The Heisenberg Hamiltonian

The matrix defined in Eq.(47) represents a general, non-relativistic spin-independent TV-electron Hamiltonian given by Eq.(2) in a specifically defined model space. The one-electron orbital space is spanned by N orthonormal localized orbitals 4 j, j = 1,2. TV and the TV-electron orbital space is one-dimensional with the basis function 

This basis function is associated with f(S, N) different spin functions 0jfcM k = 1,2./(S, N). Thus, in this case the TV-electron antisymmetric model space may be expressed as 

The procedure leading from the exact /V-electron Hamiltonian (2) to the Heisenberg Hamiltonian matrix (47) is very instructive, but it is rather lengthy. Much simpler is the use of effective Hamiltonians which in the space of N-electron eigenfunctions of S2 and S2 are represented by the same matrix. Furthermore, using the effective Hamiltonians may bring another insight into the nature of the interactions described by the model. The simplest effective Hamiltonian in the pure spin is 

Another form of the Heisenberg Hamiltonian may be obtained by means of the so called Dirac identity [45] 

As long as the model space is constructed according to Eq.(50), the matrices representing Hamiltonians defined in Eqs.(2), (17), (36), (51) and (54) axe the same (up to a constant added to all diagonal elements) and in this sense they are equivalent. In the SGA-based formalism, the form given by Eq.(51) is most convenient and this form will be used in our further discussion. 

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A priori, one might have expected a [3Fe-4S] center to give a particularly simple EPR spectrum. Contrary to what was suggested in Ref. (13), the electronic structure of this cluster, which possess three ferric sites, is not liable to be complicated by valence delocalization phenomena, so that the intersite interactions can be described by the Heisenberg Hamiltonian ... 

The competition between these two terms produces a large variety of electronic structures in molecular systems. The condition l U favors itinerant metallic states, whereas the condition t stabilizes localized insulating states. In the latter case, the Hubbard Hamiltonian is reduced to the Heisenberg 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... 

Fd n can be studied in two oxidation states. In the oxidized state the cluster has electronic spin S = H. This spin results fiom antiferromagnetic coupling of three high-spin ferric (Si = 2 = S3 = 5H) iron sites. The magnetic hyperfine parameters obtained from an analysis of the low tempo ture MSssbauer spectra have been analyzed (18) in the frmiework of the Heisenberg Hamiltonian. 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

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. 

Write the Heisenberg Hamiltonian of allyl radical and diagonalize it. Then write the wave functions for the ground and first neutral excited state. Show that the excited state has a positive a spin density on the central atom, as discussed in Chapter 1. 

In this chapter we gave a brief review of several applications of SGA to a desription of spectra of the Heisenberg Hamiltonian. There axe three topics in which using SGA is particularly useful (1) establishing relations... 

To cover the gap between them the Hubbard model Hamiltonian was quite generally accepted. This Hamiltonian apparently has the ability of mimicking the whole spectrum, from the free quasi-particle domain, at U=0, to the strongly correlated one, at U —> oo, where, for half-filled band systems, it renormalizes to the Heisenberg Hamiltonian, via Degenerate Perturbation Theory. Thence, the Heisenberg Hamiltonian was assumed to be acceptable only for rather small t/U values. 

Meanwhile in Chemistry the Heisenberg Hamiltonian, which is known as the (covalent-structure) valence-bond (VB) model, has also been applied to conjugated systems. Therefore, some parallelism can be established between ladder materials and long conjugated polymers. 

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