DFT+DMFT study of spin-charge-lattice coupling in covalent LaCoO$_3$

We study the energetics of both homogeneous and mixed spin-states incorporating structural changes of the crystal volume expansion and the Co-O bond-disproportionation (BD) in LaCoO$_3$ using the density functional theory plus dynamical mean field theory (DFT+DMFT) method. Within DFT+DMFT, the multi-configurational nature of excited spin-states exhibits smooth crossover from the low spin state during the spin-state transition and both excited spin-states are energetically almost the same while the DFT+U calcu lation of the same structure produces various meta-stable solutions with energetics depending sensitively on spin-states. The homogeneous spin-state has non-negligible occupancy probabilities of both high spin (HS) and low spin (LS) states along with $d^6$ and $d^7$ charges indicating the dynamically fluctuating nature of spin and charge states due to the Co-O covalency. We find that the energetics depend on the $d-$occupancy, $N_d$ tuned by the double counting potential, and $N_d$ close to 6.7 is consistent with the nature of the spin-state transition. The mixed spin-state under the BD structure shows that Co sites with long Co-O bonds favors HS with $d^6$ states while covalent Co sites with short bonds occupies more LS with $d^7$ states, therefore charge ordering is also induced from spin-state ordering.