## 3 Results and Discussion  
### Metal Release  
The values of \(\Delta G_{1}\) are reported in Table 1 for the five lowest energy structures at each Li\({}_{\rm x}\) along with the standard deviation (\(\sigma\)) values. It was also useful to track changes in \(\Delta G_{1}\), referred to as \(\Delta(\Delta G_{1})\). For this analysis, we reference the values of \(\Delta G_{1}\) and Li\({}_{\rm x}\) = 1.00 for a given metal to zero. The \(\Delta G_{1}\) values for all metals show an initial decrease from Li\({}_{\rm x}\) = 1.00 to Li\({}_{\rm x}\) = 0.88, a driving force favoring metal release. This initial change in \(\Delta G_{1}\) is reflected in the values of \(\Delta(\Delta G_{1})\) for Li\({}_{\rm x}\) = 0.88, shown in Figure 4, and is the greatest in the case of Ni (\(-\)1.13), followed by Co (\(-\)0.58), then Mn (\(-\)0.33). This ranking can be linked to the initial oxidation states in pristine NMC with Li\({}_{\rm x}\) = 1.00. Specifically, in pristine NMC, Ni, Mn and Co are present in the oxidation states 2+, 4+, and 3+, respectively. The initial decrease in Li\({}_{\rm x}\) destabilizes the lattice and causes Ni, the metal that has the lowest oxidation state (and, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Li\({}_{\rm x}\)** & **Ni \(\Delta G_{1}\) (eV)** & \(\sigma\) & **Mn \(\Delta G_{1}\) (eV)** & \(\sigma\) & **Co \(\Delta G_{1}\) (eV)** & \(\sigma\) \\ \hline
1.00 & 9.38 & 0.00 & 12.70 & 0.00 & 9.92 & 0.00 \\ \hline
0.89 & 8.25 & 0.33 & 12.37 & 0.35 & 9.34 & 0.20 \\ \hline
0.78 & 8.28 & 0.23 & 12.41 & 0.29 & 9.47 & 0.15 \\ \hline
0.67 & 8.40 & 0.39 & 12.48 & 0.29 & 9.64 & 0.53 \\ \hline
0.56 & 8.36 & 0.24 & 12.46 & 0.48 & 9.72 & 0.25 \\ \hline
0.44 & 8.38 & 0.59 & 12.60 & 0.65 & 9.64 & 0.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(\Delta G_{1}\) values (eV, Equation (2)) for M = Ni, Mn, or Co. The values of the standard deviation, \(\sigma\) (eV), for all configurations are also reported. Figure 3: Plot of Li-intercalation voltage, V, versus Li fraction, Li\({}_{\rm x}\). The data plotted in blue follow approach 1 (single pathway), defined in the text. In black are the voltage values using lowest E configurations across Li\({}_{\rm x}\). The data plotted in green represent the literature results for a 6 Li atom bulk cell [1]. thus, the weakest interaction with lattice oxygen), to have the largest \(\Delta(\Delta G_{1})\) value. As \(\rm Li_{x}\) is further decreased, the values of \(\Delta G_{1}\) gradually increase as the metal oxidation states increase in response to the loss of Li, resulting in stronger \(M-\)O interactions in the lattice. \(\sigma\) increases, moving towards \(\rm Li_{0.50}\), where there are the greatest number of possible lithium configurations that could be tested and, thus, the largest range of voltage values. The values of \(\Delta(\Delta G_{1})\) can also be related to changes in the slab interatomic layer spacings, \(d\), and the electronic structure of the transition metals. We monitored changes in two-layer spacings, shown in Figure 5b: \(d_{1}\) is defined as the average interplanar distance between the outermore metal oxide layers O-\(M-\)O of the slab and \(d_{2}\) is defined as the average interplanar distance between the outermost O-Li-O layer of the slab. The relationship between \(\Delta(\Delta G_{1})\) and values of \(d\) is to be expected: When Li is removed from the intercalation layers of NMC, there are changes in the values of \(d_{1}\) layer (Figure 5a, black curve) and the \(d_{2}\) (Figure 5a, blue curve). As \(\rm Li_{x}\) is reduced, there is a less positive charge in the intercalation layer to shield the ionized oxygen atoms from the adjacent layers. The oxygen layers experience a repulsion due to the like charges, which is manifested in increased values of \(d_{2}\) as \(\rm Li_{x}\) decreases. The values of \(d_{1}\) show a trend of gradual increase as \(\rm Li_{x}\) decreases. As expected from the functionality of a Li-ion battery, the transition metals oxidize as Li is removed from the inner layers.