# Chemical bonding

#### Prerequisites

In our discussion of Interatomic forces we learned that the typical potential energy between two atoms includes two parts:  an attractive part that falls off rapidly with increasing distance (proportional to $1/r^6$), and a repulsive part that is very short ranged and dominates when the atoms try to get too close. The commonly used Lennard-Jones model of this repulsion goes like $1/r^{12}$, so the total PE is modeled by the equation

$$U(r) = \epsilon \bigg[ \bigg(\frac{\sigma}{r}\bigg)^6 - \bigg(\frac{\sigma}{r}\bigg)^{12} \bigg]$$

where $\epsilon$ and $\sigma$ are constants whose values represent the energy and distance scales appropriate to the specific pair of atoms. The shape of the PE curve looks like the graph shown in the figure at the right.

At large $r$, the potential energy graph looks flat. The slope is just about zero.  Thus, atoms that are far apart feel just almost no force. At small $r$, the graph slopes very steeply down and to the right, indicating that there is a very strong repulsive force at close range.

Since the strong positive repulsion dominates at short distances, while the longer-range negative attraction dominates at longer distances, in between the graph has to turn around.  As the atoms come towards each other from far away, the potential energy slopes downward going negative, and then becomes repulsive and goes positive.  Somewhere in between, it has to reach a minimum potential energy, before going back up again.

The slope of the graph at this point is zero, so two atoms located at this distance experience no force.  The attractive and repulsive forces balance exactly.  In classical physics this is a stable equilibrium: if you move the atom away from this point in either direction, it will experience a force pushing it back towards the equilibrium point. If the atoms were classical, they could stably remain at that separation. Since energy would have to be put in to separate them to $r = \infty$, they are bound. For those atoms that bond, the value of $r$, where the potential energy is at a minimum, tells us the bond length for that particular state.

If electrons and nuclei behaved according to the rules of classical physics, atoms that have this potential energy interaction would all be able to form stable molecules, but things are a bit more complicated than that. Electrons inside atoms and molecules are described by the laws of quantum physics and this changes things dramatically.  The lowest possible energy of the pair isn't actually sitting at the bottom of the well: it's oscillating around the bottom of that well typically a little energy up from the bottom. If the potential well is not deep enough, the atoms might not be able to bind at all (e.g., He-He interaction). Which particular atoms can bond and what kind of bonds they form depends on detailed quantum calculations. We will mostly leave those discussions for chemistry and more advanced physics classes. For most of this class we will only need two results:

• For those atoms that bond, the Lennard-Jones potential provides a reasonable description of the energy of their interaction (at least for separations larger than the bottom of the well.
• Not all energies are allowed. Only particular discrete levels (quantum states) are permitted. The lowest allowed state sits near (but not at) the bottom of the well. The two interacting atoms can't "roll all the way down to the bottom of the well." Rather, they oscillate a bit around the bottom. We'll indicate this by drawing a black line across the well indicating the energy level.

Here's a link to a simulation of two atoms coming together and forming a bond.  (In this simulation, the atoms don't stay together!  Why not? What would need to happen for them to stay together?)

An example of a pair of atoms in a bound state is shown in the figure at the left by the heavy black line. This represents a particular case of a state of the total mechanical (kinetic plus potential) energy of the atoms' relative motion.

Note that the total energy of the atoms is NEGATIVE.  "Negative relative to what?", you might ask. Relative to our "zero" potential energy, which we take to be when the atoms are far apart and at rest.  So there is LESS total mechanical energy when atoms are bonded together than when they are separated. The negative value of the energy (the black line in the diagram) is referred to as the binding energy of the two-atom molecule.

This has two important consequences:

1. If atoms start out bonded together, you have to ADD an energy whose magnitude is equal to the magnitude of the binding energy just to get them back to "zero" potential energy.  "Breaking" the bond requires an input of energy.
2. In reverse: If a bond is formed (between atoms which were previously separate), the result is less potential energy than they started with, but by the principle of conservation of energy, we know this energy had to go somewhere else (it doesn't just disappear).  Thus, when bonds are formed, energy is released. (Where does the energy go when a bond is formed?  We'll get into some answers to this question later on, but in the meantime, think about this question and try to come up with some possible answers yourself.)

This is decidedly tricky! Especially since in chemistry class they typically report the bond energy of a chemical bond, NOT the binding energy. This often causes confusion.

The bond energy is what you have to add to break the bond; the binding energy is the initial energy of the state and is equal to the negative of the bond energy. Strong bonds correspond to large (positive) bond energies and deep (negative) wells. Bond energies are useful when comparing bond strengths and doing calculations in chemistry. But when we try to make sense of energy conservation and the flow of energy in a reaction here in physics, the binding energies will be much more useful.

Some caveats: Molecules are complicated! There are lots of strongly interacting particles running around and interacting dynamically. The Lennard-Jones potential (and the more accurate Morse potential) is only a simplified model for atom-atom interactions. When one has strong bonds where the electron orbitals are shared and even modified, a model describing the interaction as occurring between atoms is not really sufficient. As noted above, a better description explicitly needs to consider the structure of the atom, separating it into a nucleus and electrons. You will learn about more sophisticated phenomenological models in chemistry, and there are theoretical calculations in quantum theory that can accurately describe even complex organic molecules.

But these simple potential models allow us to talk about the energy balances involved when chemical reactions happen in a convenient way. It describes the transfer of energy accurately, even if the mechanism is not quite correct.

Ben Dreyfus 10/30/11 and Joe Redish 11/17/11 and 2/26/19

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