Shapes and energies of orbitals

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).