Why 4s fills before 3d.

Orbital energy levels for hydrogen (all subshells of a shell degenerate) morph into a multi-electron atom where s<p<d<f split and 4s drops below 3d by the (n + l) rule.

In the many-electron view, order 3d, 4s and 4p correctly by tapping the lowest first; then switch to hydrogen and find a pair that becomes degenerate (e.g. 3s and 3d).

3d fills before 4s because 3 < 4; in hydrogen 2s is lower than 2p.

• For 1s, |ψ|² is highest at the nucleus and falls away steadily; for 2s it falls to zero, rises to a small maximum and falls again. A surface where the probability density is zero is a node.
• A boundary surface diagram encloses the region with about 90% probability; 100% is impossible because |ψ|² never becomes exactly zero at any finite distance. Every s orbital is a sphere, and size grows 1s < 2s < 3s < 4s.
• Each p orbital has two lobes on either side of a nodal plane through the nucleus; the three (px, py, pz) are equal in size, shape and energy and lie along mutually perpendicular axes. There is no simple match between mₗ values and x, y, z.
• The five d orbitals (from n = 3 onwards) are dxy, dyz, dxz, dx²−y² and dz²; the first four share a shape, dz² looks different, and all five are equal in energy in the isolated atom.
• Nodes: angular nodes = l, radial nodes = n − l − 1, total = n − 1. So 2s has one radial node, 3p has one radial and one angular, and pz has the xy-plane as its nodal plane.
• In hydrogen, energy depends on n alone, so all subshells of one shell tie: 2s = 2p; 3s = 3p = 3d; and 4s, 4p, 4d and 4f are equal too. Orbitals of equal energy are called degenerate.
• In multi-electron atoms, electron-electron repulsion and shielding make energy depend on n and l: within a shell s < p < d < f, because an s electron penetrates closer to the nucleus and feels a larger effective nuclear charge (Zeff).
• The (n + l) rule: lower n + l means lower energy; for equal n + l, lower n is lower. This gives 4s < 3d, 6s < 5d and 4f < 6p. The same orbital drops in energy as Z rises: E₂ₛ(H) > E₂ₛ(Li) > E₂ₛ(Na) > E₂ₛ(K).

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