Spin-exchange integral

Experimental data on the dependence of exchange parameters (the rate of triplet-triplet energy transfer (kTT) or the spin exchange integral (Jse)) on the distance (r) between interacting centers are approximated by the following equation (Likhtenshtein 1988(a,b), 1993,1995 Likhtenshtein et al. 1982 Kotel nikov et al. 1981)... 

Figure 2.7. Dependence of the logarithm of relative parameters of the exchange interaction on the distance between the interacting centers (AR). krr is the rate constant of triplet-triplet electron transfer, and JSE is the spin-exchange integral. Index 0 is related to van der Waals contact. (Likhtenshtein, 1996) Reproduced in permission.

Note Do not use CNDO on any problem where electron-spin is critically important. Its complete neglect of atomic exchange integrals makes it incapable of dealing with these problems. 

Figure 2. The structural energy difference (a) and the magnetic moment (b) as a function of the occupation of the canonical d-band n corresponding to the Fe-Co alloy. The same lines as in Fig. 1 are used for the different structures. In (b) the concentration dependence of the Stoner exchange integral Id used for the spin-polarized canonical d-band model calculations is shown as a thin dashed line with the solid circles. The value of Id for pure Fe and Co, calculated from LSDA and scaled to canonical units, are also shown in (b) as solid squares.

Note that we must use i and not r, as variable here, because of the spin-dependence of the exchange integral, recall Section 1.3. 

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... 

As far as the size of the exchange integral J between the two open-shell centres is concerned, it can become as high as a few tenths of an eV in diradicals (see Section 3). The /-values between radicals are usually smaller than 0.01 eV. Therefore, we cannot readily expect spin-ordering temperatures higher than a few to 20 K for intermolecular spin alignment. 

Note that the exchange integral vanishes in this case. The last two terms of Eq. (B2) show a spin flip of the electrons as they are excited from orbital i to orbital a and from orbital j to orbital b. This contribution vanishes, since the Coulomb interaction between the particles does not cause a spin flip. 

The CNDO and CNDO/S methods apply the ZDO approximation to all integrals, regardless of whether the orbitals are located on the same atom or not. In the INDO method, which was designed to improve the treatment of spin densities at nuclear centers and to handle singlet-triplet energy differences for open-shell species, exchange integrals... 

The exchange integral J is found to be about 0.13 eV. A value J 0.23 eV was found using de Gennes spin disorder theory ... 

Fig. 3.8 Left-hand panel The on-site atomic energy levels for up and down spin electrons due to the exchange splitting Im where / and m are the Stoner exchange integral and local moment respectively. Right-hand panel The local magnetic moment m, as a function of //2 / where / and h are the exchange and bond integrals respectively. Compare with the self-consistent LSDA solution in the upper panel of Fig. 3.6.

Whereas Si and s2 are true one-electron spin operators, Ky is the exchange integral of electrons and in one-electron states i and j (independent particle picture of Hartree-Fock theory assumed). It should be stressed here that in the original work by Van Vleck (80) in 1932 the integral was denoted as Jy but as it is an exchange integral we write it as Ky in order to be in accordance with the notation in quantum chemistry, where Jy denotes a Coulomb integral. 

It can be evaluated by calculation of its expectation value. For this purpose, we need to calculate the exchange integral Kab as given by Clark and Davidson in Eq. (84) and then evaluate the local spin expectation values for the operators SA SA. 

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