9.5 Molecular Orbital Theory
PRE-HEALTH PREP
Molecular Orbital Theory (MO Theory) Simplified
General Chemistry 9.5
Lesson Overview
Molecular Orbital (MO) Theory describes chemical bonding by combining atomic wave functions into bonding and antibonding molecular orbitals through constructive and destructive interference. MO diagrams organize these orbitals by energy, enabling calculation of bond order and prediction of molecular stability — including the 2 distinct energy orderings that arise from sp mixing in period-two homonuclear diatomics. The theory's most notable success is its correct prediction of paramagnetism in O₂, a result that Lewis structures cannot reproduce.
Table of Contents
Why MO Theory?
Molecular Orbital (MO) Theory is the most accurate description of chemical bonding you will encounter in a general chemistry course. That is also what makes it the most demanding — and fair warning, it earns its reputation as a stumbling block for many students. Reality, it turns out, is more complicated than Lewis structures would have us believe.
Both Lewis structures and valence bond theory are enormously useful, but neither tells the complete story. MO Theory outperforms them in 2 key areas that you will explore in this lesson:
- It explains why H₂ exists as a stable diatomic molecule while helium does not form He₂.
- It correctly predicts that O₂ is paramagnetic — attracted to a magnetic field — a fact that the Lewis structure gets completely wrong.
Everything in chemistry has its utility and its place. Lewis structures remain indispensable for quickly mapping connectivity and formal charges. But when you need the best available description of molecular electronic structure, MO Theory delivers. The goal of this lesson is to make that theory genuinely understandable.
Wave Functions and Orbital Overlap
Atomic Orbitals as Wave Functions
To understand MO Theory, you first need to appreciate what atomic orbitals actually are. Every atomic orbital is a solution to the Schrödinger equation — a three-dimensional mathematical function, or wave function, that describes the probability distribution of an electron in space.
In valence bond theory, when two hydrogen atoms approach each other, their 1s orbitals overlap and the electrons are said to occupy that shared region. That picture captures something real, but it glosses over an important mathematical detail: wave functions can be combined in more than one way. MO Theory accounts for all the ways those combinations can occur.
Nodes and the Sine Function Analogy
Before tackling three-dimensional orbitals, consider a familiar two-dimensional wave: the sine function. From 0 to π (0° to 180°), the sine function takes on positive values. From π to 2π (180° to 360°), it takes on negative values. At every integer multiple of π — exactly where the function crosses from positive to negative — it equals zero. In wave function language, a point where the function equals zero is called a node.
For an atomic orbital, a node means there is zero probability of finding the electron at that location. The 1s orbital has no nodes. Every p orbital, by contrast, has a node at the nucleus, which is why an electron in a p orbital is never found precisely at the nucleus. This concept carries directly into the molecular orbitals built from them.
Sign vs. Charge: The positive and negative regions of an atomic orbital represent the mathematical sign of the wave function, not electric charge. In diagrams, these regions are often distinguished by shading rather than by + and − symbols to avoid confusion.
Constructive and Destructive Interference
When two wave functions overlap, the outcome depends on whether the overlapping regions share the same mathematical sign or opposite signs — that is, whether the waves are in phase or out of phase.
- In-phase overlap (constructive interference): Positive overlaps with positive (and negative with negative). The amplitudes add, producing a larger, amplified combined function. The electron density in the overlap region increases.
- Out-of-phase overlap (destructive interference): Positive overlaps with negative. The amplitudes cancel, producing a node — a region of zero electron density — where the two waves meet.
These 2 outcomes always occur simultaneously whenever atomic orbitals combine. You cannot have one without the other. For every constructive combination that produces a lower-energy bonding molecular orbital, there is a corresponding destructive combination that produces a higher-energy antibonding molecular orbital.
Conservation of Orbital Count: The total number of orbitals is always conserved. 2 atomic orbitals combine to give exactly 2 molecular orbitals — 1 bonding and 1 antibonding. This is a universal rule in MO Theory.
