrical rotation about the axis parallel to the O—O inter-nuclear axis below 77 K, but above this temperature the motion becomes close to an isotropic rotation probably due to a translational diffusion mechanism. 

ESR theory for determination of the number of spins or radicals in a given sample is described in detail by Poole [46]. The theory, however, does not fall within the scope of this chapter. Nonetheless, to gain knowledge about the selection of standard reference materials and experimental variables, additional references are cited [47-50]. 

Freed J H 1969 Theory of saturation and double resonance effects in ESR spectra. IV. Electron-nuclear triple resonance J. Chem. Rhys. 50 2271-2... 

Freed J 1979 Theory of multiple resonance and ESR saturation In liquids and related media Multiple Electron Resonance Spectroscopy ed M M Dorlo and J H Freed (New York Plenum) ch 3, pp 73-142... 

This could account for the paramagnetism, but esr evidence shows that the 2 cobalt atoms are actually equivalent, and X-ray evidence shows the central Co-O-O-Co group to be planar with an 0-0 distance of l3l pm, which is very close to the 128 pm of the superoxide, 02, ion. A more satisfactory formulation therefore is that of 2 Co atoms joined by a superoxide bridge. Molecular orbital theory predicts that the unpaired electron is situated in a rr orbital extending over all 4 atoms. If this is the case, then the jr orbital is evidently concentrated very largely on the bridging oxygen atoms. 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

Geometries, hyperfme structure, and relative stabilities of the different positional isomers of monodeuterated benzene cations have been studied theoretically by density functional theory, using the B3-LYP functional, and experimentally by ESR and ENDOR spectroscopy. A comparison between theoretical and experimental results at 30 K gives acceptable agreement, but further experiments on multiply deuterated species should improve the analysis by making the effects of deuteration larger. 

In addition to the magnetic differences between the deuteron and proton, however, their mass difference may also cause observable effects. A well known example is found in the theory of chemical reactions, where the so called kinetic isotope effects (KIE s) are an important source of information about reaction mechanisms. Also in the field of ESR, such effects may arise, although these have been much less studied than the KIE s. 

Malkin, V. G., Malkina, O. L., Eriksson, L. A., Salahub, D. R., 1995, The Calculation of NMR and ESR Spectroscopy Parameters Using Density Functional Theory in Modem Density Functional Theory A Tool for Chemistry, Seminario, J. M., Politzer, P. (eds.), Elsevier, Amsterdam. 

MO) with the protons in the nodal plane. The mechanism of coupling (discussed below) requires contact between the unpaired electron and the proton, an apparent impossibility for n electrons that have a nodal plane at the position of an attached proton. A third, pleasant, surprise was the ratio of the magnitudes of the two couplings, 5.01 G/1.79 G = 2.80. This ratio is remarkably close to the ratio of spin densities at the a and (3 positions, 2.62, predicted by simple Hiickel MO theory for an electron placed in the lowest unoccupied MO (LUMO) of naphthalene (see Table 2.1). This result led to Hiickel MO theory being used extensively in the semi-quantitative interpretation of ESR spectra of aromatic hydrocarbon anion and cation radicals. 

Most ESR studies of organic radicals were carried out in the 1950s and 1960s. They provided important tests of early developments in valence theory. The results of these early studies are nicely summarized in a review by Bowers.11 Applications of hyperfine splittings to structure determination are discussed in many of the texts and monographs referenced in Chapter 1. 

This is a simplified Hamiltonian that ignores the direct interaction of any nuclear spins with the applied field, B. Because of the larger coupling, Ah to most transition metal nuclei, however, it is often necessary to use second-order perturbation theory to accurately determine the isotropic parameters g and A. Consider, for example, the ESR spectrum of vanadium(iv) in acidic aqueous solution (Figure 3.1), where the species is [V0(H20)5]2+. 

Figure 3.3 Stick spectrum showing hyperfine pattern for coupling to three equivalent 59Co nuclei (1=1/2) computed to (a) first-order and (b) second-order in perturbation theory. (Adapted from ref. 7.) (c) Isotropic ESR spectrum of [PhCCo3(CO)9r in THF solution at 40°C.

In Chapter 4 (Sections 4.7 and 4.8) several examples were presented to illustrate the effects of non-coincident g- and -matrices on the ESR of transition metal complexes. Analysis of such spectra requires the introduction of a set of Eulerian angles, a, jS, and y, relating the orientations of the two coordinate systems. Here is presented a detailed description of how the spin Hamiltonian is modified, to second-order in perturbation theory, to incorporate these new parameters in a systematic way. Most of the calculations in this chapter were first executed by Janice DeGray.1 Some of the details, in the notation used here, have also been published in ref. 8. 

For cationic zeolites Richardson (79) has demonstrated that the radical concentration is a function of the electron affinity of the exchangeable cation and the ionization potential of the hydrocarbon, provided the size of the molecule does not prevent entrance into the zeolite. In a study made on mixed cationic zeolites, such as MgCuY, Richardson used the ability of zeolites to form radicals as a measure of the polarizing effect of one metal cation upon another. He subsequently developed a theory for the catalytic activity of these materials based upon this polarizing ability of various cations. It should be pointed out that infrared and ESR evidence indicate that this same polarizing ability is effective in hydrolyzing water to form acidic sites in cationic zeolites (80, 81). 

Freed, J.H. 1976. Theory of slow thumbhng ESR spectra for nitroxides. In Spin Labeling Theory and Applications, ed. LJ. Berliner. New York Academic Press. 

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