Evaluating the LJ potential

The LJ potential is a useful model, but like all models, it has limitations. To evaluate a model, we have to evaluate it in light of the purpose(s) it is meant to serve. Often people think that scientific models are only meant to predict, but models can serve other purposes. A model can explain a phenomenon without making precise predictions. For example, the model of plate tectonics explains why earthquakes happen, but we still can’t predict exactly when they will happen (we can, however, predict the general areas they will happen based on the locations of tectonic plates). Other reasons for constructing models include helping us reason about phenomena, honing our intuitions, and helping us design technologies. Sometimes there are trade-offs between different purposes of models. Most prominently, the models that are best at prediction are often the hardest to understand, making them less useful for helping us hone our intuitions and reason about a phenomenon.

In reflecting on the model we built we can ask ourselves a couple questions.

First, How is this model useful? It hold values because it:

• Allows us to better conceptualize interatomic interactions.
• Predicts real phenomenon such as thermal expansion and bond dissociation (melting/boiling).

Second, How is this model limited?

• Imprecise predictions: The LJ-potential is simple in form, but doesn't capture the precise details of real interactions between atoms, which are founded in quantum mechanics. Scientists and engineers have made some simple adjustments to improve the quantitative predictive ability of the LJ-potential, which better match physical observation.
• Unaccounted-for phenomenon: Speaking of quantum mechanics, an obvious limitation of the LJ potential is that it cannot model any quantum phenomenon. The LJ potential also ignores the existence of electrons. Instead, it treats atoms as unified wholes. This means it can tell us nothing about the electrical or optical properties of materials, as they depend on the behavior of electrons and their interactions with photons. The LJ potential is largely limited to physical and mechanical behaviors of materials.
• Isotropic Bonding: The LJ potential can only model bonds that are isotropic - the same regardless of direction - because the LJ potential only depends on the absolute distance between atoms, not their relative orientations. This sort of isotropic bonding does occur in many materials and results in close-packed structures (e.g., solid phases of group 18 elements (He, Ne, Ar...)). However, there are many different crystal structures that form in part due to the influence of bond directionality (i.e., the electron orbitals have non-isotropic distributions in spaces, imagine a $p$-orbital. The LJ potential will not be able to predict these structures.

In summary, the simplicity of the LJ potential makes it easy to understand and therefore useful for qualitatively explaining and reasoning about various mechanical properties of materials. For this reason, we will continue to use it at various points in the course. However, the simplicity of the LJ-potential means that there are many phenomena it can’t explain. For those phenomena, we will need other models. We will see some in the next chapter.