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:




The p orbitals have lobes pointing along the cartesian axes: the labels pxpy, and pz 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 px orbital.

This figure shows the projection of the px orbital along the x-axis  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:


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(z2) and d(x2-y2).

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 px orbital case above, is a way of describing the orientation dependent part of the wavefunction. That the wavefunction of the px 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, Rnl(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, Pnl(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 4πr2 arises because the radial distribution function refers to the probability of finding an electron not at a specific point in space (which equals Ψ2), but on a spherical shell of area 4πr2, 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 Rnl(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.

This following box shows the shapes of the radial wavefunctions, Rnl(r), and the radial distribution functions, Pnl(r), of the atomic orbitals.
Rnl(r) Pnl(r) n l
1s 1s 1 0
2s 2s 2 0
2p 2p 2 1
3s 3s 3 0
3p 3p 3 1
3d 3d 3 2