Molecular orbital theory (intro) Knowledge-Based Study

Study notes

1. Foundations of Molecular Orbital (MO) Theory

   While Lewis structures and Valence Bond (VB) theory provide intuitive pictures of localized electron pairs, they fail to explain key experimental observations, such as the paramagnetism of molecular oxygen ($O_2$) or fractional bond orders in resonance-stabilized species.

   Molecular Orbital (MO) Theory treats electrons as delocalized across the entire molecule rather than confined to individual atoms or localized bonds. In this model, atomic orbitals (AOs) combine mathematically through a technique known as the Linear Combination of Atomic Orbitals (LCAO) to generate a set of molecular orbitals.

   • Conservation of Orbitals: The total number of molecular orbitals formed is always equal to the number of atomic orbitals combined. Combining two atomic orbitals produces exactly two molecular orbitals.
   • Spatial Delocalization: Molecular orbitals encompass multiple atomic nuclei, allowing electrons to occupy regions of electron density that belong to the molecule as a whole.

   Constructive and Destructive Wave Interference

   Electrons exhibit wave-particle duality. When atomic orbitals overlap, their mathematical wavefunctions ($\psi$) combine either in-phase or out-of-phase:

   • Bonding Molecular Orbitals ($\sigma$, $\pi$): Formed by constructive interference (addition of wavefunctions: $\psi_{\text{bonding}} = \psi_A + \psi_B$). Electron density increases in the internuclear region between the nuclei, shielding nuclear repulsions and stabilizing the molecule. Consequently, bonding orbitals lie at a lower potential energy than the parent atomic orbitals.
   • Antibonding Molecular Orbitals ($\sigma^*$, $\pi^*$): Formed by destructive interference (subtraction of wavefunctions: $\psi_{\text{antibonding}} = \psi_A - \psi_B$). A nodal plane (a region where electron probability is zero) is formed perpendicular to the internuclear axis. The lack of electron density between the nuclei leads to increased nuclear repulsion, placing antibonding orbitals at a higher potential energy than the isolated atomic orbitals.

   Orbitals that maintain cylindrical symmetry along the internuclear axis are designated as $\sigma$ (sigma) or $\sigma^*$ (sigma-star) orbitals.

   Constructing and Filling MO Energy Diagrams

   A molecular orbital energy diagram arranges molecular orbitals vertically according to their relative energy levels, flanked by the atomic orbitals of the constituent atoms.

   Electrons are populated into molecular orbitals following the same quantum mechanical rules used for atomic orbitals:

   1. Aufbau Principle: Orbitals of lowest energy are filled first (e.g., $\sigma_{1s}$ before $\sigma^*_{1s}$).
   2. Pauli Exclusion Principle: Each molecular orbital can accommodate a maximum of two electrons, which must possess opposite spins.
   3. Hund's Rule: For degenerate orbitals (orbitals with identical energy, such as $\pi_{2p_x}$ and $\pi_{2p_y}$), electrons occupy separate orbitals with parallel spins before pairing up.

   For instance, for the hydrogen molecule ($H_2$), the two available valence electrons occupy the lower-energy bonding orbital, giving the molecular electron configuration: $(\sigma_{1s})^2$.

   Predicting Molecular Stability and Magnetism

   Molecular Orbital Theory allows us to quantitatively determine the strength of a chemical bond through the concept of Bond Order:

   $$\text{Bond Order} = \frac{N_b - N_a}{2}$$

   where $N_b$ is the number of electrons in bonding molecular orbitals and $N_a$ is the number of electrons in antibonding molecular orbitals.

   • Stability: A positive bond order indicates a stable, bound molecule. A bond order of zero (such as in $He_2$, where $N_b = 2$ and $N_a = 2$) indicates no net stabilization, predicting that the species does not form a stable covalent bond under standard conditions.
   • Bond Characteristics: Higher bond orders correlate with shorter bond lengths and greater bond dissociation energies.

   Magnetic Behavior

   • Diamagnetism: Substances with all electron spins paired are weakly repelled by magnetic fields.
   • Paramagnetism: Substances possessing one or more unpaired electrons are attracted into external magnetic fields.