The Pauli Principle and the Periodic Table

The Schrödinger equation cannot be solved exactly for polyelectronic atoms, but the hydrogen atom provides a good model on which to build an understanding of heavier atoms. The description of atomic structure in terms of orbitals is still valid and the quantum numbers n, l, and m are still useful. For H the energy depends only on n, but orbital energies for heavier atoms depend also on l (but not on m). For a given value of n, orbital energies increase with l, but the 3d and 4s energies are comparable, 4s usually being filled first; similarly 4d and 5s are close in energy with 5s usually filled first.

The Pauli Principle: No two electrons in an atom can have the same set of all four quantum numbers, n, l, m, and ms.

For atoms with more than one electron, we must take account of a fourth quantum number, ms, the electron spin quantum number, which has only two values, ms = ±1/2. An electron has a magnetic moment which can be rationalized by imagining that electrons spin about an internal axis. Since the spin can be "clockwise" or "counterclockwise", the electron's magnetic moment has two possible orientations described by the quantum number ms. We must take account of ms because of the Pauli principle, which states that every electron must have a unique set of quantum numbers, n, l, m, and ms. Thus any orbital characterized by the quantum numbers n, l, and m can accommodate at most two electrons, one with ms = +1/2, the other -1/2.