Sigma Orbitals from s-Orbital Overlap
The Sigma 1s Bonding Molecular Orbital
When two hydrogen atoms approach each other, their 1s orbitals overlap end-to-end along the internuclear axis. If both orbitals have the same sign (both positive, say), the overlapping region undergoes constructive interference: the electron density between the two nuclei is amplified, creating a region of high electron probability directly between two positively charged nuclei. This is energetically favorable — the electrons are stabilized by simultaneously attracting both nuclei.
2 additional factors lower the energy of this molecular orbital relative to the original 1s atomic orbitals:
- The electron density accumulates between the nuclei, maximizing attractive interactions with both.
- The resulting molecular orbital is larger in volume than either parent atomic orbital. By the Heisenberg Uncertainty Principle, a larger volume means lower kinetic energy.
This end-to-end overlap produces a sigma (σ) bond, and the resulting orbital is called the σ1s bonding molecular orbital.
The Sigma 1s Star Antibonding Molecular Orbital
The simultaneous destructive combination — when one 1s orbital is treated as having its negative phase — produces the opposite result. Destructive interference creates a node directly between the two nuclei. This is precisely the region where electrons would be most stabilized by both nuclei, and the node makes it forbidden territory. As a consequence:
- The electrons are excluded from the inter-nuclear region, raising the energy.
- The effective volume of the orbital is reduced, which by the Heisenberg Uncertainty Principle also raises the energy.
This higher-energy orbital is called the σ1s* antibonding molecular orbital (pronounced "sigma-one-s-star"). Placing electrons in an antibonding orbital destabilizes the molecule relative to separate atoms.
Example: Bonding vs. Antibonding at a Glance
σ1s (bonding): Constructive overlap between the nuclei → high electron density between nuclei → lower energy than the original 1s orbitals. Holding atoms together is favorable.
σ1s* (antibonding): Destructive overlap creates a node between the nuclei → zero electron density in the inter-nuclear region → higher energy than the original 1s orbitals. Holding atoms together in this orbital is unfavorable.
Sigma and Pi Orbitals from p-Orbital Overlap
The pz Orbitals — Sigma Overlap
For period-two diatomics, we must also consider the 2p orbitals. By convention, the pz orbital lies along the internuclear (bonding) axis. When two atoms approach, the pz orbitals overlap end-to-end — the same geometry as s-orbital overlap — so this interaction is also classified as sigma overlap.
Recall that each p orbital has a node at the nucleus, with opposite signs on either lobe. When the pz orbitals approach with matching signs in their overlapping lobes (in-phase), constructive interference occurs. No new nodes are introduced between the nuclei, and the resulting orbital is the σ2p bonding molecular orbital. When the lobes meet with opposite signs (out-of-phase), a new node appears directly between the nuclei, yielding the higher-energy σ2p* antibonding molecular orbital.
The px and py Orbitals — Pi Overlap
The px and py orbitals are perpendicular to the internuclear axis. When the atoms approach, these orbitals overlap side-to-side rather than end-to-end. This geometry is called pi overlap, and it is inherently weaker than sigma overlap because the sideways alignment produces less electron-density build-up between the nuclei.
- In-phase side-to-side overlap → constructive interference above and below the internuclear axis → π2p bonding molecular orbital. The node that exists along the internuclear axis was already present in the original p orbitals; no new node is created.
- Out-of-phase side-to-side overlap → destructive interference creates a new node between the nuclei → π2p* antibonding molecular orbital.
Because both px and py undergo this side-to-side pi overlap, each produces its own bonding/antibonding pair. The result is 2 degenerate (equal-energy) π2p bonding orbitals and 2 degenerate π2p* antibonding orbitals.
1 Key Rule for Identifying Bonding vs. Antibonding: A bonding molecular orbital has no new nodes introduced between the nuclei. An antibonding molecular orbital always has a new node created between the nuclei. Look for that new node — it's the defining feature of an antibonding orbital and the reason it destabilizes the molecule.
How to Read MO Diagrams
The Structure of an MO Diagram
An MO diagram (molecular orbital diagram) is an energy-level diagram that shows both the "before" and "after" pictures of bonding in a single schematic. The 2 outer columns represent the atomic orbitals of the individual atoms before bonding; the central column represents the molecular orbitals that result from their combination. Energy increases upward.
