The United Atom Method for Diatomic Term Symbols
A complementary way to determine possible electronic term symbols of a diatomic molecule is the united atom method. In this approach, the two nuclei are imagined to be brought together until they form a single “united” nucleus. The molecule is then treated as a single atom with a higher nuclear charge.
Although this limit is not physically realized, it provides a powerful way to:
- identify which molecular terms can correlate with atomic terms at short internuclear distance,
- eliminate molecular term symbols that are not physically reasonable, and
- connect molecular states to familiar atomic configurations and term symbols.
Basic Idea
Consider a diatomic molecule AB with nuclei of charges \( Z_A \) and \( Z_B \). In the united atom limit, the two nuclei merge to form a single atom with nuclear charge \( Z = Z_A + Z_B \), and the electrons occupy atomic orbitals of this united atom.
The procedure is:
- Determine the electronic configuration of the united atom with \( Z = Z_A + Z_B \).
- Find the allowed atomic term symbols \({}^{2S+1}L\) for that configuration using standard atomic rules (Hund’s rules, microstates).
- Correlate each atomic term with possible molecular term symbols \({}^{2S+1}\Lambda\) by identifying how the atomic orbital angular momentum \( L \) projects onto the internuclear axis.
From Atomic \( L \) to Molecular \( \Lambda \)
In the united atom, the total orbital angular momentum quantum number is \( L \). When the internuclear axis is introduced, this atomic angular momentum can have projections \( m_L = -L, -L+1, \ldots, +L \). Each distinct magnitude of the projection corresponds to a molecular quantum number
\[ \Lambda = |m_L| = 0, 1, 2, \ldots, L. \]
Thus, an atomic term \({}^{2S+1}L\) can correlate with multiple molecular terms \({}^{2S+1}\Lambda\), with \( \Lambda = 0 \Rightarrow \Sigma \), \( 1 \Rightarrow \Pi \), \( 2 \Rightarrow \Delta \), and so on.
Spin Multiplicity
The total electronic spin \( S \) of the united atom is preserved in the molecular correlation. Therefore, the spin multiplicity \( 2S+1 \) of the atomic term carries directly over to the molecular term symbols.
What the United Atom Method Tells You
- Which molecular term symbols can correlate with low-energy atomic states at small internuclear separation.
- Which molecular terms are forbidden because no corresponding atomic term exists.
- How degeneracies in atomic terms split into different \( \Lambda \) components as a molecular axis is introduced.
Reflection Symmetry and the Wigner–Witmer Rule
For molecular states with \( \Lambda = 0 \) (\(\Sigma\) states), there is an additional symmetry associated with reflection of the electronic wavefunction through a plane containing the internuclear axis. This symmetry is labeled by \( + \) or \( - \).
In the united atom method, the \(\Sigma^{+}/\Sigma^{-}\) designation can be determined using the Wigner–Witmer rule.
Wigner–Witmer Rule (reflection symmetry):
If a molecular
\(\Sigma\)
state arises from an atomic term with even orbital angular momentum
\( L \),
the resulting molecular state has
\(\Sigma^{+}\)
symmetry.
If it arises from an atomic term with odd
\( L \),
the resulting molecular state has
\(\Sigma^{-}\)
symmetry.
\[ L + \sum_i l_i \;\text{even} \;\Rightarrow\; \Sigma^{+}, \qquad L + \sum_i l_i \;\text{odd} \;\Rightarrow\; \Sigma^{-}. \]
This rule applies specifically to \(\Lambda = 0\) states obtained from the united atom limit. For states with \(\Lambda > 0\) (e.g. \(\Pi, \Delta\)), the \( + / - \) reflection label is not defined.
Practical use: Once the united atom term \({}^{2S+1}L\) is known, the Wigner–Witmer rule allows the \(\Sigma^{+}\) or \(\Sigma^{-}\) label to be assigned immediately, completing the molecular term symbol up to spin–orbit and parity (\( g/u \)) considerations.
Limitations
Like the separated atom method, the united atom method does not determine:
- the energetic ordering of molecular terms,
- which term is the ground state,
- spin–orbit or rotational fine structure,
- parity (\( g/u \)).
Interpretation: The united atom method provides a bottom-up view of diatomic term symbols, starting from familiar atomic structure and asking how atomic terms evolve as a molecular axis is introduced. Used together with the separated atom method, it sharply constrains the set of physically meaningful diatomic molecular terms.
