s-Orbitals

The s-orbitals are fairly easy to describe. The angular parts of the wavefunctions turn out to be constants, so the orbital has a 3D representation that is independent of angle; the net result is a sphere.

Due to the simplicity of the s-orbitals, now would be a good idea to take a qualitative look at the concept of valence shells. Valence shells are the interpretation of the energy levels of the Particle in a Box or Rigid Rotator Solutions. Both systems describe discrete energy levels.

The graph on the left is of the radial portions for the 1s (blue), 2s (green), and 3s (red) orbitals. Each of the graphs is on the same x-axis (radius), but varying y-axis values. Notice that as the radius goes to infinity, that each of three functions goes to zero. This fits in with our notion that each of the wavefunctions must be normalized, and square integrate to 1.

While the 1s orbital is always positive, the 2s orbital changes sign once, and the 3s orbital changes sign twice. At every point where the radius becomes zero, there will be no chance of finding the electron. These are radii called nodes.

The Radial Wavefunctions and their Probability Distributions – r2 ● R(r)2

The graph on the right is of the radial portions of the curved integrated with respect to radius (as in standard integration in spherical co-ordinates). They are graphed on the same x-axis, but variable y-axes. Some of the y-scales have been exaggerated so their features are easier to see. The curve of the 1s orbital reaches one maximum in its curve. The distance is known as a Bohr radius, and is the same average distance that Niels Bohr obtained when he approximately the behaviour of the Hydrogen Atom (which is 1s1). As mentioned previously, at each x-intercept, there is zero probability chance of finding the electron. In the 2s and 3s cases, the probability varies, going from regions of high and low. Because the angular elements are spherical, one can imagine this as a set of nested or concentric spheres. At each of the maxima, there is a solid layer, that alternates with a nodal sphere or an empty volume.

Here is a colored version where each of the 1s, 2s, and 3 orbitals are on the same x axis, and variable y axes. As the radius increases, the larger orbitals become more diffuse (spread out) and higher in energy.

Introduction
Introduction to Quantum Chemistry