Intermediate-spin ferrous iron in the Earth's lower mantle?

Using density functional theory $+$ self-consistent Hubbard U (DFT$+$Usc) calculations, we investigate intermediate-spin (IS) ferrous iron (Fe$^{\mathrm{2+}})$ in major lower-mantle minerals, ferropericlase (Fp) and magnesium silicate (MgSiO$_{\mathrm{3}})$ perovskite (Pv). In both minerals, two distinct types of IS Fe$^{\mathrm{2+}}$ are found. In Fp, while both types of IS Fe$^{\mathrm{2+}}$ are configured t$_{\mathrm{2g}}^{\mathrm{5}}$ e$_{\mathrm{g}}^{\mathrm{1}}$, one has a d$_{\mathrm{z2}}$ electron$_{\mathrm{\thinspace }}$and the other has a d$_{\mathrm{x2-y2}}$ electron, referred to as the IS(z$^{\mathrm{2}})$ and IS(x$^{\mathrm{2}}-$y$^{\mathrm{2}})$ state, respectively.$_{\mathrm{\thinspace }}$The IS(z$^{\mathrm{2}})$ state has an exceptionally high QS ($\ge $ 5.5 mm/s); the IS(x$^{\mathrm{2}}-$y$^{\mathrm{2}})$ state has a quite low QS (\textless 0.5 mm/s). Also, the IS(z$^{\mathrm{2}})$ state has a stronger on-site Coulomb interaction and much higher energy. In Pv, while Fe$^{\mathrm{2+}}$ substitutes Mg in the dodecahedral site, it is effectively under a distorted octahedral crystal field, and the two IS states can be characterized by their filled e$_{\mathrm{g}}$-like orbitals as well. These two IS Fe$^{\mathrm{2+}}$, in contrast to those in Fp, are energetically competitive, and they both have a small QS (\textless 1.6 mm/s). Our calculations show that all IS Fe$^{\mathrm{2+}}$ in lower-mantle minerals are unfavorable, and their QSs are all inconsistent with experiments. Therefore, IS Fe$^{\mathrm{2+}}$ is highly unlikely in the Earth's lower mantle.