But let there be spaces in your togetherness
and let the winds of the heavens dance between you.
Love one another but make not a bond of love:
let it rather be a moving sea
between the shores of your souls.
~Kahlil Gibran

Two weeks ago now, I flew to Conway, Arkansas to attend the wedding of my very good friends Vincent and Mary. This and an academic conference got in the way of blogging for a little while but I’m back. As such I decided to a post in their honor about bonding. Not human bonding, mind you, but on chemical bonding. Specifically, covalent bonding! You probably know that atoms missing electrons like to form covalent bonds with each other where they share their electrons. But why does this happen? The secret lies in the quantum mechanical nature of electrons.

This post will rely heavily on the articles I’ve previously written about quantum mechanics. You might want to check out my previous posts. These ones will be particularly helpful:

• I originally did a three part series on quantum mechanics where I first explained particle wave duality, then I explained what a matter wave is using the Bohr model of the hydrogen atom. Finally, I offered an interpretation of matter waves as probability waves.
• I’ve also discussed the Pauli exclusion principle, which forbids two fermions from existing in the same quantum state.
• And, perhaps most importantly, I’ve talked about quantum tunneling, which describes how quantum particles can do things classical particles cannot. Quantum tunneling is very similar to what I’m going to describe here.

If you don’t want to go through all of these, I will try and link to the relevant ones as they come up in discussion.

Energy

In physics, we have this thing we call energy. We usually break energy into two categories: kinetic energy and potential energy. (There are more “types,” but in the end, they can be reduced to these two types.)  Kinetic energy is a little easier to understand, so let’s talk about that first.

Roughly, Kinetic energy measures how much something moves, and how difficult it is to make that thing move or to stop it once it’s moving. In classical mechanics, the kinetic energy is given by one half the square of the velocity times the mass,

   

(Astute readers may remember that I described momentum in a similar way. I said it measured how much something is moving and how difficult it is to change an object’s motion. Kinetic energy and momentum are very much related. The biggest difference is that momentum is a vector. It has a size and a direction… and it measures the direction of motion as well as the resistance to change. On the other hand, energy is a scalar. It’s just a number, which comes from squaring the velocity. Also, although energy can be transferred between objects and transformed between potential energy and kinetic energy, momentum only describes motion. Finally, as  a rough intuition, momentum measures how difficult it is to make a small change in an object’s motion, while kinetic energy measures how many small changes are required to change the motion in a big way.)

Potential energy measures the ability to generate kinetic energy. If I’m very high up on a cliff and I jump off, I can accelerate very fast and acquire a lot of kinetic energy (which I stole from the Earth’s gravitational field). This means that my potential energy (at least from gravity) is proportional to my height. Other sources of potential energy include electric and magnetic fields, springs, and even massive particles themselves.

The total energy of a system or a particle is the sum of the kinetic energy and the potential energy. And the total amount all energy in the (classical) universe is conserved; It can’t be created or destroyed, simply passed around between particles, objects, and people.

(Experts know I’m glossing over a lot here. In truth, the distinction between kinetic and potential energies is pretty artificial. The physicist and mathematician Emmy Noether defined energy as whatever quantity a physical system possess that doesn’t change in time. In other words, it is the time-translational symmetry of the system. We’ve simply given names to the contributions to the energy like kinetic and potential energy. And indeed, energy may not be conserved for the universe as a whole. In Einstein’s general relativity, energy is not necessarily conserved.)

Of Energy and Wiggles

I’ve described what energy means in a “classical” system, where quantum effects are negligible. But in quantum mechanics, things get a bit weird (as they often do). If we understand kinetic energy, it’s simple enough to define potential energy as the ability to create kinetic energy. But what does kinetic energy mean in the quantum case? We defined it as the motion of a particle, but quantum particles don’t travel in the same way… they’re waves that exist everywhere at once. What does motion mean in this context?

We can take a hint from the wave nature of quantum particles. In quantum mechanics, the height of the wave at a given position tells us how likely it is we’ll observe a particle there. But all waves wiggle. The height rises and falls. And it just so happens that the number of times the wave wiggles over a given distance determines the energy! I’ve plotted three quantum probability functions below, each with a different energy. In this case, the quantum particle—say an electron—is confined to a box of length 1 by an infinitely strong electric field. This means that the probability of measuring the particle outside of the box is zero, and the height of the wavefunction must reflect that.

