## 1 Introduction  
Understanding correlated materials remains a problem of enduring interest in condensed matter physics. So far, lot of attention has been focused on the way a material evolves from a metallic to an insulating state [1]. Electron-electron and electron-lattice interactions are well known driving forces behind metal-to-insulator instabilities. In a Peierls system a metal becomes an insulator via lattice distortion, which usually doubles the crystal unit cell. This in turn opens a bonding-antibonding band gap at the Fermi level (\(E_{F}\)), whereby the lowering of the one-particle bonding state compensates the energy cost incurred by the lattice distortion [2]. This situation is similar to a one-dimensional spin-Peierls effect [3], in which a reduction of magnetic energy of spins on dimers also induces lattice distortions. Hence, both the electronic- and the spin-Peierls instabilies are triggered by the coupling to the vibrational excitations of the crystal, i.e., the phonons. On the other hand, the Mott-Hubbard metal-insulator transition is caused by strong Coulomb correlations that prevent double occupancy of electrons with opposite spins on the same electronic state [1]. However, in materials like VO\({}_{2}\) both scenarios seem to be cooperative [2]. As a result, concomitant quantum phase transitions involving lattice and electronic degrees of freedom become possible. The observed first-order transition occurring around \(T=340\) K [4] in VO\({}_{2}\) is dubbed Mott-Peierls, because the lattice distortion is assisted [5] by the presence of correlated electronic states in close proximity to Mottness [6]. In this regime,dynamical scatterings arising from multi-orbital (MO) electron-electron interactions should be taken into consideration to understand the physical properties intrinsic to materials showing exotic phase transitions upon external perturbations. In this work we show that a reconstructed orbital state induced by MO electronic interactions [7] also holds true for MoO\({}_{2}\). We shall notice here, however, that not only at the interfaces of oxide heterostructures [7] but also in bulk materials, both the modification of the nearby surroundings and intrinsic dynamical electron-electron interactions can significantly affect the electronic structure of \(d\)-band materials, in particular the occupation and energy position of the \(d\)-shell levels. These and related nontrivial effects are fingerprints of electronic orbital reconstruction. Within a correlated multi-band framework [8], the active \(d\)-orbitals are normally reconstructed relative to their bare, band values. This behavior brings to the fore the most fundamental aspect in correlated electronic structure calculations, i.e., how the reconstructed orbitals in quantum materials are dynamically reshaped by realistic multi-orbital electron-electron interaction effects. Interest in molybdenum oxides is particularly motivated by their application in a wide variety of fields such as flexible electronics [9], solid fuel cells [10], gas sensors, catalysis, diffusion barrier [11], and storage lithium ion batteries [12; 13; 14; 15; 16]. Importantly, MoO\({}_{2}\) is considered to be one of the most studied anode systems for lithium ion batteries [13; 16]. Both for fundamental science [17] and for applications, MoO\({}_{2}\) is relevant in view of the quantum nature of its metallic conductivity [9; 18] and the possible reduced localization of \(4d\) electrons as compared, for example, to \(3d\) electrons of VO\({}_{2}\)[19]. MoO\({}_{2}\) is isostructural with VO\({}_{2}\) at room temperature. Similar to what is found for the \(d_{xy}\) orbital of VO\({}_{2}\)[5], electronic structure calculations within the monoclinic phase show large bonding-antibonding splitting of the \(d_{x^{2}-y^{2}}\) band in MoO\({}_{2}\)[19; 20]. This behavior is consistent with metal-metal dimerization parallel to the \(c\)-axis [21].