Transferable Classical Force Field for Pure and Mixed Metal Halide Perovskites Parameterized from First-Principles

Many key features in photovoltaic perovskites occur in relatively long time scales and involve mixed compositions. This requires realistic but also numerically simple models. In this work we present a transferable classical force field to describe the mixed hybrid perovskite MAxFA1–xPb(BryI1–y)3 for variable composition (∀x, y ∈ [0, 1]). The model includes Lennard-Jones and Buckingham potentials to describe the interactions between the atoms of the inorganic lattice and the organic molecule, and the AMBER model to describe intramolecular atomic interactions. Most of the parameters of the force field have been obtained by means of a genetic algorithm previously developed to parametrize the CsPb(BrxI1–x)3 perovskite (Balestra et al. J. Mater. Chem. A. 2020, DOI: 10.1039/d0ta03200j). The algorithm finds the best parameter set that simultaneously fits the DFT energies obtained for several crystalline structures with moderate degrees of distortion with respect to the equilibrium configuration. The resulting model reproduces correctly the XRD patterns, the expansion of the lattice upon I/Br substitution, and the thermal expansion coefficients. We use the model to run classical molecular dynamics simulations with up to 8600 atoms and simulation times of up to 40 ns. From the simulations we have extracted the ion diffusion coefficient of the pure and mixed perovskites, presenting for the first time these values obtained by a fully dynamical method using a transferable model fitted to first-principles calculations. The values here reported can be considered as the theoretical upper limit, that is, without grain boundaries or other defects, for ion migration dynamics induced by halide vacancies in photovoltaic perovskite devices under operational conditions.


Classical Force field parameters
As discussed in the main text, a Buckingham potential is used to model the interactions between inorganic components in the perovskite material, i.e. Pb, I and Br, and between the inorganic components and the organic molecules.
Intermolecular interactions between the organic molecules themselves are treated via Lennard-Jones potentials. Intramolecular interactions in the organic molecules are modeled via the AMBER model. All electrostatic interactions are calculated using Coulomb potentials with point charges.
All parameters of the force field, as derived from the application of the GA and by manual fitting to DFT energies can be found in Tables S1, S2, S3, S4 and S5   Table S1. Atomic parameters and labels used in Tables S2 and S3. "MA" and "FA" refer to methylammonium and formamidinium, respectively.

Analysis of "c" parameters in Buckingham potential
There is an important aspect of the model that needs to be mentioned. When running the GA, the  The average distance between Br or I ions in our system is approximately 4.3Å. It can be observed in the plot that between 3 and 6 Å the "c" values modify the Buckingham interaction by 3% at the most, so it seems reasonable to neglect this parameter in the fitting and reduce the number of adjustable parameters. In our case, the "c" parameters were also not affecting the energies of the system appreciably, so any random and extreme mutation in the "c" parameter along the GA calculation could survive if the other parameters of its same phenotype improved the cost function of the GA.
It could be argued that two halides could get close enough so that the "c" parameter will become relevant. To rule out that possibility we ran simulations long enough to allow for diffusion of ions beyond the ballistic regime. Simulations with various realizations of the "c" parameter produced equivalent values of the diffusion coefficient (within the numerical error of the simulation). This is sort of expected, because when an halide ion moves to the position of a vacancy (which is the only possible diffusion mechanism in the time scale covered in this work), it follows a trajectory far apart from any other ion, as shown in Ref. 3 Based on these exploratory simulations, we chose then to set all the "c" parameters equal to zero and reduce the total number of free parameters in the GA, hence reducing the complexity of the FF and the procedure without losing accuracy.

Stability in the Classical Molecular Dynamics Simulations
The physical suitability of the force field parameters are checked by confirming that there is no energy drift in the CMD simulations. In Figure S2   We have prepared our own perovskite materials for comparison with the simulations. The methyl ammonium halides used in the preparation of these samples were synthesized in our labs while the rest of solvents (PbI2, PbBr2 and γ-butyrolactone) were provided by Sigma Aldrich. Table S6. Molar fraction of bromide in the prepared samples (XRD in Figure 8, main text)