Atomic and Molecular Orbitals Page

Several hydrogenic atomic orbitals, hybrid orbitals, and molecular orbitals for some small molecules are listed below. The data files for the orbitals are given in "Protein DataBank" format and the data files have a ".PDB" extension. You can view these molecules with a program that reads files which are in .PDB format. If you do not have such a program you can download one from the Internet. Instructions for downloading a molecular viewing program named RasMol are given on the Molecular Viewing Information page.

You can download the RasMol program and the .PDB files for the orbitals you want to see, or you can download the RasMol program and set it up as a "helper application" in your web browser. In the latter case, when you select one of the orbitals below, your browser will automatically launch RasMol and load the .PDB file for the orbital you selected. (At present the general access computer labs on the UofA campus have not installed RasMol as a helper application for Netscape - they may eventually do so. However, they do have the program loaded on the servers in each of the labs. You can download the .PDB files to a floppy disk and then run the program from the server using your floppy as the data file source.)

The orbital data files are fairly large, most of them are over 300KB, so it may take several seconds for RasMol to load them. Also, after the file loads there may be nothing showing on your RasMol screen, or there may be a few "scratch marks" here and there. To see the orbitals click on "Display" and select either "Ball & Stick" or "Spacefill" mode.

Once you have the orbital showing on your screen you can rotate it by moving the mouse with the left mouse button held down. (Or you can use the slide bars at the right and bottom of your screen.) You can make the orbital larger or smaller by moving the mouse with the left mouse button and the shift key held down. You can move the image around on the screen by moving the mouse with the right mouse button held down. (You can get another rotation angle by moving the mouse with the right button and the shift key held down.)

The orbitals are not all shown on the same scale (except for the 1s and 2s orbitals, which are on the same scale). For example, when you load the 2s orbital it will fill your screen the same as one of the 4f orbitals, etc. In actual fact, the 2s orbital is much smaller than the 4f orbitals.

For people who might be interested in such things, we are plotting surfaces of constant probability density. That is, we set the (absolute value of) the square of the wave function equal to a constant and plot the values of the coordinates which satisfy this equation.

Hydrogen Atomic Orbitals

The 1s orbital is not very interesting, comparatively. It has spherical symmetry and has no nodes.

The 2s orbital is a little more interesting because it has a spherical nodal surface. You can see evidence of this node although you probably can't tell that it is a node. The 2s orbital graphic here just looks like a ball within a ball.

The 2pz orbital has a nodal plane (the x-y plane). The orbital is plotted so that the z-axis is "up" on your screen, the y-axis is to your right, and the x-axis is perpendicular to the screen toward you. We only show one of the 2p orbitals. To see what the others look like you need only rotate this one 90 degrees either to the right or toward you. If you rotate the orbital toward you, you will see that the orbital has cylindrical symmetry.

We show one other p orbital, the 3pz orbital , so that you can see how orbitals of a given type (the p orbital here) change as we go to higher principal quantum numbers. In this orbital most of the electron "density" is in the two outer lobes of the orbital, but there is some electron density in the two small inner lobes. (Just for the record we remind you that this is all one orbital, not several orbitals.) This orbital has cylindrical symmetry, as do all p orbitals.

The "d" orbitals have more lobes. The orbitals are named in terms of the cartesian coordinate directions in which the lobes point. We show here one of each type of 3d orbital. The 3dz2 orbital (read "3 d z squared") has cylindrical symmetry. The other 3d orbitals, such as the 3dzx orbital (read "3 d z x") and the 3dx2-y2 orbital (read "3 d x squared minus y squared") all have the same shape, but differ in the directions in which the lobes point. You will have to rotate each of these last two orbitals to see that they each have four lobes. (There are two more 3d orbitals which are not shown here, but they have the same shapes as these last two. Their names are 3dxy and 3dyz. See if you can rotate one of the above orbitals to make it look like one of these.)

Most people, even chemists, have never seen the shapes of "f" orbitals. We show here one of each type of "f" orbital. The f orbital analogous to the dz2 orbital is the 4fz3 orbital. There is one other orbital with the same shape of the 4fz2x orbital. The 4fz(x2-y2) orbital also has a mate, and so does the 4fxy2 orbital.

We have "prettier" versions of the 2pz and 3dz2 orbitals.

