Index
Chapter 7 · Item 7.9
Application 7.B: Angular momentum of atoms and Hund's rules
Effective atomic angular momenta and term symbols
9 / 10
Guided reading

The application applies the addition of angular momenta to atoms. Closed shells do not contribute to the total angular momentum; only the incomplete shells determine the effective spin, orbital and total angular momenta.

Effective angular momenta

In complete shells the total angular momentum is zero. For incomplete shells in light atoms, the spin angular momenta first couple to produce an effective spin:

\[\vec{\hat S}=\sum_i \vec{\hat S}_i.\]

The orbital angular momenta also couple:

\[\vec{\hat L}=\sum_i \vec{\hat L}_i.\]

These interact through spin-orbit coupling:

\[\hat H=\zeta\vec{\hat S}\cdot\vec{\hat L}.\]

This is the Russell-Saunders coupling route used for light atoms.

Heavy atoms and j-j coupling

For heavy atoms, each electron couples its spin and orbital momentum to produce its own total angular momentum. These individual total angular momenta then couple:

\[\vec{\hat J}=\sum_i\vec{\hat J}_i.\]

This case is known as \(j-j\) coupling.

Hund's rules
  • First rule: fill the available orbitals one by one, respecting the Pauli exclusion principle, so that the effective spin momentum has its maximum value.
  • Second rule: once the spin is fixed, choose the maximum effective orbital momentum.
  • Third rule: the sign of \(\zeta\) determines whether the ground state has \(\vec{\hat S}\) antiparallel or parallel to \(\vec{\hat L}\).
\[\zeta>0:\ j=|l-s|,\qquad \zeta<0:\ j=l+s.\]

For shells less than half filled, \(\zeta>0\). For shells more than half filled, \(\zeta<0\).

Term symbol

The angular momenta are written in Russell-Saunders notation, known as the Term Symbol:

\[{}^{2s+1}X_j.\]

The letter \(X\) represents the effective orbital quantum number \(l\) in spectroscopic notation:

\(l\)0123456
\(X\)SPDFGHI
Example: Sm3+

The application uses \(Sm^{3+}\), with electronic configuration

\[Sm:[Xe]4f^6 6s^2\quad\longrightarrow\quad Sm^{3+}:[Xe]4f^5.\]
\(m_l\)+3+2+10-1-2-3
\(4f^5\)↑↑↑↑↑

Hund's first rule gives \(s=5/2\). Hund's second rule gives \(l=+3+2+1+0-1=+5\). Because the \(f\) shell is less than half filled, Hund's third rule gives the ground-state multiplet \(j=|s-l|=5/2\).

The multiplets listed in the application are \(j=5/2\) (ground state), \(7/2\), \(11/2\), \(13/2\) and \(15/2\). The ground state is written as

\[{}^6H_{5/2}.\]
Physical reading of the rules

The first rule follows the Pauli exclusion principle and the tendency to occupy different orbitals before pairing. The second rule selects the largest effective orbital angular momentum compatible with the spin choice.

The third rule uses the spin-orbit term: for less than half-filled shells the ground state has \(\vec{\hat S}\) antiparallel to \(\vec{\hat L}\), while for more than half-filled shells it has them parallel.

Exercise-ready boundary

This page supports compact guided exercises on: effective atomic angular momenta, Hund's rules and Russell-Saunders notation.

  • Use from this page: the definitions, physical setup, highlighted equations and quantum-number restrictions shown here.
  • Keep in Chapter 7: the complete demonstrations, long examples, historical notes and extended exercise solutions.
  • Boundary: do not introduce notation, Hamiltonians, coupling schemes or selection rules outside the sequence presented here.
Practice anchors

Use these anchors only within the material introduced on this page.

  • Focus: effective atomic angular momenta, Hund's rules and Russell-Saunders notation.
  • Conceptual check: identify the basis or angular-momentum interaction before applying the equation.
  • Key equation 1:
    \[\vec{\hat S}=\sum_i\vec{\hat S}_i,\qquad \vec{\hat L}=\sum_i\vec{\hat L}_i\]
  • Key equation 2:
    \[{}^{2s+1}X_j\]
  • Typical task: apply one central equation and state what basis, quantum numbers or physical consequence it uses.
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Source note: Original auxiliary summary for this book-app, based on Chapter 7 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for complete demonstrations, examples and exercises.