## 2 Results and Discussion
### Correlated Electronic Structure  
We now present our results. We start with the monoclinic (space group \(P2_{1}/c\)) structure of MoO\({}_{2}\) (Figure 2) with lattice constants and monoclinic angle as in Refs. [24; 36]. In agreement with previous studies [19; 20], one-electron band structure results for the LDA spectral function (Figure 1) shows partially occupied 4\(d\) orbitals with pronounced bonding-antibonding splitting in the \(x^{2}-y^{2}\)-orbital due to dimerized metal-pairs in the monoclinic crystal structure. In what follows, we employ the LDA+DMFT scheme [8] to treat local MO interactions present in MoO\({}_{2}\) and related transition metal dioxides. Consistent with experimental observations [13; 19; 20; 22], we show that the metallic phase of MoO\({}_{2}\) is correlated with appreciable changes in the electronic structure compared to LDA. Figure 1: Partial LDA densities-of-states (DOS) of MoO\({}_{2}\) within the monoclinic symmetry. Notice the splitting of LDA the bands into bonding and antibonding branches in the projected \(d_{x^{2}-y^{2}}\) DOS, a characteristic property of monoclinic materials. Figure 2: Crystal structure of MoO\({}_{2}\) of monoclinic symmetry (Mo blue, O red), atomic coordinates and cell parameters are taken from Ref. [36]. Short Mo-Mo contacts are represented as blue sticks. The monoclinic unit cell is highlighted in green. The distorted octahedral coordination of Mo by O is represented as transparent polyhedron. A 3 \(\times\) 2 \(\times\) 3 supercell was chosen. In Figures 3 and 4 we display our LDA+DMFT results for different values of \(U\) and fixed \(J_{H}=0.7\) eV. These results show how MO electron-electron interactions modifies the orbital resolved (Figure 3) and total (Figure 4) density-of-states (DOS) within the Mo\({}^{4+}\) (\(4d^{2}\)) configuration of MoO\({}_{2}\). Likewise for VO\({}_{2}\)[5], MO dynamical correlations arising from \(U,U^{\prime}\) and \(J_{H}\) lead to spectral weight redistribution over large energy scales and the formation of lower- (LHB) and upper-Hubbard (UHB) bands. Remarkable differences in the spectral weight transfer (SWT) is seen between the \(x^{2}-y^{2}\) and the \(yz,xz\) channels. In the upper panel of Figure 3 we show the LDA+DMFT DOS of the \(x^{2}-y^{2}\)-orbital, whose evolution as function of \(U\) is crucial for understanding the role of intra- and inter-orbital dynamical correlations mutually assisting the orbital reconstruction and Mott-Peierls instability in MoO\({}_{2}\). As seen, the incipient LHB at \(\omega\approx-2.4\) eV for \(U=2\) eV is transfered to higher binding energies, becoming more pronounced with increasing \(U\). On the other hand, the UHB is not clearly resolved in the \(x^{2}-y^{2}\) orbital. Indeed, we observe a sharp antibonding-like peak at energies above 3.0 eV and a shoulder feature at \(\omega\simeq 2.2\) eV, both being pushed higher in energy with increasing \(U\). Correlation effects are, however, more visible at the bonding state. This quasi-localized band centered at \(-1.2\) eV in LDA (solid line in Figure 1) is dynamically transferred to lower energies, spanning across \(E_{F}\) for 2.0 eV \(\leq U\leq 3.0\) eV. In contrast to cluster-DMFT results for VO\({}_{2}\)[5], where narrow bonding states are located below \(E_{F}\), with \(U\geq 4\) eV we find them above \(E_{F}\), yielding a pseudogaped, metallic state in \(x^{2}-y^{2}\) electronic channel, as in Figure 3.  
Figure 3: Effect of electronic correlations on the orbital-resolved LDA+DMFT density-of-states (DOS) of monoclinic MO\({}_{2}\). Worth noticing is the dynamical evolution of the \(d_{x^{2}-y^{2}}\) bonding-antibonding branch, which shows an overall shift to energies above \(E_{F}\) due to correlation-induced spectral weight transfer. Also relevant is the formation of a pseudogap-like state at energies near the Fermi level (\(E_{F}=\omega=0.0\)) for \(U=5\) eV.