O/F anion order in Pb$_2$Ti$_4$O$_9$F$_2$ stabilized by the 6$s^2$ lone pair electrons of Pb$^{2+}$

Understanding the mechanisms of anionic ordering in mixed anion compounds is a crucial factor for designing their structural and functional properties. Some oxyfluorides are known to exhibit the F$^{-}$ ordering. The ordering is accompanied by a tetragonal distortion, which is believed to be originated by the 6$s^2$ lone pairs of Bi and Pb. To elucidate the role played by the lone pair, we studied isostructural Bi$_2$Ti$_4$O$_{11}$ and Pb$_2$Ti$_4$O$_9$F$_2$ with a combination of synchrotron X-ray diffraction techniques and ab initio calculations. Bi$_2$Ti$_4$O$_{11}$ undergoes antiferroelectric-paraelectric transitions from $C2/c$ to $C2/m$. Meanwhile, Pb$_2$Ti$_4$O$_9$F$_2$ does not, because F$^-$ selectively occupies the closest anion site to Bi/Pb and reduces the distortion. The question on why a particular site is selectively occupied by F$^-$ in Pb$_2$Ti$_4$O$_9$F$_2$ becomes important to understand the role of 6$s^2$ lone pair electrons for stabilizing the anion order. Our DFT ab initio calculations reproduced the same anionic arrangement in terms of the total energy. However, this energy gain cannot be fully explained by the electrostatic energy, being an exception of Pauling's second rule. We explain the reason focusing on the steric effects of 6$s$ lone pairs.