Hamiltonian second-order effective

Spin-Hamiltonian Second order effects in solids... 

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 this section, we shall use the degenerate perturbation theory approach to derive the form of the effective Hamiltonian for a diatomic molecule in a given electronic state. Exactly the same result can be obtained by use of the Van Vleck or contact transformations [12, 13]. The general expression for the operator up to fourth order in perturbation theory is given in equation (7.43). Fourth order can be considered as the practical limit to this type of approach. Indeed, even its implementation is very laborious and has only been used to investigate the form of certain special terms in the effective Hamiltonian. We shall consider some of these terms later in this chapter. For the moment we confine our attention to first- and second-order effects only. 

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]. 

Using a perturbation satisfying V-e = Ve (such as perturbation proportional to cos 0), we obtain for the second-order effective Floquet Hamiltonian... 

We assume that the laser frequency is far from any resonance between the ground vibrational state and the excited ones, such that the partitioning is very similar to the one made with the electronic states the vibrational state w = 0) spans the Hilbert subspace and the other vibrational states 1),..., NV) the Hilbert subspace. We obtain for the second-order effective Floquet Hamiltonian connected to the ground vibrational state of the ground electronic state... 

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. 

Regardless whether the d-f interaction is predominantly exchange or admixing, the second order effect of the spin interaction in eq. (3.52) is the RKKY interaction between the lanthanide spins (Kim, 1966). The effective hamiltonian of this interaction is... 

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. 

The symmetric coupling case has been examined by using Sethna s approximations for the kernel by Benderskii et al. [1990, 1991a]. For low-frequency bath oscillators the promoting effect appears in the second order of the expansion of the kernel in coj r, and for a single bath oscillator in the model Hamiltonian (4.40) the instanton action has been found to be... 

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. 

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