## 2 Computational Methods  
DFT calculations for the bulk and surface structures were run using the open-source software, Quantum ESPRESSO [34], with the ultrasoft GBRV pseudopotential set [35] using a plane wave cutoff of 40 Ry for the wavefunction and 320 Ry for the charge density. The bulk NMC cell is a \(\surd 3\times\surd 3\) R30\({}^{\circ}\) modification of a hexagonal LCO primitive cell, as used in our previous NMC study [12]. To obtain a structural model that can accommodate a range of Li\({}_{\rm x}\) values, the lattice constant \(a\) was multiplied in-plane by 2 to obtain a \(2\surd 3\times\surd 3\) supercell, which has 3 O-Li-O inner-layers with 6 Li in each layer for a total of 18 Li. Experimental delithiation studies report that below Li\({}_{0.45}\), significant structural changes in delafossite NMC occur, such as proton insertion [2; 36; 37] and metal displacement [38; 39]. Exploratory calculations carried out below this threshold also indicated structural instability. In this study, we focused on the range Li\({}_{1.00}\)\(\rightarrow\)Li\({}_{0.44}\) for surface and metal release modeling. The bulk calculations were converged with respect to energy with a k-point grid of 6 \(\times\) 6 \(\times\) 2, folded down to 2 \(\times\) 6 \(\times\) 1 in the surface calculations. Full geometry optimizations were  
Figure 1: The total number of delithiated structures across Li\({}_{\rm x}\) is related to [PERSON]’s triangle where each row increases the total number of Li at Li\({}_{1.00}\) by one. Sums of the row equal the total number of configurations in the charge cycle. carried out until the forces were within 5 meV/atom. For the bulk system, atomic positions and lattice constants were optimized. The bulk optimized geometry was used as the basis for the construction of slab models, which obey inversion symmetry. Slab optimizations allowed for the full relaxation of all atomic positions, and all metal release calculations maintained the inversion symmetry of the slab. The exposure of NMC to water and subsequent metal release followed the pathway depicted in Figure 2. When cleaved from the bulk, NMC is Li terminated, as shown in Figure 2a, and labeled as NMC\({}_{\text{Li-term}}\). Upon exposure to water, Li is much more favorable as a Li\({}^{+}\) cation, resulting in exchange of Li/H on the surface [9]. This results in a hydroxylated termination, as confirmed using _ab initio_ thermodynamics of the isostructural LCO surface and extrapolated to NMC, as shown in Figure 2b, and labeled as NMC\({}_{\text{H-term}}\)[9]. Figure 2b,c depicts the release of an \(M-\)OH unit (from each side of the symmetric slab), which was found to be the preferred leaving group from the cathode surface in previous computational studies [3, 9, 10, 11, 12, 15]. The resulting slab with the \(M-\)OH vacancy was labeled as NMC\({}_{M-\text{OH vacancy}}\). In this study, we used the DFT + Solvent Ion model approach to calculate the energy of metal release. This allows the partitioning of the overall metal release into elementary steps, such that certain terms can be calculated purely from DFT calculations. Other elementary steps describing the aqueous chemistry were separated and the associated energies were obtained from other sources, as detailed in Ref [12]. As our focus in this study was on how Li\({}_{\text{x}}\) influences metal release, our calculations started from the Li-terminated surface and thus captured the changes in variable Li\({}_{\text{x}}\). Overall, the change in energy for the pathway depicted in going from Figure 2a,c can be expressed as:  
\[\Delta G_{1}=\frac{\left[\text{Li}_{\text{x}}\text{NMC}_{M-\text{OH vacancy}}+M_{ (s)}+\frac{1}{2}\text{O}_{2\ (g)}+3\ \text{Li}_{(s)}\right]-\left[\text{Li}_{\text{x}}\text{NMC}_{\text{Li-term}}+ \text{H}_{2\ (g)}\right]}{2} \tag{2}\]  
Equation (2) follows [PERSON]'s law of computing reaction energy differences as the sum of product energies minus the sum of reactant energy, with the factor of \(\frac{1}{2}\) considering the removal of 2 \(M\)-OH units related by symmetry.