Introduction
How do two hydrogen atoms come together to form H₂, one of the simplest and most important molecules in chemistry? The answer lies in molecular orbital theory and the way atomic orbitals combine through the linear combination of atomic orbitals (LCAO).
In this lesson, we'll build directly on the idea that orbitals are mathematical solutions that can be combined constructively and destructively. You'll see how the constructive overlap of two hydrogen 1s orbitals creates a lower-energy bonding orbital, while destructive overlap creates a higher-energy antibonding orbital. Understanding this process explains not only what a sigma bond is, but also why H₂ is more stable than two separate hydrogen atoms.
From Atomic Orbitals to Molecular Orbitals
In the previous lesson, we discussed the linear combination of atomic orbitals (LCAO). The key idea is that orbitals are simply solutions to mathematical equations. Because they are mathematical objects, they can be combined much like waves.
When two atomic orbitals interact, we must combine them in two different ways:
- Constructively
- Destructively
And because the number of orbitals going in must equal the number of orbitals coming out, two atomic orbitals must produce two molecular orbitals.
Constructive Overlap and the Sigma Bond
Let's start with two hydrogen 1s orbitals.
To combine them constructively, we overlap the orbitals so that their phases match perfectly. This is similar to two waves undergoing constructive interference.
When constructive interference occurs, the amplitude of the resulting wave increases.
In orbital language, this means the probability of finding electrons increases in the region between the two nuclei. The electron density becomes concentrated in the space connecting the hydrogen atoms.
Because electrons are more likely to be found between the nuclei, they help hold those nuclei together.
This molecular orbital is called the sigma (σ) bonding orbital.
What Is a Sigma Bond?
A sigma bond is the result of the constructive overlap of atomic orbitals.
More specifically:
- Two atomic orbitals overlap constructively.
- Electron density increases between the nuclei.
- The probability of finding electrons between the nuclei becomes greater.
- The resulting molecular orbital is lower in energy than the original atomic orbitals.
The increased electron density between the nuclei is what stabilizes the bond.
Destructive Overlap and the Antibonding Orbital
The sigma bonding orbital is only half of the story.
According to LCAO, if we start with two atomic orbitals, we must end with two molecular orbitals. Since we've already formed one through constructive overlap, the second must come from destructive overlap.
To do this, we combine the positive phase of one orbital with the negative phase of the other.
This creates complete destructive interference between the orbitals.
The result is a region between the nuclei where the amplitude collapses to zero.
Nodes and the Sigma-Star Orbital
When destructive interference produces a region with zero amplitude, that region is called a node.
Within a node:
- The wavefunction equals zero.
- There is zero probability of finding an electron.
Because there is no electron density between the nuclei, this molecular orbital does not contribute to bonding.
This higher-energy orbital is called the sigma-star (σ*) antibonding orbital.
The Two Molecular Orbitals Produced
Starting with two hydrogen atomic orbitals gives two molecular orbitals:
Sigma Bonding Orbital (σ)
- Formed by constructive overlap
- Lower in energy
- Increases electron density between nuclei
- Stabilizes the molecule
Sigma Antibonding Orbital (σ*)
- Formed by destructive overlap
- Higher in energy
- Contains a node between nuclei
- Destabilizes the molecule
This satisfies the LCAO rule that the number of orbitals remains constant.
Filling the Molecular Orbitals
After constructing the molecular orbitals, we place the electrons into them.
The two hydrogen atoms contribute a total of two electrons.
Following:
- Aufbau Principle
- Hund's Rule
- Pauli Exclusion Principle
the electrons occupy the lowest-energy orbital first.
Both electrons enter the σ bonding orbital.
No electrons occupy the higher-energy σ* antibonding orbital.
Because the electrons occupy only the lower-energy orbital, the overall energy of the system decreases.
Why H₂ Exists
The entire purpose of bonding is to lower the energy of the system.
When two hydrogen atoms combine:
- Their atomic orbitals overlap.
- A σ bonding orbital and a σ* antibonding orbital are created.
- Both electrons occupy the lower-energy σ bonding orbital.
- The overall energy decreases.
As a result, H₂ is more stable than two separate hydrogen atoms.
That increased stability is why the hydrogen molecule exists.
Looking Ahead to Pi Bonds
The same molecular orbital principles apply to p orbitals.
Just as s orbitals overlap to form sigma bonds, p orbitals can overlap to form pi (π) bonds through constructive and destructive combinations. The details of pi bonding build directly on the concepts introduced here.
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