Hund Rules

How do electrons, the tiny, mutually-repelling tenants of an atom, arrange themselves to achieve the most stable, lowest-energy state? While the Pauli exclusion principle sets a fundamental law, it is a set of elegant guidelines known as Hund's rules that governs their placement in degenerate orbitals. This article demystifies these critical rules, moving beyond rote memorization to reveal the deep quantum mechanics at play. It addresses the core problem of determining an atom's ground-state configuration, a cornerstone for predicting its chemical and physical behavior.

This article is structured to build a comprehensive understanding from the ground up. In the first chapter, Principles and Mechanisms​, we will dissect each of Hund's three rules, exploring the quantum phenomena—like exchange interaction and spin-orbit coupling—that give them their predictive power. Next, in Applications and Interdisciplinary Connections​, we will witness these rules in action, seeing how they explain everything from the magnetism of liquid oxygen and the color of gemstones to the physics of high-temperature superconductors and the engineering of quantum computers. Finally, the Hands-On Practices section provides an opportunity to apply this knowledge by working through guided problems, solidifying your ability to predict and interpret atomic properties.

Imagine you are the manager of a very peculiar apartment building—an atom. Your tenants are electrons, and they are a notoriously difficult bunch. They are all negatively charged, so they repel each other constantly. Furthermore, they are governed by a strict, non-negotiable law of the universe: the Pauli exclusion principle​, which states that no two of them can ever be in the exact same quantum state. Your job is to house them in a set of available "rooms," which we scientists call orbitals​, in a way that keeps the total energy of the building as low as possible. In other words, you want to find the most stable, most peaceful arrangement—the ground state​.

For many of the floors in your building, the rooms are all identical; they have the exact same energy. We call these degenerate orbitals. Think of the three rooms on the 'p-floor' or the five rooms on the 'd-floor'. When new electrons move in, they have a choice of which degenerate room to occupy. How do they decide? Do they pair up to save space, or do they each take a private room? And how do they orient their intrinsic magnetic poles, their spins​?

It turns out that electrons don't make these choices randomly. They follow a set of marvelously elegant guidelines that minimize their mutual repulsion and lead to the state of lowest energy. These guidelines were first formulated in the 1920s by the German physicist Friedrich Hund, and we now call them Hund's rules​. They are not fundamental laws like the Pauli principle, but rather brilliant rules of thumb derived from the deep consequences of quantum mechanics. Let’s unpack these rules, not as a dry list to be memorized, but as a journey into the strange and beautiful social lives of electrons.

Let’s say you need to house two new electrons on a floor with three empty, degenerate 'p-orbitals'. You have two primary options:

1. Put both electrons in the same orbital, forcing them to have opposite spins (one "spin-up," $\uparrow$, and one "spin-down," $\downarrow$) to satisfy the Pauli principle.
2. Put them in two different orbitals. Now, they have a choice: their spins can be opposite ($\uparrow \downarrow$) or parallel ($\uparrow \uparrow$).

Common sense, based on classical electrostatic repulsion, might suggest that a key goal is simply to keep the electrons in separate orbitals. But it offers no clear preference for how their spins should align. This is where quantum mechanics enters with a stunning twist. Hund's first rule states that the lowest energy arrangement is achieved when the total spin is maximized. For our two electrons, this means they will occupy separate orbitals and have parallel spins ($\uparrow \uparrow$).

Why on earth would this be? The answer lies in a profoundly quantum phenomenon called the exchange interaction. When two electrons have the same spin, their collective wavefunction must be antisymmetric with respect to their spatial coordinates due to the Pauli principle. What this means in plain English is that the probability of finding these two electrons very close to each other is dramatically reduced—it actually goes to zero! They are forced to stay away from each other more effectively than if they had opposite spins.

It’s like a "quantum handshake" of sorts. By aligning their spins, the electrons are implicitly agreeing to give each other a wider berth. This enforced "social distancing" reduces their electrostatic repulsion, lowering the overall energy of the atom. This energy reduction is quantified by something called the exchange integral, denoted by $K$. It's a purely quantum mechanical discount on the energy bill, and it's only available to electrons with parallel spins.

For the carbon atom, with its two $2p$ electrons, this effect is not just theoretical. The state where the spins are parallel (a "triplet" state, $^3P$) is experimentally found to be lower in energy than the most stable state with opposite spins (a "singlet" state, $^1D$). In fact, quantum mechanical calculations show that this energy difference is a direct consequence of both reduced Coulomb repulsion and this extra exchange stabilization. The result is that maximizing the number of unpaired, parallel-spin electrons—maximizing the total spin quantum number $S$—is the first and most important step to finding the ground state.