Electrons are filled into molecular orbitals using the same 3 rules as atomic electron configurations:
- Aufbau Principle: Fill from lowest energy to highest.
- Pauli Exclusion Principle: Each orbital holds a maximum of 2 electrons with opposite spins.
- Hund's Rule: Among degenerate orbitals, place 1 electron in each before pairing.
How to Calculate Bond Order
Once the MO diagram is filled, bond order quantifies the net bonding: it is the formal equivalent of the bond multiplicity you assign in a Lewis structure, but derived rigorously from the electronic structure.
Bond Order = (Bonding Electrons − Antibonding Electrons) / 2- Bond order = 1 → single bond
- Bond order = 2 → double bond
- Bond order = 3 → triple bond
- Bond order = 0 → no bond; the diatomic does not exist
- Fractional bond orders (e.g., 1/2) → weakly bonded species that may exist under special conditions
Example: H₂ — Bond Order 1
Each hydrogen atom contributes 1 electron for a total of 2. Both electrons fill the lowest-energy σ1s bonding orbital. The σ1s* antibonding orbital remains empty.
Bond order = (2 − 0) / 2 = 1
The after-picture is lower in energy than the before-picture, confirming that forming H₂ is favorable. This is exactly why hydrogen exists as a diatomic: the molecule is more stable than 2 separate hydrogen atoms.
Molecular Orbital Diagram for H2
H2 has a bond order of 1 — confirming the single covalent bond predicted by the Lewis structure.
Bond Order = (2 - 0)/2 = 1
Example: He₂ — Bond Order 0 (Does Not Exist)
Each helium atom contributes 2 electrons for a total of 4. The first 2 fill the σ1s bonding orbital; the second 2 must fill the σ1s* antibonding orbital.
Bond order = (2 − 2) / 2 = 0
The 2 bonding electrons are exactly canceled by the 2 antibonding electrons. There is no net stabilization, so He₂ has no reason to exist — and it doesn't.
Molecular Orbital Diagram for He2
He2 has a bond order of 0, meaning no net bond forms. This is why helium exists only as isolated atoms, never as a diatomic molecule.
Bond Order = (2 - 2)/2 = 0
Example: He₂⁺ — Bond Order 1/2
Removing 1 electron from He₂ leaves 3 electrons total. The σ1s bonding orbital holds 2; the σ1s* antibonding orbital holds 1.
Bond order = (2 − 1) / 2 = 1/2
A bond order greater than zero means net stabilization exists. He₂⁺ has no Lewis structure analog, but MO Theory predicts it can exist — and spectroscopic evidence confirms it does, albeit transiently. This is something Lewis structures simply cannot capture.
Period-Two Homonuclear Diatomics
Homonuclear diatomic means both atoms in the molecule are identical: O₂, N₂, F₂, and so on. Period-two homonuclear diatomics involve 1s, 2s, and 2p atomic orbitals, which overlap to produce a full ladder of molecular orbitals.
2 Different MO Diagram Patterns
Period-two homonuclear diatomics do not all use the same MO energy ordering. There are 2 distinct patterns, and which one applies depends on where the element falls in the period:
| MO Diagram Pattern | Elements | σ2p vs. π2p Energy Order |
|---|---|---|
| Standard (right side) | O, F, Ne | σ2p is lower in energy than π2p |
| sp-Mixing (left side) | Li, Be, B, C, N | σ2p is higher in energy than π2p |
Why the Diagram Switches: sp Mixing Explained
For elements to the left of oxygen in period two (Li through N), the 2s and 2p orbitals are closer in energy than they are for O, F, and Ne. MO Theory requires that overlapping orbitals be similar in energy to interact meaningfully. In oxygen, fluorine, and neon, the energy gap between 2s and 2pz is large enough that they interact only with their matching counterparts across the bond. But in the lighter elements, the gap is small enough that the 2s orbital of one atom can overlap with the 2pz orbital of the other — a phenomenon called sp mixing.
This mixing effectively raises the energy of the σ2p molecular orbital, pushing it above the π2p orbitals. The result is the switched energy ordering seen in the left-side pattern. Once you recognize which elements belong to which pattern, filling in the MO diagram follows the same rules as before.
