We present and discuss the results of calculations of ${\mathrm{SrTiO}}_3$ (100) surface relaxation and rumpling with two different terminations (SrO and ${\mathrm{TiO}}_2).\mathrm{}$ These are based on ab initio Hartree-Fock method with electron correlation corrections and density functional theory calculations with different exchange-correlation functionals, including hybrid exchange techniques. Both approaches use the localized Gaussian-type basis set. All methods agree well on surface energies and on atomic displacements, as well as on considerable increase of covalency effects nearby the surface. More detailed experiments on surface rumpling and relaxation are necessary for further testing theoretical predictions.

We present and discuss the results of calculations of surface relaxations and rumplings for the (001) and (011) surfaces of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$ and $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ using a hybrid B3PW description of exchange and correlation. On the (001) surfaces, we consider both $A\mathrm{O}$ ($A=\mathrm{Ba}$ or Pb) and $\mathrm{Ti}{\mathrm{O}}_2$ terminations. In the former case, the surface $A\mathrm{O}$ layer is found to relax inward for both materials, while outward relaxations of all atoms in the second layer are found at both kinds of (001) terminations and for both materials. The surface relaxation energies of BaO and $\mathrm{Ti}{\mathrm{O}}_2$ terminations on $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$ (001) are found to be comparable, as are those of PbO and $\mathrm{Ti}{\mathrm{O}}_2$ on $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ (001), although in both cases the relaxation energy is slightly larger for the $\mathrm{Ti}{\mathrm{O}}_2$ termination. As for the (011) surfaces, we consider three types of surfaces, terminating on a TiO layer, a Ba or Pb layer, or an O layer. Here, the relaxation energies are much larger for the TiO-terminated surface than for the Ba- or Pb-terminated surfaces. The relaxed surface energy for the O-terminated surface is about the same as the corresponding average of the TiO- and Pb-terminated surfaces on $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$, but much less than the average of the TiO- and Ba-terminated surfaces on $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$. We predict a considerable increase of the Ti-O chemical bond covalency near the $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$ and $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$ (011) surfaces as compared to both the bulk and the (001) surface.

We present and discuss the results of the calculations of surface relaxation and rumpling on ${\text{SrTiO}}_3$ (001) and (011) surfaces. We consider both SrO and ${\text{TiO}}_2$ terminations of the (001) surface, and three terminations (Sr, TiO, and O) of the polar (011) surface. The calculations are based on hybrid Hartree–Fock and density-functional theory exchange functionals by using Becke’s three-parameter method combined with the nonlocal correlation functionals of Perdew and Wang. We find that all top-layer atoms for ${\text{TiO}}_2$ and SrO-terminated ${\text{SrTiO}}_3$ (001) surfaces relax inward, with the exception of SrO-terminated surface O atoms, whereas all second-layer atoms relax outward. The surface rumpling for the TiO-terminated ${\text{SrTiO}}_3$ (011) surface, which is 11.28% of the bulk lattice constant, is considerably larger than the relevant surface rumplings for SrO and ${\text{TiO}}_2$-terminated (001) surfaces. The surface rumplings for the SrO and ${\text{TiO}}_2$-terminated (001) surfaces are in excellent agreement with relevant low-energy electron diffraction and reflection high-energy electron diffraction experimental data, and the surface relaxation energies on both surfaces are similar. In contrast, the different terminations of the (011) surface lead to large differences in relaxation energies. The O-terminated (011) surface has the lowest surface relaxation energy $\left(-1.32\text{eV}\right)$. The TiO-terminated (011) surface has a much higher surface relaxation energy of $-1.55\text{eV}$, while the Sr-terminated (011) surface has the highest surface relaxation energy $\left(-1.95\text{eV}\right)$. Our calculations indicate a considerable increase in the Ti-O bond covalency $\left(0.130e\right)$ near the TiO-terminated (011) surface relative to the bulk $\left(0.088e\right)$, which is much larger than that for the (001) surface $\left(0.118e\right)$. The Ti-O bond populations are considerably larger in the direction perpendicular to the TiO-terminated (011) surface $\left(0.188e\right)$ than in the plane $\left(0.130e\right)$.