Alright, so the electrons have spread out into different orbitals with parallel spins to get their exchange energy discount. But what if there's still a choice? Consider an ion with two electrons in the d-shell (a $d^2$ configuration), like Vanadium(III) ($V^{3+}$). The d-shell has five degenerate orbitals, so our two parallel-spin electrons have many pairs of orbitals to choose from. Which pair do they pick?

Hund's second rule provides the answer: for a given total spin, the state with the maximum possible total orbital angular momentum, $L$, will have the lowest energy. The quantum number $L$ is found by summing the individual orbital angular momentum quantum numbers ($m_l$) of the electrons. To maximize $L$, the electrons will preferentially occupy orbitals with the highest available $m_l$ values.

What's the physical intuition here? Think of the electrons as orbiting the nucleus. A high value of $L$ corresponds to a state where the electrons are, on average, orbiting in the same direction. Imagine two runners on a circular track. If they are both running clockwise, they will pass each other far less frequently than if one is running clockwise and the other counter-clockwise. By moving in a correlated, "same-direction" manner, the electrons stay apart more effectively, which further minimizes their electrostatic repulsion. It's a beautifully coordinated orbital ballet designed to keep the mutually repelling dancers as far from each other as possible.

For our $V^{3+}$ ion, the $d^2$ configuration gives rise to several terms, including two with the maximum spin: $^3F$ (which has $L=3$) and $^3P$ (which has $L=1$). Following Hund's second rule, the $^3F$ term, with its larger orbital angular momentum, corresponds to the more "spacious" orbital dance and is therefore lower in energy. Combining the first two rules gives us the lowest energy spectroscopic term​, denoted as $^{2S+1}L$. For a half-filled shell, like the $d^5$ configuration of an Fe$^{3+}$ ion, this process has a particularly elegant outcome. The five electrons occupy the five d-orbitals, one in each, all with parallel spins ($S=5/2$). The $m_l$ values ($+2, +1, 0, -1, -2$) sum perfectly to zero, giving $L=0$. The ground term is thus $^6S$.

We have now used the first two rules to find the lowest-energy term (like $^3P$ for carbon), which is a group of states with the same $S$ and $L$. But the story isn't quite over. There's a final, more subtle interaction at play: spin-orbit coupling​.

You can picture an electron's spin as making it a tiny bar magnet. At the same time, its orbit around the positively charged nucleus creates a magnetic field. The energy of the atom depends on how the electron's spin-magnet aligns with this orbital magnetic field. This tiny energy difference splits a single term into several closely spaced fine-structure levels​, each distinguished by a different total angular momentum quantum number, $J$. The value of $J$ represents the way that the total spin angular momentum $S$ and total orbital angular momentum $L$ add up as vectors.

Hund's third rule tells us which of these $J$ levels is the ground state:

• For subshells that are less than half-filled​, the level with the minimum possible value of $J$ ($J=|L-S|$) has the lowest energy.
• For subshells that are more than half-filled​, the level with the maximum possible value of $J$ ($J=L+S$) has the lowest energy.

This reversal is one of the most beautiful symmetries in atomic physics. Why does it happen? The key is the concept of electron-hole equivalence. Consider a fluorine atom, with a $2p^5$ configuration. This is a p-shell that is almost full; it's just missing one electron. We can think of this not as five electrons, but as one "hole" in an otherwise full shell. This hole behaves like a particle, but because it represents the absence of a negatively charged electron, it effectively has a positive charge.

This change in the sign of the charge carrier flips the sign of the spin-orbit interaction energy. Consequently, the energy ordering of the $J$ levels gets inverted.

Let’s see this in action. A boron atom has a $2p^1$ configuration—a single electron in a less-than-half-filled shell. Its ground term is $^2P$. Here, Hund's third rule predicts the lowest energy level has $J = |L-S| = |1 - 1/2| = 1/2$. The ground state is $^2P_{1/2}$. Now look at fluorine with its $2p^5$ configuration (one hole). The term is also $^2P$, but the shell is more than half-filled. The rule is reversed: the lowest energy level has $J = L+S = 1 + 1/2 = 3/2$. The ground state is $^2P_{3/2}$. This elegant symmetry between particles and holes is a powerful tool in physics. We can apply this logic to find the complete ground state for any atom, like carbon ($p^2$, less than half-filled, $^3P_0$) or nickel ($d^8$, more than half-filled, $^3F_4$).

Hund's rules, as we've described them, are based on a specific model of the atom called Russell-Saunders (LS) coupling​. This model assumes that the electrostatic repulsion between electrons (the physics behind rules 1 and 2) is much, much stronger than the magnetic spin-orbit interaction (the physics of rule 3). For lighter atoms like carbon, this is an excellent approximation.

