How to Visualize the Orbitals of Atoms

Atomic orbitals are a well-known concept that is already taught in chemistry classes at secondary schools. Unfortunately, they often do not go into detail about what they actually are and how they are calculated. Of course, we cannot provide a quantum mechanical derivation here, but we can highlight a few interesting aspects nonetheless.

What are atomic orbitals?

Atomic orbitals, as they are generally known, describe the residence probability of the electrons in the atomic shell. From a quantum mechanical point of view, the electrons of an atom can be found everywhere. It is just much more probable on statistical average to find electrons in an orbital.

Apart from the number of electrons, atomic orbitals have nothing to do with atomic nuclei or nuclear physics. Analytically calculable and representable are only the orbitals of the hydrogen atom, which has only one electron. Nevertheless, there are also orbitals further outside, which are "strictly" separated from each other. These are created when further energy is added to the electron by electromagnetic radiation.

On the right (or further down on the smartphone) some configuration of the hydrogen atom with the principal quantum number n = 3 are shown. In the first visualization even the two inner orbitals can be seen. Due to the fact that the color scale was restricted, they appear visually separated, in fact the probability even between the cloud-like formations is not zero.

How are they defined?

Atomic orbitals are the solution of the Schrödinger equation of the hydrogen atom. During the solution, one can separate the resulting partial differential equation into an angular and a radial part. The angular part is solved by the so-called spherical surface functions, the radial part by the radial equation, which uses the associated Laguerre polynomials. Both solutions are orthogonal polynomial systems in their respective solution spaces, so that also each linear combination of the solutions describes another solution.

Spherical functions are complex-valued polynomials, so they do not describe a physically measurable result. This problem is solved by considering the magnitude square of the solution, which is then again purely real-valued.

How to visualize atomic orbitals?

The solution of the hydrogen Schrödinger equation is three-dimensional, so we can use one of the existing 3D plotting variants. Particularly good results are achieved for these cloud-like residence probabilities with dens3d and the cloudplot option. However, one must be aware that this option is very computationally intensive. So in the code snippet below, you should not have all 9 plots active at the same time, or you might risk a crash.

We also use the clog option because the residence probability is logarithmic. But it is important to specify an interval for the color scale that stats is greater than 0. cut also allows a view into the center of the visualization.

The function describing the (complex-valued) orbitals of the hydrogen psi_H() can be found in the first lines of the code snippet. This is composed by the mentioned spherical surface functions Y() and the radial term radial(), which is defined before.

The following code snippet must be copied into a script and executed from there, because lclfunc does not work in the terminal.

Ready to give it a shot? Here's the code: