Questions: Term Symbols for d-Electron Configurations

Question 1

What is the ground-state term symbol for a d³ free ion, and which of Hund's rules determines it?

• A. ⁴F — Hund's first rule (maximize S) gives S = 3/2 (quartet), and among the possible L values for this S, Hund's second rule (maximize L) selects L = 3 (F term)
• B. ³D — the maximum multiplicity for d³ is 3
• C. ⁴P — maximum S = 3/2 is correct, but L = 1 for a less-than-half-filled shell
• D. ²G — maximize L first, then find the matching S

Answer: (A)

For d³, place three electrons in five d-orbitals (ml = +2, +1, 0, −1, −2) following Hund's first rule: maximize spin. All three electrons have parallel spins: S = 3/2, giving multiplicity 2S+1 = 4 (quartet). Hund's second rule: among all microstates with S = 3/2, maximize L. The maximum L is achieved with ml values +2, +1, 0 → L = 3, which is an F term. The ground-state term is ⁴F. This same term arises for Cr³⁺ and V²⁺ (both d³). Hund's rules are applied sequentially: first maximize S, then maximize L for that S value, then apply the third rule for J if needed.

Question 2

The number of microstates for a d² configuration is 45, and these microstates distribute among multiple term symbols whose total microstate count sums to 45.

• T. True
• F. False

Answer: True

For d², the number of microstates is C(10,2) = 45, where 10 = 2×5 is the number of available spin-orbital combinations for d-orbitals. These 45 microstates belong to the terms ³F (21 microstates), ³P (9), ¹G (9), ¹D (5), ¹S (1): total = 21 + 9 + 9 + 5 + 1 = 45. Each term symbol ²ˢ⁺¹L has (2S+1)(2L+1) microstates. The complete enumeration of terms from microstates is a combinatorial exercise that ensures every possible electronic arrangement is accounted for.

Question 3

A d⁸ configuration has the same set of term symbols as a d² configuration.

• T. True
• F. False

Answer: True

This is the hole formalism: a d-shell with n electrons produces the same term symbols as one with (10−n) electrons. d² and d⁸ both give ³F, ³P, ¹G, ¹D, ¹S. The physical reason is that 8 electrons in 10 orbitals is equivalent to 2 holes — the angular momentum coupling of the holes produces the same terms as the electrons. However, the ground state J values differ (Hund's third rule): for d² (less than half-filled), J = |L−S| is lowest; for d⁸ (more than half-filled), J = L+S is lowest. The hole formalism simplifies term-symbol derivation by reducing a d⁶, d⁷, d⁸, or d⁹ problem to its equivalent d⁴, d³, d², or d¹ problem.

Question 4

Starting from the microstates of a d² configuration, explain the systematic procedure for identifying all term symbols.

Model answer: 1) List all 45 microstates by their ml and ms values for two electrons in d-orbitals, respecting the Pauli exclusion principle. 2) Tabulate the microstates in an ML vs MS grid (ML ranges from +4 to −4, MS from +1 to −1). 3) Identify the highest ML,MS entry: ML = +3, MS = +1 → this defines a ³F term (L=3, S=1). Remove all (2×1+1)(2×3+1) = 21 microstates belonging to ³F. 4) The highest remaining entry is ML = +2, MS = +1 → ³P term (L=1, S=1), but checking carefully, the next highest is ML = +4, MS = 0 → ¹G term (L=4, S=0). Continue removing: ¹G (9), then ³P (9), ¹D (5), ¹S (1). 5) Verify: 21 + 9 + 9 + 5 + 1 = 45. All microstates are accounted for.

This systematic 'peeling' procedure works for any d^n configuration. For d³, you start with C(10,3) = 120 microstates and extract ⁴F, ⁴P, ²H, ²G, ²F, ²D(×2), ²P terms. The procedure is tedious but algorithmic — each step reduces the microstate table until empty.