## 2 Results and Discussion
### Optical Conductivity  
We now present the optical conductivity and galvanostatic, voltage-capacity profiles of MoO\({}_{2}\) battery material using the LDA+DMFT propagators for the most relevant \(4d\)-orbitals discussed above. In the limit of high lattice dimensions, the optical response is directly evaluated as convolution of the DMFT propagators [8]. For MO systems, the real part of the optical conductivity can be written as \(\sigma^{{}^{\prime}}_{a,\sigma}(\omega)=\gamma_{a}\sum_{k}\int d\omega^{\prime }\frac{f(\omega^{\prime})-f(\omega+\omega^{\prime})}{\omega}A_{\mathbf{k},a, \sigma}(\omega^{\prime}+\omega)A_{\mathbf{k},a,\sigma}(\omega^{\prime})\), where \(\gamma_{a}\equiv v_{a}^{2}\frac{2\pi\omega^{2}h}{V}\) and \(V\) is the volume of the unit cell per formula unit, \(v_{a}\) is the fermion velocity in orbital \(a\), \(A_{a,\sigma}(\mathbf{k},\omega)\) is the corresponding fully renormalized one-particle spectral function and \(f(\omega)\) is the Fermi distribution function. Within our correlated, MO scheme the complex optical conductivity is given by \(\sigma(\omega)=\sum_{a,\sigma}[\sigma^{{}^{\prime}}_{a,\sigma}(\omega)+i \sigma^{{}^{\prime\prime}}_{a,\sigma}(\omega)]\). Hence, using the Kramers-Kronig relations [39; 40], the complex dielectric function \(\varepsilon(\omega)=1+\frac{4\pi i\sigma(\omega)}{\omega}\) can be computed for the metallic state relevant to MoO\({}_{2}\), providing a microscopic scheme to study the voltage-capacity profiles of correlated battery materials [41]. However, as in our earlier study of normal-state electrodynamic responses of LaOFeAs the approximation made here is to ignore the \(\mathbf{k}\)-dependence of electron's velocity, \(v_{\mathbf{k},a}\). In this situation, following [PERSON] et al. [42], we approximate \(v_{\mathbf{k},a}\) by a single average carrier velocity \((v_{a}=v)\) for all orbitals. This assumption works well for numerical computations of optical conductivity responses of Kondo insulators (FeSi and YbB\({}_{12}\)) [42], V\({}_{2}\)O\({}_{3}\)[43], \(3d^{1}\) perovskite titanates [44], and to lithium-ion battery materials [41], supporting our approximation in \(\sigma^{{}^{\prime}}_{a,\sigma}(\omega)\) above. The observed features in optical conductivity originate from correlation induced spectral changes: Showing how this provides a compelling description of extant experimental data [22] is our focus here. In Figure 5 we show the real part of the orbital-resolved optical conductivity computed within the LDA+DMFT scheme. As expected, our formulation reveals a Drude-like peak in the coherent FL regime at \(U\leq 3\) eV. Moreover, as found in other correlated metals, large-scale two-particle SWT with increasing \(U\) is also explicit in our results. In MoO\({}_{2}\) this is linked to the reconstructed orbital bonding-antibonding state in the correlated MO problem. An additional interesting feature in our results is the damped Drude component at \(U=5\) eV in the \(yz\) orbital. This behavior is characteristic of a bad metal, and it is consistent with the emergent pseudogaped spectral function as shown in Figure 3. However, the most interesting aspect of our many-particle description are the non-FL features within the \(x^{2}-y^{2}\) orbital at \(U\geq 4\) eV. At \(U=4\) eV we observe two peaks at 0.2 eV and 3.0 eV, which can be understood as particle-hole excitations (of the two-particle Green's function related to the current-current correlation function in DMFT) from the valence band to the reconstructed bonding-antibonding bands above \(E_{F}\), see Figure 3. As expected, with increasing \(U\) these two optical peaks are shifted to higher energies, and for \(U=4.5\) eV we observe them at 0.72 eV and 3.37 eV. Similar features as shown for \(U=4.5\) eV, i.e., very narrow Drude low-energy part followed by two peaks were resolved in the optical spectra of MoO\({}_{2}\)[22].