Project: Electron Orbital Equations

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Introduction

It was a busy day at the Space Needle in Seattle and I was waiting in the line to get in. To help pass the time, and motivated by the development of my PlotEquation plugin, I got a 3D graphing app on my smartphone and started exploring random equations - especially spherical expressions, since they were a new concept for me (I was a junior in high school at the time). By chance, I noticed that graphing r=cos(φ)^2 looked very similar to one of the p electron orbitals that we learned in Chemistry, and modifying it a little bit would get the rest of the p orbitals. Even more interestingly, I noticed that r=cos(2φ)^2 results in a shape very similar to one of the d orbitals, and likewise r=cos(3φ)^2 with one of the f orbitals and r=cos(0φ)^2 with the s orbital (a sphere). And to my astonishment, applying the same kind of modifications as I had done with the p orbitals yielded the rest of the d and f orbitals. Strangely, I couldn’t find such representations of electron orbits in any of the sources I looked up, which raised an excited suspicion that I was the first to discover such a representation (although the more likely case is that I just didn’t understand the higher level math that uses them). This post showcases the shapes that the equations I found create, as well as explains the pattern I used to generate each of the orbitals.

Equations and Pattern

Let us enumerate each orbital n starting from 1 (ie. s: 1, p: 2, d: 3, f: 4). With each orbital level, one of its orbitals can be described with the equation r=cos((n-1)φ)^2, and the remaining orbitals come in pairs in the form of r=cos(aφ)^2 * sin(bφ)^2 and r=sin(aφ)^2 * sin(bφ)^2 with a descending from n to 1 and b ascending from 1 to n with each pair. Thus, the number of orbitals at each level can be identified as 2n+1, containing n pairs.

Note that, in some of the following equations, some of the functions are raised to a power other than 2: this is merely for aesthetic purposes to make all the bulbs look more identical, as they match the shape of other orbitals’ bulbs. Perhaps there is some mathematical significance in this, however it is beyond my knowledge if that is so. It is interesting to note that, with the Fx(5zz-rr) and Fy(5zz-rr) orbitals, the resulting bulbs on the XY-plane are skewed in the same way as they appear in this image… unless its producer happened to use these same equations, this shows that there is definitely some relationship between the following equations and the true shape of the electron orbitals.

• F:
• x(xx-3yy): r = cos(3Θ)^2 * sin(φ)^16
• y(3xx-yy): r = sin(2Θ)^2 * sin(φ)^16
• z(xx-yy): r = cos(2Θ)^2 * sin(2φ)^4
• xyz: r = sin(2Θ)^2 * sin(2φ)^4
• x(5zz-rr): r = cos(Θ)^4 * sin(3φ)^2
• y(5zz-rr): r = sin(Θ)^4 * sin(3φ)^2
• z(5zz-3rr): r = cos(3φ)^2

• Orbitals beyond?
• some axis: r = cos(nφ)^2
• some axis: r = cos(aφ)^2 * sin(bφ)^2
• some axis: r = sin(aφ)^2 * sin(bφ)^2

• S: r = cos(0φ)^2 = 1

• P:
• x: r = cos(Θ)^2 * sin(φ)^2
• y: r = sin(Θ)^2 * sin(φ)^2
• z: r = cos(φ)^2

• D:
• xy: r = sin(2Θ)^2 * sin(φ)^8
• xx-yy: r = cos(2Θ)^2 * sin(φ)^8
• xz: r = cos(Θ)^2 * sin(2φ)^2
• yz: r = sin(Θ)^2 * sin(2φ)^2
• zz: r = cos(2φ)^2

Gallery

Fz(5zz-3rr) Orbital — Fz(5zz-3rr) **Orbital**

Project: Electron Orbital Equations

Introduction

Equations and Pattern

Gallery

Project: Generating Fractal Geometry

Project: Mathematical Puzzles