We explore whether the topology of energy landscapes in chemical systems obeys any rules and what these rules are. To answer this and related questions we use several tools: (i)Reduced energy surface and its density of states, (ii) descriptor of structure called fingerprint function, which can be represented as a one-dimensional function or a vector in abstract multidimensional space, (iii) definition of a distance between two structures enabling quantification of energy landscapes, (iv) definition of a degree of order of a structure, and (v) definitions of the quasi-entropy quantifying structural diversity. Our approach can be used for rationalizing large databases of crystal structures and for tuning computational algorithms for structure prediction. It enables quantitative and intuitive representations of energy landscapes and reappraisal of some of the traditional chemical notions and rules. Our analysis confirms the expectations that low-energy minima are clustered in compact regions of configuration space (funnels) and that chemical systems tend to have very few funnels, sometimes only one. This analysis can be applied to the physical properties of solids, opening new ways of discovering structure-property relations. We quantitatively demonstrate that crystals tend to adopt one of the few simplest structures consistent with their chemistry, providing a thermodynamic justification of Paulings fifth rule.
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