Bond Order & Magnetism

Molecular Orbital Theory (MOT) provides a quantum-mechanical framework for understanding chemical bonding by treating electrons as belonging to the entire molecule rather than to individual atoms. Unlike the simpler Valence Bond Theory, MOT constructs molecular orbitals (MOs) by mathematically combining atomic orbitals through a process called linear combination of atomic orbitals (LCAO). Each combination produces two MOs: a bonding orbital (lower energy, stabilizing) and an antibonding orbital (higher energy, destabilizing, denoted with an asterisk, e.g., σ or π). Two of the most powerful predictions MOT makes are bond order and magnetic behavior. Bond order is a measure of the net number of bonds holding two atoms together, calculated directly from the MO electron configuration. Magnetic behavior — whether a substance is attracted to a magnetic field (paramagnetic) or repelled/unaffected (diamagnetic) — is determined by the presence or absence of unpaired electrons in the MO diagram. Crucially, MOT correctly predicts that molecular oxygen ( $\ce{O2}$ ) is paramagnetic, a fact that simpler Lewis structures fail to explain. These two predictions are not merely academic: bond order correlates directly with bond length and bond energy, making it a practical tool for understanding molecular stability. A bond order of zero means the molecule does not exist as a stable species; fractional bond orders arise in species like $\ce{O2^{-}}$ (superoxide) and $\ce{He2^{+}}$ . Understanding MOT is therefore essential for predicting whether diatomic molecules and ions can exist and how strongly their atoms are bonded together.

Key takeaways

Construct the MO Energy Diagram
Fill Electrons Using Aufbau, Pauli, and Hund's Rules
Count Bonding (Nb) and Antibonding (Na) Electrons
Calculate Bond Order
Determine Magnetic Behavior

Construct the MO Energy Diagram

Atomic orbitals from each atom are combined to form MOs. For homonuclear diatomics (same element, e.g., $\ce{O2}$ , $\ce{N2}$ ), orbitals of the same energy and symmetry combine. The order of MO energy levels for Period 2 diatomics with Z ≥ 8 (O, F, Ne) is: σ1s < σ1s < σ2s < σ2s < σ2p < π2p = π2p < π2p = π2p < σ*2p. (For Z ≤ 7, the σ2p and π2p order is reversed due to s–p mixing.)

Fill Electrons Using Aufbau, Pauli, and Hund's Rules

Place all valence electrons (or all electrons if using 1s too) into MOs from lowest to highest energy, placing at most 2 electrons per orbital (Pauli Exclusion Principle), and filling degenerate orbitals one electron at a time with parallel spins before pairing (Hund's Rule). For $\ce{O2}$ , there are 16 total electrons. After filling through σ2s, we have 8 electrons used. The remaining 8 fill: σ2p (2e), π2p (2e each = 4e), then the 2 remaining go one each into the degenerate π2p orbitals — giving two unpaired electrons.

Count Bonding (Nb) and Antibonding (Na) Electrons

Bonding electrons (Nb) are those in bonding MOs (σ, π without asterisk). Antibonding electrons (Na) are those in MOs marked with (σ, π). For $\ce{O2}$ : Nb = 10 (σ1s:2, σ2s:2, σ2p:2, π2p:4), Na = 6 (σ1s:2, σ2s:2, π2p:2). Note: the 1s core electrons largely cancel each other, so chemists often count only valence MOs, giving Nb = 8, Na = 4.

Calculate Bond Order

Apply the bond order formula: BO = (Nb − Na) / 2. For $\ce{O2}$ using all electrons: BO = (10 − 6) / 2 = 2. This matches the double bond in the Lewis structure. The bond order of 2 means $\ce{O2}$ has a strong, short bond (bond length ≈ 121 pm, bond energy ≈ 498 kJ/mol).

Determine Magnetic Behavior

Check for unpaired electrons in the final MO configuration. If any MO contains exactly one electron, the molecule is paramagnetic (attracted to external magnetic fields). If all electrons are paired, the molecule is diamagnetic. For $\ce{O2}$ , the two π*2p orbitals each hold one unpaired electron → paramagnetic. This was experimentally confirmed in 1848 by Faraday: liquid oxygen visibly deflects between magnet poles. Lewis structures show $\ce{O2}$ with all pairs, incorrectly predicting diamagnetism — MOT's success here was a landmark vindication.

The Key Equation

BO = \frac{N_b - N_a}{2}

BO — Bond Order (dimensionless): Net number of chemical bonds between two atoms; higher bond order means shorter, stronger bond. BO = 0 implies the molecule is unstable and does not form.
N_b — Number of bonding electrons (electrons): Total electrons occupying bonding molecular orbitals (σ, π). These electrons lower the energy of the molecule relative to separate atoms, stabilizing the bond.
N_a — Number of antibonding electrons (electrons): Total electrons occupying antibonding molecular orbitals (σ, π). These electrons raise the energy of the molecule, destabilizing the bond. Each antibonding electron cancels the effect of one bonding electron.

Worked Example

Problem. Use Molecular Orbital Theory to determine the bond order of $\ce{O2^{2-}}$ (peroxide ion) and predict whether it is paramagnetic or diamagnetic.

BO = \frac{10 - 8}{2} = 1

Answer. Bond order = 1 (single bond); $\ce{O2^{2-}}$ is diamagnetic.

Figure

Diagram

Origins · 1 minRobert S. Mulliken and how this was discovered

Robert S. Mulliken

1896–1986 · American

Robert Sanderson Mulliken is the founder of Molecular Orbital Theory. Working in the 1920s–1930s alongside Friedrich Hund, Mulliken developed the mathematical framework of LCAO-MO theory, introduced the concept of molecular orbitals as wavefunctions delocalized over whole molecules, and pioneered the use of MO diagrams to explain spectra, bonding, and properties of diatomic molecules. He was awarded the Nobel Prize in Chemistry in 1966 for his fundamental work concerning chemical bonds and the electronic structure of molecules. Mulliken was initially trained as a chemist, but taught himself advanced quantum mechanics largely through self-study after reading papers by Heisenberg and Schrödinger — then used those concepts to revolutionize how chemists think about bonding.

Atomic Orbitals and Quantum Numbers — MO theory is built by combining atomic orbitals (1s, 2s, 2p, etc.). Understanding their shapes, energies, and quantum numbers is a prerequisite for constructing MO diagrams.
Electron Configuration and the Aufbau Principle — Filling molecular orbitals with electrons follows the same Aufbau principle, Pauli Exclusion Principle, and Hund's Rule used in atomic electron configurations.
Lewis Structures and Valence Bond Theory — MOT is often compared to and contrasted with Lewis structures and Valence Bond Theory. Understanding why Lewis structures fail to predict O₂ paramagnetism motivates the need for MOT.
Bond Length and Bond Energy — Bond order directly predicts bond length and bond energy: higher bond order → shorter, stronger bond. This connects MOT predictions to experimentally measurable quantities.
Paramagnetism and Diamagnetism — The prediction of magnetic behavior from unpaired electrons is one of MOT's greatest achievements, directly linking electronic structure to physical properties measurable by experiment.