10 Hybridization and Molecular Orbital Theory

Two Models of Covalent Bonding

Two complementary models describe covalent bonding. Valence-bond (VB) theory focuses on overlaps between orbitals associated mainly with particular atoms. Molecular orbital (MO) theory describes orbitals that extend over the molecule. Both are models of electron distribution, not literal pictures of electrons following fixed paths. The models emphasize different features: valence-bond theory helps describe localized bonds and approximate geometry, while molecular orbital theory helps explain bonding across a molecule, including bond strength and magnetic behavior.

Hybridization, Geometry, and Orbital Overlap

In valence-bond theory, Hybridization describes how an atom’s valence s and p orbitals combine mathematically to form directional orbitals. To estimate common hybridization and arrangement, count the regions of electron density around an atom. Each single, double, or triple bond counts as one region, as does each lone pair.

• Two regions commonly correspond to $sp$ hybridization and a linear arrangement, with an ideal angle of $180^\circ$. Each carbon in $\mathrm{CO_2}$ is an example.

• Three regions commonly correspond to $sp^2$ hybridization and a trigonal planar arrangement, with an ideal angle of $120^\circ$. Each carbon in $\mathrm{C_2H_4}$ is an example.

• Four regions commonly correspond to $sp^3$ hybridization and a tetrahedral arrangement, with an ideal angle of $109.5^\circ$. Carbon in $\mathrm{CH_4}$ is an example.

These arrangements are approximate. Lone pairs and different bonding partners can change observed shapes and angles. In water, oxygen is commonly described as $sp^3$ hybridized: two hybrids hold lone pairs, and two overlap with hydrogen orbitals. The $\mathrm{H-O-H}$ angle is about $104.5^\circ$, rather than the ideal tetrahedral angle, because the electron regions do not repel equally.

A Sigma bond forms through end-to-end orbital overlap along the internuclear axis. A Pi bond forms through side-by-side overlap of parallel p orbitals. A double bond typically consists of one sigma bond and one pi bond; a triple bond typically consists of one sigma bond and two pi bonds.

In ethene, $\mathrm{C_2H_4}$, each carbon has three electron-density regions and is described as $sp^2$ hybridized. Three $sp^2$ orbitals form sigma bonds. The remaining unhybridized p orbital on each carbon overlaps side by side to form the pi part of the $\mathrm{C=C}$ bond. In methane, $\mathrm{CH_4}$, four $sp^3$ orbitals form four $\mathrm{C-H}$ sigma bonds.

Hybridization is a useful way to organize approximate geometry and localized bonds. It is a simplified bonding description, not a separate physical process atoms must undergo before bonding or a uniquely observable assignment.

Takeaway: Count electron-density regions to estimate hybridization and shape; use orbital overlap to distinguish sigma and pi bonds.

Molecular Orbitals and Electron Filling

Molecular orbital (MO) theory describes electrons as occupying molecular orbitals spread over a molecule. In the linear combination of atomic orbitals approach, atomic orbitals combine most effectively when their energies, shapes, and symmetry are compatible. Combining two atomic orbitals produces two molecular orbitals:

• A bonding molecular orbital is lower in energy and has increased electron density between the nuclei, favoring bonding.

• An antibonding molecular orbital is higher in energy and has a node between the nuclei, opposing bonding. An asterisk marks an antibonding orbital, as in $\sigma^*$ or $\pi^*$.

End-to-end overlap produces $\sigma$ and $\sigma^*$ orbitals; side-by-side overlap produces $\pi$ and $\pi^*$ orbitals. Electrons fill molecular orbitals from lower to higher energy, with at most two electrons of opposite spin in each orbital. For orbitals at the same energy, electrons occupy separate orbitals before pairing, in accordance with Hund’s rule.

For second-period homonuclear diatomic molecules, the order of the orbitals derived from the $\mathrm{2p}$ atomic orbitals changes across the series. From $\mathrm{B_2}$ through $\mathrm{N_2}$, the $\pi(2p)$ orbitals lie below $\sigma(2p)$. In $\mathrm{O_2}$ and $\mathrm{F_2}$, $\sigma(2p)$ lies below $\pi(2p)$. Using the correct ordering is important when filling an orbital diagram.

Takeaway: Molecular orbital theory links orbital energy and electron occupancy to bonding effects throughout a molecule.

Bond Order and Magnetic Behavior

Bond order summarizes the net bonding effect of occupied molecular orbitals:

Here, $N_\text{bonding}$ and $N_\text{antibonding}$ count electrons in bonding and antibonding molecular orbitals, respectively. Bond order need not be an integer. For the same pair of atoms, a larger bond order generally indicates a stronger, shorter bond; a bond order of zero predicts no net bond in this introductory model.

For $\mathrm{O_2}$, filling the valence molecular orbitals gives eight bonding electrons and four antibonding electrons:

For $\mathrm{O_2^+}$, removing one electron from an antibonding $\pi^*$ orbital leaves three antibonding electrons:

A substance is Paramagnetic if it has one or more unpaired electrons and is attracted to a magnetic field. A substance is Diamagnetic if all its electrons are paired. In $\mathrm{O_2}$, the last two electrons occupy separate, equal-energy $\pi^*$ orbitals, leaving two unpaired electrons. Thus, oxygen is paramagnetic. This illustrates a strength of molecular orbital theory: a basic Lewis structure shows an $\mathrm{O=O}$ double bond but does not account for oxygen’s two unpaired electrons. In $\mathrm{N_2}$, the occupied molecular orbitals give a bond order of $3$, and all electrons are paired, so nitrogen is diamagnetic.

Takeaway: Bond order captures the balance of bonding and antibonding electrons, while unpaired electrons explain magnetic behavior.