The Heisenberg Exchange Hamiltonian

In the previous Sections we have made considerable use of the freeon exchange Hamiltonian. In this Section we give a little history of this concept. 

Thirty years later Heisenberg showed that the new hypothesis of atomic structure was the freeon exchange interaction expressed by the freeon Heisenberg Hamiltonian, 

In this Section the ferromagnetic and the antiferromagnetic Heisenberg exchange Hamiltonians are derived from the Hiickel-Hubbard Hamiltonian in which the value of the coupling constant, z takes on values from - 1 to + 1. The M = N = 2, linear, extended Hiickel-Hubbard spectrum is plotted in Fig. 4.1. (Compare with Fig. 3.1) 

The Heisenberg Hamiltonians are derived by the application of second-order perturbation theory to the extended Huckel-Hubbard Hamiltonian at z close to -l(ferromagnet) and at z close to + 1 (antiferromagnet) respectively. 

In Section 4.4 the freeon ferromagnetic Hamiltonian is used to the develop the freeon theory of spin-waves. 

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Generalization of equation 89 to the many-electron system of a crystal is given by the Heisenberg exchange Hamiltonian... 

It was pointed out in Chapter II that the Heisenberg exchange Hamiltonian of equation 90, which can be directly related to the Weiss field parameters at T = 0°K by equation 94, is an excellent formal expression for the interactions between atomic spins (or moments) of neighboring atoms. There remains the problem of establishing the various spin-dependent mechanisms that contribute to the Jij. In general, there are two types of interaction cation- -cation and cation-anion-cation (or even cation-anion-anion-cation) interactions. 

The application of second-order perturbation theory to the Hiickel-Hubbard Hamiltonian for z close to one (Section 4.2) yields the freeon, antiferromagnetic, Heisenberg exchange Hamiltonian,... 

By analogy to the ordinary-spin Heisenberg exchange Hamiltonian the isospin exchange Hamiltonian is... 

Since the Eu chalcogenides are considered to be models for Heisenberg magnets, the isotropic exchange hamiltonian... 

Various predictive methods based on molecular graphs of Jt-systems as described in Section 3 have been critically compared by Klein (Klein et al., 1989) and can be extended to more quantitative treatments. In principle, the effective exchange integrals /ab in the Heisenberg Hamiltonian (4) for the interaction of localized electron spins at sites a and b are calculated as the difference in energies of the high-spin and low-spin states. It was Hoffmann who first tried to calculate the dependence of the M—L—M bond... 

The Hamiltonian of equation (78) corresponds to so-called Heisenberg exchange. In some systems, particularly chains of atoms, it seems that the coupling is restricted to one direction, z. Then the Hex becomes... 

The exchange interaction between two centres in orbitally non-degenerate ground states with spins Si and S2 is described by the Heisenberg Hamiltonian ... 

To model several other electronic situations (e.g., in mixed-valence compounds), the Heisenberg Hamiltonian can be augmented by specific exchange operators that accommodate, for example, double exchange, antisymmetric exchange, anisotropic exchange, or biquadratic exchange. 

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 ... 

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