Lewis suggests that two atoms with unpaired electrons may share their electrons to form a bond. In a Lewis structure, we can draw those two electrons between the symbols for the two atoms to show that a bond has formed.  Quantum mechanics offers a quantitative mathematical view of molecules, and is considered to give an accurate picture of the behaviour of atoms and molecules.  Now, instead of putting two atoms beside each other and letting two dots get together to form a bond, like in a Lewis structure, we'll think about bonding in quantum mechanical terms.

• Two hydrogen atoms are placed next to each other.
• The two atoms are surrounded by the waves formed by their electrons.
• If the two atoms are close together, these two waves overlap in between the two atoms.

What happens when waves overlap? Interference patterns develop. Think of a flat, calm pond, into which you and a friend throw two small stones, a few feet apart. The ripples that result move out in circles from the two points where the stones fell in, and when the two sets of waves meet each other, new patterns result. The circular sets of waves combine to form new waves with new shapes.

When waves meet, there are two basic combinations that result. These combinations depend on whether the two waves are in phase or out of phase with each other.

• If the waves are in phase, so that the peak of one wave lands on top of the peak of another, and the trough of one wave coincides with the trough on the other, constructive interference occurs. The waves add up, and get bigger where they overlap.
• If the waves are exactly out of phase, the peak of one falls into the trough of another, and where they overlap the two waves are canceled out. This is destructive interference.

We can apply these ideas to the electrons in to atoms that come together to make a bond.  In Lewis structures, individual electrons on two different atoms are paired up in a bond between the two atoms.  The electrons have to be combined to make a bond.  If electrons have wave properties, then there will be interference patterns that result when these atoms come together.

Before we start, note that the waves that we will be dealing with look a little different from the standing waves we looked at before.  These waves will be three-dimensional and will occupy different regions of space.  Instead of drawing phase information as peaks and troughs, we are going to use a common convention of shading to signify whether we are looking at the positive part (in terms of phase) or the negative part of a wave.  For example, we might show the crests in blue and the troughs in red,or we might show one in black and the other in white.  In a standing wave, the phases would look like this:

In a hydrogen atom, the electron is found in a spherical area around the nucleus.  We won't be able to see two distinct phases - a crest and a trough - in this wave at one time.  However, we could look at two different hydrogen atoms and illustrate whether their electrons are in phase or out of phase with each other.

When those two electrons come together and they are in phase with each other, constructive interference occurs.  They won't overlap completely; these are atoms and they take up space.  However, they will be able to sit pretty close to each other, sort of smushing their electrons together.  That constructive intereference will happen in the middles, between the atoms.

When the two electrons come together and they are out of phase, destructive interference occurs.  Again, that will happen in the middle, between the atoms.

So, in the pair of hydrogen atoms, two waves can combine to produce two possible outcomes.

• In one, the height of the combined wave is highest between the two atoms, where they are overlapping and in phase. The amplitude of the wave is greater here.
• In the other case, the combined wave is composed of two halves that are out of phase, and there is no wave at all between the two atoms, where they have cancelled out; the amplitude of this wave is zero between the two nuclei.

Those two possible combinations produce two different results.  We keep track of these outcomes using a molecular orbital interaction diagram.  The diagram says that the two orbitals can combine in two different ways: in phase or out of phase.  The diagram also tells us that the in-phase combination results in a decrease in energy when the two atoms come together.  The out-of-phase combination results in an increase in energy when the atoms come together.