2.3 Molecular Orbital Theory

Molecular Orbital Formation

Molecular Orbital (MO) Theory describes how atomic orbitals on separate atoms combine to form new orbitals that belong to the molecule as a whole. Unlike valence bond theory, which treats bonds as localized between two atoms, MO theory distributes electrons across the entire molecule. This makes it especially powerful for explaining properties like paramagnetism in $\text{O}_2$ that other bonding models can't account for.

Linear Combination of Atomic Orbitals (LCAO)

The mathematical foundation of MO theory is the Linear Combination of Atomic Orbitals (LCAO). You take the wavefunctions of atomic orbitals and either add or subtract them to produce molecular orbitals.

• Constructive interference (adding wavefunctions): produces a bonding molecular orbital with increased electron density between the nuclei. This stabilizes the molecule because the electrons are attracted to both nuclei simultaneously.
• Destructive interference (subtracting wavefunctions): produces an antibonding molecular orbital (labeled with an asterisk, e.g., $\sigma^*$) with a node between the nuclei. Electron density is pushed away from the internuclear region, which destabilizes the bond.
• Non-bonding orbitals retain their atomic character and don't contribute significantly to bonding or antibonding interactions.

Two conditions must be met for atomic orbitals to combine effectively:

1. Symmetry match: the orbitals must have compatible symmetry with respect to the internuclear axis. For example, an $s$ orbital and a $p_x$ orbital (along the bond axis) can combine, but an $s$ orbital and a $p_y$ orbital (perpendicular to the bond axis) cannot.
2. Energy match: the atomic orbitals should be reasonably close in energy. The closer in energy, the greater the interaction and the larger the splitting between bonding and antibonding levels.

Greater spatial overlap between the orbitals also leads to a larger energy gap between the bonding and antibonding MOs, which means a stronger bond.

Bond Types and Characteristics

The type of molecular orbital formed depends on the geometry of the overlap:

• Sigma ($\sigma$) bonds form through head-on overlap along the internuclear axis. These can arise from $s$-$s$, $s$-$p$, or $p$-$p$ (end-on) combinations. They have cylindrical symmetry around the bond axis.
• Pi ($\pi$) bonds form through side-by-side overlap of $p$ orbitals perpendicular to the internuclear axis. They have a nodal plane that contains the internuclear axis, which means electron density sits above and below (or in front of and behind) the bond axis rather than directly along it.
• Delta ($\delta$) bonds arise from face-to-face overlap of $d$ orbitals. These are relevant in transition metal chemistry, particularly in metal-metal multiple bonds like those in $\text{Re}_2\text{Cl}_8^{2-}$.

A double bond consists of one $\sigma$ bond plus one $\pi$ bond. A triple bond consists of one $\sigma$ bond plus two $\pi$ bonds. The $\sigma$ component is always the strongest because head-on overlap is more effective than lateral overlap.

Molecular Orbital Diagrams and Properties

Building MO Diagrams

MO diagrams show the energy levels of atomic orbitals on the left and right sides, with the resulting molecular orbitals in the center. To fill them correctly, follow these steps:

1. Identify the atomic orbitals that will participate in bonding (typically valence orbitals).
2. Arrange them by energy on each side of the diagram. For homonuclear diatomics, both sides are identical. For heteronuclear diatomics, the more electronegative atom's orbitals sit lower in energy.
3. Combine orbitals of matching symmetry and similar energy to form bonding and antibonding MOs. Each pair of atomic orbitals produces exactly one bonding and one antibonding MO.
4. Fill electrons from lowest energy to highest, applying the Aufbau principle, Pauli exclusion principle (max 2 electrons per orbital, opposite spins), and Hund's rule (fill degenerate orbitals singly before pairing).

An important detail for second-period homonuclear diatomics: For $\text{Li}_2$ through $\text{N}_2$, the $\sigma_{2p}$ orbital is higher in energy than the $\pi_{2p}$ orbitals due to $s$-$p$ mixing (the 2s and 2p energy levels are close enough to interact). For $\text{O}_2$ and $\text{F}_2$, the $\sigma_{2p}$ drops below the $\pi_{2p}$ because the larger $Z_{\text{eff}}$ increases the 2s-2p energy gap, reducing $s$-$p$ mixing. Getting this ordering wrong will give you incorrect electron configurations and wrong predictions about magnetic properties.

