Trigonal Fields

Starting from this rotated set complex orbitals and (t)3 multiplet operators may be constructed in a way which is entirely analogous to the treatment of Sect. 2. Hence the multiplets in Table 2 can be used equally well for trigonal complexes, keeping in mind that the axis of quantization is now the z axis. This implies that the subduction rules for real components in Eq. 15 have to be replaced by the appropriate S03 j. O j D3 subduction rules. In order to obtain the real forms of the (t2)3 basis functions the resulting expressions have to be multiplied once again by the pseudoscalar quantity of A2 symmetry. The appropriate product rules have been given by Ballhausen [59], For the individual orbital functions one obtains  

Here ax and e are representations of D3. The 9 and e components are resp. symmetric and antisymmetric under the C2 operator along the x direction. Combining Eqs. 47 and 48 with Eq. 9 then yields the conversion formula for the real t2g orbitals in trigonal and tetragonal frames  

The corresponding d-functions may also be expressed in the trigonal frame as follows  

In classical crystal field theory [2] the trigonal field is parametrized by means of two independent parameters v and v describing resp. the interaction between the t2g and eg shells and the splitting of the t2g shell. The v parameter was already discussed in Sect. 4.3 in connection with the trigonal zfs of the A2g ground state. Here special attention will be devoted to the v parameter which seems to dominate the doublet splittings, especially in orthoaxial trischelated 

On the other hand the octahedral levels contain the following trigonal symmetry species  

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A notable exception are chemisorbed complexes in zeolites, which have been characterized both structurally and spectroscopically, and for which the interpretation of electronic spectra has met with a considerable success. The reason for the former is the well-defined, although complex, structure of the zeolite framework in which the cations are distributed among a few types of available sites the fortunate circumstance of the latter is that the interaction between the cations, which act as selective chemisorption centers, and the zeolite framework is primarily only electrostatic. The theory that applies for this case is the ligand field theory of the ion-molecule complexes usually placed in trigonal fields of the zeolite cation sites (29). Quantum mechanical exchange interactions with the zeolite framework are justifiably neglected except for very small effects in resonance energy transfer (J30). ... 

Spin-orbit coupling in conjunction with the trigonal field leads to a zero-field splitting of the 2E levels. In the strong field limit the spin-orbit levels can be obtained by vector addition of cylindrical orbital and spin momenta. Hence the 2IT state will give rise to 2f7 3/2 and 2/7 1/2 components, comprising resp. the 2D 1/2 1) and 2D 1/2 +1) functions. The trigonal symmetries of these functions are as follows ... 

In the limit of strong negative trigonal field the parameter R takes on the value y, we approach the following principal values of g-factor g = 4 and g = 0. This case can be referred to as the fully anisotropic limit. 

FIGURE 5.26 Energy-level diagram for Co(III) complexes in an octahedral and a trigonal field. 

FIGURE 5.28 Energy-level diagram for Cr(III) complexes in octahedral and trigonal fields. 

If two octahedral-site cations share a common face, as in the corundum or NiAs structures, the cation--cation interactions may be particularly important since the cation separations are relatively small and the Ug orbital stabilized by the resulting trigonal field is directed through the common face ( o of equation 68), as shown in Figure 43(b). 

Figure 19 Orbital ordering within the t ground state due to the trigonal field in LnTiOs perovskites. (Ref. 16. Reproduced by permission of Physical Society of Japan)...

By analyzing the Zeeman pattern at various field strengths, it was possible to establish that the nominal trigonal field in this Z>3 complex was perturbed by lower symmetry components due to the local environment. A major influence was attributed as most likely arising from a tipping of the bpy ligands. 

In the next sections we describe briefly the main interactions, which are in charge of splitting of the 3d ions energy levels in crystals. These interactions include the Coulomb interaction, the crystal field interaction, the spin-orbit interaction and the JT interaction. As it was pointed out by Ham [13], the observed spin-orbit and trigonal field splittings of the orbital triplet states are significantly affected by the dynamic JT effect. 

The key point of this energy-level diagram is that the trigonal field is large with the orbital doublet term lower lying. The wavefunctions of the lower lying Kramers doublet are then of the form Ml,Ms) = 1, T, with corresponding g values ... 

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