Practice: United Atom Method & Wigner–Witmer Rule
| Problem 1 | |
|---|---|
| In the united atom method, a diatomic molecule correlates with an atomic term \( {}^{3}P \). Which molecular spin multiplicity is possible? | |
| A. Singlet only | |
| B. Triplet only | |
| C. Singlet and triplet | |
| D. Doublet only | |
| Problem 2 | |
|---|---|
| An atomic united-atom term has \( L = 1 \). Which molecular term types can correlate with this term? | |
| A. \( \Sigma \) only | |
| B. \( \Pi \) only | |
| C. \( \Sigma \) and \( \Pi \) | |
| D. \( \Sigma, \Pi, \Delta \) | |
| Problem 3 | |
|---|---|
| A united-atom configuration gives an atomic term with \( L = 1 \) and \( \sum_i l_i = 1 \), which correlates to a \( \Sigma \) molecular state. What is the reflection symmetry of this state? | |
| A. \( \Sigma^{+} \) | |
| B. \( \Sigma^{-} \) | |
| C. Both \( \Sigma^{+} \) and \( \Sigma^{-} \) | |
| D. Cannot be determined | |
| Problem 4 | |
|---|---|
| When using the united atom method, for which molecular term types does the \( \Sigma^{+}/\Sigma^{-} \) reflection label apply? | |
| A. All molecular terms | |
| B. \( \Sigma \) and \( \Pi \) only | |
| C. Only homonuclear molecules | |
| D. Only \( \Sigma \) states | |
The Separated Atom Method for Diatomic Term Symbols
One way to determine the possible electronic term symbols of a diatomic molecule is to consider the separated-atom limit, where the two nuclei are far apart and each atom behaves essentially as an isolated atom. In this limit, molecular electronic states can be constructed by combining the atomic term symbols of the two constituent atoms.
The separated atom method is most useful for understanding:
- which molecular terms are allowed in principle,
- how atomic angular momenta map onto molecular quantum numbers, and
- the correlation between atomic states and molecular states at large internuclear separation.
Basic Idea
Suppose two atoms A and B, with atomic term symbols \({}^{2S_A+1}L_A\) and \({}^{2S_B+1}L_B\), are brought together to form a diatomic molecule. In the separated-atom limit:
- The total electronic spin is obtained by vector addition: \( \mathbf{S} = \mathbf{S}_A + \mathbf{S}_B \), giving \( S = S_A + S_B,\, S_A + S_B - 1,\, \ldots,\, |S_A - S_B| \).
- Each atomic orbital angular momentum \( \mathbf{L}_A \) and \( \mathbf{L}_B \) has a projection onto the internuclear axis. These projections combine to form the molecular quantum number \( \Lambda \).
Determining \( \Lambda \)
For each atom, the possible projections of orbital angular momentum onto the internuclear axis are \( m_{L} = -L, -L+1, \ldots, +L \). The molecular projection quantum number is then
\[ \Lambda = |m_{L,A} + m_{L,B}|. \]
All distinct values of \( \Lambda \) generated in this way correspond to possible molecular term types: \( \Lambda = 0,1,2,\ldots \Rightarrow \Sigma,\Pi,\Delta,\ldots \).
Spin Multiplicity
Once the allowed values of the total spin \( S \) are determined, each value corresponds to a spin multiplicity \( 2S+1 \). Each allowed combination of \( S \) and \( \Lambda \) produces a possible molecular term symbol of the form
\[ {}^{2S+1}\Lambda. \]
Reflection Symmetry and the Wigner–Witmer Rule (Separated Atom Method)
For molecular states with \( \Lambda = 0 \) (\(\Sigma\) states), there is an additional symmetry associated with reflection of the electronic wavefunction through a plane containing the internuclear axis. This symmetry is labeled by \( + \) or \( - \).
In the separated atom method, the \(\Sigma^{+}/\Sigma^{-}\) label can be determined using the Wigner–Witmer rule, based on the parity of the atomic orbital angular momenta that combine to form the molecular state.
Wigner–Witmer Rule (reflection symmetry, separated atoms):
For a molecular
\(\Sigma\)
state formed by combining two atomic states with orbital angular momenta
\( L_A \) and
\( L_B \),
the reflection symmetry is determined by the parity of
\( L_A + L_B \):
\[ L_A + \sum_i l_{A,i} + L_B + \sum_j l_{B,j} \;\text{even} \;\Rightarrow\; \Sigma^{+}, \qquad L_A + \sum_i l_{A,i} + L_B + \sum_j l_{B,j} \;\text{odd} \;\Rightarrow\; \Sigma^{-}. \]
Here \(L_A\) and \(L_B\) are the total orbital angular momentum quantum numbers of the separated atoms (from their atomic term symbols), and \(\sum_i l_{A,i}\) and \(\sum_j l_{B,j}\) are the sums of the one-electron orbital angular momentum quantum numbers for the relevant (open-shell) electrons on atoms A and B, respectively.
This rule applies only to the odd \(\Lambda = 0\) state. For molecular states with \(\Lambda > 0\) (e.g. \(\Pi, \Delta\)), the \( + / - \) reflection label is not defined.
Consistency check: In homonuclear diatomic molecules, the separated atom and united atom Wigner–Witmer rules must give the same \(\Sigma^{+}/\Sigma^{-}\) assignment. Using both limits therefore provides a powerful cross-check when constructing complete diatomic term symbols.
What This Method Does Not Determine
The separated atom method identifies which molecular terms are symmetry-allowed, but it does not determine:
- the ordering of energy levels,
- which term is the ground state,
- fine-structure splittings or spin–orbit coupling patterns,
- parity labels (\( g/u \)).
Interpretation: The separated atom method provides a top-down way to enumerate possible diatomic molecular term symbols by correlating atomic states to molecular states. It is especially useful as a starting point for spectroscopy problems and for understanding how molecular terms evolve as the internuclear distance changes.