Exact mapping from short-ranged harmonic elastic models of spin crossover materials to Ising models with interactions at all length scales
Spin crossover (SCO) materials are reversible molecular switches found in a wide range of transition metal complexes and metal organic frameworks (MOFs). They exhibit diverse spin state orderings and transitions between them. Here we present an exact mapping from harmonic elastic models to Ising-like models with both a short-range Ising interaction that decays with a power law at large distances and a long-range (infinite-range) Husimi-Temperley interaction that is independent of distance. We apply this mapping to a simple model of SCO frameworks. This provides a microscopic justification for an Ising-Husimi-Temperley model description, which has previously only been justified on phenomenological grounds. Elastic frustration is required for non-zero Ising interactions, but whether or not the short-range interactions in the Ising model are geometrically frustrated depends on the ratio of the bulk and shear moduli, or equivalently Poisson's ratio. The long-range interaction has two origins: (i) a self-interaction on the average spin state, mediated through the coupling between the average spin state and the unit cell parameters; and (ii) an infrared divergence in the spin state-spin state coupling mediated by displacements of metals and ligands within the unit cell. %In the absence of elastic frustration these terms are equal and opposite so there is no long-range interaction. However, in general they do not cancel and there is a long-range Ising interaction. The long-range interaction is independent of the distance between metal centers, nevertheless it leads to an extensive contribution to the (free) energy. In this model the Husimi-Temperley interaction dominates transitions of spin states, whereas multistep transitions and intermediate order are observed if only the pure (power-law) Ising interaction are retained, only single-step transitions are found in the full model.