Interactive simulations & visualizations

Visualizing the beauty in physics and mathematics

---

Project maintained by zhendrikse Hosted on GitHub Pages — Theme by mattgraham

Zeger's home / Science home / Molecularphysics /

Atomic orbital scatter plots

• 2s orbital: $\psi_{2s} \propto (2-r)e^{-r/2} \Rightarrow |\psi|^2 \propto (2-r)^2 e^{-r}$
• 2p orbital: $\psi_{2p} \propto r e^{-r/2} \cos\theta \Rightarrow |\psi|^2 \propto r^2 e^{-r} \cos^2\theta$
• 2py orbital: $|\psi|^2 \propto r^2 e^{-r} \sin^2\theta \sin^2\phi$
• 3s orbital: $\psi_{3s} \propto (27 - 18r + 2r^2)e^{-r/3}$
• 3p orbital: $\psi_{3p} \propto r(6-r)e^{-r/3}\cos\theta$
• 3d orbital: $|\psi|^2 \propto r^4 e^{-2r/3} \sin^4\theta \cos^2(2\phi)$

2D scatter plots

3D scatter plots

The 3D-version uses Monte-Carlo sampling + rejection sampling based on the probability density $|\psi|^2$, so per orbital a radial distribution and angular weight is generated.

\[P(r,\theta,\phi) \propto |\psi(r,\theta,\phi)|^2\]

First we generate a radial candidate, next uniform angles, and accept the point with chance equal to weight.

Share on: