Introduction

Why do atoms form bonds, and what is a bond at the most fundamental level? Many students learn that atoms form bonds to become more stable, but the deeper explanation often feels mysterious. In this lesson, we take a step back and look at bonding through the lens of orbitals, mathematics, and wave behavior.

You'll learn how chemists describe bonds as overlaps of orbitals, why orbitals themselves are mathematical solutions, and how the linear combination of atomic orbitals (LCAO) allows those mathematical solutions to combine into new molecular orbitals. Along the way, we'll connect bonding to constructive and destructive interference, helping explain why bonding can lower the energy of a system and make molecules more stable.

What Is a Bond?

Why do atoms form bonds? Bonds form to make molecules more stable.

A molecule will do anything it can to be as stable as possible. If making a bond increases stability, the bond will form. If making a bond would make the molecule less stable, it generally will not form. Most of the molecules we encounter in chemistry exist because bonding makes them more stable than the separated atoms.

So what is a bond?

A bond is an overlap of orbitals.

Orbitals Are Mathematical Objects

In a previous lesson, orbitals were described as the solutions to very complicated mathematical equations. Those equations define a volume of space within which it is probable to find an electron.

This creates an interesting question:

If a bond is an overlap of orbitals, and orbitals are solutions to mathematical equations, how do you overlap mathematical equations?

The answer is a mathematical construct called the linear combination of atomic orbitals (LCAO).

Linear Combination of Atomic Orbitals (LCAO)

LCAO is a set of mathematical rules used to combine atomic orbitals into molecular orbitals. Although the mathematics can become quite involved, there are a few practical ideas that are important for chemistry:

• The number of orbitals going into the process must equal the number of orbitals coming out.
• Orbitals must be combined both constructively and destructively.
• New molecular orbitals result from those combinations.

In other words, if we start with two atomic orbitals, we must end with two molecular orbitals.

Thinking About Electrons as Waves

To understand how these combinations work, it helps to remember wave-particle duality. Electrons behave as both particles and waves.

Imagine two electron waves that are perfectly in phase:

• Every peak aligns with another peak.
• Every trough aligns with another trough.

When these waves are added together, they undergo constructive interference.

Constructive Interference

In constructive interference, matching peaks reinforce each other and matching troughs reinforce each other.

The result is:

• The same overall frequency
• A larger amplitude

Every peak amplifies another peak, and every trough amplifies another trough.

This constructive overlap represents one possible outcome when orbitals combine through LCAO.

Destructive Interference

LCAO also requires a destructive combination.

For destructive interference, the waves are perfectly out of phase:

• Every peak aligns with a trough.
• Every trough aligns with a peak.

When these waves are combined, the amplitudes cancel each other out.

The result is a flat line with zero amplitude.

This zero-amplitude region is called a node.

Bonds as Mathematical Combinations

If this feels confusing, that's because we're reducing chemistry down to mathematics.

A common saying is:

• Biology is applied chemistry.
• Chemistry is applied physics.
• Physics is applied math.

At this level, a bond is the result of combining mathematical descriptions of atomic orbitals.

Using LCAO:

• Orbitals combine constructively.
• Orbitals combine destructively.
• The number of orbitals is conserved.

If we start with two electrons in two atomic orbitals, we must end up with two new molecular orbitals described by two new mathematical solutions.

What a Bond Really Is

At its most fundamental level, a bond is the result of combining orbitals through the linear combination of atomic orbitals.

Because orbitals are solutions to mathematical equations, a bond can be viewed as the result of combining those mathematical solutions. The constructive and destructive combinations generate new molecular orbitals, providing the foundation for understanding why atoms bond and why bonded molecules are often more stable than isolated atoms.

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