Buckingham Catastrophe and Short-Range Repulsion

The Buckingham Potential and Its Dark Side

As you know from your work, the Buckingham potential is a common choice for modeling ionic systems like oxides. It combines a short-range exponential repulsion term (representing Pauli exclusion) with a long-range attractive dispersion term. Its general form is:

Under normal conditions, where atoms are near their equilibrium separation, this potential works well. The exponential term provides a strong, "soft" repulsive wall that prevents atoms from collapsing onto each other. However, the potential has a critical weakness at extremely short interatomic distances.

Let's analyze the behavior as the distance $r$ approaches zero:

• The repulsive term, $A\exp (-Br)$, converges to a finite positive value, $A$.
• The attractive term, $-C/r^6$, diverges to $-\infty$.

This means that at some very small separation, the attractive force, which is supposed to be a long-range effect, catastrophically overwhelms the short-range repulsion. Instead of repelling each other more strongly, two atoms that get too close will experience an immense, unphysical attraction, pulling them together with near-infinite force and sending the potential energy plummeting. This is the Buckingham catastrophe.

To understand this behavior more formally, let's turn to a short description.

This unphysical behavior is also discussed in the context of LAMMPS simulations. The following reading from a technical report describes the exact scenario you are likely facing.

Visualizing the Catastrophe

A plot of the potential energy function makes this flaw immediately obvious. Below is a comparison of several common interatomic potentials, all normalized to their equilibrium distance ($r_0$) and energy depth ($U_0$).

Pay close attention to the solid black line representing the Buckingham potential, especially on the left side of the plots where $r/r_0<1$. While potentials like Lennard-Jones (dash-dot line) correctly curve up towards positive infinity (strong repulsion), the Buckingham potential turns over and dives downwards. The peak of this curve represents the point of maximum repulsion; any closer, and the catastrophic attraction takes over.

Checking Your Simulation for the Catastrophe

Now for the most important part: how to diagnose this in your own system. The catastrophe is triggered if any two atoms governed by a Buckingham potential start the simulation closer than this critical turnover distance. Given that your instability is at the water-oxide interface, you must inspect the initial distances between water atoms and nanoparticle atoms.

Here is your diagnostic checklist:

1. Identify the Critical Distance: The exact turnover point depends on the $A$, $B$, and $C$ parameters of your cross-interaction. However, as a rule of thumb from the DTIC report and general experience, any interatomic distance below 1.5 Å for a Buckingham-style potential is in the danger zone. The region below 1.0 Å is almost certainly catastrophic.

2. Measure Your Initial Distances: Use a visualization tool like VMD or OVITO to load your initial data file (the one read by read_data before any run command).

   • Find the water-nanoparticle interface.
   • Select an oxygen atom from a water molecule (OW) that is very close to the nanoparticle surface.
   • Use the measurement tools to find the distance between this OW atom and the nearest atoms on the nanoparticle surface (e.g., Fe and O_oxide).
   • Do the same for the hydrogen atoms (HW) of that water molecule.
   • Repeat this for several water molecules at the interface to find the minimum separation distance in your entire system.

3. Analyze the Findings: If you find any water-oxide atom pairs with a separation distance in the danger zone (e.g., < 1.5 Å), you have very likely found the trigger for your simulation's instability. The minimize command or the first few dynamics steps will push these atoms into the catastrophic part of the potential, generating enormous forces and causing the simulation to crash.

This "atomic overlap" is especially common when placing a pre-built block of water around a nanoparticle, as the random orientation of water molecules can easily lead to a hydrogen or oxygen atom being too close to a surface atom.

Mitigating the Catastrophe

If you confirm that initial overlaps are triggering the catastrophe, what's next? While we will cover solutions in detail in Module 4, it's useful to know the strategies. Some force fields explicitly "fix" the Buckingham potential by adding a strongly repulsive term, such as $D/r^{24}$, to ensure the potential goes to $+\infty$ at $r=0$.

Since modifying the potential is an advanced step, the most immediate and practical solution is to ensure your initial geometry is sound. This involves carefully constructing the water-nanoparticle system to avoid any initial overlaps, which we will explore in detail later.

Conclusion

You have now reached the end of the initial triage module. You have a complete diagnostic toolkit to probe simulation instabilities, moving from high-level energy analysis down to the specific positions of individual atoms and the mathematical form of the potentials governing them.

Key Takeaways:

• The Buckingham potential contains a mathematical flaw, the "Buckingham catastrophe," where it becomes unphysically attractive at very short interatomic distances.
• This is caused by the $-C/r^6$ term overpowering the exponential repulsion as $r\to 0$.
• This instability can be triggered by initial atomic overlaps in the simulation geometry, particularly at interfaces between different materials.
• You can diagnose this by measuring the minimum atom-pair distances in your initial data file and checking if they fall within the potential's catastrophic region (typically < 1.5 Å).

By completing this check, you will have a definitive answer on whether the instability is caused by a simple (but fatal) geometric overlap. If it is, our path forward will focus on building a better initial state. If not, the problem is even more subtle and lies in the physical validity of the cross-interaction parameters themselves.

Next Lesson: We will begin Module 2, "Understanding Cross-Interaction Failures." We will explore the physical limitations of standard combining rules like Lorentz-Berthelot, especially for dissimilar materials like water and oxides, and start our journey into how more robust interaction parameters are derived.