Exchange antisymmetric

The antisymmetric term of the exchange interaction is defined through the operator 

The components of the polar vector dAB(dx, dy, dz) are related to the antisymmetric part of the exchange coupling tensor as follows [18, 19] 

In terms of Cartesian components the antisymmetric exchange term can be written as 

The cross-product can be expressed with the help of the first-order tensor product 

In transforming the spherical first-rank tensor components into Cartesian ones one can utilise the expressions 

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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 the simplest system of SA = SB = 1/2 the antisymmetric exchange interaction matrix adopts the form of... 

Fig. 10.12. Zero-field splitting of the energy levels due to antisymmetric exchange for an Sj = SB — 1/2 system.

The generalisation of the antisymmetric exchange operator, as introduced for dinuclear systems, leads to the operator... 

Reduced matrix elements of the antisymmetric exchange for a general triad1... 

Asymmetric exchange is incapable of removing this kind of degeneracy since there are no interaction elements between l, 1/2) and 0,1/2) kets. Only antisymmetric exchange couples the above states, which lead to the removal of the accidental degeneracy of the two Kramers doublets, yielding... 

Some consequences of antisymmetric exchange for trinuclear systems are discussed elsewhere [6]. 

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