Equivalent Spin-Hamiltonian

The Hamiltonian (1) is spin free, commutative with the spin operator S2 and its z-component Sz for one-electron and many-electron systems. The total spin operator of the hydrogen molecule relates to the constituent one-electron spin operators as 

Suppose denotes the common eigenfunction with quantum number S of four operators, S2, Sz, Sf and S, being commutative with each other, then we have 

Evidently, if one defines a Hamiltonian operator containing only spin operators and numerical parameters as follows 

It was Pauling and his collaborators who generalized the Heitler-London treatment of the hydrogen molecule to general polyelectronic systems and gave the rules for evaluating the Hamiltonian matrix elements [3,4], which could also 

Here i j denotes a specified nearest-neighbor pair of AOs and Ktj is the corresponding exchange integral, and (Q TLKV) is taken as the zero of energy. 

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The spin Hamiltonian for a biradical consists of terms representing the electron Zeeman interaction, the exchange coupling of the two electron spins, and hyperfine interaction of each electron with the nuclear spins. We assume that there are two equivalent nuclei, each strongly coupled to one electron and essentially uncoupled to the other. The spin Hamiltonian is ... 

The Mu spin Hamiltonian, with the exception of the nuclear terms, was first determined by Patterson et al. (1978). They found that a small muon hyperfine interaction axially symmetric about a (111) crystalline axis (see Table I for parameters) could explain both the field and orientation dependence of the precessional frequencies. Later /xSR measurements confirmed that the electron g-tensor is almost isotropic and close to that of a free electron (Blazey et al., 1986 Patterson, 1988). One of the difficulties in interpreting the early /xSR spectra on Mu had been that even in high field there can be up to eight frequencies, corresponding to the two possible values of Ms for each of the four inequivalent (111) axes. It is only when the external field is applied along a high symmetry direction that some of the centers are equivalent, thus reducing the number of frequencies. 

Recently similar doublet structures have been observed in other systems with inversion symmetry58,66). Fujimoto et al.58) used a somewhat different perturbation approach for the explanation of the 14N-ENDOR spectra in copper-doped a-glycine, whereas Brown and Hoffman66) determined the nitrogen ENDOR frequencies of Cu(TPP) and Ag(TPP) by numerical diagonalization of the spin Hamiltonian matrix for an electron interacting with a single pair of equivalent 14N nuclei. 

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. 

Gv( f) covering symmetry67. For orientations of B0 in the mirror plane S, the symmetry group of the spin Hamiltonian is < 9f = C2h(e2f). The direct product base of the nuclear spin functions of two geometrically equivalent nuclei reduces to two classes, containing six A-type and three B-type functions, respectively. Second order perturbation theory applied to H = UtHU, where U symmetrizes the base functions of the Hamil-... 

The 55Mn2+ ion, S = 5/2,1 = 5/2, was studied in a-ZnV207. The orientation dependence of the spectra at 250 GHz was measured32 using a goniometer in a quasi-optical spectrometer at 253 K. Two magnetically inequivalent, but physically equivalent, sites were detected that could be described by the same set of spin Hamiltonian parameters. The (/-factor was found to be isotropic with a value close to 2 and the ZFS was 6 GHz. 

The similarity of the preceding first-order ESR treatment to the first-order NMR treatment of two coupled protons is evident. For an unpaired electron interacting with n equivalent nuclei of spin the hyperfine coupling term in the spin Hamiltonian is... 

A new treatment for S = 7/2 systems has been undertaken by Rast and coworkers [78, 79]. They assume that in complexes with ligands like DTPA, the crystal field symmetry for Gd3+ produces a static ZFS, and construct a spin Hamiltonian that explicitly considers the random rotational motion of the molecular complex. They identify a magnitude for this static ZFS, called a2, and a correlation time for the rotational motion, called rr. They also construct a dynamic or transient ZFS with a simple correlation function of the form (BT)2 e t/TV. Analyzing the two Hamiltonians (Rast s and HL), it can be shown that at the level of second order, Rast s parameter a2 is exactly equivalent to the parameter A. The method has been applied to the analysis of the frequency dependence of the line width (ABpp) of GdDTPA. These results are compared to a HL treatment by Clarkson et al. in Table 2. 

With these data, we calculated numerically the (mi, m2 He[t m u m 2) matrix elements of the effective exchange Hamiltonian, which are presented in Table 1. Each group of matrix elements can be associated with a definite equivalent spin operator, which has the same matrix elements in the basis set as those... 

Table 1. Non -zero (ni],m2 Haflm3,m4) matrix elements of the effective exchange Hamiltonian and the equivalent spin operators (all parameters are in cm-1)...

Key Words NMR, EPR, Spin-spin coupling, Equivalent nuclides, Spin-Hamiltonian energies and eigenstates, Exchange degeneracy. 

In complete analogy with this, for a nucleus with spin Io, interacting with a set of n equivalent other nuclei, the spin-Hamiltonian term is / Oplo (opI 1 + opI2+- -+opIn), and the same table applies to yield the spin-spin multiplet (see Ref. 36, p. 247, and Ref. 5, p. 26) for nuclear spins 1=1/2. 

The principle of sample spinning has been described in Section 3.3.3. As a result of sample spinning, the interaction Hamiltonian and thus the resonance frequency (cf. eqn (3.3.6)) becomes time dependent. For the simple case of a pair ij of dipolar coupled spins i, equivalent to a quadrupolar nucleus spin 1 like with, the time-dependent spin Hamiltonian is given by [Mehl, Schl]... 

A question of recent interest is whether the two iron sites in each molecule are equivalent (58), Mossbauer effect studies (28) and ESR studies (59) of human serum transferrin could detect no difference in the spectra of the two iron sites, but recently a very careful study by Aasa (27) has shown small differences in the ESR spectrum due to a small change in the ESR parameter A. The Mossbauer data were shown to be in good agreement with a calculation by Spartalian and Oosterhuis (28) and more recently by Tsang, Boyle and Morgan (60). The spin Hamiltonian parameters for each experimental study are presented in Table 4. 

As is well known, the lattice gas model can be rewritten in terms of an equivalent Ising Hamiltonian W/sing by the transformation c, = (1 — )/2, which maps the two choices c, = 0,1 to Ising spin orientations St — 1. In our example this yields (Binder and Landau, 1981)... 

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