Shapes and energies of orbitals

Orbitals are drawn as boundary surface diagrams: a shape that encloses the region where the electron is found most of the time (say 90%).

• s orbitals are spheres, centred on the nucleus.
• p orbitals have two lobes, on opposite sides of the nucleus, pointing along the x, y or z axis (pₓ, p_y, p_z).
• d orbitals mostly have four lobes; the d_z² orbital has two lobes and a ring round its middle.

A node is a region where the probability of finding the electron is zero. An orbital has n – 1 nodes in all:

• l angular nodes (flat planes through the nucleus).
• n – l – 1 radial nodes (spherical shells).

A 2s orbital has one radial node; a 2p orbital has one nodal plane through the nucleus. A 4p orbital (n = 4, l = 1) has 1 angular node and 4 – 1 – 1 = 2 radial nodes.

In hydrogen, with only one electron, the energy of an orbital depends only on n: 2s and 2p have the same energy (they are degenerate).

In atoms with many electrons, the inner electrons shield (screen) the outer ones from the nucleus. So an outer electron feels an effective nuclear charge, Z_eff·e, less than the full charge Ze. s electrons spend more time close to the nucleus (they penetrate most), so they are held more strongly. Within the same shell, the energy order is s < p < d < f.

To decide which orbital fills first, use the (n + l) rule: the lower the value of n + l, the lower the energy. If two orbitals have the same n + l, the one with the lower n comes first. Example: 4s has n + l = 4 + 0 = 4, and 3d has 3 + 2 = 5, so 4s fills before 3d. The order is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d …