  We have computed second and third nonlinear optical susceptibilities of two
crystalline bulk tellurium oxide polymorphs: $\alpha$-TeO$_{2}$ (the most
stable crystalline bulk phase) and $\gamma$-TeO$_{2}$ (the crystalline phase
that ressembles the more to the glass phase. Third order nonlinear
susceptibilities of the crystalline phases are two orders of magnitude larger
than $\alpha$-SiO$_{2}$ cristoballite, thus extending the experimental
observations on glasses to the case of crystalline compounds. While the
electronic lone pairs of Te contribute to those large values, a full
explanation of the anisotropy of the third order susceptibility tensor requires
a detailed analysis of the structure, in particular the presence of helical
chains, that seems to be linked to cooperative non-local polarizabilty effects.
Our results demonstrate that first-principles simulations are a powerful
predictive tool to estimate nonlinear optical susceptibilitites of materials.
