How molecular dynamics works

This page builds the idea from the beginning. Knowing that force makes things accelerate is enough to start. Everything below is something to play with.

1. What is a film of atoms actually for?

Making the film is not the point. What we want are answers we cannot get any other way. They come in two kinds.

Numbers we can check against experiment. Nearly every property of a material is really an average over atoms jiggling about. Run the film long enough, average over it, and out come real measurable quantities: melting and boiling points, density, stiffness, viscosity, how tightly a drug molecule holds on to its target, how fast a reaction goes. Those averages are the actual product. The film is just how we get them.

Mechanisms nothing can show us. No instrument watches a protein fold, or lithium ions squeezing through a battery electrode, or a crack starting deep inside a lump of metal. Experiments usually tell us what happened. The film tells us how it happened.

Some things this is genuinely used for:

• Drug discovery. Does this candidate molecule stay stuck to its target protein, and how tightly? That number decides whether a drug is worth making.
• Materials. At what temperature does this alloy melt? Where does it crack, and why there?
• Batteries. How fast do ions move through this electrolyte? That sets how quickly the battery can charge.
• Biology. How does an ion channel in a cell membrane open and shut?

2. The whole method is one loop

Here is the entire algorithm. There is nothing hidden behind it.

1. Look at where all the atoms are.
2. Work out the force on each atom.
3. Use a = F / m to get each atom's acceleration.
4. Move every atom forward by a very small slice of time.
5. Go back to step 1.

That "small slice of time" is usually about one femtosecond, a millionth of a billionth of a second. Atoms vibrate incredibly fast, so if we take bigger steps we miss the motion and the simulation blows up.

3. Force is the steepness of an energy hill

Step 2 of the loop says "work out the force". Where does it come from? In almost every simulation, forces are not stored directly. Instead we store energy, and the force falls out of it.

Think of a ball on a hill. Where the hill is steep, the ball is pushed hard. Where it is flat, nothing pushes it. And the push is always downhill. That is the whole rule:

The minus sign just means "downhill". If we know the energy of every arrangement of atoms, we know every force, everywhere. So the real job of a molecular simulation is to answer one question: given where the atoms are, what is the energy?

4. The energy between two atoms

Now we need actual energy values. Start with the simplest case in all of chemistry: two atoms of argon, a gas that does not react with anything.

They feel two competing effects.

• They pull together when far apart. Even neutral atoms attract each other faintly, because their electrons wobble in step. This is weak and fades fast with distance.
• They push apart hard when too close. Two atoms cannot sit in the same place. Squeeze them and they shove back extremely violently.

Add those together and we get a curve with a dip in it. The bottom of the dip is the comfortable distance, where the atoms would happily sit. This is the Lennard-Jones potential, and it is the single most used model in all of molecular simulation.

5. Many atoms: just add up every pair

With more than two atoms, the simplest assumption is that the total energy is the sum over every pair of atoms, one pair at a time:

That is called a pair potential. It is a guess, and a surprisingly good one for simple things like argon. Below are a few hundred argon atoms doing nothing but obeying that rule.

Nobody programmed "become a liquid" or "form a crystal" into this. Those behaviours simply appear once we add up pairs and let the atoms move.

6. Temperature is just how fast the atoms are moving

"Temperature" sounds like its own thing, but in a simulation it is only a measure of motion. Add up the kinetic energy of every atom, divide by the number of atoms, and that is the temperature.

Not every atom moves at the same speed. They constantly collide and trade energy, so at any moment there is a spread, a few crawling, a few racing, most somewhere in the middle. That spread always settles into the same characteristic shape.

7. Getting a real number out of the film

Section 1 promised that the point of all this is averages we can compare with the real world. Here is one of them, done properly.

Pick a moment. Remember where every atom is. Then keep watching, and measure how far each atom has wandered from its remembered spot. Average the square of that distance over all the atoms. That is the mean squared displacement, and it settles the question "solid or liquid?" without anybody having to squint at the picture.

• In a solid each atom is trapped by its neighbours. It rattles in place but never goes anywhere, so the curve rises a little and then flattens off.
• In a liquid atoms slide past one another and keep wandering, so the curve climbs steadily and never flattens.

The slope of that climb is the diffusion coefficient, a number we can look up in a table and check. This is a measurement, not a picture.

8. Where "add up the pairs" falls apart

Pair potentials work well for argon. For metals they fail, and the reason is worth understanding, because it is the same reason machine learning became useful later.

In a metal the outer electrons are not tied to single atoms. They are shared across the whole piece of metal, like a pooled resource. So how strongly an atom is held depends on how many neighbours it has in total, not on each neighbour separately.

Add a neighbour to a lonely atom and we get a strong new bond. Add a neighbour to an atom that already has eleven and we barely gain anything, the shared electrons are already spread thin. Bonds get weaker as atoms get more crowded.

A pair potential cannot express that. It counts every bond separately, so twelve neighbours must be worth exactly twelve times one neighbour. The fix is the embedded atom method (EAM): on top of the pair sum, each atom gets an extra energy that depends on the total electron density it sits in, and that extra term grows like a square root rather than a straight line.

We can check which one is right with a real measurement. A vacancy is a missing atom inside a crystal. The energy needed to make one is compared below with the energy needed to boil a single atom away entirely (the cohesive energy).

A pair potential is not slightly off here. It is wrong by roughly a factor of three, and no amount of re-tuning its numbers fixes it, the shape of the model is what is wrong. Adding one many-body term repairs it.

Copper values: cohesive energy 3.54 eV, vacancy formation energy about 1.28 eV.

9. So the whole field is one question

Every model on this page is an attempt to answer the same question: given the positions of the atoms, what is the energy? Once we can answer it, the loop in section 2 does the rest.

The answers get better and more expensive as we go down this list:

Notice the pattern. Every model above is a shape that a human chose, with a few numbers fitted to data. Choose the shape well and it works in the situations it was designed for. Step outside those situations and it breaks, and we cannot fix it by adjusting the numbers.

Aayush Neupane
CS Researcher & Engineer · MS CS, University of Kansas

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