However, as we move down the periodic table to heavier elements, the atom's nucleus becomes more highly charged. This forces the inner electrons to orbit at blistering speeds, approaching a fraction of the speed of light. As Einstein's theory of relativity tells us, strange things happen at these speeds. For our atom, the main consequence is that the spin-orbit interaction becomes dramatically stronger.

In very heavy atoms, like lead (Pb), the spin-orbit coupling can become as strong as the electrostatic repulsion between electrons. When this happens, the neat separation of concerns in LS coupling breaks down. The atom is better described by a different model known as jj-coupling, where each electron's spin and orbital momentum couple together first, before they all couple to each other.

We can see the experimental signature of this breakdown. In pure LS coupling, the energy gaps between the fine-structure levels of a term should obey a simple relationship called the Landé interval rule. For the $np^2$ configuration of Group 14 elements, this rule predicts the ratio of the energy gaps, $\mathcal{R} = (E(^3P_2) - E(^3P_1)) / (E(^3P_1) - E(^3P_0))$, should be $2$. For carbon, the experimental ratio is about $1.64$, already showing some deviation. For silicon, it's $1.9$. But by the time we get to lead, the experimental ratio is a mere $0.3621$! This dramatic departure from the ideal value is a clear sign that the simple, elegant picture of LS coupling is no longer the whole story. The rules haven't been broken, but we've pushed them to their limit and found the boundary where a deeper, more complex layer of reality begins to emerge. And that, of course, is where the next adventure in physics begins.

We have seen that Hund's rules are a tidy set of instructions for electrons arranging themselves within the "house" of an atom's orbitals—a sort of quantum mechanical building code. But to leave it there would be like learning the rules of chess and never watching a game. The real magic, the stunning checkmates and brilliant strategies, unfold when these simple rules play out on the grand stage of molecules, crystals, and even our own engineered quantum devices. We find that they are the secret architects behind some of nature's most profound and useful phenomena, from the invisible pull of a magnet to the vibrant color of a gemstone.

Let's begin with the most direct consequence of these rules. An atom with unpaired electrons behaves like an infinitesimal compass needle. Hund's first rule, the mandate to maximize spin, is essentially a recipe for creating these tiny magnets. Consider the manganese ion, $Mn^{2+}$, with its five $d$-electrons. Nature could pair them up, but Hund's rule insists on spreading them out, one in each of the five $d$-orbitals, all with their spins aligned. The result is a total spin of $S = 5/2$, giving the ion a robust magnetic moment. The same principle explains why a neutral chromium atom, with its peculiar $[Ar] 4s^1 3d^5$ configuration, is also strongly magnetic, boasting six unpaired electrons. This isn't just an atomic curiosity; it's the very basis of paramagnetism.

This principle extends beyond single atoms into the realm of chemistry. One of the classic puzzles of introductory chemistry is the oxygen molecule, $\text{O}_2$. A simple drawing of its Lewis structure shows all electrons neatly paired, suggesting it should be diamagnetic—indifferent to a magnetic field. Yet, if you pour liquid oxygen between the poles of a strong magnet, it hangs suspended, clearly attracted! The solution lies in a more sophisticated molecular orbital picture, where the two highest-energy electrons must occupy a pair of degenerate $\pi^*$ orbitals. Here, a molecular version of Hund's rule takes over. Rather than crowding into one orbital, the electrons occupy separate orbitals with parallel spins, leaving the $\text{O}_2$ molecule with two unpaired electrons and making it paramagnetic, just as experiment shows. It is a beautiful, textbook example of a simple quantum rule explaining a dramatic macroscopic effect.

But the rules dictate more than just magnetism; they shape how atoms dance with light. The arrangement of electrons determines not just the ground state but the entire ladder of possible excited states. This "energy ladder" is an atom's unique fingerprint, defining the colors of light it can absorb or emit. Take the Europium ion, $Eu^{3+}$. Applying all three of Hund's rules to its six $f$-electrons leads to a remarkable prediction: the total angular momentum in the ground state is zero, $J=0$. This means it has no net magnetic moment at low temperatures—a striking, non-intuitive result. However, this non-magnetic ion possesses a rich structure of excited states. When it absorbs energy, it can re-emit it as very pure, sharp lines of light. This property makes europium compounds a workhorse of technology, providing the critical red color in everything from early color television screens to modern fluorescent lighting. Its utility as a phosphor is a direct consequence of the specific energy-level structure that Hund's rules ordain.

Now, what happens when you bring a whole congress of these characterful atoms together, packing them into a solid crystal? They begin to talk to each other. The simple story of an isolated ion gives way to the complex, cooperative symphony of the solid state. Here, Hund's rules are still the first chapter, setting the magnetic character of each individual ion, but now new interactions between neighbors come into play, leading to collective order.

