  We study energetics and the nature of both homogeneous and mixed spin (MS)
states in LaCoO$_3$ incorporating structural changes of the crystal volume
expansion and the Co-O bond disproportionation (BD) during the spin-state
transition using the density functional theory plus dynamical mean field theory
(DFT+DMFT) method. DFT+DMFT predicts that energetics of both excited spin
states are almost the same while DFT+U calculations of the same structures
energetically favor the MS states and produce various metastable solutions
whose energetics depend sensitively on final spin states. Within DFT+DMFT, the
homogeneous spin state in the expanded crystal volume shows the
multiconfigurational nature with non-negligible occupancy probabilities of both
high spin (HS) and low spin (LS) states along with $d^6$ and $d^7$ charge
configurations indicating the dynamically fluctuating nature of spin and charge
states due to the Co-O covalency. The nature of the MS state under the BD
structure reveals that Co sites with the long Co-O bonds develop a Mott
insulating state and favor HS with a $d^6$ configuration, while more covalent
Co sites with the short Co-O bonds occupy more LS states with a $d^7$
configuration and behave as a band insulator, as a result, charge ordering is
induced in the BD structure from the spin-state ordering. We also find that
both energetics and electronic structure sensitively depend on the Co-O
covalency effect, which can be tuned by changing the double counting potential
and the resulting $d-$occupancy ($N_d$), and $N_d$ close to 6.7 is consistent
with the nature of the spin-state transition. Our results show that structural
changes during the spin-state transition can play an important role in
understanding energetics and electronic structure of LaCoO$_3$.
