For example, consider the biologically important molecule porphyrin:

Porphyrin is a flat molecule that can act as a *tetradentate*
(i.e., four nitrogen "teeth") *chelating* molecule
for metals such as Mg, Fe, Zn, Ni, Co, Cu, and Ag
(i.e., has "claw" that can grab and hold these metals). The red arcs represent
the electrostatic interaction of the nitrogen's electrons.
If the central metal is iron, the resulting iron porphyrin *complex* is the
the heme group. There are four heme groups in hemoglobin in your blood.
Heme is also the key part of enzymes like peroxidase, catalase, and cytochrome c.
(In most heme compounds the iron picks up a top and a bottom atom producing
a 6-coordinate, octahedral complex, rather than the square-planar 4-coordinate
bare porphyrin.)
A magnesium porphyrin complex is the heart of chlorophyll, the active molecule
of photosynthesis.

Our aim in this section is to figure out the effect of the electrostatic
cage on the energy levels of the central metal atom. We begin by considering
a octahedral metal cage: 6 charges located at: **r**_{1}=(*h*,0,0),
**r**_{2}=(-*h*,0,0); **r**_{3}=(0,*h*,0), **r**_{4}=(0,-*h*,0);
and **r**_{5}=(0,0,*h*), **r**_{6}=(0,0,-*h*).
We consider the electric potential field near the origin, which
as usual can be expanded in *Y*_{lm}. We use *Mathematica*
for this task:

For small *r*:

For large *r*:

It is perhaps interesting to see how well this approximation works. (For our below calculation, a "good" approximation by the truncated series is not needed...all the [missing] higher terms produce zero result.)

Here is a set of exact isopotentials in the *x-y* plane:

Contours are on the integer and half-integer values, except for the special contour at "6", for which I've provided additional contours at 6±.05

Of course by symmetry it looks exactly the same on the *x-z* plane,
so instead we plot isopotentials for the plane that includes the *z*
axis and the line *x*=*y* (which we call *xy*):

Now if we look at the potential and the approximate potential (in red)
along the *x*=*y*=*z* (the "111" direction) there is
no visible difference between the potential and the approximation. (In fact
differences are less than 0.1.):

If we look (in the *x-y* plane) along the line *x*=*y*
(and *z*=0) we can see a noticeable deviation between approximate
(red) and exact (black) and a jump at *r*=1 in the approximate where we
switch between formulas:

If we go right through on of the charges (e.g., by going along the
*x* axis, the differences are large:

So lines that are far from charges are fairly accurate, but as you approach a
charge the approximation breaks down, but only in a limited range (e.g., .5<*r*<1.5).

Here are our contours for our approximation for the electric potential:

Our game here is to see how the *d* orbitals are affected by the
octahedral **E** field using first order (degenerate) perturbation theory.
Thus we need to calculate the 5×5 interaction matrix involving the
above electric potential and our 5 *d* orbitals (i.e., *m*_{i}=2,1,0,-1,-2).
We have typical terms like:

where *r*_{<} is the smaller of *r* and
*h* and *r*_{>} is the larger of *r* and
*h*.

From our previous results with 3-*Y* integrals:

The *l* must satisfy the triangle inequality and add to an even
number (so for us: *l*=0,2,4) and the *m*'s must "add up":
*m*_{1}=*m*+*m*_{2}. In this problem the
smallest *l*s we found in our expansion are 0 and 4. The *l*=0 term
(6/*h* & 6/*r*) must
be diagonal. (*Y*_{00} is a constant that can be taken from
the integral. From the above: non-zero integrals must have
*m*_{1}=*m*_{2}; for those integrals normality
tells us the result is 1. Since the result is independent of *m*s,
it's a constant value down the diagonal.) Diagonal terms move all the *d* levels
up or down together; they do not affect the level *splitting*
which is what I'm aiming to calculate.

The three *l*=4
terms: *Y*_{40} and *Y*_{4±4}, share
the same radial integral, which becomes part of the multiplicative
shift for all levels (much as the charge at the octahedral vertices
is part of the overall scale factor).
*Y*_{40} can't mix different *m*s, but it does
have different values down the diagonal:

(*Y*_{4,+4}+*Y*_{4,-4}) mixes
*m*s that differ by 4: *m*=-2 and *m*=2. The
resulting matrix is:

The interaction matrix (ignoring the overall factor of the radial integral
and and overall shift up from the *l*=0 term) is:

We can use *Mathematica* to find the `Eigensystem`
(also easy to do by hand):

The result is that there are three degenerate lower lying states
("-2/3") and two degenerate higher states ("1"). The *m*=±2
have mixed. The mixed states lack symmetry, so they
are harder to display. The lower lying state (**v**_{1}, called by chemists
*d*_{xy}) avoids the
interaction by having its lobes point in the *xy* direction. Here is what
it looks like in the *x-y* plane:

Here is what it looks like in the *z-xy* plane:

The higher lying state (**v**_{5}, called by chemists
*d*_{x2-y2}) strongly feels the
interaction because its lobes point at charges in the *x*
and *y* directions. It is exactly like the above *d*_{xy}
orbital except rotated by 45°. Here is what
it looks like in the *x-y* plane:

Here is what it looks like in the *z-x* plane:

The complex normalization [1/(2^{½}*i*)] of **v**_{1}
is designed to produce a real-valued function. In similar fashion one can make
linear combinations of the *m*=±1 states that are real-valued but lack
symmetry. What chemists call *d*_{xz}
is (-*Y*_{21}+*Y*_{2-1})/2^{½}. It's zero on
the *x-y* and *z-y* planes. We display it on the *x-z*
plane:

The other linear combination:
(-*Y*_{21}-*Y*_{2-1})/(2^{½}*i*)
called *d*_{yz} is just like the above orbital, but rotated by
90°. (In fact the four orbitals:
*d*_{x2-y2}, *d*_{xy}, *d*_{yz},
*d*_{xz} are identical except for orientation.)
The *Y*_{2±1} orbitals are low lying ("-2/3")
orbitals; as you can see they avoid pointing at the charges.

The amount of the splitting between these levels depends on the amount of the
charge at the octahedron vertices. That of course depends on what molecule
(called a *ligand*) is at those sites. Chemists have noticed a *spectro-chemical series*
of ligands in which the energy jump gets increasingly large:
(weak ligands) I^{-}, Cl^{-}, F^{-}, OH^{-},
H_{2}0, NH_{3}, SO_{3}^{-2}, NO_{2}^{-2},
CN^{-} (strong ligands).

One final example: ruby consists of a small amount of Cr^{+3} in an octahedral
sites in a Al_{2}O_{3} crystal. It's red color comes from the absorption
of light by Cr^{+3} *d* electrons...Al_{2}O_{3} has no color.
The Al_{2}O_{3} crystal plays the role of the ligands.