@Lind_2012_Mater_NTEReview; @Takenaka_2012_NTEReview; @Miller_2009_NTEReview], a growing number of materials have been identified as exhibiting *negative thermal expansion* (NTE): the phenomenon whereby a material shrinks, rather than expands, on heating. NTE is found in several framework material families, such as those represented by SiO$_2$ [@NTE_quartz_data; @NTE_quartz_theory], Cu$_2$O [@Cu2O_Tiano; @Cu2O_Sanson; @Cu2O_Gupta; @Leila_Cu2O], ScF$_3$ [@ScF3_discovery; @ScF3_phonons], ZrW$_2$O$_8$ [@ZrW2O8_Science_1996; @Pryde_1996; @ZrW2O8_structure_1999; @Cao_etal_2002_PRL_ZrW2O8NTEFrustratedModes; @Cao_etal_2003_PRB_ZrW2O8NTEXAFS], Sr$_2$W$_3$O$_{12}$ [@Sc2W3O12_Evans], Zn(CN)$_2$ [@Goodwin_Zn(CN)2; @Hong_Zn(CN)2] as well as in a number of hybrid metal-organic framework materials [@MOFs_Han_2007; @MOFs_Dubbeldam_2007; @MOFs_Wu_2008].

Our understanding of positive thermal expansion is based on the nature of interatomic bonding. The potential energy between a pair of bonded atoms is anharmonic and asymmetric about the position of the energy minimum, with the slope of the potential energy function being shallower in the direction of greater interatomic separation. As a result, when the material is heated and the energy of the atoms increases, the average separation of each bonded pair also increases. This implies an overall expansion of the bulk material on heating; that is *positive* thermal expansion, making *negative* thermal expansion a counter-intuitive and unexpected phenomenon.

Our qualitative understanding of NTE in framework materials is based on the fact that, in NTE systems, the energy cost of stretching interatomic 