Heisenberg spins

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... 

Rodriguez, J. H., Wheeler, D. E., McCusker, J. K., 1998, Density Functional Studies of a Heisenberg Spin Coupled Chromium-Semiquinone Complex and its Chromium-Catechol Analog , J. Am. Chem. Soc., 120, 12051. 

The pure RKKY interaction is isotropic, and the canonical spin glass systems are therefore often referred to as Heisenberg spin glasses. However, some anisotropy is also present in those systems originating from dipolar interaction and interaction of the Dzyaloshinsky-Moriya (DM) type [73]. The latter is due to spin-orbit scattering of the conduction electrons by non-magnetic impurities and reads... 

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). 

The local expressions obtained in the decomposition procedure can then conveniently be employed in a Heisenberg Spin Hamiltonian, which reads for a pair interaction... 

Though the exact solution of the Pauhng-Wheland VB model (or the positive-./ Heisenberg spin Hamiltonian) is generally a nontrivial matter, there are a number... 

As illustrated above, the microscopic explanation of observed magnetic properties hinges on the construction of an appropriate model. In most instances, simplifications have to be weighed and phenomenological models can be employed, such as the Heisenberg spin Hamiltonian. 

For organic spin systems, one frequently assumes applicability of Heisenberg spin behavior, in which all interactions can be reasonably modeled by pairwise exchange interactions. A typical Heisenberg spin Hamiltonian for exchange Jy between various spin sites i and j, with spin quantum numbers S, and Sj, is given in the following equation ... 

For simplicity, we will assume that unpaired electrons can be assigned to specific spin orbitals, as part of the Heisenberg spin approximation in which the... 

Here 0, is the isospin describing the C, orbital, Iy is the orbital exchange constant, and Jq is the Heisenberg spin-exchange constant. This Hamiltonian can describe many of the magnetic properties of TDAE-C60. 

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... 

Figure 5 Diamagnetic rings currents, Nx z), of half-filled Pariser-Parr-Pople models for regular polygons with D h symmetry. The dashed line at 2 = U/4 t0 = 1.17 corresponds to standard parameters z = 0 is the Hiickel limit of free electrons, while z 1 is the strong-correlation limit of antiferromagnetic Heisenberg spin chain with vanishing ring currents [50].

It is easily seen that (28) has a form of uniform one-dimensional Heisenberg spin Hamiltonian with a well-known spectrum. Therefore the ground state spin of our lattice has a minimal value at p <1 and a maximal value at p > 1. 

For negative values of t, we obtain the uniform Heisenberg spin chain with antiferromagnetic coupling which has a nondegenerate singlet ground state. 

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