
Shapes of atomic orbitals
The wavefunction for a given atomic orbital has a
characteristic mathematical expression.
The wavefunctions for the l = 0 levels, the s
orbitals, depends only on the distance of the electron from
the nucleus. These orbitals are therefore spherically
symmetric.
The mathematical expression for the 1s orbital of Hydrogen is
where a is a constant known as the Bohr radius, and
its value is 52.9 pm 
The value of N in the wavefunction of the 1s atomic
orbital is calculated from the normalization
condition, [Ψ(x,y,z)]^{2}
= 1, which is a result of the Born interpretation of the wavefunction.
The wavefunctions for the l = 1 levels, the p
orbitals, are not spherically symmetric. They have the
forms below.
These are the expressions for the p orbitals: 
p_{x} 


p_{y} 


p_{z} 


The p orbitals have lobes pointing along the cartesian
axes: the labels p_{x}, p_{y},
and p_{z} refer to the axes
along which the orbitals point. If the functions for these orbitals
are plotted in two dimensions, they have the forms as shown below
for the p_{x} orbital.
This figure shows the projection of the p_{x} orbital along the xaxis 

The signs show the relative
phases of the orbital: these are important in bonding, as
only the overlap of orbitals of the same phase leads to
bond formation. 
The wavefunctions for the l = 2 levels, the d orbitals, are more complicated still:
d_{xy} 


The different d orbitals all depend on more than
the one cartesian direction of the p orbitals, and are given the
labels d(xy), d(xz),
d(yz), d(z^{2})
and d(x^{2}y^{2}). 
Radial Wavefunctions and Radial Distribution Functions
The method of describing the shape of an orbital in
terms of its projection of its wavefunction along an axis, as in
the p_{x} orbital case above, is a way of describing the
orientation dependent part of the wavefunction. That the wavefunction
of the p_{x} orbital is orientationally dependent means
that its projection is not the same along each of the cartesian
axes.
The electronic wavefunction, as we have seen above,
describes the distribution of the electron positions in terms of
the distance of the electron from the nucleus, r, and the orientaion
of the electron relative to the nucleus. We can separate the wavefunction
into an orientationally dependent part,
known as the angular wavefunction,
and an orientationally independent
part, which is known as the radial wavefunction.
The Radial Wavefunction, R_{nl}(r),
depends only on the distance of the electron from the nucleus,
and is characterized by the values of the principal and orbital
angular momentum quantum numbers, whereas the angular
wavefunction, Y, depends on the angles of the electron
from the nucleus, and is characterized by the values of the orbital
angular momentum and magnetic quantum numbers.
The s orbitals consist only of a radial part to
the wavefunction, and Y = 1. The angular wavefunctions for other
types of orbitals are complicated mathematical expressions, so
generally only the shapes of the orbitals, as shown above, are
important.
Another important function we need to consider if
the Radial Distribution Function, P_{nl}(r).
This is defined as the probability that an electron in the orbital
with quantum numbers n and l will be found at a distance r from
the nucleus. It is related to the radial wavefunction by the following
relationship:
; normalized by
The factor 4πr^{2}
arises because the radial distribution function refers to the
probability of finding an electron not at a specific point in
space (which equals Ψ^{2}),
but on a spherical shell of area 4πr^{2},
at a distance r from the nucleus. The integral results from the
fact that the total probability of finding the electron is one,
as it must be found somewhere around the nucleus.
The number of radial nodes,
where the sign of the R_{nl}(r) changes, in the radial
wavefunction, is equal to n  l  1.
The number of maxima in the radial distribution
function is equal to n  1.

