explain wigner witmer rules

Sonar Pro

Answer: The Wigner-Witmer rules are correlation rules in diatomic molecular spectroscopy that link the electronic states of a diatomic molecule to the orbital and spin angular momenta of its separated atoms, with specific applications for determining the +/- symmetry of Σ states.

Wigner and Witmer's 1928 work on diatomic molecules consists of two parts: the first derives an exact diatomic eigenfunction using symmetry operators like time translation, spatial translation, rotation, parity, and nuclear exchange, separating Euler angles into nuclear and electronic rotations.[1] The second part, which gained prominence, provides the correlation rules connecting molecular electronic states (denoted by term symbols like ^{2S+1}Λ) to the atomic states of the separated atoms, including rules for spin and orbital angular momentum coupling.[1][3][4] A key application is determining the +/- symmetry for Σ states (Λ=0) in diatomic term symbols: Σ states are symmetric (Σ⁺) or antisymmetric (Σ⁻) under reflection through a plane containing the internuclear axis; non-Σ states (Π, Δ, etc.) are doubly degenerate with both symmetries.[2] When an odd number of Σ states arise (e.g., from United Atom or Separated Atom methods), they pair as Σ⁺/Σ⁻, and the remaining one's symmetry follows the Wigner-Witmer rule based on whether the sum of atomic quantum numbers (like l + s) is even or odd, per a standard table.[2] These rules have been reformulated and extended to ionic states and Mulliken's correlations, aiding spectroscopy and potential energy diagrams.[3][4]

Kimi K2.5

Answer: The Wigner-Witmer rules are selection rules that govern how atomic term symbols correlate with molecular electronic states when atoms combine to form diatomic molecules, based on the conservation of total spin and the projection of orbital angular momentum onto the internuclear axis.

These rules, developed by Eugene Wigner and E. E. Witmer in 1928, apply the principles of group theory to determine which molecular states can arise from specific atomic states as two atoms approach each other. The core principle is that the projection of the total electronic orbital angular momentum onto the internuclear axis (quantum number Ω = Λ + Σ) must be conserved, where Λ (0, 1, 2...) corresponds to molecular states labeled Σ, Π, Δ, etc., and Σ represents the projection of total spin. When two atoms with term symbols $^{2S_1+1}L_1$ and $^{2S_2+1}L_2$ combine, the molecular state term symbol takes the form $^{2S+1}\Lambda_{\Omega}^{+/-}$ or $^{2S+1}\Lambda_{\Omega}^{(g/u)}$ (for homonuclear diatomics). The total spin S results from vector coupling of individual spins $S_1$ and $S_2$, giving possible values from $|S_1 - S_2|$ to $S_1 + S_2$. Similarly, the projection Λ equals the algebraic sum of the projections of individual orbital angular momenta ($\Lambda = |m_{l1} \pm m_{l2}|$). These rules are crucial for constructing correlation diagrams connecting separated atoms to united atoms, predicting permitted dissociation channels, and interpreting molecular spectra. For example, two ground-state hydrogen atoms ($^2S$) can only form molecular $^1\Sigma_g^+$ and $^3\Sigma_u^+$ states of H$_2$, demonstrating the rule that only specific spin multiplicities and symmetries are allowed based on the atomic parents.

Grok 4.20

Answer: The Wigner-Witmer rules specify which molecular electronic states can correlate with given atomic states in diatomic molecules based on conservation of angular momentum, spin, and parity.

Developed by Eugene Wigner and E. E. Witmer in 1928, these rules apply group theory and quantum mechanics to predict allowed combinations of atomic terms (characterized by L, S, J) that can form molecular terms (such as 1Σg+, 3Πu, 2Δg) when two atoms approach each other. They ensure that the projection of orbital angular momentum along the molecular axis (Λ), total spin (S), and inversion symmetry (g/u) are conserved during correlation. In practice, the rules are used to construct correlation diagrams for potential energy surfaces, helping identify dissociation limits in spectroscopy. For instance, two ground-state oxygen atoms (3P) can correlate to several O2 states like 3Σg−, 1Δg, and 1Σg+, but not all symmetries are permitted. Violations would imply forbidden processes or require additional perturbations like spin-orbit coupling. These principles are foundational in molecular physics and chemistry for interpreting electronic spectra, predicting reaction pathways, and understanding selection rules in photodissociation. They do not replace full quantum calculations but provide symmetry-based constraints that must be satisfied.

