## 1 Introduction
An additional governing factor that was observed when investigating metal release from newly proposed formulations of ternary CMOs was the spin paring in the lattice where formulations that have poor anti-ferromagnetic spin coupling of the surface metals lead to a more favorable metal release [15]. In summary, our previous work conducted in close collaboration with experimentalists demonstrates the success of the modeling methodology to capture experimentally observed trends in metal release and the ability to connect the energetic trends to governing chemical properties, such as the oxidation states of transition metals in the CMOs. In all the previous computational studies of metal release from CMOs, a fully lithiated stoichiometry was considered [10, 11, 12, 13, 15]. We refer to the fractional content of Li in the CMO as Li\({}_{\rm x}\), and for the remainder of the paper use \"NMC\" to denote the equistoichiometric material with variable Li content as Li\({}_{\rm x}\)(Ni\({}_{1/3}\)Mn\({}_{1/3}\)Co\({}_{1/3}\))O\({}_{2}\). A side view ball-and-stick representation of NMC is shown in the computational methods section, which also shows the Li intercalation layers interspacing the transition metal oxide layers in the delafossite structure. To better relate to realistic materials, it is important for the modeling work to capture that ideal Li content that is hardly, if not ever, achieved. This component should be considered for CMO reactivity, such as metal release, because: (1) The operation of the Li-ion battery is redox driven, and changes in Li\({}_{\rm x}\) alter the oxidation states of the lattice transition metals. The metal oxidation state has been shown to be a thermodynamic driving force for metal release and relates broadly to CMO stability and reactivity. (2) Batteries can be disposed of at any given charge state, which means the CMOs present in electronic waste exhibit variable Li\({}_{\rm x}\). (3) Finally, experimental techniques cannot easily quantify Li\({}_{\rm x}\) due to its small size in dissolution studies. However, synthesis conditions are known to lead to variations in Li\({}_{\rm x}\)[6,10]. Metal release is a cross-cutting mechanism that has importance and implications beyond environmental and biological impacts. For example, the standard synthesis procedure for CMOs includes conditions where relative humidity can cause metal release and Li loss, leading to decreased cathode functionality [16,17]. Limited experimental research has been conducted at the electrode level due to the sensitivity of measurements required to detect Li in the lattice [18]. This is an excellent avenue for modeling due to its atomistic control of the structure where we can remove individual Li atoms to see how transition metal oxidation states and release values change. In considering variable Li\({}_{x}\), it is expected that removal of a Li atom from the inner layers of NMC is accompanied by the oxidation of a lattice transition metal. Equistoichiometric NMC has oxidative events that occur in thirds of the Li fractions where Ni\({}^{2+}\)/Ni\({}^{3+}\) is over the Li fractions Li\({}_{1.00}\)\(\rightarrow\)Li\({}_{0.66}\) followed by Ni\({}^{3+}\)/Ni\({}^{4+}\) over the next third and finally Co\({}^{3+}\)/Co\({}^{4+}\)[19]. To understand how Li content relates to cathode functionality, we follow on previous computational approaches to calculate redox potentials for transition metals and voltage profiles [20; 21; 22; 23; 24; 25]. The details of the calculation of the lithium intercalation voltage, \(V\), are given in Equation (1). Equation (1) uses the calculated total energies (\(E\)) of (NMC) at two different fractional values (\(a\) and \(b\), where \(a>b\)) of x, referred to as _Li\({}_{a}\)NMC_ and _Li\({}_{b}\)NMC_. The numerator can be interpreted as the change in energy for removing some number of Li atoms from NMC. \(E(Li_{(s)})\) is the energy of a single bulk metal Li atom, which is multiplied by the number of Li atoms removed between the two structures, represented by (\(a-b\)).