Spin Hamiltonian matrix

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. 

Fig. lOa-c. Higher order splittings in symmetry planes Single crystal nitrogen ENDOR spectrum of Cu(TPP) diluted into (H20)Zn(TPP) with Bo normal to the porphyrin plane B0 = 327.7 mT. a) Observed spectrum. (Adapted from Ref. 66) b) Transition frequencies obtained by numerical diagonalization of the full spin Hamiltonian matrix (Four nitrogen nuclei). (Ref. 68) c) First order frequencies, (Eq. (3.10))... 

To get accurate numerical results, the spin densities can be deduced from the diagonalization of the spin-Hamiltonian matrix (9). To calculate the spin density of a given atom, one only has to sum up the weights (in this case, the squared coefficients) of all determinants, in which this atom is occupied by an a spin, while subtracting the weights of determinants in which this atom has a (3 spin (consult the case of allyl radical in Exercise 7.1). 

Heisenberg portion of the Hamiltonian in earlier work (9) for the delocalization terms we assume that a single parameter B characterizes resonance interactions between each pair of sites. Hence, the spin Hamiltonian matrix becomes ... 

Notice that in the transformed basis set the z-component of the Zeeman operator is no longer diagonal there are off-diagonal matrix elements of the type P i Hf Pj).) The full spin Hamiltonian matrix Hy is then diag-onalised. The obtained eigenvalues ,-(/ ) are expanded in powers of B, yielding the van Vleck coefficients. 

The complete or truncated spin Hamiltonian matrix is then formed, e.g. 

The coupling of the local microstates into the proper molecular states is provided by the diagonalisation of the spin Hamiltonian matrix. For the zero-field case the diagonalisation matrix represents an orthogonal transformation and its matrix elements relate to the combination of the Clebsch-Gordan coefficients. 

The zero-field energies of a cluster are obtained by constructing the spin Hamiltonian matrix, which should be diagonalised when non-zero off-diagonal matrix elements occur. 

Table 7.8 Spin Hamiltonian matrix for the octahedral quartet irrep. A common factor of 1/VT5 is absorbed into the /-parameters...

In the brute-force technique, for each chosen orientation (0,(p) of B, one diagonalizes the spin-Hamiltonian matrix for a large munber of values of B to find the resonant magnetic-field value. It requires exorbitant computer time, especially... 

The simulation procedure presented here, among others, is eminently exploitable to estimation of spin-Hamiltonian parameters from a powder spectrum by the least-squares fitting (LSF) procedure in conjunction with numerical diagonalization of the spin-Hamiltonian matrix similar to that proposed by Misra (1976) in context... 

A similar set of equations can be written for the case of a pair of / = spins. When the cross-products between spin Hamiltonian matrix elements of different... 

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