Core Electrons in Period-Two Diagrams: The 1s electrons of period-two atoms are core electrons. When both σ1s and σ1s* are completely filled, they contribute zero to the bond order (equal bonding and antibonding electrons cancel). Many textbooks and instructors omit these core levels from the diagram entirely and focus only on the valence MOs — a completely valid shortcut. Confirm with your professor which convention your course uses.
Molecular Orbital Filling Diagram for Li2, Be2, B2, C2, and N2
The molecular orbital filling diagram for Li2 , Be2, B2, C2, and N2 is different than that of O2, F2, and Ne2 due to sp mixing. The energy of the 2s and 2pz orbitals are much closer in energy for Li, Be, B, C, and N than for O, F, and Ne. This results in significant overlap of these orbitals that contributes to the molecular orbital picture. The ultimate result is the inversion of the σ2p and π2p molecular orbitals in the MO diagram.
Worked Examples — Valence MOs Only
Molecular Orbital Filling Diagram for O2, F2, and Ne2
The molecular orbital filling diagram for O2 , F2, and Ne2 is different than that of the rest of the elements in the 2nd period due to a lack of sp mixing.
Example: O₂ — Bond Order 2, Paramagnetic
Oxygen: 1s² 2s² 2p⁴. Using the standard (right-side) diagram and counting valence electrons only:
- Four 2s electrons → fill σ2s (2 e⁻) and σ2s* (2 e⁻): net contribution = 0
- Eight 2p electrons → fill σ2p (2 e⁻), π2p (4 e⁻, two degenerate orbitals each taking 2), π2p* (2 e⁻ — one in each degenerate orbital by Hund's Rule)
Bonding electrons from 2p: σ2p (2) + π2p (4) = 6. Antibonding electrons from 2p: σ2s* (2) + π2p* (2) = 4 total. But counting only 2p region: 6 bonding, 2 antibonding.
Overall valence bond order = (8 bonding − 4 antibonding) / 2 = 2 → double bond. Consistent with the Lewis structure.
Two electrons occupy the π2p* orbitals, one in each — both unpaired. O₂ is therefore paramagnetic.
Molecular Orbital Diagram for O2
O2 has a bond order of 2 (a double bond) and — crucially — two unpaired electrons, making it paramagnetic. This is MO theory's signature success: Lewis structures cannot explain why oxygen is attracted to a magnet.
Bond Order = (8 - 4)/2 = 2
Example: F₂ — Bond Order 1, Diamagnetic
Fluorine: 1s² 2s² 2p⁵. Compared to O₂, each fluorine atom contributes 1 additional 2p electron — 2 more electrons total. These fill the remaining π2p* orbital, completing that pair.
Bond order = (8 − 6) / 2 = 1 → single bond. Consistent with the Lewis structure. All electrons are now paired → F₂ is diamagnetic.
Molecular Orbital Diagram for F2
F2 has a bond order of 1, consistent with its Lewis structure single bond. All electrons are paired, so fluorine gas is diamagnetic.
Bond Order = (8 - 6)/2 = 1
Example: Ne₂ — Bond Order 0, Does Not Exist
Neon: 1s² 2s² 2p⁶. Every valence molecular orbital is completely filled — bonding and antibonding alike.
Bond order = (8 − 8) / 2 = 0. No net bonding. Ne₂ does not exist, for exactly the same reason He₂ does not exist.
Molecular Orbital Diagram for Ne2
Ne2 has a bond order of 0 — bonding and antibonding electrons cancel completely. Like He2, noble gas diatomics do not form.
Bond Order = (8 - 8)/2 = 0
Example: N₂ — Bond Order 3, Diamagnetic
Nitrogen: 1s² 2s² 2p³. N₂ uses the sp-mixing (left-side) diagram, where the π2p orbitals are lower in energy than σ2p. Counting valence electrons from 2p (6 total):
- First 4 electrons fill the 2 degenerate π2p bonding orbitals (2 in each).
- Last 2 electrons fill the σ2p bonding orbital.
No antibonding 2p electrons are occupied. Bond order (valence only) = (8 bonding − 2 antibonding from 2s region) / 2 = 3 → triple bond. Consistent with the Lewis structure. All electrons are paired → N₂ is diamagnetic.
