Biradicals spin Hamiltonian

In this section we consider the spin Hamiltonian appropriate to a biradical with weak dipolar coupling and see how ESR spectra of such species should appear. Obviously, it is possible to find triradicals, tetraradicals, etc. treatment of such species is similar, though of course somewhat more complicated. 

The spin Hamiltonian for a biradical consists of terms representing the electron Zeeman interaction, the exchange coupling of the two electron spins, and hyperfine interaction of each electron with the nuclear spins. We assume that there are two equivalent nuclei, each strongly coupled to one electron and essentially uncoupled to the other. The spin Hamiltonian is ... 

Zimmerman and Kutateladze calculated spin-orbit coupling in linear l,n-diyl biradicals (n = 3 to 8) at the CASSCF(4,4)/3-21G level with the Breit-Pauli spin-orbit Hamiltonian and analyzed it using natural bond orbitals. For diyls with an even number of carbons, symmetry forces zero SOC, whereas for those with an odd number of carbons the main terms have like signs and add, providing a nonvanishing net SOC. The maximum SOC is found at 90° orientation... 

A general operator in either the spin or the orbital space can be written in terms of the angular momentum operator for a particle of spin 1/2, represented by the Pauli matrices. Casting the Hamiltonian operator in this form provides a natural identification of a perfect biradical as the reference system, and of three linearly independent types of fundamental perturbation covalent, magnetizing, and polarizing. 

Finally, we consider the spin-orbit coupling term in the Hamiltonian, which acts in a space defined as the direct product of the spin and the geminal space. Within the framework of the simple model, we find that in a perfect biradical the triplet state spin-orbit couples only to the highest-energy singlet state 82-... 

In systems with multiple magnetic sites, the number of energy differences between the different spin functions is not always enough to determine all the /-values. An obvious example is the three-center/three-electron case as depicted in Fig. 3.5. The Hamiltonian has three different /-values while the quartet and the two doublet states only define two energy differences. The effective Hamiltonian theory described in Chap. 1 is a more general approach to extract /-values, because it not only uses the energies but also information contained in the wave function. To illustrate the procedure we first treat a simple biradical model with two 5 = 5 spins and after that focus on the three center problem. 

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