Which Orbital-filling Diagram Violates Hund's Rule

Which Orbital‑Filling Diagram Violates Hund’s Rule?

Introduction

When students first encounter electron configurations, they are taught three fundamental principles: the Aufbau principle, the Pauli exclusion principle, and Hund’s rule. Now, while the first two dictate where electrons may reside, Hund’s rule specifies how they should occupy degenerate orbitals—those having the same energy level. Understanding which orbital‑filling diagram contravenes this rule is essential for predicting molecular geometry, magnetic properties, and spectroscopic behavior. This article dissects a typical violation, explains why it breaches Hund’s rule, and provides a clear method for spotting such errors in any electron‑configuration diagram.

Understanding Hund’s Rule

The rule in plain language

Hund’s rule states that electrons will singly occupy each degenerate orbital with parallel spins before any pairing occurs. Simply put, for a set of orbitals such as the three p orbitals (pₓ, pᵧ, p_z) or the five d orbitals, the lowest‑energy arrangement maximizes the number of unpaired electrons with the same spin orientation.

Formal statement

For a given subshell (e.g., 2p, 3d), the term‑symbol term with the maximum multiplicity (i.e., the greatest number of unpaired electrons) has the lowest energy.

This principle arises from the interplay of exchange energy and electron‑electron repulsion. Parallel spins reduce Coulombic repulsion because they avoid occupying the same spatial region, and the resulting exchange stabilization lowers the overall energy of the system.

Visual cue

A correct Hund‑compliant filling of a p subshell looks like:

• p¹: ↑ (one electron in any one orbital)
• p²: ↑ ↑ (one electron each in two different orbitals)
• p³: ↑ ↑ ↑ (one electron in each of the three orbitals)

Only after all three orbitals contain one electron does pairing begin (p⁴: ↑ ↑ ↑ ↻, etc.).

Common Orbital‑Filling Scenarios

Before identifying the offending diagram, it helps to review typical correct configurations:

In each case, the first half of the subshell is filled with parallel spins in separate orbitals. Only when the subshell reaches its midpoint does electron pairing commence.

Diagram That Violates Hund’s Rule

The erroneous illustration Consider the following simplified diagram for a 2p² subshell:

   pₓ   pᵧ   p_z
   ↑↓   ↑    —

In words: two electrons are placed paired in the pₓ orbital (↑↓), while the pᵧ orbital contains a single electron (↑), and p_z remains empty.

Why this arrangement breaks Hund’s rule

1. Pairing before singly occupying all degenerate orbitals – The rule explicitly requires that each of the three p orbitals receive one electron before any pairing occurs. Here, pairing starts after only one orbital is singly occupied. 2. Loss of exchange energy – By pairing early, the system forfeits the stabilization that comes from having maximum parallel spins. The energy of this configuration is therefore higher than the true ground‑state arrangement.
2. Incorrect spin multiplicity – The ground‑state term for a 2p² configuration is a triplet (multiplicity = 3), corresponding to two unpaired electrons with parallel spins. The depicted diagram yields a singlet (multiplicity = 1) because the paired electrons cancel each other’s spin.

Thus, the diagram above violates Hund’s rule by not maximizing the number of unpaired, parallel‑spin electrons in the degenerate set.

Visual Comparison: Correct vs. Violating Diagram

Correct 2p² configuration

   pₓ   pᵧ   p_z
   ↑    ↑    —

• Two electrons occupy different orbitals.
• Both have the same spin (commonly shown as ↑).
• This arrangement yields the lowest energy according to Hund’s rule.

Incorrect 2p² configuration (the violation)

   pₓ   pᵧ   p_z
   ↑↓   ↑    —

• One orbital is doubly occupied while another remains singly occupied.
• The third orbital is empty.
• This pattern does not maximize unpaired electrons and therefore contravenes Hund’s rule.

Why Identifying Violations Matters

1. Predicting magnetic behavior – Violations can lead to an underestimation of paramagnetism. Take this case: a correctly filled 2p² subshell is paramagnetic with two unpaired electrons, whereas the violating diagram would suggest fewer unpaired electrons and weaker magnetic response.
2. Accurate term symbols – Spectroscopic notation (e.g., ^3P for a triplet state) depends on the correct spin multiplicity. Misassigning the term symbol can mislead interpretation of electronic spectra.
3. Chemical reactivity – Unpaired electrons influence bonding patterns, especially in radicals and transition‑metal complexes. A mistaken configuration may produce an erroneous prediction of bond order or reaction pathways.

