The octahedral crystal field

A second approach to the bonding in complexes of the d-block metals is crystal field theory. This is an electrostatic 

Crystal field theory is an electrostatic model which predicts that the d orbitals in a metal complex are not degenerate. The pattern of splitting of the d orbitals depends on the crystal field, this being determined by the arrangement and type of ligands. 

If the electrostatic field created by the point charge ligands is spherical, the energies of the electrons in the 3d orbitals are raised uniformly 

If the electrostatic field created by the point charge ligands is octahedral, the energy of the electrons in the 3d orbitals that point directly at the ligands is raised with respect to that in the spherical field, while the energy of the electrons in the orbitals that point between the ligands is lowered with respect to the spherical field 

The subscript g means gerade and the subscript u means ungerade. Gerade and ungerade designate the behaviour of the wavefunction under the operation of inversion, and denote the parity (even or odd) of an orbital. 

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We first examine the relationships between electron structure and the emission and absorption spectroscopy of metal complexes. Transition metal complexes are characterized by partially filled d orbitals. To a large measure the ordering and occupancy of these orbitals determines emissive properties. Figure 4.2 shows an orbital and state diagram for a representative octahedral MX6 d6 metal complex where M is the metal and X is a ligand that coordinates or binds at one site. The octahedral crystal field of the ligands splits the initially degenerate five atomic d-orbitals by an amount... 

Sugano and Tanabe have calculated the energy of the states deriving from the 3d" ions (from n = 2to = 8)asa function of the octahedral crystal field strength. 

Value of the octahedral crystal-field splitting derived from the position of the levels forCe3+ in prismatic coordination using a point-charge mo e. ... 

The above simple picture of solids is not universally true because we have a class of crystalline solids, known as Mott insulators, whose electronic properties radically contradict the elementary band theory. Typical examples of Mott insulators are MnO, CoO and NiO, possessing the rocksalt structure. Here the only states in the vicinity of the Fermi level would be the 3d states. The cation d orbitals in the rocksalt structure would be split into t g and eg sets by the octahedral crystal field of the anions. In the transition-metal monoxides, TiO-NiO (3d -3d% the d levels would be partly filled and hence the simple band theory predicts them to be metallic. The prediction is true in TiO... 

Fig. 5 Energies of relevant terms in the octahedral crystal field...

Expressed as fractions of the octahedral crystal field splitting parameter, A0. 

Second, the octahedral crystal field splitting parameters, values of which are higher for smaller sites, are expected to decrease in the same order as eq. 

Expressed as fractions of the octahedral crystal field splitting parameter, A0 hs and Is are high-spin and low-spin configurations, respectively. 

Fig. 10. Variation of the da ratio of LaSrB04 (B = Cr, V, Fe) with the octahedral crystal field stabilization energy, ACf, and the optical electronegativity of the B ions in B2Oj compounds.

Figure 5 The molecular orbital or ligand field picture of metal-ligand bonding in an MLe complex. Compare this picture with Figure 1 to see how the octahedral crystal field splitting pattern (in the box) is still present in the MO model...

The calculation reveals that the octahedral crystal field removes the fivefold orbital degeneracy of the free ion state, yielding a threefold degenerate state at —4Dq and a twofold degenerate state at +6Dq. The results of the crystal-field calculation are collected in Table 4. The representations spanned by the crystal-field states are determined by an examination of the transformation properties of, ..., in the point group O, since it is known that d wave functions are even under inversion and Oh is O y<. i. In Oh, these become and Eg, respectively. In this manner, it has been determined that the octahedral crystal field removes the orbital degeneracy of the free ion state, resulting in the formation... 

Many of the results obtained for the octahedral crystal-field calculation can be used for the description of the magnetic properties of ions in tetrahedral environments. If an 5 4 axis of the tetrahedron is taken as the axis of quantization (being collinear with the C4 axis of quantization of the octahedron), and the C3 axes of the tetrahedron are collinear with the C3 axes of the octahedron, then the crystal-field potential energy... 

On the basis of crystal field theory, the octahedral crystal field potential Dq is as follows ... 

Fig. 8.21. Correlation diagram for the triplet terms of d2 configuration in the octahedral crystal field (a) free ion terms (H° includes F ) (b) weak field limit (H = V ) (c) intermediate field (H = + F,cf) (d) strong field limit (H1 = K ) (e) strong field configurations (H° includes P1 ).

As in the case of Cs2ZrCl6 Pa , the expected effect of the octahedral crystal field on d orbitals is apparent on the different values of of the 5f 6d(t2 ) and 5f 6d(eg) manifolds. The energy difference between the upper 6 /(eg) states... 

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