Bond Order

Bond order quantifies the net bonding in a molecule:

$\text{Bond Order} = \frac{(\text{number of bonding electrons}) - (\text{number of antibonding electrons})}{2}$

• A bond order of 0 means the molecule is not stable (e.g., $\text{He}_2$).
• Higher bond orders correspond to shorter, stronger bonds. $\text{N}_2$ has a bond order of 3, consistent with its very short bond length (109.8 pm) and high dissociation energy (945 kJ/mol).
• Fractional bond orders are possible. For instance, $\text{O}_2^+$ has a bond order of 2.5.

Homonuclear vs. Heteronuclear Diatomics

Homonuclear diatomics ($\text{O}_2$, $\text{N}_2$, $\text{F}_2$) have symmetric MO diagrams because both atoms contribute identical orbitals at the same energy. The bonding and antibonding MOs have equal contributions from each atom.

Heteronuclear diatomics ($\text{CO}$, $\text{HF}$, $\text{NO}$) have asymmetric diagrams. The more electronegative atom's orbitals sit lower in energy, so the bonding MOs have more character from that atom, while the antibonding MOs have more character from the less electronegative atom. This unequal contribution is what gives heteronuclear diatomic bonds their polar character.

For example, in $\text{HF}$, the fluorine 2p orbital is much lower in energy than the hydrogen 1s orbital. The bonding MO is predominantly fluorine in character, which is consistent with the large electronegativity difference and the polar nature of the H-F bond.

Magnetic Properties

MO theory provides a straightforward way to predict magnetic behavior:

• Paramagnetic: the molecule has one or more unpaired electrons and is attracted into a magnetic field. $\text{O}_2$ is the classic example. Its MO diagram shows two unpaired electrons in the degenerate $\pi^*_{2p}$ orbitals, which Lewis structures completely fail to predict.
• Diamagnetic: all electrons are paired, and the molecule is weakly repelled by a magnetic field. $\text{N}_2$ is diamagnetic because all of its electrons are paired in the MO diagram.

This is one of the major successes of MO theory over Lewis structures and valence bond theory. The experimental observation that $\text{O}_2$ is paramagnetic was unexplained until MO theory came along.

Advanced Molecular Orbital Concepts

Delocalized Bonding

In many molecules, electrons aren't confined to a bond between just two atoms. Delocalized bonding describes electrons spread over three or more atoms, which lowers the overall energy and increases stability.

Benzene ($\text{C}_6\text{H}_6$) is the textbook example. Rather than alternating single and double bonds, the six $\pi$ electrons are delocalized across all six carbon atoms in a ring. Resonance structures attempt to capture this, but the real electronic structure is a single hybrid of all contributing forms.

Hückel molecular orbital theory applies specifically to planar, conjugated $\pi$ systems. It predicts that a cyclic, planar, fully conjugated molecule is aromatic (extra stable) if it contains $4n + 2$ $\pi$ electrons, where $n$ is a non-negative integer. Benzene with 6 $\pi$ electrons ($n = 1$) satisfies this rule. A system with $4n$ $\pi$ electrons (like cyclobutadiene with 4) is antiaromatic and destabilized.

HOMO, LUMO, and Frontier Orbital Theory

Two molecular orbitals are especially important for reactivity:

• HOMO (Highest Occupied Molecular Orbital): the highest-energy orbital that contains electrons. This is where a molecule donates electron density.
• LUMO (Lowest Unoccupied Molecular Orbital): the lowest-energy empty orbital. This is where a molecule accepts electron density.

Frontier molecular orbital (FMO) theory focuses on HOMO-LUMO interactions to predict how molecules react. A reaction is favorable when the HOMO of one reactant overlaps effectively with the LUMO of another, and when the energy gap between them is small. FMO theory explains regioselectivity (where on a molecule a reaction occurs) and is central to understanding pericyclic reactions and transition metal reactivity.

Band Theory and Extended Systems

When MO theory is extended to very large numbers of atoms (as in metals and semiconductors), the discrete molecular orbital energy levels merge into continuous energy bands. The overlap of many atomic orbitals creates a valence band (filled) and a conduction band (empty or partially filled). The size of the band gap between them determines whether a material is a conductor (no gap or overlapping bands), semiconductor (small gap), or insulator (large gap).

Photochemical processes also connect to MO theory: when a molecule absorbs light of the right energy, an electron is promoted from the HOMO to the LUMO. The energy of this transition corresponds to the HOMO-LUMO gap and determines what wavelengths of light a molecule absorbs, which directly relates to its color and photochemical reactivity.