Hund’s rule predicts that electrons filling a set of equal-energy orbitals will spread out with their spins aligned in the same direction before any orbital gets a second electron. This gives the atom’s ground state the highest possible total spin, and the rule is remarkably reliable for predicting the lowest-energy arrangement of electrons in atoms. The physical reason behind it, though, is not what most introductory courses teach, and there are newly discovered molecular systems where the rule breaks down in ways that could reshape display technology.
What the Rule Actually Says
Friedrich Hund proposed the rule in 1925, and it actually comes in three parts, though most people only encounter the first. The first rule says that for a given electron configuration, the lowest-energy arrangement has the maximum total spin. In practice, this means electrons occupy empty orbitals one at a time, all spinning the same way, before any of them pair up in the same orbital. The second rule says that once you have maximized spin, the lowest energy comes from maximizing total orbital angular momentum. The third rule governs how spin and orbital angular momentum combine, and it flips depending on whether a set of orbitals is less than or more than half-filled.
The first rule is by far the most broadly useful and the one people usually mean when they say “Hund’s rule.” A detailed treatment of the rule’s formulation describes it as stating that “the deepest lying term corresponds to the highest possible value of the total spin.”1Advances in Quantum Chemistry. Theoretical Interpretation of Hund’s Rule The second rule follows from minimizing the energy associated with how the electrons’ orbital motion couples to their spins. Analysis of the physical basis for Hund’s rule has shown that the maximum-spin requirement arises from minimizing the energy of electron-electron repulsion alone, while the maximum orbital angular momentum requirement comes from minimizing spin-orbit interaction energy.2American Journal of Physics. Physical Basis for Hund’s Rule
Why Electrons Prefer to Spread Out
If you have ever seen the rule explained, you probably heard something like this: electrons with the same spin direction cannot be in the same orbital (by the Pauli exclusion principle), so they stay farther apart on average, which reduces the energy from their mutual repulsion. This story is intuitive and neat. It is also substantially wrong.
The reality is more subtle. When electrons have parallel spins, quantum mechanics forces them to avoid each other in a very specific way, creating what physicists call a “Fermi hole,” a region of space around each electron where the other electron is unlikely to be found. This changes not just how much the electrons repel each other, but also how close they can get to the nucleus. In the high-spin state, the electrons can actually sit closer to the nucleus on average, which lowers their energy through stronger attraction to the positive nuclear charge. The gain from increased nuclear attraction more than compensates for any change in electron-electron repulsion.
Detailed computational studies of simple systems like helium-like atoms confirm this picture. In these systems, the probability of finding an electron shifts toward regions closer to the nucleus in the triplet (high-spin) state compared to the singlet (low-spin) state, and the structure of the Fermi holes governs how this redistribution occurs.3Journal of Physics B: Atomic, Molecular and Optical Physics. Origin of the first Hund rule and the structure of Fermi holes in two-dimensional He-like atoms and two-electron quantum dots
The Textbook Explanation Gets It Backwards
The traditional story that electrons in the high-spin state repel each other less is not just an oversimplification. In many cases, the opposite is true: electrons in the triplet state repel each other more, not less. The high-spin arrangement wins because the increased nuclear attraction outweighs the increased repulsion.
This has been confirmed by careful calculations on real molecules. Multi-reference calculations on the oxygen molecule and the carbon dimer show that the electron-electron repulsion in the triplet state is actually greater than in the singlet state, while there is more electron-nuclear attraction in the triplet than in the singlet. The authors of that study explicitly noted that their findings are “contrary to the traditional explanation of Hund’s rule.”4Chemical Physics. Singlet-triplet energy component differences in homonuclear diatomics: A multi-reference configuration interaction study of Hund’s rule The kinetic energy differences between the two spin states tend to go the same direction as the repulsion differences, while the nuclear attraction energy goes the opposite direction and dominates.
This matters for anyone trying to build real intuition about atomic structure rather than just memorize a mnemonic. The correct picture is about how the Pauli exclusion principle reshapes where all the electrons sit relative to the nucleus, not about electrons simply “getting out of each other’s way.”
How Reliable Is Hund’s Rule
For predicting the ground state of an atom, Hund’s rule is extremely reliable. Across the periodic table, the vast majority of ground-state electron configurations follow it precisely. You can use it to predict the ground-state term of any element’s valence electrons with high confidence, which is why it remains a staple of chemistry and physics education.
Where the rule becomes shaky is in predicting the ordering of excited states. The rule was originally meant as a guide for identifying the lowest-energy term of a given electron configuration. A comprehensive treatment of its reliability found that while “Hund’s rule is highly reliable so far as the ground state is concerned,” it is “not nearly as dependable for the ordering of higher terms in the configuration.”5Advances in Quantum Chemistry. Theoretical Interpretation of Hund’s Rule Trying to use it as a general ordering principle for all energy levels of a configuration often leads to wrong predictions. The rule is a ground-state champion, not a universal ranking system.
