  Metal halide perovskites exhibit a materials physics that is distinct from
traditional inorganic and organic semiconductors. While materials such as
CH3NH3PbI3 are non-magnetic, the presence of heavy elements (Pb and I) in a
non-centrosymmetric crystal environment result in a significant spin-splitting
of the frontier electronic bands through the Rashba-Dresselhaus effect. We
show, from a combination of \textit{ab initio} molecular dynamics,
density-functional theory, and relativistic quasi-particle \textit{GW} theory,
that the nature (magnitude and orientation) of the band splitting depends on
the local asymmetry around the Pb and I sites in the perovskite structure. The
potential fluctuations vary in time as a result of thermal disorder and a
dynamic lone pair instability of the Pb(II) 6s$^{2}$6p$^{0}$ ion. We show that
the same physics emerges both for the organic-inorganic CH3NH3PbI3 and the
inorganic CsPbI3 compound. The results are relevant to the photophysics of
these compounds and are expected to be general to other lead iodide containing
perovskites.
