Hamiltonians Heisenberg

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... 

Much attention has been paid to Monte Carlo simulations of magnetic ordering, and its variation with temperature. Such models assume a particular form for the magnetic interactions, e.g. the Ising or Heisenberg Hamiltonian (see e.g. Binder... 

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. 

Nevertheless, the calculation of is an important issue. In experiment, it is considered as an empirical parameter fitted to experimental data so that the corresponding Heisenberg Hamiltonian describes the experimentally observed magnetic behavior (100,101). Although it would be more desirable from a quantum chemical point of view to directly calculate experimentally accessible properties, e.g., the magnetic susceptibility (102), the quantum chemical calculation of Ky provides a means to compare experimental and calculated results—though in a somewhat indirect fashion. 

When kT is large with respect to the energy gap, the population of each level is just one over the number of levels or functions. When kT is small with respect to the energy separation, then only the lowest level is occupied. The new energy levels S are linear combinations of the S, Ms > functions of each metal ion. The functions and energies can be calculated by the simple Heisenberg Hamiltonian, that for a dimer is... 

M. Said, D. Maynau, J. P. Malrieu, and M. A. G. Bach,/. Am. Chem. Soc., 106, 571 (1984). A Nonempirical Heisenberg Hamiltonian for the Study of Conjugated Hydrocarbons— Ground-State Conformational Studies. 

When both 10ns of a Cr dimer are in the electronic ground state, the exchange interaction can be represented by the well-known Heisenberg Hamiltonian... 

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

S2 possesses the eigenvalues S(S + 1). For a spin-1/2 dimer (S can be 0 or 1), the same values are obtained from the derivation of the Bleaney-Bowers equation. For spin systems in an external magnetic field, the Zeeman operator Hmag = -g/ BB S accounts for Zeeman splitting. The isotropic Heisenberg Hamiltonian for multiple spin centers can be expanded by adding the individual coupling pairs ... 

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. 

Starting with a Heisenberg Hamiltonian in which Zeeman and exchange interactions (limited to nearest neighbors) are accumulated, the susceptibility is derived as follows ... 

Unlike the theory discussed in Chapter 3, which relies on VB structures that are eigenfunctions of both the Sz and S2 operators, Heisenberg—Hamiltonian theory uses, as basis functions, VB determinants that are eigenfunctions of the Sz operator only. The reader has by now some background on the VB... 

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. 

The corresponding Heisenberg Hamiltonian derives from the connectivity of the allyl structure and is shown in the scheme below the structure. This matrix... 

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