Spin Hamiltonian application

One of the problems to extract structural information from EPR lies in the correct simulation of the experimental spectra. Misra536 reviewed spin Hamiltonians applicable to exchange-coupled Mn complexes, and described techniques for simulation of EPR spectra. Various structural models for the Mm-cluster in PS II were presented. 

The theory of stationary ENDOR transition frequeneies is well understood. In the framework of metalloprotein applications we consider one metal ion (or more) in the center of a coordination sphere in which ligands like protons and nitrogen nuclei are in interaction distance with the ion. Shown in Figure 1 are sketches of three different iron-sulfur clusters in proteins and their immediate environment that are of relevance for the present report. The Spin Hamiltonian applicable to this situation contains metal ion terms indexed as M and ligand terms (indexed L) ... 

As has been shown (Kaptein, 1971b, 1972a) by application of perturbation theory (Itoh et al., 1969), the spin Hamiltonian in equation (17) can be obtained for S and T radical pairs. 

A wide variety of ID and wD NMR techniques are available. In many applications of ID NMR spectroscopy, the modification of the spin Hamiltonian plays an essential role. Standard techniques are double resonance for spin decoupling, multipulse techniques, pulsed-field gradients, selective pulsing, sample spinning, etc. Manipulation of the Hamiltonian requires an external perturbation of the system, which may either be time-independent or time-dependent. Time-independent... 

The EPR spectrum is a reflection of the electronic structure of the paramagnet. The latter may be complicated (especially in low-symmetry biological systems), and the precise relation between the two may be very difficult to establish. As an intermediate level of interpretation, the concept of the spin Hamiltonian was developed, which will be dealt with later in Part 2 on theory. For the time being it suffices to know that in this approach the EPR spectrum is described by means of a small number of parameters, the spin-Hamiltonian parameters, such as g-values, A-values, and )-values. This approach has the advantage that spectral data can be easily tabulated, while a demanding interpretation of the parameters in terms of the electronic structure can be deferred to a later date, for example, by the time we have developed a sufficiently adequate theory to describe electronic structure. In the meantime we can use the spin-Hamiltonian parameters for less demanding, but not necessarily less relevant applications, for example, spin counting. We can also try to establish... 

To illustrate how a spin Hamiltonian can be calculated from MO s, we shall consider the dl configuration in D4h symmetry. The results will be applicable for Cu(II) complexes by considering the d9 configuration a... 

To be distinguished, the use of HDVV paradigm in systems with firm covalent bonds may really reveal the extension to which the model itself is applicable and to suggest the further necessary terms. This is indeed the case encountered in our approach. We found that for an improved description of selected set of states, the spin Hamiltonian has to be expanded with new terms, proposed here as the intercentric generalization of the traditional biquadratic terms. We propose and use here the following extended form ... 

For organic spin systems, one frequently assumes applicability of Heisenberg spin behavior, in which all interactions can be reasonably modeled by pairwise exchange interactions. A typical Heisenberg spin Hamiltonian for exchange Jy between various spin sites i and j, with spin quantum numbers S, and Sj, is given in the following equation ... 

Spin Hamiltonian Valence Bond Theory and its Applications to Organic Radicals, Diradicals, and Polyradicals... 

The spin-Hamiltonian VB theory rests on the same principles as the qualitative theory presented in Chapter 3, with some further simplifying assumptions. This chapter describes the method and focuses on its qualitative applications. 

The spin-Hamiltonian VB theory is a very simple and easy-to-use semiempi-rical tool that is based on the molecular graph. It is consistent with the VB theory described in Chapter 3, albeit with some simplifying assumptions and a more limited domain of application. Typically, this theory deals with the neutral ground or excited states of conjugated molecules or other homonuclear assemblies with one electron per site. For large systems, it reproduces the results of PPP full Cl, while dealing with a much smaller Hamiltonian matrix. 

An even more quantitative application of VB theory can be developed from the realization that the nearest-neighbor VB model as developed, for example, by Pauling [10], can be mapped exactly onto a Heisenberg spin Hamiltonian [17]. The Heisenberg spin Hamiltonian has long been used to study the interaction between magnetic atoms in transition metal compounds and other paramagnetic substances [18], and can be written most simply as... 

We now have a formula for constructing the density matrix for any system in terms of a set of basis functions, and from Eq. 11.6 we can determine the expectation value of any dynamical variable. However, the real value of the density matrix approach lies in its ability to describe coherent time-dependent processes, something that we could not do with steady-state quantum mechanics. We thus need an expression for the time evolution of the density matrix in terms of the Hamiltonian applicable to the spin system. 

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