Hund's Rule and Ground-State Configurations
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You should be familiar with the quantum numbers \(n\), \(l\), \(m_l\), and \(m_s\), and understand that electrons occupy atomic orbitals defined by these numbers. You should also know the Madelung rule for orbital energy ordering.
Why this matters
Hund's rule explains why atoms have unpaired electrons and why magnetic properties emerge from electronic structure.
While the Aufbau principle tells us the order in which orbitals are filled, it does not specify how electrons are distributed among degenerate orbitals (orbitals with the same energy). Hund's rule provides this missing piece: electrons occupy separate degenerate orbitals with parallel spins before pairing up. This maximizes total spin and minimizes electron-electron repulsion, leading to the most stable ground state. Understanding this is crucial for predicting magnetic behaviour, chemical reactivity, and the shapes of the periodic table's blocks.
The Idea in Plain Language
Imagine you have three empty chairs in a row (representing three degenerate p-orbitals) and four people (electrons) who want to sit down. If the people are polite and want to minimise crowding, the first three will each take a separate chair. The fourth person has no choice but to sit next to one of the others. Hund's rule is the "politeness" rule for electrons: they prefer to spread out into separate orbitals rather than crowd into the same one. This spreading out reduces the repulsive forces between the negatively charged electrons, making the atom more stable. Additionally, electrons in different orbitals with parallel spins (same direction) are slightly more stable than those with opposite spins due to quantum mechanical exchange energy effects. Therefore, the ground state is the arrangement with the maximum number of unpaired electrons having parallel spins.
Formal Statement and Rationale
Hund's rule states that for a given electron configuration, the term with the maximum multiplicity (maximum total spin \(S\)) has the lowest energy. In practical terms for filling degenerate orbitals (such as the three \(p\) orbitals or five \(d\) orbitals), electrons occupy each orbital singly with parallel spins before any orbital is doubly occupied.
This rule arises from two main factors. First, electron-electron repulsion is minimized when electrons are in different spatial orbitals, as they are, on average, farther apart. Second, the Pauli exclusion principle and the antisymmetry requirement of the wavefunction lead to an exchange energy stabilization when electrons have parallel spins. This exchange energy is a quantum mechanical effect that lowers the total energy of the system. Consequently, the ground state configuration is the one that maximizes the number of unpaired electrons with parallel spins.
Applying Hund's Rule to Specific Subshells
Let us apply this rule to the \(p\), \(d\), and \(f\) subshells. The \(p\) subshell has three orbitals (\(m_l = -1, 0, +1\)). The \(d\) subshell has five orbitals (\(m_l = -2, -1, 0, +1, +2\)). The \(f\) subshell has seven orbitals.
Consider the \(p\) subshell filling sequence:
- \(p^1\): One electron in any one of the three orbitals. All configurations are equivalent. Spin is \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
- \(p^2\): Two electrons must occupy different orbitals with parallel spins. There are three possible spatial arrangements, but all have the same energy and total spin \(S=1\).
- \(p^3\): Three electrons occupy all three orbitals singly with parallel spins. This is a half-filled subshell, which is particularly stable.
- \(p^4\): The fourth electron must pair up with one of the existing electrons. The spin of the paired electron is opposite to the first one. Total spin \(S=\frac{1}{2}\).
For the \(d\) subshell, the same logic applies. A \(d^5\) configuration (half-filled) has five unpaired electrons with parallel spins, maximizing stability. A \(d^{10}\) configuration (fully filled) has all electrons paired.
Connection to Magnetic Properties
The number of unpaired electrons in an atom or ion directly determines its magnetic behaviour. Atoms with unpaired electrons are paramagnetic; they are attracted to an external magnetic field. Atoms with all electrons paired are diamagnetic; they are weakly repelled by a magnetic field.
