MO theory: A new perspective

As we know, Valence Bond (VB) theory visualizes molecules as a collection of atoms coming together to “share” specific pairs of electrons, forming localized bonds. Molecular Orbital (MO) theory, however, introduces a radical shift in perspective by treating the molecule as a single, unified entity. In this model, once the nuclei are positioned, the electrons no longer “belong” to any individual atom; instead, they settle into molecular orbitals that extend across the entire structure.

MO theory: the core concept

The core idea is the linear combination of Atomic  Orbitals (LCAO). When atoms get close enough for their electron waves to overlap, their atomic orbitals (AO’s) combine mathematically to form molecular orbitals (MO’s).

Conservation of orbitals: number of MO’s formed is equal to number of AO’s combined.

Constructive Interference: When 2 AO’s wave functions add together mathematically, they create a Bonding MO (BMO). This MO’s energy is lower than the original  AO’s energy making it more stable.

Destructive Interference: When 2 AO’s cancel each other out (i.e subtraction of wavefunction), they create a Antibonding MO (ABMO). This MO’s energy is higher than the original AO’s making it more unstable.

The MO’s are classified based on their symmetry and how the atomic orbitals overlap:

Sigma Orbitals: Formed by the head-on overlap (i.e. on the same internuclear axis). They are symmetrical around the internuclear axis.

Pi Orbitals: Formed by the side-by-side overlap of p orbitals. These have electron density above and below the internuclear axis.

WHAT IS IT’S USE

While VB theory is highly intuitive for visualizing localized bonds and molecular geometry, if falters when describing electronic phenomena that require a more holistic view.

Magnetic nature: VB model cannot accurately predict the magnetic property of certain molecule, but this is possible using MO theory.

Resonance: unlike VB model which imagine the molecule just flipping back and forth between states, the MO theory simply says that the electrons are in a giant orbital across the molecule (i.e. it treats the molecule as one single unit).

Electronic Spectroscopy and Excited State: VB theory can describe a molecule in ground state but falters when asked about what happens after it absorbs light. The MO theory introduces the HOMO (Highest Occupied MO) and LUMO (Lowest Unoccupied MO), MO theory says that when UV light is absorbed the electron jumps from HOMO to LUMO. The magnitude of this energy gap determines the frequency of light absorbed, which in turn determines the spectroscopy and colour of the molecule.

Bond Fractionalization (Odd-Electron molecule): Since VB theory relies on pairs of electrons, it can’t easily explain a bond held together by odd number of electrons. In MO theory, a bond doesn’t have to be a whole number, it can be 0.5, 1.5, etc. MO theory uses bond order.

THE MATH BEHIND MO THEORY

In MO theory, we transition from the particle model to a wave model of electron. By applying Linear Combination of Atomic Orbitals (LCAO), we treat individual atomic waves as overlapping waves which undergo mathematical interference. Hence, when 2 waves combine, they can either reinforce one another through constructive interference to form more stable MO or cancel each other out through destructive interference to form an unstable MO.

We approximate the molecular wavefunction as a weighted sum of N atomic orbitals :

Where the weights represent how much each atomic orbital contributes to the molecular orbital.

Using this new molecular orbital, we calculate its energy using the Rayleigh Ratio:

Now we have a Energy function, which we will partially differentiate with the weight to find the lowest energy as well as get the weights so as to find the final molecular orbital wavefunction.

This gives us a group of equations, where its determinant is used to find the energy value. Which when inserted back into the equation allows us to find the value of the weights, Hence giving us the final Molecular Wavefunction, which describes the new molecule as a whole.

This molecular Wavefunction tells you exactly where the electron can most likely be found across the whole nuclear framework.