MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 16: The Shape of Atomic Orbitals
Certain functions are known to describe the shapes of atomic orbitals, which are building blocks for
describing the electron charge clouds of atoms and molecules. The different types of shapes are distinguished
by letters and numbers:
l = 0 s-orbital
l = 1 p-orbital
l = 2 d-orbital
l = 3 f-orbital
...
...
For each type there are 2l + 1 subtypes. These are distinguished by a number ml and its values range from
ml = -l, . . . , l. Two formulas have been found that yield functions Pml
l that describe the -dependence of
orbital shapes; they are
P0
l (z) =
1
2ll!
dl
dzl
(z2
- 1)l
, Pml
l (z) = (1 - z2
)ml/2 dm
dzm
P0
l (z).
After one finds these in terms of z, z is replaced by cos  ( is one of the usual spherical coordinates , , ,
and ranges in value from 0 to ).
Problems:
1. Find Pml
l (cos ) for l = 0 to l = 3, including all appropriate choices of ml.
2. Find the nodal structure. (That is, as  varies from 0 to , at how many places does each function take
on a value of zero?)
3. Evaluate the overlap of any two of the functions. That is, find

0
Pml
l (cos )P
ml
l (cos ) sin d.
Contributor: Dr. Clifford Dykstra
Department of Chemistry, IUPUI