Electronic configurations describe which orbitals are occupied, but they do not fully specify the quantum state of a polyelectronic atom. To understand atomic spectra, we must determine how individual electron angular momenta combine to produce the allowed total angular momentum states. This is done using the concepts of microstates and term symbols.
Microstates
A microstate is a specific assignment of quantum numbers \(m_l\) and \(m_s\) for every electron in a given electronic configuration. Each microstate represents a distinct quantum state that satisfies the Pauli Exclusion Principle.
For example, in a \(p^2\) configuration:
- each electron can occupy one of the three \(p\) orbitals (\(m_l=-1,0,+1\)),
- each electron has spin \(m_s=\pm \tfrac{1}{2}\),
- no two electrons may have the same \(m_l\) and \(m_s\).
Enumerating all allowed combinations of \(m_l\) and \(m_s\) gives the complete set of microstates for that configuration.
Total angular momentum from microstates
Each microstate has well-defined values of:
- total orbital angular momentum projection \(M_L = \sum m_l\),
- total spin angular momentum projection \(M_S = \sum m_s\).
By grouping microstates according to their \(M_L\) and \(M_S\) values, we can identify the possible values of the total orbital angular momentum \(L\) and total spin angular momentum \(S\).
The allowed values of \(L\) and \(S\) define the atomic terms associated with the configuration.
Term symbols
Atomic terms are summarized using term symbols of the form
\[ {}^{2S+1}L_J. \]
Here:
- \(S\) is the total spin quantum number,
- \(2S+1\) is the spin multiplicity,
- \(L\) is the total orbital angular momentum (\(L=0,1,2,3,\ldots\) corresponding to \(S,P,D,F,\ldots\)),
- \(J\) is the total angular momentum quantum number.
For given values of \(L\) and \(S\), the allowed values of \(J\) are
\[ J = L+S,\, L+S-1,\, \ldots,\, |L-S|. \]
Why microstates matter
Microstates provide a systematic way to determine:
- all allowed term symbols for a configuration,
- the degeneracy of each term,
- which terms can appear in atomic spectra.
This approach ensures that the Pauli Exclusion Principle is respected and that no physically allowed states are omitted.
Example: microstates and terms for a \(p^2\) configuration
A \(p\) subshell has \(m_l=-1,0,+1\) and \(m_s=\pm\tfrac12\). Choosing two electrons subject to the Pauli Exclusion Principle gives \(\binom{6}{2}=15\) allowed microstates.
| \(m_l(1)\)▴▾ | \(m_l(2)\)▴▾ | \(m_s(1)\)▴▾ | \(m_s(2)\)▴▾ | \(M_L\)▴▾ | \(M_S\)▴▾ | Term▴▾ |
|---|---|---|---|---|---|---|
| -1 | 0 | +1/2 | +1/2 | -1 | +1 | \(^3P\) |
| -1 | +1 | +1/2 | +1/2 | 0 | +1 | \(^3P\) |
| 0 | +1 | +1/2 | +1/2 | +1 | +1 | \(^3P\) |
| -1 | 0 | +1/2 | -1/2 | -1 | 0 | \(^3P\) |
| -1 | +1 | +1/2 | -1/2 | 0 | 0 | \(^3P\) |
| 0 | +1 | +1/2 | -1/2 | +1 | 0 | \(^3P\) |
| -1 | 0 | -1/2 | -1/2 | -1 | -1 | \(^3P\) |
| -1 | +1 | -1/2 | -1/2 | 0 | -1 | \(^3P\) |
| 0 | +1 | -1/2 | -1/2 | +1 | -1 | \(^3P\) |
| -1 | +1 | +1/2 | -1/2 | 0 | 0 | \(^1D\) |
| 0 | +1 | +1/2 | -1/2 | +1 | 0 | \(^1D\) |
| -1 | 0 | +1/2 | -1/2 | -1 | 0 | \(^1D\) |
| -1 | +1 | -1/2 | +1/2 | 0 | 0 | \(^1D\) |
| 0 | +1 | -1/2 | +1/2 | +1 | 0 | \(^1D\) |
| 0 | 0 | +1/2 | -1/2 | 0 | 0 | \(^1S\) |
Note: In cases where a microstate could belong to more than one term symbol (for example, ML = 1, MS = 0 could be part of 3P or 1D) one need only mark one in the "Term" column. In fact, the resulting wavefunctions describing the Russell-Saunders states will involve linear combinations of these ambiguous case. As such, the "Term" column is only for bookkeeping purposes.
Check: The 15 microstates group into \(^3P\) (9), \(^1D\) (5), and \(^1S\) (1), consistent with \((2L+1)(2S+1)\).
Big idea: microstates connect electronic configurations to term symbols by showing how individual electron angular momenta combine to produce the total angular momentum states observed in atomic spectroscopy.
The Hole Rule
When dealing with partially filled subshells, especially those that are more than half filled, it is often simpler to describe the system in terms of the absence of electrons rather than their presence. This idea is formalized in the hole rule.
The hole rule
A hole is an unoccupied spin–orbital in an otherwise filled subshell. Instead of counting the properties of the electrons, we can equivalently count the properties of these holes.
The hole rule states that:
- A subshell with \(n\) electrons more than half filled has the same term symbols as the complementary subshell with \(N-n\) electrons, where \(N\) is the maximum occupancy of the subshell.
- The total orbital angular momentum \(L\) and total spin \(S\) of the holes are the same as those of the equivalent electron configuration.
For example, a \(p\) subshell can hold \(6\) electrons. The configurations \(p^2\) and \(p^4\) therefore correspond to one another through two holes:
\[ p^4 \;\longleftrightarrow\; \text{two holes in a } p^6 \text{ subshell} \;\equiv\; p^2. \]
Both configurations give rise to the same set of term symbols: \(^3P\), \(^1D\), and \(^1S\).
Why the hole rule is useful
The hole rule greatly simplifies the analysis of term symbols for nearly filled subshells, avoiding the need to enumerate large numbers of microstates.
It also provides a clear physical explanation for the reversal of \(J\) level ordering in Hund’s third rule.
Big idea: electrons and holes carry the same angular momentum information, so a nearly filled subshell can be treated as a small number of holes in an otherwise filled shell, with identical term symbols but reversed energy ordering.