5 - Describing the landscape

Summary

For an N-atom system, including models of bulk material with N atoms in a periodically repeated supercell, the potential energy is a 3N-dimensional function. To refer to a potential energy hypersurface we must embed the function in a 3N + 1 dimensional space where the extra dimension corresponds to the ‘height’ of the surface.

There are two immediate problems with trying to use such a high-dimensional function in calculations. The first is that it is hard to visualise, and the second is that the number of interesting features, such as local minima, tends to grow exponentially with N. In this chapter we first consider how the number of stationary points grows with the size of the system (Section 5.1), and then discuss how the PES can be usefully represented in graphical terms. Simply plotting the energy as a function of one or two coordinates for a high-dimensional function is usually not very enlightening, and can be rather misleading. A very different approach to reducing the 3N + 1 dimensions down to just two uses the idea of monotonic sequences (Section 5.2), and was introduced by Berry and Kunz (1,2). Subsequently, the utility of disconnectivity graphs was recognised by Becker and Karplus (3), and a number of examples have been presented, as discussed in Section 5.3.