Orbital Energy Diagrams

In chemistry, understanding why atoms bond to form molecules with specific shapes and properties is a fundamental goal. While simple models offer a starting point, they fail to explain many critical phenomena, from the unique magnetism of oxygen to the precise geometry of water. This gap in understanding arises from a deeper question: what happens to electrons when they are no longer confined to individual atoms but belong to the molecule as a whole? This article delves into Molecular Orbital (MO) theory, a powerful framework that answers this question through the use of orbital energy diagrams. In the following chapters, you will first explore the foundational "Principles and Mechanisms" behind constructing these diagrams, learning the rules of symmetry, energy matching, and s-p mixing. Subsequently, the "Applications and Interdisciplinary Connections" chapter will demonstrate how these diagrams serve as a predictive tool, explaining everything from bond strength and molecular shape to the effects of light and relativity on chemical behavior.

To understand how a handful of atoms, governed by the strange laws of quantum mechanics, assemble themselves into the vast and complex world we see, we need a better picture than the simple "stick-and-ball" models of high school chemistry. We need to ask a deeper question: when atoms form a molecule, where do their electrons go? The answer is that they cease to belong to any single atom and instead enter a new set of states that span the entire molecule. These new states are called ​​molecular orbitals (MOs)​​, and the diagrams that map their energies are our key to unlocking the secrets of chemical bonding, structure, and reactivity.

Imagine two atoms approaching each other. Their electron clouds, which we call atomic orbitals (AOs), begin to overlap. Just as sound waves can interfere constructively (getting louder) or destructively (canceling out), these orbital "waves" can combine. This idea is captured in a powerful approximation called the ​​Linear Combination of Atomic Orbitals (LCAO)​​. When two atomic orbitals combine, they don't just merge; they split, creating two new molecular orbitals: a lower-energy ​​bonding orbital​​ where the electron density pulls the nuclei together, and a higher-energy ​​antibonding orbital​​ where a node between the nuclei pushes them apart.

But which atomic orbitals can participate in this dance? There are two fundamental rules.

First, the partners must have compatible ​​symmetry​​. Think of a hydrogen atom ($H$) approaching a fluorine atom ($F$) to make hydrogen fluoride ($HF$). Let's align them along the z-axis. The spherical $1s$ orbital of hydrogen has what we call $\sigma$ symmetry—it looks the same from all directions around the bond axis. Fluorine's $2p_z$ orbital also has $\sigma$ symmetry. These two can interact. However, fluorine's $2p_x$ and $2p_y$ orbitals have $\pi$ symmetry; they have a node along the bond axis. The hydrogen $1s$ orbital sits right in this node, meaning there is zero net overlap. They are symmetry-mismatched. As a result, the $2p_x$ and $2p_y$ orbitals of fluorine find no dance partners on the hydrogen atom. They enter the molecule unchanged in energy, forming what we call ​​non-bonding molecular orbitals​​. These are essentially the MO theory equivalent of lone pairs.

The second rule is that the interacting atomic orbitals must have similar ​​energy​​. The closer they are in energy, the stronger the interaction and the more pronounced the splitting into bonding and antibonding levels. This brings up a crucial starting point: why are the starting atomic orbitals at the energies they are? Why, for instance, are the valence orbitals of fluorine so much lower in energy than the $1s$ orbital of hydrogen? It's not because of some vague "desire" to be like a noble gas. The reason is the raw, unshielded power of the nucleus. A fluorine nucleus has a charge of $+9$, while a hydrogen nucleus is just $+1$. Even accounting for the shielding from inner electrons, the valence electrons in fluorine feel a much stronger pull towards the nucleus—a higher ​​effective nuclear charge ($Z_{\text{eff}}$)​​. This powerful electrostatic attraction is what makes fluorine so electronegative and dramatically lowers the energy of its atomic orbitals. Knowing these starting energies is the first step in constructing any meaningful orbital energy diagram.

With these rules, we can start building diagrams for simple molecules. Let's take the homonuclear diatomics of the second period. A first pass might suggest a single, universal energy ordering for all of them: the $\sigma$ orbitals from the $2s$ AOs, followed by the orbitals from the $2p$ AOs ($\sigma_{2p}$ and $\pi_{2p}$), and then their antibonding counterparts. Head-on overlap ($\sigma$) is stronger than side-on overlap ($\pi$), so naively we'd expect the $\sigma_{2p}$ bonding orbital to be lower in energy than the $\pi_{2p}$ orbitals.

