## 1 Introduction
The quantity (\(a-b\)), thus, also represents the compensating number of electrons involved in the delithiation, which appears in the denominator multiplied by Faraday's constant:  
\[V=\frac{\left[E(Li_{b}NMC)+E\Big{(}Li_{(s)}\Big{)}\times(a-b)\right]-\left[E( Li_{a}NMC)\right]}{(a-b)\times F} \tag{1}\]  
Previous modeling studies detailing the voltage profile of cathode materials typically use small (<6 Li atom) single metal oxide bulk structures where the removal of a single Li atom results in a 16.67% vacancy. A more realistic, but computationally more prohibitive, approach uses a bulk supercell involving more Li atom sites, allowing for model geometries with different/lower vacancy densities. However, this introduces multiple unique ways (configurations) to remove Li for a given Li\({}_{x}\). Previous studies that looked at the voltage profile do not sample all configurations, nor discuss the selection of configurations. Typically, a non-exhaustive subset of possible configurations are tested and the one with the lowest energy is used in the voltage calculations. The exploration of the full configurational space would result in large datasets. For such large datasets, efforts to associate calculated trends with underlying reactivity descriptors tend to require machine learning algorithms [26; 27]. The total number of possible configurations for modeling the charge cycle of a material with n lithium sites follows [PERSON]'s triangle. Using Figure 1 as an example: The rows give the total number of lithium atoms in the fully lithiated structure, starting with 0, and the sum of the rows give the total number of lithium configurations over a charge cycle. For a material with zero lithium atoms, the Li fraction cannot be changed, so there is only one configuration. For a material with one lithium atom, the structure can either be fully lithiated or delithiated; thus, there is one possible configuration for each. Given 2\({}^{n}\) scaling, a structure with two Li atoms has four total configurations. Figure 1 also shows how the configurational space is the largest when approaching Li\({}_{0.50}\) where this is an even number of Li and vacancy sites allowing for the most rearrangement. When the total number of Li atoms is sufficiently large, it is not feasible to compute all possible configurations. For example, in this study, the NMC surface supercell employed has 18 Li atoms in the fully occupied intercalation layers, which gives a total of 262,144 possible Li configurations. Of course, this mathematical framework does not consider the local chemical environment. To form a chemically guided approach to predicting the most stable Li configurations for a given Li fraction, it is first necessary to obtain energetic information (from computational chemistry) for an adequate sampling of the configuration space. From that information, it may be possible to identify key descriptors (aspects of geometry or electronic structure that govern energetic trends) such that a reduced sampling can capture the relevant configurations. In addition to Li\({}_{\rm x}\), it is also important to consider the energy barrier, E\({}_{\rm b}\), for the Li diffusion in the CMO. The values of E\({}_{\rm b}\) are computationally tractable, and there are two known pathways in the delafossite structure [20; 28; 29; 30; 31; 32; 33]. At higher lithium concentrations, the octahedral site mechanism is taken, forcing lithium to pass between two oxygen atoms. Once lithium vacancies are present, a lower energy path is available where lithium can move around the oxygen atom by tetrahedral site hopping [8]. The diffusion pathway has been shown to play a key role in determining the recharge rate of the cathode. In the present study, the values of the diffusion energy barriers are used to interpret relative energies and energy ranges calculated in the configurational space of NMC structures with varying Li\({}_{\rm x}\).