Second Order ENDOR Frequencies

Contents Introduction. - ENDOR-Instrumentation. - Analysis of ENDOR Spectra. - Advances ENDOR Techniques. - Interpretation of Hyperfine and Quadrupole Data. - Discussion of the Literature. - Concluding Remarks. - Appendix A Abbreviations Used in this Paper. - Appendix B Second Order ENDOR Frequencies. - Appendix C Relations Between Nuclear Quadrupole Coupling Constants in Different Expressions of Hq (Sect.5.2). - References. - Subject Index. 

First order ENDOR frequencies of nonequivalent nuclei or of pairs of magnetically equivalent nuclei are given by Eq. (3.3) which is derived from the direct product spin base. To obtain correct second order shifts and splittings, however, adequate base functions have to be used. We start the discussion of second order contributions with the most simple case of a single nucleus and will then proceed to more complex nuclear spin systems. 

In this section analytical expressions for ENDOR transition frequencies and intensities will be given, which allow an adequate description of ENDOR spectra of transition metal complexes. The formalism is based on operator transforms of the spin Hamiltonian under the most general symmetry conditions. The transparent first and second order formulae are expressed as compact quadratic and bilinear forms of simple equations. Second order contributions, and in particular cross-terms between hf interactions of different nuclei, will be discussed for spin systems possessing different symmetries. Finally, methods to determine relative and absolute signs of hf and quadrupole coupling constants will be summarized. 

In transition metal complexes, proton hfs are normally < 20 MHz so that the corresponding second order contributions, which amount to < 10 kHz, may usually be neglected. For nitrogen ligands, however, the second order corrections produce frequency shifts up to 200 kHz. Since hf interactions of central ions can amount to several hundred megacycles, the terms in AE become very important for a correct description of the ENDOR spectra. 

Since the shifts produced by the cross-terms between ligand and central ion nuclei are often significant, the strikingly broad ENDOR lines ( 1 MHz) observed for transition metal ions61,62) may be traced back, at least in part, to unresolved splittings due to second order interactions of numerous ligands with the central ion. ENDOR frequencies up to second order for an I = 1 nucleus in the presence of a second ligand nucleus with a spin K = 1 are tabulated in Appendix B, Eqs. (B4). 

In second order, however, eight ENDOR frequencies are obtained for each ms-state. The transition frequencies tabulated in Appendix B, Eqs. (B 5) are again described by al5 a2 and a3 defined in (3.12). If the hfs is resolved in the EPR spectrum, the number of induced transitions depends on the mp-value of the saturated line in the EPR quintet. For mF = 0 six transitions, for mF = 1 four transitions, and for mF = 2 one transition are observed in the ENDOR spectrum of each ms-state62). 

For nuclei with A > vn, the first term in (3.25) is the dominant one, so that the observed ENDOR frequencies c(ms) are rather insensitive to the signs of Aj. Since the second order term 2 in (3.11) is dependent on the signs of Ay and thus of Aj (3.12), the relative signs of the latter may be found by including higher order contributions in the fitting process. 

In all single-crystal studies, the variation in resonance frequency or magnetic field is studied as a function of the orientation of the crystal in the magnetic field. A spin Hamiltonian of appropriate form is then solved and the parameters adjusted to fit the calculated variation with the experimental data. Most errors in doing this occur because approximate solutions of spin Hamiltonians are used for systems for which the approximations are not justified. Second-order effects are often very important in analyzing single-crystal ESR and ENDOR measurements. 

Fig. 18. 35 GHz cw ENDOR signal from O (/ = f) of nitrile hydratase in 35% enriched H2 0, taken at the high-held edge ( 3) of the EPR envelope. The quintet shown represents the v+ branch of the 0 ENDOR pattern that is centered at the O Larmor frequency, 7.3 MHz (filled triangle). The quadrupole splittings and hyperfine values calculated are estimated using second-order perturbation theory in the nuclear quadrupole interaction for an O nucleus. Conditions 12,650 G, 34.92 GHz, 0.16 mW microwave power, 1 G modulation amplitude, 0.5 MHz/sec if scan speed, 2 K. (Adapted from Jin et al. )...

For spin Hamiltonians of systems with low symmetry, expressions for ENDOR transition frequencies to second order have been given by several authors " In most of these papers the spin Hamilton operator chosen is not general enough to describe the observed ENDOR spectra of transition metal complexes. For instance, some authors consider only one single nucleus make use of the assumption oi even neglect... 

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