Atomic Orbitals, Isosurfaces, and Quantum Numbers

In order to understand orbitals we proposed to begin analyzing and working with the mathematical expressions for the single-electron spatial wavefunctions. We introduced physical chemistry students to orbitals mainly from a quantitative approach, building implicit functions from the wavefunctions and visualizing them as isosurfaces. We had observed that students worked very successfully with each mathematical expression using the MAPLE software. The purpose of our analysis was to highlight the relation among the images or isosurfaces of hydrogenlike atomic orbitals and the three quantum numbers n, l and m (more precisely |m|). In this way we studied the relation between the symmetry properties of the orbital wave function Ψ and the magnetic quantum number m and the nodal surfaces. We carried out a detailed mathematical analysis of the polar expressions of four selected wave functions equations. We worked, for example, with n = 5 orbitals because they present various types of nodes, thus showing very interesting and less known examples. In all the examples we designed isosurfaces, or contour representations of the implicit expression of the orbital wave function under study. We described in detail four examples, which may be taken as a help guide to further working. The MAPLE software is quite efficient in showing a three-dimensional perspective, we may also perform adequate rotations which are very useful in the analysis of the symmetry properties and clearly show the nodal surfaces.