  DFT calculations have been used to provide insights into the origin of the
colossal positive and negative thermal expansion in Ag$_3$[Co(CN)$_6]$. The
results confirm that the positive expansion within the trigonal basal plane and
the negative expansion in the orthogonal direction are coupled due to the
existence of a network defined by nearly-rigid bonds within the chains of
Co--C--N--Ag--N--C--Co linkages. The origin of the colossal values of the
coefficients of thermal expansion arise from an extremely shallow energy
surface that allows a flexing of the structure with small energy cost. The
thermal expansion can be achieved with a modest value of the overall
Gr\"{u}neisen parameter. The energy surface is so shallow that we need to
incorporate a small empirical dispersive interaction to give ground-state
lattice parameters that match experimental values at low temperature. We
compare the results with DFT calculations on two isostructural systems:
H$_3$[Co(CN)$_6$], which is known to have much smaller values of the
coefficients of thermal expansion, and Au$_3$[Co(CN)$_6$], which has not yet
been synthesised but which is predicted by our calculations to be another
candidate material for showing colossal positive and negative thermal
expansion.
