Hamiltonian Heisenberg spin

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... 

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 ... 

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... 

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. 

To reasonably limit the focus here it the survey is primarily of many-body solution techniques as applied to a particular VB model, the covalent-space Pauling-Wheland VB model, represented by the Heisenberg spin Hamiltonian... 

Use of the Heisenberg spin Hamiltonian (equation 1) to represent the energy difference of the singlet and triplet spin states is easily demonstrated. Two spins, Sj and Sj, can be added to produce a maximum spin of 5 max = Sj + Sj, and lower values - 1, max - 2 down to a minimum of Si - Sj. When the two spins are both one-half, the two possible values for the total spin are 5 tot = 1 and 5 tot = 0, the spin triplet and singlet, respectively. To evaluate the energies of these states from equation f, it is necessary to know the value of Si Sj for the two states. This can be found by evaluating the vector sum of the spins and employing the basic quantum rule... 

The easiest case to investigate is the zero-bandwidth or atomic limit here, the standard Hubbard hamiltonian (with V = 0) is equivalent to a Heisenberg spin hamiltonian [24] ... 

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