Spin eigenvalues orbit coupling

The six elements can be used to generate the n-electron ligand field energies, to which interelectronic repulsion and spin-orbit coupling, which are already defined on the dxy, dX2, dyz, dx2 y2, dz2 basis, are added. Alternatively, the five eigenvalues can be used, but an additional term is necessary to define the correct linear combinations of the dx2 y2 and dz2 orbitals on which to evaluate interelectronic repulsion and spin-orbit coupling. Either way, there are still six total, or five spectroscopically independent, terms in the ligand field potential matrix. 

In the case (b) limit the spin-orbit coupling constant A is zero and the eigenvalues in equation (9.75) become... 

Figure 9.18. Plot of some eigenvalues for a molecule in a2 IT state against the spin-orbit coupling...

While the chemical interpretation of the e parameters is a matter of real concern to us, there are also several other difficulties which are, however, more apparent than real. Consider the question of the calculation of magnetic properties in transition metal complexes - paramagnetic susceptibilities and e.s.r. g values. In contrast to the study of eigenvalues for optical transition energies, these require descriptions of the wavefunc-tions after the perturbation by the ligand field, interelectron repulsion and spin-orbit coupling effects. In susceptibility calculations it is customary to use Stevens orbital reduction factor k in the magnetic moment operator... 

For example, a CSF written as 3300> indicates that the first two orbitals are both doubly occupied. This CSF is a four-electron singlet. The ket 3120> indicates that the first orbital is doubly occupied the second orbital is singly occupied and coupled to the lower-level orbitals to increase the spin, in this case giving a three-electron doublet. Finally the singly occupied third orbital is coupled to decrease the overall spin, in this case giving a four-electron singlet. The spin eigenvalue of any ket. denoted by S or by b , may be determined simply by the difference in the number of = 1 step vector entries and the number of dj = 2 step vector entries. In fact, the sum of the step vector entries is the same for all CSFs with the same total spin and number of electrons . The CSF expansion space may easily be limited to include only those terms that contribute to the correct overall spin. An important property of CSFs constructed in this manner is that they form an orthonormal expansion basis. 

In the presence of spin-orbit coupling, each vibrational state of the state can potentially mix with each vibrational state of the T2 state. The resulting energies and wavefunctions of the mixed states can be obtained by finding the eigenvalues and eigenvectors of a secular matrix with elements given by... 

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