Crystal Field Theory

Background

Why is a ruby red? The mineral corundum is a crystalline form of alumina: Al2O3. A pure crystal of corundum is colorless. However, if just 1% of the Al3+ ions are replaced with Cr3+ ions, the mineral becomes deep red in color and is known as ruby(Al2O3:Cr3+). Why does replacing Al3+ with Cr3+ in the corundum structure produce a red color? Ruby is an allochromatic mineral, which means its color arises from trace impurities. The color of an idiochromatic mineral arises from the essential components of the mineral. In some minerals the color arises from defects in the crystal structure. Such defects are called color centers.

The mineral beryl is a crystalline beryllium aluminosilicate with the chemical formula Be3Al2Si6O18. A pure crystal of beryl is colorless. However, if just 1% of the Al3+ ions are replaced with Cr3+ ions, the mineral becomes green in color and is known as emerald (Be3Al2Si6O18:Cr3+). Why does replacing Al3+ with Cr3+ in corundum produce a red mineral (ruby) while replacing Al3+ with Cr3+ in beryl produces a green mineral (emerald)?
	Beryl on albite, from Shigar Valley, Pakistan (4.9 x 3.7 x 3.2 cm). ©Rob Lavinsky (irocks.com), used by permission.	Emerald on white marble, from Panjshir Valley, Afghanistan (2.1 x 1.7 x 1.1 cm). ©Rob Lavinsky (irocks.com), used by permission.

Principles

Crystal Field Theory was developed in 1929 by Hans Bethe to describe the electronic and magnetic structure of crystalline solids. The theory was further developed through the 1930's by John Hasbrouck van Vleck. Crystal Field Theory describes the interaction between a central metal ion that is surrounded by anions. A quantum mechanical description of the metal ion is employed, with attention focused on the valence shell d, s, and p orbitals. The surrounding anions are typically treated as point charges.

The essential insight of Crystal Field Theory is that the geometry of the negatively charged point charges influences the energy levels of the central metal ion. Consider the 3d orbitals of a first-row transition metal. A spherical distribution of negative charge surrounding the metal ion affects each of the five 3d orbitals in the same way and consequently all five 3d orbitals have the same energy. But what happens if the negative charge is not distributed spherically?

Experiment

This exercise depicts the various 3d orbitals for a first-row transition metal. A set of negative charges (white spheres) are positioned around the metal center using one of four geometries: linear, square planar, tetrahedral, and octahedral. (Obviously other geometries are possible, but these four geometries are the most common.) The diagram at the right shows the absolute energy of the individual orbitals (E) or the energy difference from the average (spherical field) energy (ΔE). When the negative charges are infinitely far away (approximated by the maximum displacement in this exercise), all 3d orbitals have the same energy (ΔE = 0 and E = 0).

Use the controls to vary the distance between the metal center and the negative charges. Carefully observe how the energies of the orbitals change as the distance becomes smaller and smaller. Answer the questions below and explain the observed behavior. Bear in mind that an orbital represents the distribution of electron density.

1. How does the orbital energy change as the negative charges get closer to the metal center?
2. For the linear geometry, why is the 3d_z² orbital more strongly affected by the surrounding negative charge than the 3d_xy orbital?
3. For the square planar geometry, why is the 3d_x²-y² orbital more strongly affected by the surrounding negative charge than the 3d_xy orbital?
4. For which of the four geometries is the change in E greatest when the negative charge is close to the metal center? Why?
5. For each geometry a specific splitting pattern is observed in the ΔE plot. Explain each pattern.
6. Compare the splitting patterns (ΔE plot) for the tetrahedral and octahedral complexes. Is the splitting in the ΔE plot greater for the tetrahedral or octahedral geometry? Explain the observed behavior.
7.  

so, where from comes the concept of t2g and eg?

The Crystal field theory considers ionic bond arising purely from electrostatic interaction between metal atom (positively charged) and ligands ( negatively charged). The transition metal ion is considered to be positive charge equal to its oxidation state. The central metal atom is surrounded by negatively charged ligands. The ligands were considered as point charges.

Consider the complex Potassium hexacyanoferrate(II) whose chemical formula is K₄[Fe(CN)₆]. Now in this molecule the anion is a complex species with an overall charge of -4 ([Fe(CN)₆]^4-). Thus iron in this complex carries a charge of +2. Here the Fe⁺² ion is surrounded by six cyanide ions. The electronic configuration of Fe is 4s²3d⁶, therefore the electronic configuration of Fe⁺² ion will be 3d⁶. In the absence of ligands, all the 5 d orbitals in Fe⁺² ion are degenerate. However, when the ligands approach the central metal ion, the energy of all the d-orbitals is raised. These 5 d-orbitals are then split in two sets of t_2g and E_g set. As CN is a strong ligand, it causes the pairing of electrons. So all the t_2g orbitals are occupied with electrons.