Nanoscale size effects in α-FAPbI3 evinced by large-scale ab initio simulations

Nanoscale size effects in α-FAPbI3 evinced by large-scale ab initio simulations | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Nanoscale size effects in α-FAPbI3 evinced by large-scale ab initio simulations Ursula Rothlisberger, Virginia Carnevali, Lorenzo Agosta, Vladislav Slama, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5730287/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 24 Jul, 2025 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Abstract Formamidinium-lead-iodide (FAPbI 3 ) has established itself as the state of the art for high solar-energy conversion efficiency in perovskite-based solar cells. FAPbI 3 has a rich phase diagram, and it has been noted that long-range correlation between organic and lattice dipoles can influence phase transitions and, consequently, optoelectronic properties. In this regard, system size effects can play a crucial role for an appropriate theoretical description of FAPbI 3 . In this context, we perform a systematic study on the structural and electronic properties of the photoactive phase of FAPbI 3 ( α -FAPbI 3 ) as a function of system size. Utilizing ab initio molecular dynamics at 300 K and first-principles calculations, we demonstrate that the selection of the computational system/setup must satisfy three criteria concurrently to ensure an accurate theoretical description: the (correct) value of the band gap, the extent (or the absence of) structural distortions, and the zeroing out of the total dipole moment. We demonstrate that the net dipole moment vanishes as the system size increases due to PbI 6 octahedra distortions rather than due to FA + rotations. Additionally, we show that thermal band gap fluctuations are predominantly correlated with octahedral tilting. The optimal agreement between simulation results and experimental properties for FAPbI 3 is only achieved by system sizes approaching the nanoscale. Physical sciences/Chemistry/Theoretical chemistry/Computational chemistry Physical sciences/Physics/Condensed-matter physics/Structure of solids and liquids Figures Figure 1 Figure 2 Introduction Metal halide perovskites (MHP) are one of the most promising classes for the photoactive layer (AL) of photovoltaic (PV) materials. 1 , 2 Thanks to their remarkable optoelectronic properties such as a high absorption coefficient, tunable bandgap, high charge carrier mobility, and low exciton binding energy, perovskite-based solar cells (PSCs) lead to high photo conversion efficiencies (PCE) and solar cell performance. Perovskites have the chemical formula ABX 3 , where in the case of MHPs, A is an organic or inorganic cation, B is a divalent metal cation, and X is an anion of the halogen group. Efficient and stable PSCs require uniform and defect-free perovskite thin films at large areas, which can improve charge transport, suppress non-radiative energy loss, and minimize device degradation pathways. 3 , 4 So far, among all MHPs, formamidinum-lead-iodide (FAPbI 3 ) in the cubic symmetry phase ( α -FAPbI 3 ) yields the best PV performance. The reported experimental band gaps for this phase are in the range of 1.45–1.51 eV, 2 ,5–7 which is close to the ideal single-junction Shockley-Queisser band gap of 1.31 eV. 8 Besides, compared with other MHPs such as methylammonium lead iodide (MAPbI 3 ), FAPbI 3 exhibits an improved thermal stability due to higher activation energies for thermal degradation. 9 These two factors help to establish FAPbI 3 as the most promising AL among the MHPs for single-junction PSCs with PCE record of 26.7%. 10 However, the pursuit of enhanced stability and performance in FAPbI 3 has prompted experimentalists to seek theoretical support to elucidate the pivotal mechanisms involved in surface passivation and charge transport, which are critical to guide the design of new PSCs. 11 – 14 In response to this need, theorists have developed surface and bulk models of FAPbI 3 with an appropriate level of accuracy that can mimic the electronic and structural features of FAPbI 3 films. Nevertheless, numerous other salient issues persist in their theoretical intricacy. These include phenomena such as nucleation from solution, crystal growth, halide segregation in mixed halide MHPs, and defect formation, which still need to be resolved at the atomistic level. To address these collective mechanisms, simulations are required that employ an adequate level of accuracy while permitting a substantial increase in the number of atoms. Because of the system size requirements, the type of simulation techniques used to simulate FAPbI 3 phases have been mostly in the framework of classical molecular dynamics (CMD) and Monte Carlo (MC) simulations based on empirical force fields, 15 – 18 or density functional tight binding approaches. 