  Combining a precise ab initio electron band structure calculation of the TiO2
rutile structure with the temperature evolution of the Ti mean-square
displacements, we reproduce a puzzling temperature increase of the electric
field gradient at Ti sites in TiO2, observed experimentally. Our method employs
a procedure of averaging two quadrupole electron density components (L = 2)
inside a sphere vibrating with the Ti nucleus at its center, where the key
factor introducing the temperature dependence is the square root of the
Debye-Waller factor. Although the Debye-Waller factor always reduces the
corresponding Fourier component, in TiO2 due to the interplay between terms of
opposite signs, it results in a net increase of the whole sum with temperature,
leading to the growth of the electric field gradient. Quantitatively, we find
that the increase of electric field gradient is only half of the experimental
value, which we ascribe to anharmonic effects or a strong oxygen position
influence. In addition, our method reproduces the unusual temperature
dependence of the asymmetry parameter eta, which first decreases with
temperature, goes to zero and then increases.
