Crystal field splitting of d orbitals

In the trigonal point group 3, the axis of quantization is chosen along the three-fold axis, while the x and y axes may be selected anywhere in the plane perpendicular to the z axis. In the point group 3m, which occurs for many distorted octahedral complexes, there are also vertical mirror planes. The relation to the cubic axes is described by the transformation 

Since this transformation is unitary, it applies to both the axes and the coordinates x, y, and z. The new z axis along the body diagonal of the cube, 

The pseudo-octahedral eg orbitals of Eq. (10.2) have z components, so they are lifted out of the xy plane. 

For a transition metal element with more than one d electron, the atomic energy levels are more complex. As the electrons interact with each other, the 

3 Energy levels arising from the e2 electron configuration. 

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Figure 2.10 Crystal field splittings of d orbitals of a central ion in complexes having different geometries. The subscripts to A refer to the geometries.

The crystal field splittings of d orbitals that arise from tetrahedral, tetragonal, and octahedral arrangements of ligands have been described. 

Since the splitting parameter in the tetrahedral field is smaller than in the octahedral field, the tetrahedral field is always a weak field, Aptetrahedral field, the highest values of CFSE correspond to d and d configurations. Figure 3.9 presents the comparative crystal field splitting of d orbitals of the central ion in complexes of geometry tetrahedral, octahedral, tetragonal, and square-planar. 

FIC U RE 3.9 Crystal field splitting of d orbitals of central ion in complexes with geometries ... 

The crystal field splitting of f orbitals is around 1/100 that of d orbitals. This means that the position and intensity of the absorption lines are also very weakly affected by ligands, in contrast with complexes of a given transition metal, such as nickel, where the change in colour from green to violet as H2O molecules are replaced by NH3 on adding ammonia to [Ni(H20)6] " is clearly visible to the naked eye. In a few cases, notably Nd +, certain transitions are influenced by ligand and environment (the hypersensitive transitions ). 

The approach we have adopted for the d configuration began from the so-called strong-field limit. This is to be contrasted to the weak-field scheme that we describe in Section 3.7. In the strong-field approach, we consider the crystal-field splitting of the d orbitals first, and then recognize the effects of interelectron repulsion. The opposite order is adopted in the weak-field scheme. Before studying this alternative approach, however, we must review a little of the theory of free-ion spectroscopy... 

Figure S6.2 Crystal field splitting of the energy of five d orbitals when the ion is placed in a site with octahedral symmetry. The magnitude of the splitting, A or 10Dg, depends upon the size of the octahedral site and the charges on the surrounding ions.

A low-spin to high-spin transition relates to the crystal field splitting of the d-orbitals in an octahedral or tetrahedral crystal field. However, even in cases where the energy difference between two spin states is much larger, electronic transitions are observed. An atom with total spin quantum number S has (22 + 1) orientations. In a magnetic field the atom will have a number of discrete energy levels with... 

Localized d or/electrons retain their one-atom manifolds, except that states arising from different d" or f are split by crystal field and spin-orbit coupling. Multiplet splittings due to spin-orbit coupling are larger than crystal-field splittings of Af levels the converse is the case for 3d" levels. The difference in energy between d"(f) and d" (/ 1) manifolds corresponds to the amount of free atom U that is decreased due to interatomic interaction in the solid. 

Figure 9.8 Splitting of d orbitals in an octahedral crystal field.

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