N₂ makes up roughly 78% of the atmosphere you are breathing right now — a triple bond makes it extraordinarily stable and chemically inert under ordinary conditions.
Molecular Orbital Diagram for N2
N2 has a bond order of 3 — the highest of the period-two diatomics. All electrons are paired (diamagnetic), and the triple bond makes nitrogen gas remarkably stable and chemically inert.
Bond Order = (8 - 2)/2 = 3
Period-Two Diatomics Summary Table:
| Molecule | Bond Order | Bond Type | Magnetic Behavior | Exists? |
|---|---|---|---|---|
| Li₂ | 1 | Single | Diamagnetic | Yes |
| Be₂ | 0 | None | — | No |
| B₂ | 1 | Single | Paramagnetic | Yes |
| C₂ | 2 | Double | Diamagnetic | Yes |
| N₂ | 3 | Triple | Diamagnetic | Yes |
| O₂ | 2 | Double | Paramagnetic | Yes |
| F₂ | 1 | Single | Diamagnetic | Yes |
| Ne₂ | 0 | None | — | No |
Paramagnetism vs. Diamagnetism in MO Theory
One of the most striking triumphs of MO Theory is its correct prediction of the magnetic behavior of molecules — something Lewis structures cannot do. The distinction between paramagnetic and diamagnetic substances comes down entirely to unpaired electrons.
- Paramagnetic: The substance has 1 or more unpaired electrons. Unpaired electron spins do not cancel, generating a net magnetic moment. The substance is attracted toward a magnetic field.
- Diamagnetic: All electrons are paired. Opposite spins cancel, leaving no net magnetic moment. The substance experiences a very weak repulsion from a magnetic field.
The Lewis structure of O₂ places all electrons in bonds and lone pairs — everything appears paired. A student relying solely on Lewis structures would predict that O₂ is diamagnetic. But liquid oxygen, when poured between the poles of a strong magnet, is visibly attracted and clings to the magnet. This observation is completely inconsistent with the Lewis structure prediction.
MO Theory resolves the contradiction immediately. Filling the MO diagram for O₂ reveals 2 electrons occupying the degenerate π2p* antibonding orbitals — 1 in each, unpaired, by Hund's Rule. Those 2 unpaired electrons make O₂ paramagnetic, in full agreement with experiment. This is precisely the kind of insight that elevates MO Theory above earlier bonding models.
Quick Test for Magnetic Behavior: After filling an MO diagram, scan every orbital. If even 1 orbital contains a single (unpaired) electron, the molecule is paramagnetic. If all orbitals hold exactly zero or 2 electrons, the molecule is diamagnetic.
7 Key Takeaways
- Atomic orbitals are wave functions. When they overlap, the combinations can be constructive (in-phase) or destructive (out-of-phase), producing a bonding molecular orbital and an antibonding molecular orbital simultaneously. The total number of orbitals is always conserved.
- Bonding MOs are lower in energy than the parent atomic orbitals (constructive overlap increases electron density between the nuclei). Antibonding MOs are higher in energy (destructive overlap creates a node between the nuclei).
- End-to-end overlap produces sigma (σ) orbitals (from s or pz). Side-to-side overlap produces pi (π) orbitals (from px and py), which come in degenerate pairs.
- Bond order = (bonding electrons − antibonding electrons) / 2. A bond order of zero means the diatomic does not exist. A positive bond order corresponds to a stable (or at least metastable) species.
- 2 MO diagram patterns exist for period-two diatomics. O, F, and Ne use the standard ordering (σ2p below π2p). Li, Be, B, C, and N use the sp-mixing ordering (σ2p above π2p) due to the smaller 2s–2p energy gap in those lighter elements.
- MO Theory correctly predicts paramagnetism. Any molecule with unpaired electrons in its MO diagram is paramagnetic. O₂ has 2 unpaired electrons in its π2p* orbitals — a fact the Lewis structure misses entirely but MO Theory handles naturally.
- Know your course expectations. Some courses require you to memorize which period-two elements use which diagram; others provide the diagram and expect you to fill it in correctly. Confirm with your instructor what you will be responsible for on exams.