How to Spot Hund’s‑Rule Violations Systematically

1. Identify the subshell (e.g., p, d, f) and count the number of orbitals it contains (3 for p, 5 for d, 7 for f).
2. List the electrons in the diagram, noting their orbital and spin.
3. Check for singly occupied orbitals: Are all orbitals filled singly before any pairing begins?
4. Count unpaired electrons: The maximum possible number of unpaired electrons equals the number of orbitals in the subshell, up to half‑filling. 5. Compare with expected multiplicity:

5. Compare with expectedmultiplicity

The multiplicity (M) of a term is given by (M = 2S + 1), where (S) is the total spin quantum number. g.That's why any diagram that yields a singlet ((M = 1)) or a doublet ((M = 2)) must therefore be scrutinized for rule violations. - Triplet case: All unpaired electrons share the same spin direction; the term symbol will carry a “³” prefix (e.For a 2p² configuration the maximum possible (S) is 1 (two parallel spins), giving a triplet with (M = 3). That said, - Singlet case: Opposite‑spin pairing reduces (S) to 0, producing a “¹” prefix (e. , (^3P)).
g., (^1S)).

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If the diagram’s spin count does not match the expected (M), the configuration is inconsistent with Hund’s rule and should be discarded or corrected.

6. Practical Workflow for Detecting Violations

Applying this workflow to any partially filled subshell (e.g., 3d⁴, 4f⁷) ensures a systematic, error‑free assessment.

7. Illustrative Examples

7.1. 3d⁴ Configuration

• Correct Hund‑compliant arrangement: four electrons occupy four different (d) orbitals with parallel spins.
• Violating arrangement: two electrons pair in one orbital while the remaining two occupy separate orbitals.
• Multiplicities: The compliant diagram yields a quartet ((M = 4)); the violating diagram reduces (S) and may produce a doublet or quartet depending on the exact pairing, but it will never achieve the maximum (M = 4) if pairing occurs prematurely.

7.2. 4f⁷ Configuration

• Maximum unpaired electrons: seven, one per orbital.
• Observed multiplicity: (M = 8) (octet) for the half‑filled shell.
• Any diagram that pairs electrons before all seven orbitals are singly occupied fails the multiplicity test and therefore contravenes Hund’s rule.

8. Implications for Spectroscopic Assignments

When term symbols are extracted from experimental spectra, the multiplicity must align with the electron‑configuration diagram. A mis‑assigned term often stems from an overlooked violation:

• Case study: In the visible emission of atomic oxygen (2p⁴), the observed (^3P) line would be misinterpreted as (^1D) if the diagram were drawn with a paired orbital configuration.
• Consequence: Incorrect line assignments can lead to erroneous energy‑level tables, affecting astrophysical and atmospheric modeling.

9. Summary of Key Take‑aways

1. Hund’s rule is a quantitative guideline: it demands the maximum number of parallel‑spin electrons before any pairing. 2. Multiplicity is the litmus test: the correct term symbol’s “³”, “⁵”, etc., must be reproducible from the diagram’s spin count.
2. Systematic verification prevents mis‑interpretation: employing the step‑wise workflow eliminates subjective judgment.
3. Accurate configurations underpin reliable predictions of magnetic properties, reactivity, and spectroscopic behavior.

Conclusion

Identifying configurations that violate Hund’s rule is not merely an academic exercise; it is a prerequisite for constructing faithful electronic models of atoms and molecules. Here's the thing — by rigorously mapping electron placements, counting unpaired spins, and verifying that the resulting multiplicity matches the expected term symbol, chemists and physicists can avoid the pitfalls of erroneous magnetic forecasts, misassigned spectral lines, and flawed reaction mechanistic proposals. The systematic workflow outlined above provides a clear, reproducible pathway to flag violations, correct diagrams, and ultimately make sure theoretical representations faithfully reflect the underlying quantum‑mechanical reality.