Combined Computational and Experimental Investigation on the Nature of Hydrated Iodoplumbate Complexes: Insights into the Dual Role of Water in Perovskite Precursor Solutions

Water is generally considered an enemy of metal halide perovskites, being responsible for their rapid degradation and, consequently, undermining the long-term stability of perovskite-based solar cells. However, beneficial effects of liquid water have been surprisingly observed, and synthetic routes including water treatments have shown to improve the quality of perovskite films. This suggests that the interactions of water with perovskites and their precursors are far from being completely understood, as water appears to play a puzzling dual role in perovskite precursor solutions. In this context, studying the basic interactions between perovskite precursors in the aqueous environment can provide a deeper comprehension of this conundrum. In this context, it is fundamental to understand how water impacts the chemistry of iodoplumbate perovskite precursor species, PbIx2–x. Here, we investigate the chemistry of these complexes using a combined experimental and theoretical strategy to unveil their peculiar structural and optical properties and eventually to assign the species present in the solution. Our study indicates that iodide-rich iodoplumbates, which are generally key to the formation of lead halide perovskites, are not easily formed in aqueous solutions because of the competition between iodide and solvent molecules in coordinating Pb2+ ions, explaining the difficulty of depositing lead iodide perovskites from aqueous solutions. We postulate that the beneficial effect of water when used as an additive is then motivated by its behavior being similar to high coordinative polar aprotic solvents usually employed as additives in one-step perovskite depositions.


S1. Equilibrium dissociation constants of aqueous iodoplumbates from a grand-canonical formulation of solutes in aqueous solution
To calculate the equilibrium dissociation constants of iodoplumbates in aqueous solution, we employ a grand-canonical formulation of solutes in aqueous solution. 1 Considering the generic dissociation reaction: PbI x z (aq) ⇄ PbI x−1 z+1 (aq) + I − (aq) (S1) we can define as: The formation Gibbs free energy of a solute X in a charge state q is given by Eq.
where 0 < η < 1 is the Kirkwood coupling parameter. 6,7 For each value of η, the time-averaged vertical energy difference between the reactant and the product < ∆ > is calculated. The free energy change of the reaction ΔA is given by the integration of the < ∆ > values calculated at varying η: In this way, the free energy differences reported in Eq. (S7) can be evaluated, as follows: and In Eqs. (S10-S13), ∆ dI PbI (aq) and ∆ I − (aq) are the thermodynamic integral associated with the removal of an iodide from a PbI (aq) iodoplumbate and from I − (aq), ∆ zp PbI (aq) and ∆ zp I − (aq) are the zero-point motion corrections, which accounts for the error due to the classical treatment of the nuclei in DFT-MD simulations.
Thermodynamic integrals are here calculated by adopting the Marcus approximation since it has been proved to provide accurate results in previous studies. 1,8,9 In this method, we consider two values of the Kirkwood coupling parameter (0 and 1), thus giving: (S15)

S5
Energy differences at = 0 are those related with the vertical detachment of an iodide anion from the simulation cell. The average value is calculated from 50 snapshots equally spaced in time from the trajectory achieved for each iodoplumbate: PbI + , PbI2, PbI3 − and PbI4 2− .
The sampling of energy differences at = 1 corresponds to the vertical insertion of an iodide anion. In this case, we insert an I − close to the PbI −1 +1 (aq) iodoplumbate for < ∆ dI PbI (aq) > 1 (in a void in liquid water for < ∆ dI I − (aq) > 1 ). Then, we perform a structural relaxation in which all atoms except the inserted I − are fixed. This is done for 15-30 snapshots.
The zero-point motion correction ∆ zp PbI (aq) is calculated as the difference in zero-point energies between PbI (aq) and PbI −1 +1 (aq), which correspond to the zero-point energy associated with iodide in the PbI (aq) complexes. Therefore, for each case, we calculate the three vibrational frequencies of the related normal modes and the zero-point energy as: where ℎ is the Planck constant and the frequency of the i-th normal mode. For all the studied complexes, the correction terms are negligible (< 30 meV) due to the low frequencies of the associated vibrational modes (< 140 cm -1 ). Likewise, ∆ zp I − (aq), calculated from the vibrational frequencies of the I − (aq) complex, is negligible.
Therefore, we evaluate the dissociation constants from this final simplified expression: From

S2. Structural analysis of aqueous iodoplumbates
Solvated PbI + shows an average Pb−I distance of 3.14 Å, Figure S1, with a secondary distribution centered at 3.55 Å representing a minority of structural configurations in which iodide is less bonded to the metal cation. Solvated PbI2 has a cis arrangement of the PbI2 moiety geometry with an average I-Pb-I angle of 95.5 o , Figure 1b in the main text. The distribution of Pb−I distances is broadened and the small peak encountered for aqueous PbI + sharpens, a clear hint to the weakening of the Pb−I interaction as the number of coordinating I − is increased, Figure S1. The trend observed in the solvation of Pb 2+ , PbI + , and PbI2 can be extended to aqueous PbI3 − and PbI4 2− . The first solvation shell is more rigid and cannot accommodate a large number of water molecules. As a consequence, the global coordination of Pb 2+ is reduced to six-fold with nO equal to 2.93 (86%) and 1.93 (91%), for PbI3 − and PbI4 2− , respectively. Consistently, the first peak of the Pb−O RDF is shifted towards a higher distance, 2.75 Å. In both cases, Pb 2+ is found to be coordinated in a distorted octahedral geometry, Figure 1b  .77 x 10 -1 6.04 x 10 -1 5.25 x 10 -1 6.31 x 10 -2 1.12 x 10 -3 6.46 x 10 -6 1.12 The concentration of species in a PbI2 solution are evaluated assuming that all the PbI2 is solvated, thus assuming that, in the concentration range here considered, the precipitation equilibrium of PbI2 can be neglected. Concentrations of aqueous PbI4 2are negligible and not reported in Table S2.

S16
Iodide in aqueous solution coordinates on average 4.8 water molecules, as inferred from the integration of the first peak of the I-O radial distribution reported in Figure S6, with hydrogen atoms of surrounding water molecules pointing towards the anion, Figure S7. The I-H (I-O) radial distribution function peaks at 2.58 (3.54) Å, in line with previous simulations. 13