  The appearance of microcracks in CeO$_2$ buffer layers, as used in buffer
layer architectures for coated superconductors, indicates the presence of
stress between this buffer layer and the substrate. This stress can originate
from the differences in thermal expansion or differences in lattice parameters
between the CeO$_2$ buffer layer and the substrate. In this article, we study,
by means of \textit{ab initio} density functional theory calculations, the
influence of group IV doping elements on the lattice parameter and bulk modulus
of CeO$_2$. Vegard's law behavior is found for the lattice parameter in systems
without oxygen vacancies, and the Shannon crystal radii for the doping elements
are retrieved from the lattice expansions. We show that the lattice parameter
of the doped CeO$_2$ can be matched to that of the La$_2$Zr$_2$O$_7$ coated NiW
substrate substrate for dopant concentrations of about $5\%$, and that bulk
modulus matching is either not possible or would require extreme doping
concentrations.
