Hund's Rules and Ground-State Terms

Introduction

Deriving every allowed term from a microstate table is informative but slow. For a first prediction of a free ion's ground term, Hund's rules provide a shorter route. The result is useful when an octahedral or tetrahedral field later splits that parent term, yet it is not automatically the actual ground-state label of a complex. This page separates the atomic rule, the electron-count shortcut, and the limits imposed by a strong ligand field.

Core explanation

Hund's first rule selects the term with the greatest total spin S, or equivalently the largest multiplicity 2S+1, among terms of one electron configuration. Parallel spins tend to lower electron-repulsion energy through the antisymmetry of the spatial wavefunction. The rule concerns terms of the same free-ion configuration , not arbitrary changes of oxidation state or electron count. For d², the triplets outrank singlets in the first comparison; for d⁵, five parallel spins give S=5/2 and multiplicity six.

Hund's second rule chooses the largest orbital angular momentum L among terms tied for maximum S. For d², both ^3F and ^3P have S=1, but F has L=3 whereas P has L=1, so ^3F is the ground term. For d³, ^4F is lower than ^4P within the quartet set. This ordering reflects the electron–electron interaction in the approximately spherical free ion. It is not an assertion that every F-like crystal-field state lies below every P-like state after ligands bind.

The free-ion ground terms from d¹ to d⁹ are ^2D, ^3F, ^4F, ^5D, ^6S, ^5D, ^4F, ^3F and ^2D. The sequence is symmetric about d⁵: d¹/d⁹, d²/d⁸, d³/d⁷ and d⁴/d⁶ have corresponding LS ground terms. Counting holes in an otherwise full d shell explains the symmetry. A full d¹⁰ subshell has L=S=0; removing one electron produces a hole with the same allowed L and S magnitudes as one electron. The analogy predicts term sets too, although spin–orbit J ordering reverses between less-than-half-filled and more-than-half-filled shells according to Hund's third rule.

A quick box method uses columns labelled m l=+2,+1,0,−1,−2. Put up-spin electrons singly into distinct columns before pairing, choosing arrangements consistent with maximum S and largest M L. For d³, positions +2,+1,0 give M L=3 and M S=3/2, identifying ^4F. For d⁴, +2,+1,0,−1 yield M L=2 and S=2, giving ^5D. For d⁵, every column is singly occupied and their m l sum is zero, giving ^6S. The analogous hole count handles d⁶–d⁹. This is a shortcut to the ground term, not a substitute for a full term table when excited terms matter.

Hund's third rule concerns the lowest J level within an LS term: for a less-than-half-filled subshell, the smallest J lies lowest; for a more-than-half-filled subshell, the largest J lies lowest. It is best applied only after L and S are known, and in situations where LS coupling is a suitable approximation. Ligand-field splitting and low-symmetry effects often make a J label less useful for a transition-metal complex than its point-group state label.

Most importantly, strong-field ligands can change the preferred spin arrangement. A free d⁶ ion has ^5D as its ground term, but an octahedral low-spin d⁶ complex such as [Fe(CN)₆]⁴⁻ has a singlet ^1A₁g ground state. The free-ion Hund prediction is the starting parent for weak-field correlation, not a universal prediction for coordinated ions. The competition between pairing energy and Δₒ must be evaluated for the actual complex.

Step-by-step reasoning

First determine the metal oxidation state and d count. If asked for a free-ion ground term, maximise S, then maximise L among terms of that S. Use the d¹–d⁵ pattern or its hole partner for d⁶–d⁹. Add J only if asked for a free-ion spin–orbit level. If asked about a complex , assess geometry, ligand-field strength and spin state before assigning a molecular term.

Visual explanation

Draw a horizontal sequence d¹ through d⁹. Write ^2D, ^3F, ^4F, ^5D, ^6S at d¹–d⁵ and mirror the first four labels backwards at d⁶–d⁹. Above the line, show spin increasing from one to five unpaired electrons then decreasing. Below it, pair d¹ with d⁹ and d² with d⁸ by curved arrows labelled “electron–hole correspondence.”

Real-world analogy

Selecting a free-ion ground term is like ranking teams first by a primary score and then by a tie-breaker. Maximum spin is the primary comparison; maximum L breaks a tie. A ligand field changes the contest itself, so the free-ion winner cannot simply be carried over without checking the new energy terms.

Real-world example

Cr³⁺ is d³. Its free-ion ^4F ground term splits in an octahedral field, with ^4A₂g normally the ground state for an octahedral d³ configuration. The spin multiplicity remains four in this first approximation, but the capital A₂g label describes the complex's symmetry and is not a free-atom L letter.

Why?

Why does d⁵ have an S rather than a high-L ground term? Maximising spin places one parallel-spin electron in each of the five m l orbitals. Their projections +2,+1,0,−1,−2 sum to zero, so the maximum-spin sextet has L=0 and letter S.

Common misconception

“Hund's rule says electrons always stay unpaired in a complex.” It predicts a free-ion term within a configuration. A sufficiently large ligand-field splitting can make pairing in lower orbitals favourable, producing a low-spin ground state despite the free-ion high-spin parent.

Worked example

Find the free-ion ground term of Co²⁺. Neutral cobalt has nine valence electrons conventionally counted as 3d⁷4s²; removing two for Co²⁺ leaves d⁷. Its hole partner is d³, whose maximum-spin term is ^4F. Therefore Co²⁺ has free-ion ^4F, with S=3/2 and L=3. Allowed J values are 3/2, 5/2, 7/2 and 9/2. As d⁷ is more than half-filled, Hund's third rule places the largest J, 9/2, lowest within the free-ion ^4F manifold under LS coupling.

Quick check

1. Which d count has free-ion ground term ^6S? Answer: d⁵; five parallel spins give S=5/2 and the summed orbital projection is zero.

Exam focus

Write “free ion” or “complex” beside the question before applying rules. Give the whole ^(2S+1)L term, not just the number of unpaired electrons. Explain electron–hole symmetry only for LS terms; do not assume identical J ordering or identical ligand-field spectra.

Advanced insight

The simplified explanation that parallel spins “repel less” is shorthand for a quantum-mechanical exchange effect: antisymmetry forces same-spin electrons to avoid one another in their spatial probability distribution. Hund's rules summarise the usual ordering of terms in an isolated shell; they are not independent classical forces acting on electrons.

Summary

For free d ions, maximise S and then L to select a ground term. The sequence d¹–d⁹ is ^2D, ^3F, ^4F, ^5D, ^6S, ^5D, ^4F, ^3F, ^2D. Complexes require an additional ligand-field and spin-state analysis.

Practice questions

1. Predict the free-ion ground terms of Ti³⁺ and Ni²⁺. Answer: Ti³⁺ is d¹ and has ^2D. Ni²⁺ is d⁸, the two-hole partner of d², and has ^3F. 2. Why does a low-spin d⁶ complex not contradict the free-ion ^5D ground term? Answer: The complex has a ligand-field energy splitting and possible pairing advantage absent from the free ion. Hund's rules rank free-ion terms, while the complex ground state follows the combined energy balance. 3. What is the third-rule J ordering for a less-than-half-filled ^3F term? Answer: S=1 and L=3 permit J=2,3,4. For a less-than-half-filled shell the smallest J, 2, lies lowest within that term under LS coupling. 4. For d², why is ^3F favoured over ^3P? Answer: Both have maximum S=1, but ^3F has larger L=3 than the L=1 of ^3P, so Hund's second rule selects ^3F.