The Spin Hamiltonian

The Hamiltonian for a single spin was given in Eq. 2.4. For an N-spin system, Hi may be thought of as containing two parts  

which describes the interaction of each nuclear spin with the imposed magnetic field. is the sum of N terms of the sort given in Eq. 2.4, with each 

In this and subsequent expressions we drop Planck s constant h in order to convert to the more useful unit of hertz, rather than joules. To further simplify the notation, we substitute the Larmor relation, Eq. 2.14, into Eq. 6.3 to get 

which describes the coupling between all possible pairs of nuclei. In accord with the discussion in Section 5.1, each such interaction is described by a scalar product of the spin operators, and the sum covers all possible pairwise interactions  

Equation 6.5 is just the expanded form of a vector dot product and is more useful for later computations. For N nuclei the sum includes N(N — 1) terms. For example, for a system of four nuclei there are four terms in and six terms in 

Perturbational Expansion and Decomposition of Spin Interaction Energies 

At moderate to high magnetic fields (i.e., Ho 0. T), the electron Zeeman interaction dominates the EMR spectrum, which at its simplest may be described by a single transition hv=ge eHo, where ge and Pe are the so-called g-value and the Bohr magneton (A fundamental unit of the electron s magnetic moment, ettHtrie = 9.274 X 10 J T ), respectively. In atoms and molecules the g-value is replaced by a tensor and deviates from the scalar quantity of 2.0023 for a free electron. The 

EMR spectra correspond to transitions among the electronic Zeeman levels, subject to selection rule Amg = 1, Amj = 0 (for all rtij of the system), and it follows that these comparatively large transition energies are difficult to correlate with weak molecular interactions or subtle changes in the nuclear hyperfine energies due to substituent effects. 

In effect, ENDOR and ESEEM spectra permit one to conceptually drop the electronic Zeeman term fi-om the spin Hamiltonian and work on an energetic scale that is comparable to NMR spectroscopy. Both techniques can be used to obtain specific terms of the spin Hamiltonian provided that one has the means to experimentally deconvolute the spectroscopic transition energies. The primary difference between ENDOR and ESEEM resides with the manner in which the electronic 

Zeeman contribution is effectively taken out of the spectrum. In the remainder of this section, the chemically relevant spin-Hamiltonian terms will be described in the context of their chemically relevant origin. 

The Hamiltonian appropriate for the description of the spin properties of the triplet system is 

- nuclear spin operator of nucleus i D - fine structure tensor 

The sum in the Hamiltonian runs over all nuclei, intra- and intermolecular, coupled to the electron spin. 

The third part of the spin-Hamiltonian describes the interaction of the electron spin with the surrounding nuclear spins. For the a-protons in pentacene (the protons bound to the carbons in positions a, y, s and symmetry equivalent positions, see Fig. 1) the principal axes of the hyperfine tensor coincide with the zero-field tensor axes and the Hamiltonian describing the hyperfine interaction for such a proton can be written Hhf = lAxlSxIx + diAyySyly + SjA SJ. Here Axx = —91 MHz, Ayy = 29 MHz, and A.z = — 61 MHz are empirically determined constants. We have 

In the presence of an applied magnetic field the new eigenstates are linear combinations of the zero-field states and the hyperfine interaction may lead to nonvanishing matrix elements in first order. This results in a considerable broadening of the magnetic resonance lines. 

The energy levels for the states represented by the spin Hamiltonian can be written 

Where D and E are parameters which describe atomic energy level splitting from axial and non-axial interactions, the subscripts x, y and z identify the components of the atomic spin S and the nuclear spin I, fi and are the Bohr magneton and the nuclear magneton, B is the magnetic field, g and gfi are the electronic and nuclear -factors, A is the hyperfine coupling tensor, e is the charge on the proton and V and t are the principal component and the asymmetry parameter of the EFG. 

For electrons of the nucleus atom, l z=[—e/(4wo)] 3 cos d— l r . The quadrupole interaction is considered in more detail in Chapters 2 and 3. The flnal term describes the direct interaction of a magnetic field B on the nuclear dipole moment. 

The general form of the magnetic hyperflne interaction used in the 

For a paramagnetic ion in an external field B, the total effective field Bgg felt by the nucleus is given by 

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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... 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

The spin Hamiltonian operates only on spin wavefunctions, and all details of the electronic wavefunction are absorbed into the coupling constant a. If we treat the Fermi contact term as a perturbation on the wavefunction theR use of standard perturbation theory gives a first-order energy... 

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

The spin Hamiltonian is thus generated. In particular it can be used to examine the Tq-S mixing of electron spin states and its relationship to the distributions of populations of nuclear spin states. The total spin Hamiltonian is given in equation (15) which contains both electron and nuclear terms. 

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. 

The most important contributions to the spin Hamiltonian can be expressed as one-electron operators, and it will be shown that tl matrices Hf and Hf, vanish, as long as the reference state is computed up to one order of perturbation smaller than these matrices. Thus,... 