Hund’s Rule in Everyday Chemistry
If you have ever wondered why iron is magnetic or why transition metal compounds come in vivid colors, Hund’s rule is part of the answer. The rule governs how electrons fill the five d-orbitals in transition metals. An iron atom has four unpaired electrons in its d-orbitals because Hund’s rule dictates that electrons spread out with parallel spins first. Those unpaired electrons are what give iron its magnetic moment.
In coordination chemistry, things get more interesting because the environment around a metal ion can compete with Hund’s rule. When a metal atom sits inside a molecule surrounded by other atoms or groups (ligands), those ligands create an energy difference between d-orbitals that were previously equal. If that energy difference (the crystal field splitting) is large enough, it becomes cheaper for an electron to pair up in a lower-energy orbital than to jump to a higher-energy one with its spin aligned. The result is a low-spin complex where Hund’s rule appears to lose. But it has not really lost; it is just that the conditions changed. The orbitals are no longer equal in energy, and Hund’s rule only applies to orbitals of the same energy. The interplay between the crystal field splitting and Hund’s coupling drives transitions between high-spin and low-spin states in materials with partially filled d-shells.6PubMed. High-spin to low-spin and orbital polarization transitions in multiorbital Mott systems
This high-spin versus low-spin distinction has practical consequences. It affects the color, magnetic behavior, and chemical reactivity of metal complexes. Hemoglobin, for instance, changes its spin state when oxygen binds to the iron center, and that change is connected to the protein’s ability to pick up and release oxygen cooperatively.
Singlet-Triplet Inversion and the OLED Connection
One of the most exciting recent developments involves molecules that violate Hund’s first rule outright. In normal systems, the triplet state (with parallel electron spins) sits below the singlet state (with antiparallel spins) in energy. That is exactly what Hund’s rule predicts. But in certain organic molecules, researchers have found cases where the singlet drops below the triplet, an arrangement called singlet-triplet inversion.
This inversion is rare, but it has been observed in several classes of molecules. Derivatives of a bicyclic hydrocarbon called calicene, for example, exhibit Hund’s rule violations in excited states involving charge transfer between their ring systems. The inversion can be tuned by attaching different chemical groups to the molecule.7PubMed Central. Singlet-Triplet Inversions in Through-Bond Charge-Transfer States A recent review described singlet-triplet inversion as “a rare phenomenon, where, in opposition to Hund’s first rule, singlet electronic states are stabilized relative to their triplet counterparts.”8PubMed. Singlet-Triplet Inversion
Why does anyone care? Because of OLEDs, the organic light-emitting diodes found in high-end phone and television displays. In a standard OLED, electrical current generates both singlet and triplet excited states. Only singlets emit light efficiently in simple fluorescent emitters, and since quantum statistics dictate that triplets outnumber singlets three to one, roughly 75% of the energy is wasted. Phosphorescent and thermally activated delayed fluorescence (TADF) emitters work around this by harvesting triplet energy, but they have their own limitations in stability and efficiency. A molecule with an inverted singlet-triplet gap could potentially convert all its excited states into light-emitting singlets without the stability problems of current approaches. That same review identified the discovery of organic molecules exhibiting singlet-triplet inversion as presenting “exciting new technological opportunities, such as addressing stability issues in organic light-emitting diodes.”9PubMed. Singlet-Triplet Inversion
The search for these materials has accelerated. Researchers have developed design rules based on orbital interactions in certain classes of hydrocarbons that can both minimize the exchange energy between frontier orbitals and maximize an effect called dynamic spin polarization. Using this approach, many molecules violating Hund’s first rule in their excited states have been identified, substantially expanding the pool of potential emitters with inverted singlet-triplet gaps.10Matter. Bottom-up design and virtual screening of organic molecules with inverted singlet-triplet gaps
Artificial Atoms in Quantum Dots
Hund’s rule was formulated for real atoms, but it also shows up in artificial ones. Semiconductor quantum dots are nanoscale structures that confine electrons in a tiny region of space, creating discrete energy levels that mimic an atom’s shell structure. When you add electrons one at a time to a quantum dot, they fill these artificial shells in a pattern that looks remarkably like the periodic table, including obeying Hund’s rule. The few-electron ground states in these artificial atoms show atomic-like properties, with shell structure and Hund’s rule filling.11Physica E: Low-dimensional Systems and Nanostructures. Electronic states in quantum dot atoms and molecules
Things change when you couple two quantum dots together, creating an artificial molecule. As the dots are brought closer together, their electrons interact and the energy landscape shifts. In vertically coupled quantum dots, the normal atomic filling patterns hold when the dots are far apart or very close together, but at intermediate distances something different happens. Calculations have shown that Hund’s rule breaks down for certain electron counts in these coupled systems.12PubMed. Molecule-type phases and Hund’s rule in vertically coupled quantum dots The artificial molecule develops its own set of states that do not follow the same filling logic as isolated artificial atoms.
