Questions: Atomic Term Symbols and LS Coupling Scheme

Question 1

Carbon's ground state is represented as ³P₀. What does the superscript '3' signify?

• A. The principal quantum number n = 3 of the outermost electrons
• B. The spin multiplicity 2S+1 = 3, meaning total spin S = 1 (two electrons with parallel spins)
• C. The number of electrons in the 2p subshell
• D. The total angular momentum quantum number J = 3

Answer: (B)

The superscript in ²ˢ⁺¹Lⱼ is the spin multiplicity 2S+1. For ³P₀, 2S+1 = 3 means S = 1 — two electrons with aligned (parallel) spins. The letter P gives L = 1 (total orbital angular momentum), and the subscript 0 gives J = 0. Note that J = 0 is the lowest J level for ³P because carbon's 2p² subshell is less than half full (normal multiplicity, lowest J is lowest energy).

Question 2

Which of the following spectral transitions is forbidden by the electric dipole selection rules for LS-coupled atoms?

• A. ³P₁ → ³D₂ (ΔL = 1, ΔS = 0, ΔJ = 1)
• B. ¹S₀ → ¹P₁ (ΔL = 1, ΔS = 0, ΔJ = 1)
• C. ³P₁ → ¹S₀ (ΔS = −1)
• D. ²P₁/₂ → ²S₁/₂ (ΔL = 1, ΔS = 0, ΔJ = 1)

Answer: (C)

The selection rule ΔS = 0 requires that the total spin quantum number cannot change in an allowed electric dipole transition. In ³P₁ → ¹S₀, the transition goes from S = 1 to S = 0, so ΔS = −1, violating this rule. This is why singlet-triplet transitions are forbidden (or very weak) in LS coupling. The other options all have ΔS = 0 and satisfy ΔL = ±1, ΔJ = 0 or ±1, making them allowed.

Question 3

Term symbols like ³P₀ describe the quantum state of individual electrons within an atom.

• T. True
• F. False

Answer: False

Term symbols describe the state of the atom as a whole, not individual electrons. The letters and numbers (L, S, J) refer to the *total* orbital angular momentum, *total* spin, and *total* angular momentum resulting from the combined contributions of all electrons in the atom. This is a key conceptual point of LS coupling: individual electron labels lose meaning — what matters is the collective quantum state.

Question 4

For an atom whose outermost subshell is more than half full, the J-level with the highest value of J lies lowest in energy (inverted multiplet rule).

• T. True
• F. False

Answer: True

Correct — this is the inverted multiplet rule. For shells less than half full, the lowest J lies lowest (normal multiplet, as in carbon's ³P₀ ground state). For shells more than half full, the highest J lies lowest in energy. This arises from the sign of the spin-orbit coupling constant, which flips between less-than-half-full and more-than-half-full subshells. For example, oxygen (2p⁴, more than half full) has the ³P₂ level as its ground state.

Question 5

Why do only the electrons in incompletely filled subshells matter when determining the term symbol of an atom?

Model answer: Completely filled subshells (like 1s², 2s², 2p⁶) have all magnetic quantum numbers m_l and spin projections m_s symmetrically occupied. Their contributions to total orbital angular momentum L and total spin S cancel exactly: the vector sum of all l⃗ᵢ and s⃗ᵢ is zero. Only electrons in partially filled subshells have unbalanced angular momenta that contribute to L and S — so the term symbol is determined entirely by the incomplete subshell configuration.

This is why the term symbol for carbon (1s²2s²2p²) depends only on the two 2p electrons: the filled 1s² and 2s² subshells contribute L = 0 and S = 0. The same logic applies across the periodic table: noble gas core electrons are spectroscopically inert, and the term symbol reflects only the valence electrons in partially filled subshells.