D3.2 Multi-electron Atoms

The ideas already developed about quantum numbers, orbitals, and sizes and shapes of electron-density distributions apply to all atoms. However, when there are two or more electrons in an atom, there are repulsive forces between the electrons in addition to the attractive forces between electrons and the nucleus. These repulsions affect electron energies.

For example, the energy levels in a He⁺ ion (which, like H, has a single electron) are significantly lower than those in a H atom because of the stronger Coulomb’s law attraction between the one electron and the 2+ charge of the He nucleus. However, in a neutral He atom, which has two electrons, electron-electron repulsions between the electrons raise energy levels significantly compared to He⁺, and a He atom is not as stable as we might have expected.

For atoms with many electrons, the effect of electron-electron repulsions differs for different subshells. Therefore orbital energy depends on both n and ℓ quantum numbers. For the same value of n (the same shell), as ℓ increases the energy also increases. Thus s-subshell electrons have lower energy than p-subshell electrons, which are lower than d-subshell electrons, and so forth. Orbitals within the same subshell (for example 2p_x, 2p_y, and 2p_z) all have the same energy; orbitals that have the same energy are said to be degenerate.

The Austrian physicist Wolfgang Pauli formulated what is now called the Pauli exclusion principle:

• Each electron in an atom must have a different set of values for the four quantum numbers.
  • If two electrons share the same orbital (have the same n, ℓ, and m_ℓ), then their spin quantum numbers m_s must have different values; we say the two electrons have opposite spin.
• Because m_s can only have two values, +½ or -½, no more than two electrons can occupy the same orbital.

By applying the Pauli exclusion principle, the arrangement of electrons in any multi-electron atom can be determined by recognizing that the ground state of an atom has all of its electrons in orbitals with the lowest energies possible.