Orbitals, Subshells and Quantum Numbers

The principal quantum number n (1, 2, 3, ...) sets the shell: within a comparable orbital family, larger n usually means a larger average distance; n alone does not rank every many-electron orbital energy. The angular momentum quantum number l runs from 0 to n - 1 and names the subshell: l = 0 is s, l = 1 is p, l = 2 is d, l = 3 is f. The magnetic quantum number m_l runs from -l to +l and picks one orbital inside the subshell. The spin quantum number m_s is +1/2 or -1/2.

An orbital is a quantum-mechanical state with a spatial probability distribution, not a path followed by an electron like a planet. The first three quantum numbers identify a spatial orbital; the fourth labels an electron's spin state in that orbital. Two electrons can therefore share n, l and m_l only if their m_s values differ.

Larger n generally means a more extended orbital within a related family. In a hydrogen-like atom, energy depends on n alone; in many-electron atoms it also depends on subshell and interactions, so n alone cannot rank every orbital energy.

A subshell with a given l holds 2l + 1 orbitals: s has 1, p has 3, d has 5, f has 7. Every orbital holds at most two electrons, so s holds 2, p holds 6, d holds 10 and f holds 14. A whole shell n holds n squared orbitals and 2n squared electrons, so shell 3 has 9 orbitals and 18 electrons.

A 4d subshell has l = 2 and m_l values −2, −1, 0, +1, +2: five orbitals and ten electron places. The numeral 4 does not mean four orbitals. To count an entire n = 4 shell, add the s, p, d and f orbital counts, 1 + 3 + 5 + 7 = 16, then double for thirty-two electrons. That capacity is not the number of electrons actually present in every period-4 atom. Empty allowed orbitals are still allowed even when a ground-state configuration does not occupy them.

An s orbital is spherical, a p orbital is a two-lobed dumbbell along one axis, and most d orbitals are four-lobed cloverleaves. A node is a surface where the probability of finding the electron is zero. An orbital has l angular nodes (flat planes or cones) and n - l - 1 radial nodes, for a total of n - 1 nodes.

Orbital pictures often color lobes differently to indicate the sign or phase of the wavefunction, not positive and negative electric charge. A p orbital's two lobes belong to one orbital and together hold at most two electrons. For 3p, one angular nodal plane and one radial node give two total nodes. For 3d, l = 2 gives two angular nodes and 3 − 2 − 1 = 0 radial nodes, again totaling two. A 2s orbital has no angular nodes but one radial node. Count nodes from n and l rather than counting colored lobes.

First check that n is a positive integer. Next test 0 ≤ l ≤ n − 1, then −l ≤ m_l ≤ l with integer m_l, then m_s = ±1/2. Each allowed range depends on the earlier choice.

For (3, 1, −2, +1/2), n and l permit a 3p subshell, but m_l = −2 is outside −1, 0, +1, so the set is impossible. For (4, 2, −2, −1/2), all four values are allowed and describe one electron in a 4d orbital. For (2, 0, 0, 0), the spin value alone makes it invalid.

Allowed does not mean occupied in the ground state of a particular atom. A valid 4d quantum-number set does not prove a carbon atom has a ground-state 4d electron. To test two electrons together, also apply Pauli: identical values of all four quantum numbers are forbidden, even though either set is individually allowed.