This simple picture works beautifully for $O_2$ and $F_2$. But when we examine $B_2$, $C_2$, and $N_2$, experiment tells us this ordering is wrong. Nature has a beautiful subtlety we missed: the molecular orbitals themselves can interact!

Just as atomic orbitals can combine, molecular orbitals of the same symmetry can also mix. In these diatomic molecules, both the $\sigma_{2s}$ and $\sigma_{2p}$ bonding orbitals have the same symmetry type ($\sigma_g$). This allows them to interact in a process called ​​s-p mixing​​. This interaction causes the two orbitals to "repel" each other in energy: the lower-energy $\sigma_{2s}$ is pushed down, and more importantly, the higher-energy $\sigma_{2p}$ is pushed up.

Why is this effect pronounced in $N_2$ but negligible in $F_2$? It comes back to our second rule: the strength of interaction depends on energy similarity. As we move across the second period from boron to fluorine, the effective nuclear charge increases. This pulls the spherical $2s$ orbital, which penetrates closer to the nucleus, down in energy much more rapidly than it pulls down the $2p$ orbitals. For fluorine, the energy gap between the $2s$ and $2p$ AOs is huge. Consequently, the resulting $\sigma_{2s}$ and $\sigma_{2p}$ MOs are also far apart in energy and mix very little. The "naive" ordering holds. But for nitrogen, the initial $2s-2p$ gap is much smaller. This allows for strong s-p mixing, which pushes the $\sigma_{2p}$ orbital's energy up so much that it rises above the $\pi_{2p}$ orbitals. This isn't an arbitrary rule to be memorized; it's a direct and elegant consequence of the interplay between nuclear charge and orbital symmetry.

Now we see that building these diagrams requires some care. But the reward is immense. Once we have the correct energy level scheme and fill it with the molecule's valence electrons (following the Aufbau principle and Hund's rule), we can predict real, measurable properties with stunning accuracy.

First, we can quantify the strength of a chemical bond using the ​​bond order​​, defined as:

where $N_{\text{bonding}}$ and $N_{\text{antibonding}}$ are the number of electrons in bonding and antibonding orbitals, respectively. For the cyanide ion, $\text{CN}^-$, with 10 valence electrons, the MO diagram (which resembles that of its isoelectronic cousin $N_2$) fills up to the $\sigma_{2p}$ orbital, giving 8 bonding and 2 antibonding electrons. Its bond order is $\frac{1}{2}(8-2) = 3$, correctly predicting its strong triple bond. Conversely, for the hypothetical molecule $\text{Be}_2$, with 4 valence electrons, we would place two in the $\sigma_{2s}$ bonding orbital and two in the $\sigma^*_{2s}$ antibonding orbital. The bond order is $\frac{1}{2}(2-2) = 0$. The theory predicts there is no net bond, which is why diatomic beryllium is not a stable molecule under ordinary conditions.

Perhaps the most dramatic success of MO theory is its prediction of magnetism. A molecule is ​​paramagnetic​​ if it has unpaired electrons and will be drawn into a magnetic field. It is ​​diamagnetic​​ if all its electrons are paired and it is weakly repelled by a magnetic field. Consider the $B_2$ molecule. A simple Lewis structure would give it a double bond, with all electrons paired. But MO theory, accounting for s-p mixing, places the two highest-energy electrons into the degenerate $\pi_{2p}$ orbitals. According to ​​Hund's rule​​, which tells us to maximize electron spin, these two electrons will occupy separate orbitals with parallel spins. Our theory thus makes a bold prediction: $B_2$ is paramagnetic. And experiment confirms it! The same logic triumphantly explains the well-known paramagnetism of everyday oxygen, $O_2$. For $O_2$, we use the "unmixed" diagram, and with 12 valence electrons, the last two must be placed, unpaired, into the degenerate $\pi^*_{2p}$ antibonding orbitals. These correct predictions, which were failures for simpler bonding theories, are what cemented MO theory as a cornerstone of modern chemistry.

Our diagrams have explained bonding and magnetism. But what about molecular shape? Why is $BeH_2$ linear, while water, $H_2O$, is bent? Once again, MO theory provides a beautifully intuitive answer. Instead of a static diagram, we can create a ​​Walsh diagram​​, which plots the energy of each molecular orbital as we change a geometric parameter, like the bond angle.

