  The spin state transition in LaCoO$_3$ has eluded description for decades
despite concerted theoretical and experimental effort. In this study, we
approach this problem using fully charge self-consistent Density Functional
Theory + Embedded Dynamical Mean Field Theory (DFT+DMFT). We show from first
principles that LaCoO$_3$ cannot be described by a single, pure spin state at
any temperature. Instead, we observe a gradual change in the population of
higher spin multiplets with increasing temperature, with the high spin
multiplets being excited at the onset of the spin state transition followed by
the intermediate spin multiplets being excited at the metal insulator
transition temperature. We explicitly elucidate the critical role of lattice
expansion and oxygen octahedral rotations in the spin state transition. We also
reproduce, from first principles, that the spin state transition and the
metal-insulator transition in LaCoO$_3$ occur at different temperature scales.
In addition, our results shed light on the importance of electronic entropy in
driving the spin state transition, which has so far been ignored in all first
principles studies of this material.