Now the thing is, the energy depends on the wiggles per unit volume. So if we made the box longer, all three particles would lose energy because they’d all still have the same number of wiggles, but there would be fewer wiggles per cubic meter. Let’s make a note of that. It’ll be important later.

The Electron and The Nucleus

In the past, I’ve given you Niels Bohr’s description of the atom, which demonstrates that electrons in atoms can only have specific kinetic energies. This is because only an integer number of wiggles fit around a circle if you want the wave of a particle to agree with itself after you go 360 degrees around the circle. The same holds true for most quantum systems.

For simplicity, lets look at the hydrogen atom in a different light, though. Let’s imagine that a hydrogen nucleus (i.e., a proton) exerts an attractive force on the electron and ignore that the electron can orbit around the nucleus. In other words, let’s imagine one-dimensional atoms.

Because the proton is so much heavier than the electron, we can basically think of it as stationary. The attractive force between the electron and the proton gives the electron some potential energy depending on its position in space. To make the whole problem easier to visualize, let’s make an analogy with gravity. On Earth, when we’re high up, we have a lot of potential energy and when we’re down low we have very little. We can describe the potential energy of the electron by plotting as if it were a height above the ground (or a depth below the ground). In the case of the hydrogen atom, that potential energy looks something like this:

In our little approximation, the electron is more strongly attracted to the proton the closer it gets to the proton. If it overlapped with the proton, it’d be infinitely attracted to it. However, particles in quantum mechanics have a minimum energy they’re allowed to have. In this case, that minimum is the Bohr energy. If we plot the probability distribution for an electron in the lowest energy state of the atom, it looks something like this:

If the electron were classical, it couldn’t go farther away from the center of the atom than the classical turning points, which I’ve marked with big black dots. This is because the electron doesn’t have enough energy to “climb” out of the potential energy well and leave. But in quantum mechanics, the electron can exist in places it’s classically forbidden to be. This is very similar to quantum tunneling. This uniquely quantum behavior is critical to explaining how atomic bonds work.

One Electron, Two Nuclei

Imagine we take our hydrogen atom and move it next to a proton. Now there are two potential wells like the one above. If the wells are far enough apart, the electron only sees its own proton, it’s own hydrogen atom, as shown below.

However, if we move the new proton close enough to the hydrogen atom, the potential energy profile changes. It starts to look something like the figure below.

And now the true quantum nature of the electron comes into play. Remember when I said that quantum particles can exist where they classically should not? Well when we bring the two “potential wells” together, the electron in the left well has some probability of existing between the two wells. And, indeed, even if it started in the left well, the electron will ooze into the right well so that it spends about half its time in each well. Then the picture of the wavefunction look something like this:

But now something funny has happened. The electron used to have one wiggle in some amount of space. Now it has more wiggles, but it also has a lot more space. The result? The electron has lost a lot of energy!

This is why atoms bond. The electrons in an an atom want to be in the lowest energy state they can, and adding another atom lets the electrons lose some energy. There’s an optimal distance between the two nuclei which gives the electron a minimal energy. And this is what controls the length of atomic bonds.

Two Electrons?

Usually each atom in a covalent bond has an electron, not just one of the atoms. Fortunately, so long as there are only two electrons in the shared orbit (the one where the bonding happens), the electrons don’t see each other at all. Each electron chooses a “spin,” which has to do with the magnetic field an electron produces. (Spin will be the subject of a later article, I promise.) There are two possible spins and, so long as each electron has a different spin, they don’t see each other. However, if more electrons appear, we get a problem because there are only two possible spins and the third electron must choose a spin that’s already been taken. Then the electrons repel and the bond breaks.

See for Yourself?

The physics education group at the University of Colorado at Boulder has developed a simulation of a quantum particle bound by two potential wells. Click on the image below to see it in action! For the atomic bonding case, change the toggle on the right from “square” to “1D coulomb.”

Questions? Comments? Insults?

This post is a bit technical so if you have any questions please do ask! And if you’re a physical chemist and you know better than me, pipe up!