Why do we care about the small world properties of atomic clusters?

This summary is to try to keep a bigger picture in mind before getting too specific too quickly: a brief survey of a topic of interest in a text. It is specifically in the context of global optimization. I first wrote this post a few months back. Most of meat is obtained from Wales’s Energy Landscapes and Doye and Massen’s “Characterizing the network topology…atomic clusters”. Finding the global minimum and local minima on an energy landscape (a purely theoretical thing determined from pure quantum-mechanical models or hybrid quantum-classical models) gives us cues as to how a set of molecules will structurally stabalize. Practically, this information is invaluable to molecular structure prediction, which helps us better understand how molecules are actually built.

The issue lies in finding the global minimum. I don’t pretend to deeply know this area or detail it here by any means, but I may soon! Finding the global optimum in general is an NP-hard problem in computational complexity. In other words, “there is no known algorithm that is certain to solve the problem within a time that scales as a power of the system size”. So what do we do? Exploit structural properties of a given atomic cluster to drastically improve the running time of the algorithm. (Wales, “Global optimisation”)

This subfield in itself is very active with many different algorithms suited for all sorts of structures. But as far as small worlds go, it seems a basin hopping global optimization approach that could take advantage of the relatively scale-free properties of the landscape. But as implied (Wales, “Small worlds”), we should take caution in considering the cluster we’re applying these algorithms to, how exactly the global optimization search is improved, and whether the small world results apply generally enough to a broader class of molecules or not:

It is not yet clear how general this result may be for other potential energy surfaces [than Lennard-Jones]. For example, the connectivity of the constrained polymer conformations was found to exhibit a Guassian rather than a power law distribution for a noninteracting lattice polymer. For the Lennard-Jones polymer…Doye has found that the connectivity distribution does not really follow a power law, but the scaling is stronger than exponential…If low energy minima are generally highly connected then they should be easier to locate in global optimisation searches…However, the best way to utilise this information more efficiently is not yet clear.

Though it has been a few years since that was written, it seems to still hold true from my cursory understanding. Most of the work that has been done seems to be applied to Lennard-Jones clusters. Needless to say, I’m motivated to dig a little deeper.

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