Multielectron Atoms and the Periodic Table

Multielectron atoms are described by treating each electron as moving in an averaged field of the nucleus and the other electrons, and filling the resulting orbitals subject to the Pauli principle reproduces the periodic table.

Definition

A multielectron atom is an atom containing two or more electrons whose mutual repulsion prevents an exact solution; it is modelled by assigning each electron an orbital in a self-consistent averaged potential, with the overall state constrained by the antisymmetry of the many-electron wavefunction.

Scope

This topic covers the approximate treatment of atoms with more than one electron: the central-field approximation, screening and effective nuclear charge, the Hartree and Hartree–Fock self-consistent methods, electron configurations, and the angular-momentum coupling schemes (LS and jj) that give rise to atomic terms. It explains how the Pauli exclusion principle and the ordering of subshell energies build up the structure of the periodic table.

Core questions

• How can an atom with many interacting electrons be described approximately?
• What is the central-field approximation and how does screening modify the nuclear charge?
• How do the Pauli principle and subshell energies produce the layout of the periodic table?
• How do individual electron angular momenta couple into total atomic terms?

Key concepts

• Central-field approximation
• Screening and effective nuclear charge
• Slater determinants and exchange
• Hartree–Fock method
• LS and jj coupling
• Electron configurations and subshells

Key theories

Central-field approximation

Each electron is treated as moving independently in a spherically symmetric average potential due to the nucleus and the other electrons, reducing the many-body problem to a set of one-electron orbitals labelled by n and l.

Hartree–Fock self-consistent field

The averaged potential is determined self-consistently from antisymmetrized (Slater-determinant) wavefunctions, iterating until the orbitals reproduce the field that generates them and respecting electron exchange.

Pauli principle and the Aufbau of the periodic table

No two electrons may share all four quantum numbers, so subshells fill in order of increasing energy, and the periodic recurrence of outer-shell configurations explains the chemical periodicity of the elements.

Clinical relevance

The electronic structure of many-electron atoms determines chemical bonding and reactivity across chemistry and materials science, and self-consistent-field methods developed for atoms are the conceptual ancestors of the computational electronic-structure methods used to design molecules and materials.

History

Mendeleev's 1869 periodic table organized the elements empirically by chemical behaviour. Its physical basis came with Bohr's shell ideas and, decisively, Pauli's 1925 exclusion principle, which explained why shells close. Hartree (1928) and Fock (1930) then developed the self-consistent-field methods that made quantitative calculation of many-electron atoms possible.

Key figures

• Wolfgang Pauli
• Douglas Hartree
• Vladimir Fock
• Dmitri Mendeleev

Seminal works

• pauli1925
• bransden2003
• cowan1981

Frequently asked questions

Why does the 4s subshell fill before 3d in many atoms?

Because of screening and orbital penetration, the 4s orbital can lie lower in energy than 3d in neutral atoms, so it fills first; this ordering is approximate and reverses for many ions, which is why the rule has well-known exceptions.

What is the difference between LS and jj coupling?

LS (Russell–Saunders) coupling, valid for lighter atoms, couples all orbital momenta together and all spins together before combining them; jj coupling, more accurate for heavy atoms with strong spin–orbit interaction, couples each electron's spin and orbital momentum first.