  The spin crossover of iron in Fe$^{3+}$-bearing bridgmanite, the most
abundant mineral of the Earth's lower mantle, is by now a well-established
phenomenon, though several aspects of this crossover remain unclear. Here we
investigate effects of disorder, iron concentration, and temperature on this
crossover using ab initio LDA + U$_{sc}$ calculations. The effect of
concentration and disorder are addressed using complete statistical samplings
of coupled substituted configurations in super-cells containing up to 80 atoms.
Vibrational/thermal effects on the crossover are addressed within the
quasiharmonic approximation. The effect of disorder seems quite small, while
increasing iron concentration results in considerable increase in crossover
pressure. Our calculated compression curves for iron-free, Fe$^{2+}$-, and
Fe$^{3+}$-bearing bridgmanite compare well with the latest experimental
measurements. The comparison also suggests that in a close system, Fe$^{2+}$
present in the sample may transform into Fe$^{3+}$ by introduction of Mg and O
vacancies with increasing pressure. As in the spin crossover in ferropericlase,
this crossover in bridgmanite is accompanied by a clear volume reduction and an
anomalous softening of the bulk modulus throughout the crossover pressure
range. These effects reduce significantly with increasing temperature. Though
the concentration of [Fe$^{3+}$]$_{Si}$ in bridgmanite may be small, related
elastic anomalies may impact the interpretation of radial and lateral velocity
structures of the Earth's lower mantle.
