← Labs|Radial Probability Distribution — Hydrogen-like Orbitals

Select Orbitals

Atomic Number Z = 1

HKr

r max = 30 a₀

Show node positions

Key Formula

P(r) = r² |R_nl(r)|²

Radial nodes = n − l − 1
Total nodes = n − 1
r_mp ≈ n²a₀/Z (for s)

Radial Probability Density P(r) = r² |R_nl|² for Z = 1

1sn=1, l=0, nodes=0

No radial nodes. One smooth peak very close to nucleus. Most penetrating orbital.

Node r: no nodes · Maximum — closest to nucleus

2sn=2, l=0, nodes=1

One radial node creates inner and outer lobe. Inner lobe penetrates — 2s penetrates more than 2p.

Node r: 2.0a₀ · High — inner lobe penetrates

2pn=2, l=1, nodes=0

No radial nodes. Peaks farther from nucleus than 2s. Three degenerate px, py, pz.

Node r: no nodes · Lower than 2s

Penetration Order

ns > np > nd > nf

Why 4s before 3d?

4s penetrates inner shells better than 3d → lower effective Z_eff → lower energy. After 3d fills, 3d drops below 4s.

Anomalies: Cr, Cu

Cr: [Ar]3d⁵4s¹ (not 3d⁴4s²) Cu: [Ar]3d¹⁰4s¹ (not 3d⁹4s²) Half-filled/fully-filled d is extra stable.

Slater Rules (σ)

Z_eff = Z − σ • Same n: 0.35 (0.30 for 1s) • (n−1): 0.85 for s/p, 1.0 for d/f • (n−2) and below: 1.0 Higher n_eff = less penetration

Lanthanide Contraction

4f poor penetration → high Z_eff for 4f electrons → small lanthanide radii. 5d ≈ 4d in size due to this.

JEE Focus

• Nodes: n−l−1 radial, l angular • Degenerate: same n+l (fill order) • Aufbau: fill lowest n+l first • Hund: maximize spin in degenerate