Spin-Orbit Coupling
Once the allowed atomic terms have been identified, we must understand how these terms split into multiple energy levels. One of the most important interactions responsible for this splitting is spin–orbit coupling.
Origin of spin–orbit coupling
Spin–orbit coupling arises from the interaction between an electron’s orbital angular momentum and its spin angular momentum. In the electron’s reference frame, the motion of the electron around the nucleus creates an effective magnetic field, which interacts with the magnetic moment associated with the electron’s spin.
This interaction slightly shifts the energies of atomic states depending on how the spin and orbital angular momenta are aligned.
Total angular momentum
In the presence of spin–orbit coupling, the total angular momentum \( \mathbf{J} \) is formed by coupling the total orbital angular momentum \( \mathbf{L} \) and the total spin angular momentum \( \mathbf{S} \):
\[ \mathbf{J} = \mathbf{L} + \mathbf{S}. \]
For a given term characterized by \(L\) and \(S\), the allowed values of the total angular momentum quantum number are
\[ J = L+S,\; L+S-1,\; \ldots,\; |L-S|. \]
Each value of \(J\) corresponds to a distinct energy level.
Consistency of total degeneracies: the \(p^2\) example
A useful check on term-symbol analysis is to verify that the total number of quantum states is conserved when spin–orbit coupling is introduced. For a given electronic configuration, the number of microstates must be the same whether we count them using \((2L+1)(2S+1)\) for each term or using \((2J+1)\) for the spin–orbit–split levels.
For a \(p^2\) configuration (which results in 15 microstates), the allowed terms are \(^1D\), \(^3P\), and \(^1S\). Spin–orbit coupling further splits these terms into \(^1D_2\), \(^3P_2\), \(^3P_1\), \(^3P_0\), and \(^1S_0\).
| Term | \(L\) | \(S\) | \((2L+1)\) | \((2S+1)\) | \((2L+1)(2S+1)\) | Spin–orbit levels | \((2J+1)\) |
|---|---|---|---|---|---|---|---|
| \(^1D\) | 2 | 0 | 5 | 1 | 5 | \(^1D_2\) | 5 |
| \(^3P\) | 1 | 1 | 3 | 3 | 9 | \(^3P_2\) | 5 |
| \(^3P_1\) | 3 | ||||||
| \(^3P_0\) | 1 | ||||||
| \(^1S\) | 0 | 0 | 1 | 1 | 1 | \(^1S_0\) | 1 |
| Total | 15 | Total | 15 | ||||
This consistency confirms that spin–orbit coupling redistributes the same set of microstates among different \(J\) levels without creating or destroying states.
Energy splitting of terms
Spin–orbit coupling lifts the degeneracy of a term \({}^{2S+1}L\) by splitting it into multiple \({}^{2S+1}L_J\) levels. This fine structure splitting is typically much smaller than the energy differences between different terms, but it is readily observed in high-resolution atomic spectra.
The relative energies of the \(J\) levels depend on how \( \mathbf{L} \) and \( \mathbf{S} \) are aligned. States with parallel or antiparallel alignment can differ slightly in energy.
Connection to spectroscopy
Because each term splits into several \(J\) levels, atomic transitions often appear as closely spaced groups of lines rather than as single sharp features. These multiplets are a direct signature of spin–orbit coupling.
Understanding how spin–orbit coupling splits terms is essential for applying selection rules and interpreting atomic spectra.
Big idea: spin–orbit coupling links electron spin and orbital motion, splitting atomic terms into fine-structure levels labeled by \(J\), and providing the next level of structure beyond electronic configurations and term symbols.
Once spin–orbit coupling has split an atomic term into multiple \(J\) levels, we must determine how these levels are ordered in energy. This ordering is described by a set of empirical guidelines known as Hund’s rules.
Hund’s rules
Hund’s rules predict the relative energies of atomic terms and fine-structure levels that arise from a given electronic configuration. They are based on minimizing electron–electron repulsion and maximizing exchange stabilization.
-
First Hund’s rule:
For a given electronic configuration, the term with the
largest spin multiplicity
(\(2S+1\))
lies lowest in energy.
(Electrons with parallel spins avoid each other more effectively, reducing repulsion.) - Second Hund’s rule: For terms with the same spin multiplicity, the term with the largest total orbital angular momentum (\(L\)) lies lowest in energy.
-
Third Hund’s rule:
For a given term
\({}^{2S+1}L\),
the ordering of the
\(J\)
levels depends on whether the subshell is
less than or more than half filled:
- If the subshell is less than half filled, the level with the smallest \(J\) lies lowest in energy.
- If the subshell is more than half filled, the level with the largest \(J\) lies lowest in energy.
Applying Hund’s rules: qualitative example
For a \(p^2\) configuration, the allowed terms are \(^3P\), \(^1D\), and \(^1S\). Hund’s first rule predicts that \(^3P\) lies lowest in energy.
The \(^3P\) term is then split by spin–orbit coupling into \(^3P_2\), \(^3P_1\), and \(^3P_0\). Because a \(p\) subshell is less than half filled in \(p^2\), Hund’s third rule predicts that \(^3P_0\) is the lowest-energy fine-structure level.
Why Hund’s rules matter
Hund’s rules allow ground-state term symbols and fine-structure level orderings to be predicted without solving the full many-electron Schrödinger equation. They are essential for interpreting atomic spectra and understanding periodic trends in atomic structure.
Big idea: Hund’s rules connect microstate counting and spin–orbit coupling to observable energy level orderings, providing a practical roadmap from electronic configurations to real atomic spectra.