Molecular Orbital Theory

Core Concepts & Principles

While valence bond theory uses localized hybrid orbitals on specific atoms, Molecular orbital theory (MO theory) treats electrons as delocalized across the entire molecule using quantum mechanical wave functions ($\Psi$). MO theory successfully explains phenomena that Lewis structures fail to address, such as the paramagnetism of oxygen ($O_2$).

Formation of Molecular Orbitals

When atoms combine to form molecules, their atomic orbitals mathematically combine through the linear combination of atomic orbitals (LCAO):

• In-phase combination (Constructive interference): Waves reinforce each other, concentrating electron density directly between the nuclei. This forms lower-energy bonding orbitals.
• Out-of-phase combination (Destructive interference): Waves cancel out, creating a node (zero electron density) between the nuclei. This forms higher-energy antibonding orbitals, denoted with an asterisk ($*$).

Types of Molecular Orbitals

• $\sigma$ (sigma) orbitals: Formed by end-to-end overlap of $s$ or $p$ atomic orbitals along the internuclear axis ($\sigma_s$, $\sigma_s^*$, $\sigma_p$, $\sigma_p^*$).
• $\pi$ (pi) orbitals: Formed by side-by-side overlap of parallel $p$ atomic orbitals, resulting in electron density above and below the internuclear axis ($\pi$ and $\pi^*$).
• Degenerate orbitals: Orbitals that share identical energy levels (such as the $\pi_{2py}$ and $\pi_{2pz}$ sets).

s-p Mixing

In second-period homonuclear diatomic molecules ($Li_2$ through $N_2$), the energy difference between $2s$ and $2p$ atomic orbitals is small, leading to s-p mixing. This interaction lowers the energy of the $\sigma_s$ and $\sigma_s^*$ orbitals while raising the energy of the $\sigma_p$ orbital above the $\pi_p$ set. For heavier elements ($O_2$, $F_2$, $Ne_2$), the $2s$-$2p$ energy gap is larger, s-p mixing is negligible, and the standard orbital energy order is maintained.

Band Theory in Solids

In solid materials containing $10^{23}+$ atoms, massive numbers of molecular orbitals overlap to form continuous energy bands rather than discrete levels:

• Valence band: The lower-energy band formed by filled bonding orbitals.
• Conduction band: The higher-energy band formed by empty antibonding orbitals.
• Band gap: The energy difference between valence and conduction bands, which dictates whether a solid is an insulator (large gap), semiconductor (moderate gap), or electrical conductor (negligible gap).

Problem-Solving Routines & Methods

Practice & Concept Checks

Key Terms & Vocabulary