The Spin Hamiltonian and Ligand-Field Theory

For an ion in a crystal field which gives a nondegenerate ground state and no nearby excited states, g can be obtained from Eq. (74). The values 

The obvious deficiency of crystal-field theory is that it does not properly take into account the effect of the ligand electrons. To do this a molecular-orbital (MO) model is used in which the individual electron orbitals become a linear combination of the atomic orbitals (LCAO) belonging to the various atoms. Before going into the general problem, it is instructive to consider the simple three-electron example in which a metal atom with one /electron is bonded to one ligand atom whose orbital contains two electrons. Two MO s are formed from the two atomic orbitals 

The general approach to the problem is to construct the molecular orbitals ipj 

The spin Hamiltonian can be obtained from the MO s in a manner similar to that used in Sec. III. In this case the parameters of the spin Hamiltonian are determined by the Cy/ s of the ground and excited state MO s as well as by the values of (E0 — E ), f, and r 3 av. In a complete calculation the values of cjt and (E0 — En) would be found by minimizing the total energy, but this is a difficult computation and has been attempted only infrequently. The most notable attempt in this direction is the calculation by Shulman and Sugano (24,25) on KNiF3. The general practice has been to determine values of (E0 — En) from optical spectra, from atomic spectra, and -3 av from free-ion wave functions and to use these values plus the experimental values of the spin Hamiltonian parameters to determine the values of the Cy/ s. 

Before turning to the problem of calculating the spin Hamiltonian, we should consider how the results can be interpreted with respect to the bonding. From the spin Hamiltonian parameters, we shall obtain the values for Cji of the antibonding orbitals, and we need to know what relation these have to the Cj, of the bonding states. One approach to this problem, which seems fruitful, is the gross-population analysis developed by Mulliken (26). Consider first our three-electron example. In this case we split up the term 

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