Exchange interaction operators

The isotropic exchange discussed so far represents only a portion of the total exchange interaction operating in binuclear and oligonuclear systems. 

The expressions for the matrix elements of the exchange interaction operators in the uncoupled basis set are collected in Table 11.1. 

The operator from Eq. (1.5) we shall call the double exchange interaction operator. Its non-Heisenberg form is obvious. The operator represents interaction between rigid spins /S +1/2 and S. Just in this form we shall use it considering the crystal. 

Schematically, the double exchange interaction can be written in the form of a spin Hamiltonian operator (working on the system spin AB —for short, S) as...

The sum over coulomb and exchange interactions in the Fock operator runs only over those spin-orbitals that are occupied in the trial VF. Because a unitary transformation among the orbitals that appear in F leaves the determinant unchanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to choose such a unitary transformation to make the Ey matrix diagonal. Upon so doing, one is left with the so-called canonical Hartree-Fock equations ... 

The inner core electrons occupy closed shells. The only exchange part of the two-electron Breit interaction between the valence, outer core and inner core electrons, Bf and P/c, gives non-zero contribution. The contributions from Bfy and P/c, are quite essential for calculation at the level of chemical accuracy (about 1 kcal/mol or 350 cm for transition energies). This accuracy level is, in general, determined by the possibilities of modern correlation methods and computers already for compounds of light elements. Note, that the contribution from the exchange interaction is not smaller than that from the Coulomb part [29]. The inner core electrons can be considered as frozen in most physical-chemical processes of interest. Therefore, the effective operators for P/ and P/c acting on the valence and... 

Superexchange describes interaction between localized moments of ions in insulators that are too far apart to interact by direct exchange. It operates through the intermediary of a nonmagnetic ion. Superexchange arises from the fact that localized-electron states as described by the formal valences are stabilized by an admixture of excited states involving electron transfer between the cation and the anion. A typical example is the 180° cation-anion-cation interaction in oxides of rocksalt structure, where antiparallel orientation of spins on neighbouring cations is favoured by covalent... 

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

We can also introduce another effective operator, one that explicitly includes the exchange interaction of electrons. For this purpose, we shall have, in averaging, to take into account the dependence of the electrostatic interaction of two particles on their total spin. For a pair of equivalent electrons the number of various singlet states (S12 = 0) is given by... 

The 1.7% Co2+ ZnO DMS-QDs in Figure 34 were also examined by Zeeman spectroscopy in transmission mode. An average band-edge Zeeman shift of 53 cm 1/7 over the range of 0-7 T ( Fig. 35) was measured. The Zeeman data were analyzed in the mean-field approximation using Eq. 11, where x is the dopant mole fraction and (Sz) is the expectation value of the Sz operator of the spin Hamiltonian. 7/oa and iVop quantify the exchange interactions between the dopant and unpaired spins in the conduction (CB) and valence bands (VB), respectively (19, 159-161). 

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