A fascinating drama often unfolds inside these materials—a tug-of-war between competing energies. In a crystal, surrounding ions create an electric field (the "crystal field") that splits the energies of the $d$-orbitals. An electron now has a choice: obey Hund's rule and maximize its spin by occupying a higher-energy orbital, or disobey it and pair up with another electron in a lower-energy orbital to save energy. When the crystal field splitting is weak, Hund's rule wins, and we get a "high-spin" state with many unpaired electrons. When the field is strong, the energy cost is too high, and the electrons pair up in a "low-spin" state. This choice between high-spin and low-spin has dramatic consequences for a material's magnetism, color, and reactivity. In some cases, like the $Co^{2+}$ ion in an octahedral field, the crystal environment doesn't completely erase the free-ion character. While the magnetic moment is dominated by the spin of its three unpaired electrons, a significant contribution from orbital motion survives, a subtle reminder that the atom's environment is just as important as its internal rules.

This interplay gives rise to the grand phenomena of collective magnetism. In a material like Manganese Oxide ($\text{MnO}$), the $Mn^{2+}$ ions are strongly magnetic, just as we discussed. Yet at low temperatures, the material as a whole has no net magnetic moment. The reason is that each ion's magnetic moment is aligned perfectly antiparallel to its neighbors. This antiferromagnetism arises from a quantum process called "superexchange." For an electron to "hop" between two manganese ions via the oxygen atom separating them, the path of least resistance—and thus lowest energy—is available only when the manganese core spins are opposed. The Pauli exclusion principle effectively creates a high-energy roadblock for the parallel-spin case. So, the system collectively chooses to be antiferromagnetic to lower its overall energy.

Conversely, in other materials, the same drive to lower energy causes all the spins to align parallel, creating a strong ferromagnet. In mixed-valence manganites (containing both $Mn^{3+}$ and $Mn^{4+}$ ions), an effect called "double exchange" occurs. It is energetically much easier for an electron to hop from an $Mn^{3+}$ to an $Mn^{4+}$ ion if their large "core spins" (set by Hund's first rule) are pointing in the same direction. By aligning ferromagnetically, the crystal opens up a conductive superhighway for electrons, lowering their kinetic energy. This intimate link between magnetism and electrical conductivity is the origin of "colossal magnetoresistance," where applying a magnetic field can dramatically change a material's resistance.

The story does not end with explaining the natural world. In the last few decades, we have learned to use these fundamental principles as a blueprint for designing new technologies. The rules that govern a europium atom in a rock also govern an "artificial atom," a tiny semiconductor structure called a quantum dot. When we inject a few electrons into a quantum dot, they arrange themselves in its quantized energy shells, faithfully obeying the same Aufbau principle and Hund's rules, forming spin-polarized states just as they would in nature's elements.

This predictive power is at the heart of quantum engineering. The nitrogen-vacancy (NV⁻) center in diamond, a leading candidate for building quantum computers, is a point defect whose utility stems from its electronic structure. A molecular-orbital analysis reveals that the ground state is a spin-triplet—it has a net spin of $S=1$—because the filling of the defect's orbitals follows a Hund's-like imperative to maximize spin. This stable, built-in magnetic moment can be manipulated with lasers and microwaves, serving as a robust quantum bit, or qubit. Even the design of modern phosphors and lasers relies on these rules. Efficient upconversion, where low-energy light is converted to high-energy light, is achieved by co-doping a material with two different rare-earth ions, like Ytterbium ($Yb^{3+}$) and Erbium ($Er^{3+}$). The process works because the de-excitation energy of an excited Yb³⁺ ion—a value we can calculate precisely from its spin-orbit split levels determined by Hund's rules—is beautifully resonant with an absorption energy of the $Er^{3+}$ ion, allowing for efficient energy transfer.

Even at the frontiers of condensed matter physics, these century-old rules are providing new insights. In a new class of materials dubbed "Hund's metals," the local exchange coupling, $J_H$, is so strong that it becomes the star of the show. It powerfully enforces a local high-spin configuration on each atom, and in doing so, it introduces strong correlations that hinder the movement of electrons, making them behave as if they have a much larger mass. This effect, driven by Hund's rule, is thought to be a key ingredient in understanding exotic phenomena like high-temperature superconductivity in iron pnictides. The physical origin of this crucial exchange energy, $K$ (or $J_H$ in solids), can be traced back to its molecular roots, visible even in the energy splitting between the singlet and triplet excited states of a simple molecule like formaldehyde.

From the magnetism of liquid oxygen to the design of a quantum computer, from the color of a television screen to the mysteries of superconductivity, we see the same simple set of rules at play. Hund's rules are far more than an atomic bookkeeping device; they are a deep and unifying principle, a thread of logic that connects the quantum mechanics of a single atom to the rich and complex tapestry of the world we see and build.