  We have theoretically investigated the elastic properties of three
topologically identical zeolitic imidazolate frameworks: ZIF-4, ZIF-62 and
TIF-4, by means of ab initio calculations. The ZIFs are a subset of metal
organic frameworks (MOFs), whose versatile functionality is providing exciting
opportunities in solid state science. Understanding the relationship between
structure and elastic properties of hybrid materials is making it possible to
develop a new generation of functional materials. In this work we have
determined the bulk ($K$), shear ($G$) and Young's moduli ($E$), together with
Poisson's ratio($\upsilon$), and the universal elastic anisotropy index
($A$$^{\textbf{U}}$) from the computed elastic coefficients. Tensorial analysis
of the elastic constants reveals unusual and highly anisotropic elastic
behaviour. All three materials exhibit low Young's and shear moduli. In
addition, their flexibility incorporates regions of negative Poisson's ratio
(NPR) and negative linear compressibility (NLC). The elastic properties of
these ZIF crystals are are affected by the substituted organic linkages.
