Hamiltonian second-order effective spin

This spin-spin constant, a, should not be confused with the fine-structure constant a = e2/fic. The direct spin-spin contribution to the A doubling, a, cannot be separated from a second-order spin-orbit term. All contributions are combined in the ov parameter, which is the coefficient of the term in the effective Hamiltonian (Brown et al. 1979 Brown and Merer, 1979) with a AE = -AA = 2 selection rule,... 

Figure 7.5. The three terms in the effective second order hamiltonian obtained from the Hubbard model in the limit t < U. (a.) Thehopping term, which allows spin-up or spin-down electrons to move by one site when the neighboring site is unoccupied, (b) The exchange term, which allows electrons to exchange spins or remain in the same configuration, at the cost of virtual occupation of a site by two electrons of opposite spin (intermediate configuration), (c) The pair hopping term, which allows a pair of electrons with opposite spins to move by one site, either with or without spin exchange, at the cost of virtual occupation of a site by two electrons of opposite spin (intermediate configuration). Adapted from Ref. [87].

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

Our analysis thus far has assumed that solution of the spin Hamiltonian to first order in perturbation theory will suffice. This is often adequate, especially for spectra of organic radicals, but when coupling constants are large (greater than about 20 gauss) or when line widths are small (so that line positions can be very accurately measured) second-order effects become important. As we see from... 

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. 

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

Thus far, we have investigated the various contributions to the effective Hamiltonian for a diatomic molecule in a particular electronic state which arise from the spin-orbit and rotational kinetic energy terms treated up to second order in degenerate perturbation theory. Higher-order effects of such mixing will also contribute and we now consider some of their characteristics. 

The separation between adjacent spin components is therefore -2a/5 which equates with 2A(] ) from the effective Hamiltonian. Hence A(1> = -a/5 or -83.4 cm-1, using fFe = 417 cm-1. The value obtained from experiment is -77.3 cnr1 although in practice it is difficult to model the spin-rotation levels of FeH with an effective Hamiltonian because of large spin-orbit perturbations [38]. For molecules like FeH, one would expect second- and higher-order contributions to A to be significant. 

In the present work, ID QE, QCPMG, single-pulse MAS and QCPMG-MAS experiments for analysis of multi-site jump processes involving quadrupolar nuclei will be examined theoretically and by simulations. For this purpose, the Hamiltonian includes the first as well as secular second-order terms for both the CSA- and EFG-tensors, the second-order EFG-CSA mix-term and for spin-1 also the secular third-order term for the EFG-tensor. In addition effects of finite rf-pulses will be explored. Previously this has been presented for either half-integer quadrupolar nuclei70 or spin-1 nuclei71 only. 

Except for some quadrupolar effects, all the interactions mentioned are small compared with the Zeeman interaction between the nuclear spin and the applied magnetic field, which was discussed in detail in Chapter 2. Under these circumstances, the interaction may be treated as a perturbation, and the first-order modifications to energy levels then arise only from terms in the Hamiltonian that commute with the Zeeman Hamiltonian. This portion of the interaction Hamiltonian is often called the secular part of the Hamiltonian, and the Hamiltonian is said to be truncated when nonsecular terms are dropped. This secular approximation often simplifies calculations and is an excellent approximation except for large quadrupolar interactions, where second-order terms become important. 

One of the results obtained for tetrahedral centers formed by 3d ions is that one for Mn " " (3d -configuration) in ZnS [47]. The splitting of the " Ti orbital triplet of Mn + ion was analyzed using the second-order effective spin-Hamiltonian and comparing the calculated splittings with the observed ones. The lowest estimate for the JT energy in ZnSMn " " was obtained to be 750 cm [47]. 

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