Let's imagine taking a generic linear $AH_2$ molecule and slowly bending it. How do the orbital energies change? The lower, main bonding orbitals slightly increase in energy because the orbital overlap that forms the bonds is maximized at $180^\circ$. But something magical happens to one of the central atom's $p$ orbitals. In the linear geometry, the $p$ orbital that lies in the plane of the future bend is non-bonding with respect to the hydrogens. But as the molecule bends, the hydrogen atoms move off that orbital's nodal plane, allowing for a new, favorable, in-phase bonding interaction to develop. This causes the energy of this specific MO (labeled $3a_1$) to plummet dramatically as the molecule bends.

Now, the preferred geometry is simply the one that gives the lowest total energy for all the electrons.

• For $BeH_2$, with only 4 valence electrons, those electrons occupy the orbitals that get destabilized by bending. The newly stabilized $3a_1$ orbital is empty. To keep its energy low, the molecule stays linear.
• For a molecule with 6 or 8 valence electrons, like $\text{BH}_2^-$ or $H_2O$, there are electrons available to occupy this rapidly stabilizing $3a_1$ orbital. The energy payoff from dropping two electrons into this orbital is so great that it easily overcomes the slight destabilization of the lower orbitals. The molecule bends to achieve a lower energy state. In this way, Walsh diagrams derive the familiar VSEPR rules from deeper, orbital-based principles.

We have seen how rules of symmetry and energy govern the construction and interpretation of our diagrams. But where do these rules themselves come from? To see the full picture, we must zoom out and watch the entire process of bond formation, from two infinitely separated atoms to a single, unified molecule. This journey is traced by a ​​correlation diagram​​, which plots electronic state energies against the internuclear distance, $R$.

These diagrams are governed by one of the most profound principles in quantum chemistry: the ​​Wigner-von Neumann non-crossing rule​​. It states that two potential energy curves—two electronic states—of the exact same symmetry cannot cross as we vary a parameter like $R$. When they approach each other, they seem to "repel," leading to an "avoided crossing." States of different symmetry, however, are free to cross.

This single rule is the deep origin of everything we have discussed. The s-p mixing that inverts the orbital order in $N_2$ is nothing more than an avoided crossing between two curves of the same $\sigma_g$ symmetry. The reason HF's $\pi$ orbitals are non-bonding is because they have a different symmetry from any available orbital on hydrogen, allowing their energy curves to proceed independently. The Walsh diagram's predictions are a map of how orbital symmetries change with angle, dictating which crossings are avoided and which are allowed. The correlation diagram shows that the simple MO energy level scheme we draw at a fixed bond length is just one vertical slice of a much richer, more dynamic landscape, one that connects the properties of isolated atoms to the final, intricate reality of the molecule. It is a beautiful testament to the unifying power of symmetry in the quantum world.

Now that we have painstakingly learned the rules of the game—how to construct these curious diagrams of energy levels for molecules—the real fun begins. What good are they? You might be tempted to think this is just an elaborate accounting scheme, a set of abstract rules to satisfy an exam. But nothing could be further from the truth. These diagrams are not just bookkeeping; they are a window into the very soul of a molecule. They are a kind of crystal ball that, once you learn how to read it, allows you to predict a molecule’s personality: its strength, its shape, its color, how it will react when you shine light on it, and even how it behaves under conditions so extreme they bend the laws of physics themselves. Let us now take a journey through the vast and beautiful landscape of phenomena that these simple diagrams illuminate.

At its heart, chemistry is about the dance of bonds being formed and broken. Our molecular orbital (MO) diagrams give us a front-row seat to this performance by providing a single, powerful number: the bond order. It’s a simple calculation—half the difference between the number of electrons in bonding "glue" orbitals and antibonding "anti-glue" orbitals—but its consequences are profound.