GPT-5.4

Answer: The Wigner–Witmer rules are selection and correlation rules in molecular spectroscopy that determine which diatomic molecular term symbols can arise from separated atomic states by conserving angular-momentum projection and spin symmetry.

The Wigner–Witmer rules are mainly used for diatomic molecules. They answer this question: if two atoms approach each other in particular atomic electronic states, what molecular electronic states are allowed when the molecule forms? The idea is to connect atomic quantum numbers to molecular term symbols. In atoms, relevant labels include orbital angular momentum and spin. In a diatomic molecule, these are reorganized into quantities appropriate to the molecular axis, especially the projection of orbital angular momentum onto the internuclear axis, written as Λ, and the total spin S. The molecular state is then labeled by a term symbol such as ^1Σ, ^2Π, ^3Δ, and so on. The core Wigner–Witmer logic is: 1. The molecular spin multiplicity must be compatible with the way the two atomic spins couple. If the atomic spins combine to total spin S, then the molecule can only have term symbols with that same S. 2. The projection quantum number along the molecular axis must come from the allowed combinations of the atomic orbital angular momenta. In practice, Λ is obtained from the possible sums and differences of the atomic magnetic quantum number projections onto the molecular axis. This determines whether Σ, Π, Δ, Φ, etc. states can occur. 3. Reflection and inversion symmetries may further restrict the allowed states, especially for homonuclear diatomics, where g/u symmetry and +/− labels for Σ states matter. A practical way to use the rules is correlation: start from two separated atoms with known atomic terms, then list all molecular terms consistent with spin coupling and allowed angular-momentum projections. For example, if the resulting projection is Λ = 0, 1, 2, the corresponding molecular symmetries are Σ, Π, and Δ. If total spin is S = 1, those become triplet states such as ^3Σ, ^3Π, ^3Δ. These rules do not usually give precise energies by themselves. Instead, they tell you which molecular electronic states are symmetry-allowed and therefore which potential energy curves and spectroscopic transitions are plausible. So, in short, the Wigner–Witmer rules are a symmetry-based bookkeeping method for deriving allowed diatomic molecular term symbols from the quantum states of the separated atoms.

Gemini 3.1 Pro

Answer: The Wigner-Witmer rules are a set of quantum mechanical principles used to determine the permissible electronic states of a diatomic molecule that can be formed from two separated atoms in specific atomic states.

The Wigner-Witmer rules serve as a critical bridge in molecular spectroscopy and quantum chemistry, connecting the discrete quantum states of isolated atoms at an infinite distance to the bound electronic states of the resulting diatomic molecule. When two atoms approach one another, their spherical symmetry is broken, and their individual angular momenta couple together along the newly formed internuclear axis. The first primary rule governs spin angular momentum. The total spin quantum number (S) of the resulting molecule is determined by the vector addition of the individual atomic spins (S1 and S2). The molecular spin can take any value in integer steps from the absolute difference of the atomic spins to their sum: |S1 - S2| <= S <= (S1 + S2). This dictates the possible spin multiplicities (2S + 1) of the molecule. The second primary rule deals with orbital angular momentum. In a diatomic molecule, the relevant quantum number is Lambda, which represents the absolute value of the projection of the total orbital angular momentum along the internuclear axis. The rules specify that Lambda is formed by combining the projections of the atomic orbital angular momenta (ML1 and ML2), yielding possible molecular states designated by Greek letters such as Sigma (Lambda=0), Pi (Lambda=1), or Delta (Lambda=2). Furthermore, the rules account for molecular symmetry. For homonuclear diatomic molecules (composed of two identical isotopes of the same element), Wigner-Witmer rules dictate whether the resulting molecular state will be 'gerade' (even) or 'ungerade' (odd) with respect to inversion through the center of mass, based on the parities and identical nature of the separated atomic states. They also determine the reflection symmetry (+ or -) for Sigma states. Ultimately, these rules are essential for mapping potential energy curves. By knowing exactly which molecular states correlate to which atomic dissociation limits, physicists and chemists can interpret complex molecular spectra, predict the outcomes of photodissociation events, and understand non-adiabatic transitions like curve crossings.

