## 3 Results and Discussion
### Voltage Modeling  
Using Equation (1), the lithium intercalation voltage, \(V\), was calculated using the DFT total energy information for the NMC slabs as a function of Li\({}_{\text{x}}\). As mentioned earlier, delithiating NMC beyond Li\({}_{0.44}\) leads to lattice instabilities in the experiment. In exploratory calculations, the modeling of Li\({}_{\text{x}}\) structures in this range showed Ni displacement where Ni falls into the tetrahedral site of the lithium channel. These occurrences cause large changes in the DFT energy and non-sensical voltage predictions. This displacement of nickel is indicative of the known preference for a structural transformation of the delafossite structure into the spinel structure, causing the voltage prediction to fail. The changes in coordination environment of these Ni atoms can be tracked through calculated changes in their electronic structure using PDOS analysis. Figure 8 shows this comparison of two  
Figure 7: (**a**) Importance values calculated through the RFT model for the descriptors used in training the model from left to right: sum of the magnetic moments of six neighboring metals to vacancy site, \(\Sigma\rho_{\text{s}}\), bond valance sum of the removed metal, BV_vac, the number of valance electrons that the vacancy metal has, Charge_vac, and the magnetic moment of the vacancy site, \(\rho_{\text{s}}\). (**b**) \(\Delta G_{1}\) predictions of RFT model with the training data set shown in black and the testing dataset shown in red. Ni atoms from the same structure at Li\({}_{\rm x}\) = 0.22. The first is when Ni is in the octahedral site (Figure 8a) expected for the delafossite structure and when Ni occupies a tetrahedral site (Figure 8b). At a low value of Li\({}_{\rm x}\), the charge balance and redox potentials suggest that all lattice Ni will be present in the 4+ oxidation state. The octahedral PDOS shows a close match between the spin-up and spin-down spin density above and below E\({}_{\rm F}\), indicating a d\({}^{6}\) configuration where there would be three up and spin-down electron in the t\({}_{\rm 2g}\) orbitals of an octahedral field split. The tetrahedral site nickel shows a distinctly different electronic structure, reflecting that the coordination environment changed. All the spin-down densities are below E\({}_{\rm F}\), while the spin-up densities are shifted above E\({}_{\rm F}\), indicating a d\({}^{7}\) configuration in which all spin-down electrons fill the tetrahedral field split and the e\({}_{\rm g}\) is also filled with spin-up electrons. When Ni relaxes into the tetrahedral site, as seen in sufficiently low Li\({}_{\rm x}\) structures, there is a change in oxidation state, where it is now a Ni\({}^{3+}\) cation that is more stable in a tetrahedral environment than a Ni\({}^{4+}\) cation. We next examined the range of DFT total energy values, \(\Delta E\), for different configurations of a given Li\({}_{\rm x}\). The goal was to identify trends in the lowest energy configurations to guide and constrain future modeling. Depicted in Figure 9 are the calculated values of \(\Delta E\) as a function of Li\({}_{\rm x}\), where the total energy of the lowest energy configuration at each Li fraction was set to zero. On the left-hand side, the structures that have nickel atom displacements were included, and on the right-hand side, the baseline structure was set to the lowest energy structure, where nickel remains in the transition metal layer. The range of \(\Delta E\) values is smallest at the endpoints corresponding to Li\({}_{100}\) and Li\({}_{0.00}\) and similar to the pascal triangle example in Figure 1, depicted in Figure 9b by the dashed black line. As Li\({}_{\rm x}\) increases towards the intermediate lithium fractions, the total number of possible configurations increases, accompanied by larger values of \(\Delta E\).