Consider a few familiar faces from the air we breathe and its neighboring elements: dinitrogen ($N_2$), dioxygen ($O_2$), and difluorine ($F_2$). Anyone who has worked in a chemistry lab knows that nitrogen gas is remarkably inert; its triple bond is one of the strongest known in chemistry. Fluorine gas, by contrast, is ferociously reactive, tearing other molecules apart. Oxygen is the "Goldilocks" molecule, reactive enough to sustain life but not so crazed as fluorine. Why? The MO diagrams tell the story perfectly. Filling the orbitals for $N_2$ gives a bond order of 3. For $O_2$, we add two more electrons, which must go into antibonding orbitals, reducing the bond order to 2. For $F_2$, two more electrons fill the antibonding levels further, dropping the bond order to a mere 1. The calculated bond orders of 3, 2, and 1 line up precisely with the experimentally measured bond dissociation energies, which is the energy you need to supply to break the molecules apart. The stronger the bond order, the more stable the molecule. The diagram doesn’t just get the trend right; it explains why the trend exists—it all comes down to how many electrons are holding things together versus pulling them apart.

This predictive power becomes even more striking when we start modifying molecules. What happens if you pluck an electron out of a molecule? Common sense might suggest that removing a piece of the electronic "glue" should always weaken a bond. But nature is more clever than that. Consider nitric oxide ($\text{NO}$) and its cation, $\text{NO}^+$. Our MO diagram for $\text{NO}$ shows that the highest-energy electron, the last one we put in, resides in an antibonding orbital. So, when we ionize the molecule to form $\text{NO}^+$, we are removing a troublemaker! By taking away an electron that was actively working to weaken the bond, the overall bond order actually increases from 2.5 to 3. The theory predicts, therefore, that the bond in $\text{NO}^+$ should be stronger and shorter than in $\text{NO}$. And it is. This is a beautiful example of the non-intuitive, yet perfectly logical, insights that MO theory provides.

The model’s subtlety shines when comparing molecules with the same number of electrons, known as isoelectronic species. Carbon monoxide ($CO$) and dinitrogen ($N_2$) both have 14 electrons. Yet $N_2$ is famously inert, while $CO$ is a reactive ligand that binds strongly to metals like the iron in your hemoglobin (with deadly consequences). Why the difference? In the symmetric $N_2$ molecule, the atomic orbitals from each nitrogen atom are at the same energy. In $CO$, the oxygen atom is more electronegative than carbon, meaning its atomic orbitals start at a lower energy. This energy mismatch skews the resulting molecular orbitals. The Highest Occupied Molecular Orbital (HOMO) of $CO$ ends up being higher in energy and more localized on the carbon atom than the HOMO of $N_2$. A higher-energy HOMO means the electrons are more "eager" to react, explaining why CO is a much better electron donor in chemical reactions than $N_2$. The diagram makes this invisible difference in electronic character plain to see.

You might still be skeptical. These diagrams are elegant, sure, but are they real? Can we actually "see" these energy levels? The answer, wonderfully, is yes. An experimental technique called Photoelectron Spectroscopy (PES) is like a tiny, precise hammer we can use to knock electrons out of a molecule one by one. By measuring the energy required to eject an electron from each orbital, we can map out the molecule's energy level structure. The energy needed to ionize an electron from a specific orbital is, to a good approximation, simply the negative of the orbital's energy—an idea known as Koopmans' theorem.

So, when a PES experiment is performed on a molecule like hydrogen sulfide ($H_2S$), it yields a series of distinct signals. Each signal corresponds to the ionization from a different molecular orbital. The highest energy signal corresponds to ejecting an electron from the most stable, lowest-energy MO. The lowest energy signal comes from ejecting an electron from the least stable, highest-energy MO (the HOMO). By matching the experimental ionization energies to the energies predicted by our diagrams, we find stunning agreement. We can experimentally confirm, for instance, that the non-bonding $1b_1$ orbital in $H_2S$ is indeed its HOMO, requiring the least energy (about 10.5 eV) to ionize. This technique also beautifully resolves a puzzle: the first ionization energy of $N_2$ is higher than that of $O_2$. This seems backward, as oxygen atoms hold their electrons more tightly than nitrogen atoms do. The MO diagrams provide the answer: the HOMO of $N_2$ is a strongly bonding $\sigma$ orbital, low in energy. The HOMO of $O_2$, however, is a higher-energy, antibonding $\pi^*$ orbital. It is simply easier to remove an electron from this less stable, antibonding orbital, and so the ionization energy of $O_2$ is lower.

Perhaps the most dramatic application of orbital energy diagrams is in predicting the three-dimensional shape of a molecule. To do this, we use a special kind of diagram called a Walsh diagram. Instead of being static, a Walsh diagram is a sort of "movie" that shows how the energy of each molecular orbital changes as we bend the molecule.

