Spin Hamiltonian description

The most common approach to the interpretation of EPR and Mossbauer spectra of siderophores is the spin Hamiltonian formalism. The wavefimctions are parameterized in terms of a few coupling constants that arise in the spin Hamiltonian description of the electronic states. In this approach, the crystal field potential is generally described by a series of spherical harmonics. The corresponding operators are tabulated. ... 

In the spin Hamiltonian description, the presence of double exchange can be described by the introduction of a new parameter ... 

Computer synthesis of EPR spectra from a parametric spin Hamiltonian. Description of programs (MAGNSPEC). 

The complete spin Hamiltonian for a description of EPR and ENDOR experiments is given by... 

CIDNP involves the observation of diamagnetic products fonned from chemical reactions which have radical intemiediates. We first define the geminate radical pair (RP) as the two molecules which are bom in a radical reaction with a well defined phase relation (singlet or triplet) between their spins. Because the spin physics of the radical pair are a fiindamental part of any description of the origins of CIDNP, it is instmctive to begin with a discussion of the radical-pair spin Hamiltonian. The Hamiltonian can be used in conjunction with an appropriate basis set to obtain the energetics and populations of the RP spin states. A suitable Hamiltonian for a radical pair consisting of radicals 1 and 2 is shown in equation (B1.16.1) below [12]. 

MMVB is a hybrid force field, which uses MM to treat the unreactive molecular framework, combined with a valence bond (VB) approach to treat the reactive part. The MM part uses the MM2 force field [58], which is well adapted for organic molecules. The VB part uses a parametrized Heisenberg spin Hamiltonian, which can be illustrated by considering a two orbital, two electron description of a sigma bond described by the VB determinants... 

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

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. 

How do we know or decide what terms to put in the spin Hamiltonian This is a question of rather far-reaching importance because, since we look at our biomolecular systems through the framework of the spin Hamiltonian, our initial choice very much determines the quality limits of our final results. In other branches of spectroscopy this is sometimes referred to as a sporting activity. We are guided (one would hope) by a fine balance of intellectual inspection, (bio)chemical intuition, and practical considerations. In a more hypochondriacal vein, one could also call this the Achilles heel of the spectroscopy a wrong choice of the model (the spin Hamiltonian) will not lead to an accurate description of nature represented by the paramagnetic biomolecule. 

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. 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

In scheme (9) we could have substituted on the diagonal S(S -i-1)/2 for As a-ASb and BSa-BSb- As it stands, the matrix is a hybrid because it contains matrix elements on the off-diagonal and operators on the diagonal. The present form of (9) is, however, despite its awkward form, a more useful starting point for setting up a spin Hamiltonian for the description of trinuclear and tetranuclear clusters containing delocalized dimers. 

In present-day quantum chemistry the Heisenberg Spin Hamiltonian is widely applied for the description of magnetic coupling in transition-metal clusters and may read in the case of a many-electron system,... 

However, magnetic coupling behavior may be more complex to model than possible with a simple isotropic Heisenberg Spin Hamiltonian as defined in Eq. (79) and several recent studies set out to improve this description by modification of this Hamiltonian (86-89). 

In molecules, the interaction of surrogate spins localized at the atomic centers is calculated describing a picture of spin-spin interaction of atoms. This picture became prominent for the description of the magnetic behavior of transition-metal clusters, where the coupling type (parallel or antiparallel) of surrogate spins localized at the metal centers is of interest. Once such a description is available it is possible to analyze any wave function with respect to the coupling type between the metal centers. Then, local spin operators can be employed in the Heisenberg Spin Hamiltonian. An overview over wave-function analyses for open-shell molecules with respect to local spins can be found in Ref. (118). 

Electron spin resonance (ESR) measures the absorption spectra associated with the energy states produced from the ground state by interaction with the magnetic field. This review deals with the theory of these states, their description by a spin Hamiltonian and the transitions between these states induced by electromagnetic radiation. The dynamics of these transitions (spin-lattice relaxation times, etc.) are not considered. Also omitted are discussions of other methods of measuring spin Hamiltonian parameters such as nuclear magnetic resonance (NMR) and electron nuclear double resonance (ENDOR), although results obtained by these methods are included in Sec. VI. 

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 ... 

It took several decades for the effective Hamiltonian to evolve to its modem form. It will come as no surprise to learn that Van Vleck played an important part in this development for example, he was the first to describe the form of the operator for a polyatomic molecule with quantised orbital angular momentum [2], The present formulation owes much to the derivation of the effective spin Hamiltonian by Pryce [3] and Griffith [4], Miller published a pivotal paper in 1969 [5] in which he built on these ideas to show how a general effective Hamiltonian for a diatomic molecule can be constructed. He has applied his approach in a number of specific situations, for example, to the description of N2 in its A 3 + state [6], described in chapter 8. In this book, we follow the treatment of Brown, Colbourn, Watson and Wayne [7], except that we incorporate spherical tensor methods where advantageous. It is a strange fact that the standard form of the effective Hamiltonian for a polyatomic molecule [2] was established many years before that for a diatomic molecule [7]. 

The quantum description of N coupled protons in Hilbert space is given by a spin Hamiltonian of dimension 2 equalling the number of direct product spin-i states. Two experimental tools have been used for the decoupling of spin interactions, RF irradiation and MAS. In the following, any discussion of sample spinning assumes MAS conditions. MAS effectively eliminates the CSA and DDfcetero interaction between protons and other spin-i nuclei. 

Jansen and van der Avoird (1985) have also made spin-wave calculations as described earlier. The RPA equations with the effective spin Hamiltonian (140), averaged over the translations and librations, could be solved analytically for any wave vector q. The optical (q = 0) magnon frequencies emerging from these calculations are 6.3 and 20.9 cm-1, in reasonable agreement with the experimental values 6.4 and 27.5 cm-1. This agreement is very satisfactory if we realize that the spin Hamiltonian has been obtained from first principles, with none of its parameters fitted to the magnetic data. We conclude that the RPA model, both for the lattice modes and the spin waves, when based on a complete crystal Hamiltonian from first principles, yields a realistic description of several properties of solid O2 that were not well understood before. 

To determine these parameters accurately and rigorously, the experimental ESR spectrum should be compared to a computer-simulated spectrum calculated using trial parameters, and a convenient mathematical representation and description of the ESR spectrum should be provided by use of the operator spin Hamiltonian (Wertz and Bolton, 1972). In practice, the g-values and hyperline and superhyperfine constants, A, can be obtained relatively simply, although not rigorously, by direct computation from data derived accurately from the experimental ESR spectrum and from spectrometer setting values used in the measurement, according to conventional equations (Senesi, 1992). 

Let us remark that in crystals consisting of aromatic molecules, to which the theory of Sternlicht and McConnell (26) was applied, the excited triplet states are not three-fold degenerate even when an external magnetic field is absent. Due to the dipole spin-spin interaction between electrons the degeneracy is totally or partially removed, depending on the symmetry of the excited state wavefunction. By a phenomenological description of this splitting the so-called Spin-Hamiltonian is usually applied... 

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