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Atomic orbital scatter plots
📌 Recommended background information: LibreTexts Chemistry
- 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.
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