Spin states and hyperfine interactions of iron in (Mg,Fe)SiO$_{3}$ perovskite under pressure

We have found several metastable equilibrium sites for substitutional iron in (Mg,Fe)SiO$_{3}$ with the guidance of first-principles phonon calculations. In the relevant energy range, there are two distinct sites for high-spin (HS), one for low-spin (LS), and one for intermediate-spin (IS) iron. The two competing HS sites have different iron nucleus quadrupole splittings (QS) due to their different $d$-orbital occupancies. At low pressure, the HS site with QS of 2.3 mm/s is more stable, while the HS site with QS of 3.3 mm/s is more favorable at higher pressure. The crossover from low-QS to high-QS site occurs between 4 and 24 GPa, depending on the exchange-correlation functional and on the inclusion of Hubbard $U$. Our calculation supports the notion that the transition observed in recent M\"ossbauer spectra corresponds to an atomic site change rather than a spin-state crossover.