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 $0leq x leq 0.25$ and a roughly de
creasing 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.
The issues raised in the comment by T.A. Manz are addressed through the presentation of calculated atomic charges for NaF, NaCl, MgO, SrTiO$_3$ and La$_2$Ce$_2$O$_7$, using our previously presented method for calculating Hirshfeld-I charges in Solids
[J. Comput. Chem.. doi: 10.1002/jcc.23088]. It is shown that the use of pseudo-valence charges is sufficient to retrieve the full all-electron Hirshfeld-I charges to good accuracy. Furthermore, we present timing results of different systems, containing up to over $200$ atoms, underlining the relatively low cost for large systems. A number of theoretical issues is formulated, pointing out mainly that care must be taken when deriving new atoms in molecules methods based on expectations for atomic charges.
In this work, a method is described to extend the iterative Hirshfeld-I method, generally used for molecules, to periodic systems. The implementation makes use of precalculated pseudo-potential based charge density distributions, and it is shown that
high quality results are obtained for both molecules and solids, such as ceria, diamond, and graphite. The use of such grids makes the implementation independent of the solid state or quantum chemical code used for studying the system. The extension described here allows for easy calculation of atomic charges and charge transfer in periodic and bulk systems.
The crystal structure of Lanthanum Cerium Oxide (La$_2$Ce$_2$O$_7$) is investigated using textit{ab initio} density functional theory (DFT) calculations. The relative stability of fluorite- and pyrochlore-like structures is studied through comparison
of their formation energies. These formation energies show the pyrochlore structure to be favored over the fluorite structure, apparently contradicting the conclusions based on experimental neutron and X-ray diffraction (XRD). By calculating and comparing XRD spectra for a set of differently ordered and random structures, we show that the pyrochlore structure is consistent with diffraction experiments. For these reasons, we suggest the pyrochlore structure as the ground state crystal structure for La$_2$Ce$_2$O$_7$. %we show that among the structures considered in this work, the pyrochlore geometry is clearly favorable over the disordered fluorite geometry.
