  We report on investigations of the complex magnetostructural and spin-state
transitions in the breathing pyrochlore LiFeCr$_{4}$O$_{8}$ by means of
magnetization, M\"ossbauer spectroscopy, and density functional theory (DFT)
calculations. Three transitions corresponding to the ferrimagnetic transition
at $T_N\sim94$ K, the spin-gap transition at $T_{SG}\sim50$ K, and the
magnetostructural transition at $T_{MS}\sim19$ K were observed from the
$\chi$(T) curve, whereas only $T_N$ and $T_{MS}$ were evidenced for the Fe site
from our M\"ossbauer measurements, suggesting that the spin-gap transition is
absent at the Fe site. This indicates that the spin-gap transition is an effect
of the breathing Cr$_4$ lattice, in agreement with our DFT calculations from
which we see nearly decoupled electronic states for the FeO$_4$ and CrO$_6$
units. From the temperature dependence of the hyperfine magnetic field we also
observed a spin-state transition for the Fe spins at $T_{MS}$ consistent with
earlier neutron diffraction measurements. These local characteristics are
believed to be important for a complete understanding of the complex
magnetostructural coupling effects observed in similar systems.