A related frontier involves superatoms, clusters of real atoms that collectively behave like a single giant atom with their own shell structure. In ordinary metal clusters, the superatom orbitals span multiple atoms and do not usually follow Hund’s rule the way real atoms do. But adding a transition metal atom to the center of a cluster of lighter metal atoms can change the picture dramatically, enhancing the exchange splitting between spin-up and spin-down electrons in the superatom shells and producing Hund’s rule filling. The magnetic moment and the size of the splitting can be tuned by choosing different central atoms.13PubMed Central. Hund’s rule in superatoms with transition metal impurities This opens a path toward designing nanoscale magnets with controllable properties.
Hund Metals and Strongly Correlated Materials
In the last two decades, Hund’s rule has found a second life in condensed matter physics through the concept of “Hund metals.” These are metallic materials where the tendency of electrons to follow Hund’s rule, specifically the coupling that favors maximum spin, creates unusually strong correlations between electrons without driving the material into an insulating state.
In a normal metal, electrons move more or less independently. In a Mott insulator, electrons are so strongly repelled by each other that they get stuck in place. Hund metals sit in between. The Hund coupling forces electrons to maintain the same spin alignment as they move through the material, which makes them sluggish and strongly interacting but still able to conduct electricity. As a description of this physics puts it, the Hund coupling “forces the electrons to keep a collective configuration with the largest possible spin as they hop around, hence inducing strong correlations.”14Physics Today. The Hund-metal path to strong electronic correlations In this framework, Mott insulators, heavy fermion compounds, and Hund metals are all correlated materials, but their correlations arise from fundamentally different blocking mechanisms: charge blocking, hybridization blocking, and spin blocking, respectively.
The Hund metal concept has become important for understanding iron-based superconductors, ruthenates, and other multiorbital materials that do not fit neatly into older theoretical categories. Research distinguishing “Mottness” from “Hundness” in correlated metals has found that Hund’s coupling leads to a separation of energy scales: orbital degrees of freedom get screened at much higher temperatures than spin degrees of freedom, and this separation is especially pronounced in Hund-dominated systems.15PubMed Central. Signatures of Mottness and Hundness in archetypal correlated metals Mott and Hund systems look similar in some respects but show contrasting behavior at intermediate energies, meaning they respond differently to experimental probes like photoemission or optical conductivity.
Common Misconceptions Worth Clearing Up
Beyond the electron-repulsion misunderstanding discussed earlier, several other misconceptions about Hund’s rule circulate widely. One is that the rule is a law of nature in the same sense as conservation of energy. It is not. Hund’s rule is an empirical guideline that works extraordinarily well for atomic ground states but has well-documented exceptions in excited states, molecules, and engineered systems. Treating it as inviolable leads to confusion when encountering the perfectly real exceptions in transition metal chemistry or molecular excited states.
Another misconception is that the rule says something about how electrons “fill up” orbitals in time, as if there is a sequence where the first electron goes into one orbital, the second goes into the next, and so on. Electrons do not fill orbitals sequentially. Hund’s rule describes the final arrangement that has the lowest energy, not a process. The box-and-arrow diagrams used to teach electron configurations are snapshots of the ground state, not choreography.
A subtler misunderstanding involves confusing the first rule with the second and third rules, or applying the first rule’s logic when the second or third rule is what matters. In introductory courses, students often come away thinking Hund’s rule is just about maximizing unpaired electrons. That gets the first rule right but misses that the second rule (maximizing orbital angular momentum) and the third rule (total angular momentum) are needed to pin down the precise ground-state term. For lighter elements where spin-orbit coupling is weak, the first rule dominates and the others are fine-tuning. For heavier elements, where spin-orbit effects become large, the third rule becomes critical for getting the right answer, and applying just the first rule can lead you astray.
Finally, some treatments extend Hund’s rule to situations where it was never meant to apply, like comparing states from different electron configurations. The rule is designed to order the energy levels within a single configuration, a fixed assignment of electrons to a specific set of orbitals. Comparing the energies of two different configurations requires a different analysis entirely, and shoehorning Hund’s rule into that comparison produces nonsense. When you see a claim like “Hund’s rule explains why chromium has the configuration [Ar] 3d⁵ 4s¹ instead of [Ar] 3d⁴ 4s²,” be skeptical. That is a question about the relative energies of different configurations, and Hund’s rule does not directly address it.