  Fluorite CeO$_2$ doped with group IV elements is studied within the DFT and
DFT+U framework. Concentration dependent formation energies are calculated for
Ce$_{1-x}$Z$_x$O$_2$ (Z= C, Si, Ge, Sn, Pb, Ti, Zr, Hf) with $0\leq x \leq
0.25$ and a roughly decreasing trend with ionic radius is observed. The
influence of the valence and near valence electronic configuration is
discussed, indicating the importance of filled $d$ and $f$ shells near the
Fermi level for all properties investigated. A clearly different behavior of
group IVa and IVb dopants is observed: the former are more suitable for surface
modifications, the latter are more suitable for bulk modifications.\\ \indent
For the entire set of group IV dopants, there exists an inverse relation
between the change, due to doping, of the bulk modulus and the thermal
expansion coefficients. Hirshfeld-I atomic charges show that charge transfer
effects due to doping are limited to the nearest neighbor oxygen atoms.
