  Spin crossover of iron is of central importance in solid Earth geophysics. It
impacts all physical properties of minerals that altogether constitute $\sim
95$ vol\% of the Earth's lower mantle: ferropericlase [(Mg,Fe)O] and Fe-bearing
magnesium silicate (MgSiO$_3$) perovskite. Despite great strides made in the
past decade, the existence of intermediate-spin (IS) state in ferrous iron
(Fe$^{2+}$) (with total electron spin $S=1$) and its possible role in the
pressure-induced spin crossover in these lower-mantle minerals still remain
controversial. Using density functional theory $+$ self-consistent Hubbard $U$
(DFT$+U_{sc}$) calculations, we investigate all possible types of IS states of
Fe$^{2+}$ in (Mg,Fe)O and (Mg,Fe)SiO$_3$ perovskite. Among the possible IS
states in these minerals, the most probable IS state has an electronic
configuration that significantly reduces the electron overlap and the iron
nuclear quadrupole splitting (QS). These most probable IS states, however, are
still energetically disfavored, and their QSs are inconsistent with
M\"{o}ssbauer spectra. We therefore conclude that IS Fe$^{2+}$ is highly
unlikely in the Earth's lower mantle.
