The effect of off-diagonal density matrix in DFT+DMFT for Li$_2$MnO$_3$

Li$_2$MnO$_3$ has garnered much attention as one of the new-generation battery material, due to the high capacity and low cost. In the present work, we performed density functional theory (DFT)+$U$ and dynamical mean field theory (DMFT) calculations with continuous time quantum Monte Carlo impurity solver to study the electronic properties of Li$_2$MnO$_3$. Due to the nature of monoclinic $C2/m$ symmetry, the off-diagonal terms in the $d$-orbital block Hamiltonian (and $d$-orbital density matrix) are large, which results the large suppression of the energy gap due to the underestimation of the crystal-field splitting. We diagonalize the Mn $d$ block in the full $p-d$ Hamiltonian by applying unitary rotation matrix, and obtained an energy gap of 0.8eV, although it is still smaller than the experimental gap of 2.1 eV even with the large $U$. In the $p$-$d$ model, a small double counting energy is essential to reduce the $p$-$d$ hybridization, thus to obtain the experimental gap. We show that the low-energy ($d$-only basis) model is efficient to study the electronic structure of Li$_2$MnO$_3$, since the Wannier basis is the hybridized state of Mn $d$ and O $p$ orbitals. These results suggest the correct way to investigate the low-symmetry materials using DFT+ DMFT method and to our knowledge, there is no systematic study of the effect of the off-diagonal terms so far. We also find that the antiferromagnetic ground state $\Gamma_{2u}$ is stable with $U \leq 2$ within DFT+$U$, which is much smaller than widely used $U$=5 eV.