Crystal Field Splitting

Transition metals, with their d orbitals available and partially populated with electrons, bond to ligands through their lone pairs.

For octahedral complexes, the ligands approach along the axes and these electron pairs repel the d electrons on the metal.

 

The ligands here are approaching in an octahedral geometry.  Which d orbitals are in medium blue?    This repulsion leaves some d orbitals ( medium blue) at a higher energy than others (it costs more energy to put an electron near the ligand pairs than it does somewhere else).   Notice that the lightest blue d orbitals all point toward the middle of an edge. 

Can you see which orbitals have higher energy and which have lower energy?

 First the lower energy orbitals, can you see that they are all equivalent?

The higher energy orbitals are harder to understand , it is not obvious that they are equivalent in energy.  But,

Don't forget the donut! -Homer Simpson

The website from the University of the West Indies Chemistry Department in Kingston, Jamaica offers animated d orbitals which you can rotate about to investigate the geometry.
http://wwwchem.uwimona.edu.jm:1104/courses/CFTpt2.html
 


This same reasoning can be used for other shapes.  In order to apply it, we need to have a picture of where the d orbitals are located with respect to a cube. 
 

Tetrahedral Geometry:

Compare the following figure in your book with that of the d orbitals we saw above:             
Which orbitals are now closer/farther away from the ligands?  More discussion

To make things easier, remember that you can rotate the collection of orbitals to any orientation you wish.  Try rotating around the z axis by 45 degrees.

square planar discussion

Given the relative energies in the table below, sketch an energy level diagram for coordination numbers 2,4,5,6,8 and 9.  Can you rationalize (explain) why the orbitals are in the order they are?

Relative energy levels of d orbitals for various geometries .
C.N. Structure dz2 dx2-y2 dxy dxz dyz
1 linear 5.14 -3.14 -3.14 0.57 0.57
2 linear 10.28 -6.28 -6.28 1.14 1.14
3 trigonal -3.21 5.46 5.46 -3.86 -3.86
4 tetrahedral -2.67 -2.67 1.78 1.78 1.78
4 square planar -4.28 12.28 2.28 -5.14 -5.14
5 trigonal bipyramidal 7.07 -0.82 -0.82 -2.72 -2.72
5 square pyramidal 0.86 9.14 -0.86 -4.57 -4.57
6 octahedral 6.00 6.00 -4.00 -4.00 -4.00
6 trigonal prismatic 0.96 -5.84 -5.84 5.36 5.36
7 pentagonal bipyramidal 4.93 2.82 2.82 -5.28 -5.28
8 cubic -5.34 -5.34 3.56 3.56 3.56
8 square antiprismatic -5.34 -0.89 -0.89 3.56 3.56
9 trigonal prismatic with equatorials -2.25 -0.38 -0.38 1.51 1.51
12 icosohedral 0.00 0.00 0.00 0.00 0.00
From  Huheey, Keiter and Keiter, Inorganic Chemistry, Principles of Structure and Reactivity, 4th ed., Harper Collins, New York, 1993, p.405.
Information From :
Zuckerman, JJ, J. Chem Ed., 1965, 42, 315
Krishnamurthy, R. and W.B. Schaap,  J. Chem Ed., 1969, 46, 799
 

Molecular orbital treatment

Not shown here are the degeneracies of the levels.
For example, each of the ligand levels has a degeneracy of at least six, one per ligand.

The eg and t2g levels corresponding to crystal field theory are emphasized.

The metal atom s and p orbitals shown are the next level up from the ground state valence shell.