Let's start with a simple triatomic molecule. Is it linear or is it bent? The answer depends on its electrons. Consider the radical anion of carbon dioxide, $\text{CO}_2^-$. The neutral $CO_2$ molecule is linear. When we add an extra electron to make the anion, where does it go? It goes into the Lowest Unoccupied Molecular Orbital (LUMO) of $CO_2$. The Walsh diagram for $CO_2$ shows something remarkable: as the molecule bends, one of the components of this LUMO is dramatically stabilized—its energy plummets. Since the single added electron occupies this orbital, the entire molecule can lower its total energy by bending. And so, $\text{CO}_2^-$ is bent!. The same principle explains why a molecule like beryllium dihydride ($BeH_2$), with four valence electrons, is linear, but adding just one more electron to make $\text{BeH}_2^-$ forces it to become bent.

This reasoning gives us one of the most powerful explanations in chemistry, showing why the simple VSEPR rules taught in introductory classes work. Consider ammonia ($NH_3$) and borane ($BH_3$). VSEPR tells us $BH_3$ is planar and $NH_3$ is pyramidal, but it doesn't give a deep energetic reason. A Walsh diagram does. For an $AH_3$ molecule, pyramidalization strongly stabilizes one key orbital—the orbital that becomes the lone pair. In ammonia, which has eight valence electrons, this orbital is occupied by two electrons. Their powerful stabilization upon bending is the driving force that contorts the molecule into its characteristic pyramidal shape. In borane, which only has six valence electrons, this crucial orbital is empty. There is no energetic gain to be had from bending; in fact, other occupied orbitals are slightly destabilized. The molecule therefore finds its lowest energy state in the flat, planar geometry. VSEPR gives us a rule; MO theory gives us the reason.

The reach of MO theory extends far beyond these foundational concepts, deep into the most advanced and fascinating areas of modern science. It turns out that for very heavy elements, like gold and platinum, the electrons move so fast as they orbit the massive nucleus that the effects of Einstein's theory of relativity become important. These relativistic effects cause the $s$ orbitals to contract and become much more stable (lower in energy), while the $d$ orbitals expand and become less stable.

This has bizarre and beautiful consequences. The famous color of gold is a direct result of this relativistic shifting of orbital energies. It also explains the surprisingly strong bond in the gold dimer, $Au_2$. In lighter atoms like silver, the valence $s$ and $d$ orbitals are far apart in energy and don't interact much. But in gold, relativity pushes the $6s$ and $5d$ orbitals so close in energy that they mix extensively when forming the molecular orbitals of $Au_2$. This strong $s$-$d$ mixing creates a very stable, low-energy sigma bonding orbital, leading to a much stronger bond than would be expected otherwise. Our MO diagram, when adjusted for relativity, correctly predicts a stable, closed-shell molecule, explaining the unexpected tenacity of the $Au_2$ bond. Even the complex bonding in the octahedral complexes of inorganic chemistry can be understood by applying these same principles of symmetry and energy matching, and we can even use them to explore hypothetical scenarios, like what would happen if relativistic effects became even more extreme.

Finally, what happens when a molecule absorbs a particle of light? It enters an excited electronic state. And what happens next—does it fluoresce, simply fall apart, or twist into a new shape? The fate of the molecule is governed by the intricate landscape of its potential energy surfaces—maps that are themselves built from molecular orbital considerations.

Sometimes, the energy surfaces of two different electronic states—say, a stable, high-energy "Rydberg" state and an unstable, repulsive "valence" state—approach each other as the molecule twists or bends. Quantum mechanics forbids states of the same symmetry from crossing. Instead, they "avoid" each other, mixing their characters in the process. A molecule excited to the stable surface can suddenly find itself at one of these "avoided crossings" and get shunted onto the repulsive surface, causing the molecule to fly apart in a process called predissociation. These avoided crossings, which are a direct consequence of how orbital energies change with geometry (the very essence of a Walsh diagram), act as switches on the railroad tracks of chemical reactions, redirecting molecules down pathways of dissociation and transformation that would otherwise be inaccessible.

From the simple strength of a nitrogen molecule to the golden gleam of a precious metal and the fleeting dance of a molecule shattering under light, the humble orbital energy diagram is our guide. It is a testament to the power of a good idea—a simple, elegant physical model that unifies a vast and complex universe of chemical phenomena. It is, in short, one of the most beautiful and useful tools we have for understanding the world.