Claude Opus 4.7

Answer: The Wigner–Witmer rules are symmetry-based selection rules that determine which molecular electronic states of a diatomic molecule can correlate with given electronic states of its two separated atoms.

The Wigner–Witmer rules, formulated by Eugene Wigner and E. E. Witmer in 1928, address a fundamental question in molecular quantum mechanics: when two atoms (each in a specified electronic state) come together to form a diatomic molecule, which molecular electronic states can arise? Conversely, when a diatomic molecule dissociates, into which atomic states can it break apart? The rules answer this by exploiting the conservation of symmetry (angular momentum, spin, and reflection/inversion symmetries) as the internuclear distance varies from infinity to a bonded equilibrium. The key ingredients are: 1. **Total orbital angular momentum projection (Λ).** Each separated atom has orbital angular momentum quantum number L_A and L_B with projections M_{L_A} and M_{L_B} along the internuclear (z) axis. Because only the component along this axis is conserved when the atoms approach, the molecular quantum number Λ = |M_{L_A} + M_{L_B}| labels the molecular state as Σ (Λ=0), Π (Λ=1), Δ (Λ=2), etc. All allowed combinations of M_{L_A} and M_{L_B} (each ranging from −L to +L) generate the possible Λ values, with appropriate multiplicities. 2. **Total spin (S).** The atomic spins S_A and S_B couple vectorially: the allowed molecular spin quantum numbers are S = |S_A − S_B|, |S_A − S_B|+1, …, S_A + S_B. This gives the multiplicity 2S+1 of the resulting molecular terms (singlets, triplets, etc.). 3. **Reflection symmetry (+/−) for Σ states.** For Σ states, one must determine whether the electronic wavefunction is symmetric (Σ⁺) or antisymmetric (Σ⁻) under reflection in a plane containing the internuclear axis. Wigner and Witmer gave explicit counting rules: e.g., for two unlike atoms with term symbols giving L_A and L_B, the number of Σ⁺ and Σ⁻ states can be worked out from the parities of L_A and L_B and the combinations of M values summing to zero. 4. **Gerade/ungerade (g/u) symmetry for homonuclear molecules.** When the two atoms are identical, the molecular states must be classified under inversion through the molecular center. If the atoms are in different electronic states, each molecular term appears twice, once as g and once as u. If the atoms are in the same state, the distribution between g and u depends on whether the atomic state is even or odd, and is given by specific counting rules. **How to apply them in practice:** Take the atomic term symbols (e.g., two ground-state O atoms, each ³P). List all allowed (M_{L_A}, M_{L_B}) pairs; the distribution of M_{L_A}+M_{L_B} values gives the count of Σ, Π, Δ, … molecular states. Combine spins S_A=1 and S_B=1 to get S = 0, 1, 2 (singlets, triplets, quintets). Apply the reflection rule to split Σ into Σ⁺/Σ⁻, and for homonuclear O₂ apply the g/u rule. The result is the famous correlation: O(³P)+O(³P) gives rise to 18 molecular states, including the X³Σg⁻ ground state of O₂ and the a¹Δg and b¹Σg⁺ excited states, among others. **Significance.** The Wigner–Witmer rules are the foundation of correlation diagrams between separated-atom and united-atom (or molecular) limits. They tell spectroscopists and quantum chemists which dissociation channels a given molecular state can adiabatically connect to, guide the construction of potential energy curves, and underpin the understanding of predissociation, photodissociation, and recombination processes. They are strict only to the extent that the invoked symmetries (axial symmetry, spin conservation in the absence of spin–orbit coupling, inversion for homonuclear systems) are good quantum numbers; spin–orbit interaction can mix states of different S, relaxing the spin part of the rules.

Final Summary

The AI Roundtable unified around these rules as the essential symmetry-based bridge connecting isolated atoms to diatomic molecules. Claude Opus 4.7 and Gemini 3.1 Pro detailed how conserving spin and orbital angular momentum allows chemists to predict allowed potential energy curves, while GPT-5.4 described the system as a critical "bookkeeping method" for molecular spectroscopy. The group noted that these rules are foundational for mapping how atoms like oxygen combine to form specific molecular states like O2.

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