## 2 Method
### Analytical Model  
The interactions between LCO ions involve both long-range Coulombic forces and short-range pairwise interactions. The Buckingham potential is employed to describe the short-range interactions, defined as follows:  
\[E=Ae^{-\ r/\rho}-\frac{C}{r_{ij}{}^{6}} \tag{1}\]  
where \(r_{ij}\) represents the distance between atoms \(i\) and \(j\)[8; 9; 23]. The parameter \(C\) governs the attractive forces, while \(\rho\) defines the repulsive interactions. The parameter \(A\) represents the magnitude of short-range Pauli repulsion due to overlapping electron clouds. A higher value of \(A\) leads to a stronger repulsive force at short interatomic distances. The parameter \(\rho\) governs the decay length of this repulsion; a smaller \(\rho\) implies a more rapidly decaying repulsive interaction. Together, \(A\) and \(\rho\) define the steepness and position of the repulsive wall of the potential. The parameter \(C\) accounts for van der Waals attraction. In cases where \(C\) = 0, as seen in the Li-O interaction, long-range attraction is dominated by Coulombic interactions, rather than van der Waals forces. The parameters \(A\), \(\rho\), and \(C\) are empirical constants, with their values provided in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline
**SOC** & **Configurations** \\ \hline
0.58 & 627,264 \\
0.50 & 853,776 \\
0.41 & 627,264 \\
0.33 & 245,025 \\
0.25 & 48,400 \\
0.16 & 4356 \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 2: Representative initial energy curve of all possible configurations at 0.5 SOC, arranged in ascending order.  
To account for electronic polarization effects, the adiabatic core-shell model is implemented, which is an effective approach for simulating the dielectric properties of ceramic oxides [8; 25]. The core-shell model has been widely utilized in previous studies to investigate layered cathode materials [8; 26]. In this model, the core, representing the heavier component, governs pairwise interactions, while the shell, carrying the highest charge, contributes to the system's polarizability. The total charge of a core-shell pair is the sum of the core charge (\(q\)) and the shell charge (\(\gamma\)). The shell is connected to the core via a spring force \(\mathrm{F_{s}}\), defined by the following:  
\[\mathrm{F_{s}}=\frac{\mathrm{k}}{2}r^{2} \tag{2}\]  
where k is a parameterized spring constant, and \(r\) is the distance between the core and the shell. The parameters for the core-shell model, including charges and spring constants for each ion, are summarized in Table 3[8]. Since the core-shell model parameters for Co\({}^{4+}\) are not available in existing references, we estimated the shell charge (\(\gamma\)) by maintaining the same core charge (0.96) as used for Co\({}^{3+}\), resulting in a shell charge of 3.04 to preserve the total formal charge of +4. Additionally, the spring constant (k) was assumed to be the same as that of Co\({}^{3+}\) to ensure consistency within the core-shell model.  
The selected structure from all possible configurations was equilibrated for 50 picoseconds (ps) at 300 K using the NPT ensemble. The initial stress values of all equilibrated structures were averaged, and only those structures with initial stress values close to the average were selected for stretching. A timestep of 1 femtosecond (fs) and periodic boundary conditions were applied throughout the simulation. For the stress-strain response, 0.5% stretching was applied to the selected structures using the NVT ensemble and equilibrated over 50 ps. The final stress value and its contributing factors were determined by calculating the difference between the final virial stress and the initial virial stress. In this study, the stress components are categorized as follows: 'pairwise' represents the short-range interatomic forces modeled by the Buckingham potential, 'bond' refers to the core-shell spring interaction, and 'long-range' corresponds to Coulombic electrostatic interactions.