## 3 Results and Discussion
### Voltage Modeling
The two horizontal lines at 0.136 eV and 0.544 eV represent the energy barriers for different lithium diffusion pathways for the tetrahedral site (\(E_{\rm a,1}\), red) and octahedral site mechanism (\(E_{\rm a,2}\), cyan), respectively. Figure 9a shows the configurations that transition to the spinel phase and cause energy shifts that make the configurational energies unrelatable through the lithium diffusion pathway (circled in blue). In the case of the right-hand side, there are many low energy structures below the lithium diffusion barriers, which give a sensible voltage curve with a limited number of optimized structures. Overall, the analysis of \(\Delta E\) and comparison to  
Figure 8: PDOS of the Ni d-states in NMC at Li\({}_{\rm x}\) = 0.22 for (**a**) Ni atom in O\(-\)\(M-\)O layer and (**b**) Ni that dropped into the Li channel, resulting in a tetrahedral coordination. Color scheme follows that of Figure 6 fillings. diffusion barriers further justifies the static calculations from Li\({}_{\text{x}}\) = 1.00 down to 0.44. This further explains why the calculations for \(V\) can become non-monatomic and unreliable for some Li\({}_{\text{x}}\) configurations lower than 0.44. The study of \(\Delta E\) as a function of Li\({}_{\text{x}}\) provides guidance in the calculation and interpretation of \(V\). We calculated the values of \(V\) based on Equation (1) and using all possible combinations between two neighboring lithium fractions, and then we refined the dataset based on the outcomes of the above analyses. Figure 10 shows the voltage curve for all datasets, including the full (\"all configs.,\" black), configuration energies below \(E_{\text{a,2}}\) (cyan), configuration energies \(E_{\text{a,1}}\) (red) and another computational voltage curve of a small bulk system for comparison (blue). As shown in Figure 10, \(\sigma\) in \(V\) is the smallest for configurations that are below \(E_{\text{a,1}}\). Because these structures exhibit a relatively small value of \(\Delta E\), they will yield similar \(V\) values when compared to neighboring Li\({}_{\text{x}}\). For structures that fall outside either of the \(E_{\text{a}}\), neighboring structures are no longer related by the energy of Li diffusion, thus produce a voltage value outside of the expected voltage range and can produce non monotonic changes in voltage. In summary, the non-monatomic changes in \(V\) can be traced to instances where the changes in Li\({}_{\text{x}}\) induce structural deformations in the slab, in which case Equation (1) is not applicable as it is meant to represent the differential limit of small changes in Li\({}_{\text{x}}\). We propose structural features that relate the lowest energy configurations for a given Li\({}_{\text{x}}\) can be used to select configurations _a priori_ and reduce the total number of configurations that need testing. For example, in the case of removing six lithium atoms, it would be very unfavorable to remove all the lithium atoms from the same layer as doing so results in the most oxygen-oxygen repulsion. Using a radial distribution function, we looked at the Li-Li distribution to determine which lithium configurations would give more favorable energies. By comparing %\(\Delta g\) values between high and low energy configurations, we tracked the changes in Li proximity going across lithium fractions, see Table 2.  
Figure 9: (**a**) Relative energies of lithium configurations with the lowest energy configuration set to zero at each Li\({}_{\text{x}}\). Large changes in energy are observed at low Li\({}_{\text{x}}\) because of Ni displacement, indicated by the blue circle. (**b**) Relative energies with Ni displacement structures removed. T\({}_{\text{d}}\) and Oh\({}_{\text{h}}\) Li diffusion energy barriers are shown as red and cyan horizontal lines, respectively, to compare the configurations that are relatable by Li diffusion. The black dashed lines help to show the spread in energies where the largest deviations are around Li\({}_{0.50}\).