Spin-Hamiltonian mapping

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... 

An alternative mapping between the electronic and spin Hamiltonians can be carried out by employing broken-symmetry (BS) states [14-16]. The triplet states I ft) and I U ) of the dimer are referred to as the high spin states IHS), and the Neel states I fi ) and I if ) as the broken-symmetry states IBS)... 

In a computational study, the values of Ja and J e can be obtained by mapping the total energies of different long-range ordered magnetic arrangements onto an Ising spin Hamiltonian of the form... 

Just as the magnetic state energies can be mapped onto an Ising spin Hamiltonian, so may the energies of various crystal field split states be mapped onto the Kanamori Hamiltonian [89]. For an isolated metal site in which bonding and magnetic interactions are neglected, this takes the form... 

Figure 51 Total charge and spin density maps for MgO [Li]° as obtained at the UHF level and with various DFT Hamiltonians. The section illustrated is parallel to the (100) plane through the defect. The separation between contiguous isodensity curves is 0.01 and 0.001 bohr for the electron charge and spin density, respectively. The density range is 0.01-0.1 (charge) and —0.01-0.01(spin map). Continuous, dashed, and dot-dashed lines denote positive, negative, and zero values.

The stated aim of this review is to demonstrate that elassical analyses of physieal organie ehemistry are feasible with respect to complex systems such as supported metal catalysts through the application of advanced EMR spectroscopic techniques and determining the relevant spin Hamiltonian parameters via the Zeeman-dependent hyperfine spectrum. The principles of analysis were outlined in the preceding section and entail replicate collection of ESEEM or ENDOR spectra by incremental steps and mapping the trajectory of peak positions. Deconvolution of peaks may be made either by traditional tau-suppression in the stimulated echo pulse sequence or via advanced pulse sequences such as HYSCORE (2-D ESEEM, Hofer, 1994). Mapping of spectral peak position as it varies depending on the Zeeman field is very important to the accurate determination of hyperfine terms. 

With this exact mapping, we can transform to the Hamiltonian Hxxz representing the spin-1/2 quantum XXZ model [113,114], written by... 

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)... 

Ogata et al. attacked the same nucleic acid conformation problem, but replaced the buildup scheme of Lucasius with a local filter that is equivalent to the use of a rotamer library. In both cases, these methods must deal with the fact that this is an underconstrained problem because several of the dihedrals have no NOEs associated with them. Schuster earlier treated a simple model of RNA to predict three-dimensional (3D) conformations, using a variant on a spin-glass Hamiltonian as his fitness function. The simple model used allowed for the analysis of the complexity of the fitness landscape, couched in terms of the genotype-to-phenotype mapping. 

For distances ry > D/B) l with D = (g di e+)p and d,- the dipole operator, the dipole-dipole interaction in Equation 12.4 between two polar molecules can be mapped onto the effective spin interaction Hamiltonian The first... 

Hence, for the Ft complex, the scalar-relativistic DKH Hamiltonian and the spin-orbit ZORA-Hamiltonian recover relativistic effects on the electron density of the valence region found in four-component calculations. This is particularly remarkable since the DKH density has not been corrected for the picture-change error in these calculations. The difference density maps for the Ft complex clearly show the importance of the scalar-relativistic contraction of the inner shells, while spin-orbit effects can be neglected for this closed-shell complex (otherwise, the DKH results would deviate much more from the fourspin-orbit-coupling-induced splitting of the d shell is small but noticeable. 

---