![Picture from [1]. Blue squares are occupied 8a sites, red stars are occupied 16c sites.](https://s3-us-west-2.amazonaws.com/secure.notion-static.com/d66b62df-5922-4a19-94ce-3552859477cd/Untitled.png)
Part of LTO's open-circuit voltage.
The authors of [1] assume $ε_{8a}$ and $ε_{16c}$ to be the Lithium intercalation energies on 8a and 16c sites, respectively, and $J$ to be the interaction energy between Lithium ions occupying neighbouring 8a and 16c sites, i. e. two sites that share a face. They model the potential energy of a particle with $3L$ lattice sites for Lithium intercalation ($L$ sites of 8a type and $2L$ sites of 16c type) $H_C(N)$ where $N$ is the number of Lithium atoms intercalated the sum of energies of all 8a and 16c sites occupied ($ε_{8a}$ and $ε_{16c}$, respectively), plus the number of pairs of neighbouring occupied 8a and 16c sites multiplied by $J$.
The model decides whether there is a Lithium atom in every individual site in the lattice, unlike Lattice Gas Model. This demands a different method for obtaining Lithium atom configuration that minimises the potential energy: the authors use Markov chain Monte Carlo stochastic simulation (using the Metropolis algorithm) of a particle with 1200 sites (i. e., $L = 400$) whereas for Lattice Gas Model that is numerical optimisation of the expression of the Gibbs free energy (modelled as potential energy plus configurational entropy) of a system with less than 10 layers, with aggregate stoichiometries of layers as optimisation variables.
Note that the stochasticity of Metropolis simulation (where Boltzmann factor is used as the probability of accepting a randomly generated update) plays the same role as configuration entropy term in the analytical expression of free Gibbs energy in Lattice Gas Model, so we shouldn't additionally include entropy into the $H_C(N)$ energy expression that guides the simulation.
The chemical potential of a Lithium atom intercalation $\mu$ (i. e., the open-circuit voltage divided by the charge of Li+ ion: $1e$) can be calculated as $\overline{H_C(N+1)} - \overline{H_C(N)}$, where $\overline{H_C(N)}$ is the final $H_C(N)$ of a Monte Carlo simulation run averaged over several dozens of runs (the authors used 36–72), after performing about a million steps in each run, with the temperature initially set to 3000K (which allows almost any Lithium ion "jumps" in the lattice) and then gradually decreased towards normal temperatures (e. g., 300K). This trick is called simulated annealing, it helps the simulation to converge on the globally minimal energy state. A real cell, of course, doesn't need to be heated to 3000K to achieve open-circuit potential: it can do this at a normal temperature when rested for a few hours.
The authors didn't publish the variance within the outcomes of the sets of 36–72 Monte Carlo simulation runs that they averaged to obtain the open-circuit potential function of LTO.
This Monte Carlo approach to calculating open-circuit voltage takes vastly more computation than the approach used with Lattice Gas Model. The latter is not applicable to LTO because there are no separate layers in the LTO structure like in graphite. The positive side is that the fine-grained model helps to better understand the dynamics of the material when visualised:
Picture from [1]. Blue squares are occupied 8a sites, red stars are occupied 16c sites.
To validate the hypothesis that the energetic model with just three parameters: $ε_{8a}$, $ε_{16c}$, and $J$ can be used, the authors of [1] used DFT to "more truthfully" compute the potential energies of about a hundred configurations of $\mathrm{Li_{1-2}Ti_{2}O_4}$ particles with several hundred lattice sites for Lithium atoms and various populations of 8a and 16c sites.
Personally, I don't understand what is said in this sentence, but quote it here for a reference:
The Vienna Ab initio Simulation Package (VASP) was used for the DFT calculations, in which we adopted the Perdew, Burke, and Ernzerhof formula as the exchange-correlation functional and the plane-wave basis with a projector-augmented wave scheme as the basis set.
The authors found values for $ε_{8a}$, $ε_{16c}$, and $J$ energy parameters (using multiple linear regression) so that the values of the potential energy expression $H_C(N)$ lay within approximately ±5% of the potential energy calculated using DFT for all tested particle configurations. Qualitatively, this means that the potential energy of a Lithium ion in the LTO particle depends primarily on whether the adjacent 8a and 16c sites are occupied, whereas the contribution of second-nearest neighbours, and the mean-field depending on the local stoichiometry is much smaller.
The authors also found that this does not hold for $\mathrm{Li_{4/3-7/3}Ti_{5/3}O_4}$, evidently because the potential energy of a Lithium ion at a 16c site depends significantly on whether it is surrounded by 16d sites occupied by Lithium or Titanium.
Another thing to note about the above conclusion is that although the total potential energy of all Lithium intercalated into a particle predicted by the energetic model described on this page and predicted by DFT differ by at most 5%, the variance among individual Lithium ions could have been higher: for instance, the energetic model could have significantly underestimated the energy of Lithium ions at 16c sites whose vertex-adjacent 16c sites are also occupied, and underestimated the energy of Lithium in other situations. However, since at 0% and 100% almost all Lithium atoms occupy 8a and 16c sites (as explained here), respectively, and the DFT and the energetic model predictions are still quite similar at these extremes, there seems to be no room for significant "hidden" variance.