## 2 Results and Discussion
### Voltage-Capacity Using LDA+DMFT  
Optical spectroscopy experiments are important for characterizing charge dynamics in solids [40]. Specifically, they measure how particle-hole excitations propagates in the system, uncovering the detailed nature of the excitation spectrum itself. Motivated by our earlier studies on two-particle responses of lithium-ion battery materials [41] in this work we show that coherent and incoherent propagation of particle-hole pair excitations built from correlated electronic states are also applicable to understanding charge/discharge experiments of anodic MoO\({}_{2}\) battery material [16; 27]. To benchmark a microscopic description of quantum capacity, we recall that capacitance of a flat circular disk of radius \(R\) is given by \(C=8 Re\)[45], with \(\varepsilon\) being the dielectric constant. To make progress, we assume \((i)\) that linear relation with a constant slope holds true for moderate values of voltage \(V\), i.e., \(C(V)\approx\varepsilon(V)\), as well as \((ii)\) that the energy of electron-hole pair excitation probed in optical spectroscopy is close to the potential change of the battery material during charge/discharge, i.e., \(\omega\cong V\). With these assumptions we have established [41] an analogy between electrodynamics and charging experiments as long as \(C(V)\cong|\varepsilon(\omega)|\). Thus, one can use our results for the complex dielectric function \(\varepsilon(\omega)=1+\frac{4\pi i\sigma(\omega)}{\omega}\) (not shown) to clarify intrisic features seen in the voltage profile of charge/discharge rate capabilities [16; 27]. Let us now discuss the implications of our results for a microscopic understanding of voltage-capacity profiles of MoO\({}_{2}\) battery material. We shall first recall that in experiments the charge-discharge cycling is limited to the potential window between 0.5 to 2.5 V [16; 27]. However, the two main charge plateaus in galvanostatic charge and discharge curves seen in experiments are mostly located at voltage range between 1.3 and 1.8 V [15; 16; 27]. Hence, as displayed in Figure 6, the MoO\({}_{2}\) electrodes have two potential plateaus at about 1.5 V and 1.3 V for lithium insertion [16; 27], and the voltage drop between these two plateaus is usually ascribed to a phase transition from monoclinic to a orthorhombic lattice structure upon Li insertion [27]. Moreover, it is also noteworthy that MoO\({}_{2}\) nanoplates [46] and nanorods [47] have excellent electrochemical and cycling performance when the discharge cutoff voltage is set close to 1.0 V [46]. With this caveats in mind, in Figure 6 we show our results for voltage versus capacity of discharged MoO\({}_{2}\) using our relation \(C(V)\cong|\varepsilon(\omega)|\) for the quantum capacity of battery materials [41]. Figure 6: Rate capability of stoichiometric MoO\({}_{2}\) within LDA+DMFT in the potential (\(V\)) window relevant for future battery applications. Experimental voltage-capacity profiles of MoO\({}_{2}\) taken from Refs. [27] (triangles) and [16] (diamond) are shown for comparison. (The experimental data was shifted upward to coincide with theory at low specific capacities.) Notice the good qualitative agreement between the experimental data and the LDA+DMFT results for \(U=3.5\) eV. As seen in Figure 6, stoichiometric MoO\({}_{2}\) has large potential versus capacity traces at two characteristic values between 1.5 to 0.9 V for 3.0 eV\(\leq U\leq 4.0\) eV; and good qualitative agreement with experiments at finite capacity values is obtained using the LDA+DMFT results for \(U=3.5\) eV. Our description in Figure 6 thus implies that characteristic voltage-capacity profiles of MoO\({}_{2}\) anode material [16; 27] are originated from intrinsic optical spectral weight transfer in \(\sigma(\omega)\) (from where the quantum capacity \(C(V)\) is computed) [41] due to particle-hole excitations within the \(x^{2}-y^{2},yz,xz\) channels of MoO\({}_{2}\).