The simplest iron-sulfur centers, which were first discovered in ru-bredoxins, consist of one iron ion coordinated by a distorted tetrahedron of cysteinyl sulfur atoms. This environment provides a weak ligand field giving a spin equal to and 2 when the ion is Fe(III) and Fe(II), respectively. It also determines the splitting of the ground spin manifold, and consequently the characteristics of the EPR spectrum. This splitting is generally described in the framework of the spin Hamiltonian ... 

The EPR spectra of cell walls saturated with copper has been fitted to the numerical solutions of the spin hamiltonian describing the EPR lineshape of cupric ions. Two simulations have been performed. The first one (Fig. 4.a) considers that all uronic acids of the cell walls are similar the best fit is rather poor. The second one assumes existence of two populations of exchange sites with different parameters. In this case, the optimization is much better and confirms the existence of two different types of uronic acids in the cell wall (Fig. 4.b). 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

The spin-Hamiltonian concept, as proposed by Van Vleck [79], was introduced to EPR spectroscopy by Pryce [50, 74] and others [75, 80, 81]. H. H. Wickmann was the first to simulate paramagnetic Mossbauer spectra [82, 83], and E. Miinck and P. Debmnner published the first computer routine for magnetically split Mossbauer spectra [84] which then became the basis of other simulation packages [85]. Concise introductions to the related modem EPR techniques can be found in the book by Schweiger and Jeschke [86]. Magnetic susceptibility is covered in textbooks on molecular magnetism [87-89]. An introduction to MCD spectroscopy is provided by [90-92]. Various aspects of the analysis of applied-field Mossbauer spectra of paramagnetic systems have been covered by a number of articles and reviews in the past [93-100]. 

For the evaluation of magnetically split Mossbauer spectra within the spin-Hamiltonian formalism, the purely -dependent Hamiltonian must be extended by an appropriate... 

The spin-Hamiltonian formalism is a crutch in the sense that it is a parameterized theory, but it provides a common theoretical frame for the various experimental techniques with a minimum number of adjustable parameters that describe the essential physics of the system under investigation. Even more important is the fact that the same parameters can be derived relatively easily from quantum chemical calculations. Therefore, theoreticians appreciate the concept as a convenient place to rest in the analysis of experimental data by theoretical means [123, 124]. 

Orbitally degenerate grormd states, in general, cannot be treated in the spin-Hamiltonian approach. In this case, SOC has to be evaluated explicitly on an extended basis of spin-orbit functions. However, in coordination chemistry and bioinorganic chemistry, this is only of marginal importance, because the metal centers of... 

Finally, the spin Hamiltonian also contains contributions from the magnetic and quadrupole hyperfine interactions, Hhf and Hq where... 

As discussed in Sect. 6.2, the electronic states of a paramagnetic ion are determined by the spin Hamiltonian, (6.1). At finite temperamres, the crystal field is modulated because of thermal oscillations of the ligands. This results in spin-lattice relaxation, i.e. transitions between the electronic eigenstates induced by interactions between the ionic spin and the phonons [10, 11, 31, 32]. The spin-lattice relaxation frequency increases with increasing temperature because of the temperature dependence of the population of the phonon states. For high-spin Fe ", the coupling between the spin and the lattice is weak because of the spherical symmetry of the ground state. This... 

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... 

Blinc R (2007) Order and Disorder in Perovskites and Relaxor Ferroelectrics. 124 51-67 Boca R (2005) Magnetic Parameters and Magnetic Functions in Mononuclear Complexes Beyond the Spin-Hamiltonian Formalism 117 1-268 Bohrer D, see Schetinger MRC (2003) 104 99-138 Bonnet S, see Baranoff E (2007) 123 41-78... 

When an electron is placed in a magnetic field, the degeneracy of the electron spin energy levels is lifted as shown in Figure 1.1 and as described by the spin Hamiltonian. ... 

When one or more magnetic nuclei interact with the unpaired electron, we have another perturbation of the electron energy, i.e., another term in the spin Hamiltonian ... 

The eigenfunctions of the spin Hamiltonian [eqn (1.7)] are expressed in terms of an electron- and nuclear-spin basis set ms, mr), corresponding to the electron and nuclear spin quantum numbers ms and mr, respectively. The energy eigenvalues of eqn (1.7) are ... 

Once a hyperfine pattern has been recognized, the line position information can be summarized by the spin Hamiltonian parameters, g and at. These parameters can be extracted from spectra by a linear least-squares fit of experimental line positions to eqn (2.3). However, for high-spin nuclei and/or large couplings, one soon finds that the lines are not evenly spaced as predicted by eqn (2.3) and second-order corrections must be made. Solving the spin Hamiltonian, eqn (2.1), to second order in perturbation theory, eqn (2.3) becomes 4... 

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... 

As illustrated in Chapter 2, ESR spectra of radicals in liquid solution can usually be interpreted in terms of the spin Hamiltonian ... 

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