Paramagnetism arises because the magnetic moments of the unpaired electrons align with the external field. The strength of the paramagnetic response grows with the number of unpaired electrons. For example, oxygen (\(O_2\)) is paramagnetic because its molecular orbital configuration has two unpaired electrons. In contrast, nitrogen (\(N_2\)) is diamagnetic because all its electrons are paired.
By correctly applying Hund's rule to determine the ground-state configuration, we can predict whether a substance will be paramagnetic or diamagnetic. This is a powerful experimental test for the correctness of our electronic structure models.
The result
The building-up principle
The ground-state electron configuration of an atom is obtained by filling the atomic orbitals of the previous result one electron at a time, subject to three rules: (1) the Madelung rule: orbitals fill in order of increasing \(n+l\), and for orbitals sharing the same \(n+l\), in order of increasing \(n\); (2) the Pauli exclusion principle: no orbital holds more than two electrons, and those two must have opposite spin (\(m_s=+\tfrac12\) and \(m_s=-\tfrac12\)); (3) Hund's rule: within a set of degenerate orbitals (equal \(n\) and \(l\), differing only in \(m_l\)), electrons occupy separate orbitals with parallel spin before any orbital is doubly occupied.
A common misconception is that Hund's rule is a strict law that must always be followed. In reality, it is a rule of thumb that works well for light elements but can fail for heavier elements where relativistic effects and spin-orbit coupling become significant. Additionally, in transition metal complexes, the crystal field splitting energy can sometimes exceed the pairing energy, leading to low-spin configurations that violate the simple Hund's rule prediction for free ions. Always consider the context: free atoms vs. ions in a ligand field. C
Worked example
Problem. Write the ground-state configuration of sulfur (\(Z = 16\)), draw the 3p box diagram, and give the number of unpaired electrons and the total spin \(S\). Is the free sulfur atom paramagnetic?
- Fill in Madelung order: \(1s^2\,2s^2\,2p^6\) (10 electrons), \(3s^2\) (12), leaving 4 for 3p.
- Hund's rule: the first three 3p electrons go into separate orbitals with parallel spins; the fourth must pair: \(\uparrow\downarrow\ \ \uparrow\ \ \uparrow\).
- Unpaired electrons: 2, each with \(m_s = +\tfrac12\), so \(S = 2 \times \tfrac12 = 1\) and spin multiplicity \(2S + 1 = 3\) (a triplet).
- Unpaired electrons make the free atom paramagnetic.
Units check. Electron counts must add to \(Z\): \(2 + 2 + 6 + 2 + 4 = 16\). The same pattern (\(p^4\): two unpaired) holds for oxygen, selenium and tellurium.
Three quick tests
- How many unpaired electrons are in a neutral nitrogen atom (\(Z=7\))?
Answer
3 unpaired electrons. Configuration: \(1s^2 2s^2 2p^3\). The three \(2p\) electrons occupy separate orbitals with parallel spins. - Is a neutral iron atom (\(Z=26\)) paramagnetic or diamagnetic?
Answer
Paramagnetic. Configuration: \([\mathrm{Ar}] 3d^6 4s^2\). In the \(3d^6\) subshell, there are 4 unpaired electrons (5 orbitals, 6 electrons: \(\uparrow\downarrow, \uparrow, \uparrow, \uparrow, \uparrow\)). - Which is more stable: a half-filled \(d\) subshell (\(d^5\)) or a fully filled \(d\) subshell (\(d^{10}\))?
Answer
Both are particularly stable compared to other configurations due to symmetry and exchange energy. However, \(d^{10}\) is generally more stable than \(d^5\) in many contexts because it has no unpaired electrons and is spherically symmetric, but \(d^5\) is exceptionally stable for high-spin systems due to maximum exchange energy. The question is tricky; both are "anomalous" in stability. For free ions, \(d^5\) high-spin is very stable. \(d^{10}\) is also very stable. The key is that both are more stable than intermediate configurations like \(d^4\) or \(d^9\).
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Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.