Hybrid Orbitals

"Mixtures" of atomic orbitals, called "hybrid orbitals" are sometimes useful in discussiing molecular shapes and bonding. The three most common types of hybrid orbitals are called the "sp hybrids," the "sp2 hybrids," and the "sp3 hybrids." There are two sp hybrid orbitals and we show one of them. There are three sp2 hybrid orbitals of which a typical one is shown. There are four sp3 hybrid orbitals of which one is shown. In all cases the other orbitals of each set have the same shape but "point" in different directions in space.

The two sp hybrids are arranged on a straight line with their major "lobes" pointing 180 degrees away from each other. You can see this by viewing them all at the same time.

The three sp2 hybrid orbitals are arranged in a plane pointing 120 degrees apart. You can see this by viewing them all at the same time.

Likewise the four sp3 hybrid orbitals are arranged in a tetrahedral manner. You can see this by viewing them all at the same time.

Simple Molecular Orbitals

Molecular orbitals, as the name implies, are orbitals representing the motion of electrons in molecules. A molecular orbital encompasses more than one nucleus. We give examples here of a few of the simplest molecular orbitals. First we list some molecular orbitals for the N2 molecule, and then the six simplest pi molecular orbitals for benzene. (Benzene is C6H6. If you have forgotten the structure of benzene you can check it out on the Molecules and Crystal structures page.) In these graphics the nuclei are represented by small - very small - orange dots. (In actual fact, the nuclei are so small that you should not even be able to see them on the scale of these pictures. However, the nuclei are easier to see in Space Fill mode than in Ball & Stick mode.)

Molecular Orbitals for the N2 Molecule

The 2s-sigma orbital as a bonding orbital. Note that there is a build-up of electron density between the nuclei. This orbital is "occupied" by two electrons in the N2 molecule. The 2s-sigma star orbital is an "antibonding" orbital. This orbital is not occupied in the ground-state N2 molecule. For this orbital, not only is there no build-up of electron density between the nuclei, there is a node between them.

The 2p-sigma orbital is a bonding orbital which can be approximated by two 2p orbitals with their electron density lobes pointing toward each other. This orbital is occupied by two electrons in the ground-state N 2 molecule. The 2p-sigma star orbital is an antibonding orbital. This orbital is not occupied in the ground-state N2 molecule. Note that there is a node between the nuclei.

The 2p-pi orbital is a bonding orbital. It is approximately two 2p orbitals aligned side-by-side. It has a nodal plane just like p atomic orbitals do. There is another 2p-pi orbital that has the same shape as this one. You can see what it looks like by rotating this one by 90 degrees toward you. Both of these orbitals are occupied in the ground-state N2 molecule. The 2p-pi star orbital is an antibonding orbital. It has a node between the nuclei and a nodal plane containing the nuclei. There is another 2p-pi star orbital with the same shape as this one in the N2 molecule. You can see what it looks like by rotating this one by 90 degrees toward you. Neither of these orbitals is occupied in the ground- state N2 molecule.

Pi Molecular Orbitals for Benzene

The six lowest energy pi molecular orbitals for benzene are relatively easy to construct and show. Since there are only six pi electrons in the benzene molecule only the lowest three pi orbitals will be occupied. We will label the orbitals by their symmetry designation (for people who care about the symmetry designations).

The lowest pi orbital of benzene is called the A2u orbital. This orbital is occupied by two electrons in the lowest energy state of benzene. Notice that this orbital has a nodal plane in the plane containing the nuclei and no other nodes. (You can tell the relative energies of a set of molecular orbitals - or atomic orbitals, for that matter - by counting nodal surfaces. The more nodes an orbital has the higher energy it has. If you like, you can test this statement on the pi molecular orbitals of benzene.)

The second lowest energy pi orbitals of benzene are designated E1g so we will call them E1g-1 and E1g-2. They are degenerate (i.e., they have the same energy) so that all the remaining four pi electrons occupy these orbitals.

Next up (in energy) is another set of two degenerate orbitals, the E2u orbitals, E2u-1 and E2u-2. These orbitals are unoccupied in the ground electronic state of benzene.

The highest (in energy) of these six benzene pi orbitals is the B2g orbital. This orbital is unoccupied in the ground state of benzene.

Last updated 30 May 97
W. R. Salzman
Department of Chemistry
University of Arizona
salzman@arizona.edu