17 , 18 In this regard, also force-matched force fields and machine-learning potentials trained on ab initio molecular dynamics (AIMD) data are promising solutions because they allow CMD to be performed with AIMD accuracy, provided the physics is correctly described in the training data set. 19 – 21 For the generation of sufficiently accurate reference data, it is evident that an adequate and appropriate description of FAPbI 3 at the quantum level is of paramount importance. In this context, there is an important amount of computational literature on FAPbI 3 - and MHPs in general - where different periodic models and levels of theory are used, sometimes leading to contradictory results for fairly fundamental properties such as band structure and band gap, 22 – 24 suggesting a lack of predictive ability. Several computational methods have been adopted to determine the band gap of FAPbI 3 , spanning from density functional theory (DFT) with different generalized gradient approximations, meta functionals or (range-separated) (meta) hybrid functionals, van der Waals functionals, to the GW approximation of many-body perturbation theory, or combinations of them. In addition, spin-orbit coupling (SOC) is usually implemented in presence of Pb/Sn atoms, which leads to a significant lowering of the band gap. In the context of lead perovskites, the combination with hybrid functionals has been empirically demonstrated to effectively mitigate the impact of SOC on the band gap. 25 A fairly comprehensive summary of the band gap calculated for FAPbI 3 , depending on the level of theory used, is provided in this review 26 . Apart from the employed theoretical method, the band gap can also be influenced by the choice of system size. In fact, part of the variations in the computed FAPbI 3 properties with different levels of theories might also be due to different choices of system sizes, and in particular, the use of small supercells, which imposes an ordered molecular orientation of ferroelectric multidomains, which is not observed experimentally. 27 – 30 There are some examples in literature in which larger systems sizes are adopted obtaining interesting results. Carignano et al. 31 utilize fairly large FAPbI 3 supercells to analyze the cation dynamics with AIMD. As demonstrated by Ma et al., 32 the electronic structure of MAPbI 3 must be studied at the nanoscale to ensure an accurate capture of its properties. A similar approach was adopted by Wiktor et al. 33 for CsMX 3 (M = Sn, Pb; X = Cl, Br, I). Evidence has also been found for electron-phonon coupling in FAPbI 3 as a cause of long-range optical phonon modes, indicating large size effects. 34 . Using CMD, Maheshwari et al. 35 demonstrated that phase transitions in hybrid halide perovskites are driven by a complex interplay between dipole–dipole interactions between organic cations and the metal halide lattice, resulting in the formation of large organized domains of organic cations. This has significant implications for the electronic structures of these materials. In this work, we want to address the key questions: what are the system size and level of theory needed to properly capture the structural and electronic properties of FAPbI 3 and, is the static zero Kelvin description sufficient or are simulations at finite temperature required? These are crucial questions from both theoretical and experimental perspectives. Using AIMD at 300 K and first-principles calculations, we characterized the structural and electronic properties of α -FAPbI 3 with increasing size of the simulation cell at the DFT level with both the PBE and PBE0 functionals. SOC was also considered when allowed by memory and/or computational resources. Our simulations show that it is necessary to use sufficiently large simulation cells (at least a 768-atom cell) in which all degrees of freedom (atomic positions and simulation cell) are fully relaxed in order to obtain an non-distorted α -FAPbI 3 structure, a converged band gap and a small overall dipole moment. For 0 K calculations, it is also essential to set up an initial system configuration in which the FAs cations are randomly oriented according to the 3-fold symmetry (Supplementary Fig. 1). Furthermore, with large simulation cells (from 2592 atoms upwards), PBE is able to describe the electronic band gap in good agreement with the experimental value when calculated as thermal average over finite-temperature equilibrated AIMD snapshots. We also show that the net dipole moment of the system goes to zero as the cell size increases and this is related to long-range effects due to the PbI 6 octahedral tilting. Finally, we demonstrate that thermal band gap oscillations are mainly related to the octahedral tilting. Results Characterization of α -FAPbI 3 at 0 K The obtained band gaps for different theoretical methods and computational schemes are reported in Table 1 as a function of system size ranging from the minimal primitive cell (12-atoms) to larger supercells containing increasing even numbers of primitive cells. The periodicity imposed by the size of the supercell has to be compatible with the periodicity in the octahedral tilting, 36 while the 12-atoms cell that is not compatible with this condition (but is still an often used setup in FAPbI 3 calculations) was included as a reference. At 0 K, we ran two series of calculations using as the initial structure α -FAPbI 3 from the material project database. 37 In the first case (relax), only the atomic positions were optimized keeping the simulation cell fixed to cubic, while in the second (vc-relax), both the atomic positions and the lattice parameters were optimized. We also set up two initial configurations: ( i ) all − aligned : FA molecules are kept aligned with their dipole moments pointing all in the same direction in order to have a meaningful comparison between the 12-atoms cell and all the other supercells; ( ii ) pseudo − random : FAs are pseudo-randomly oriented (from the 96-atom supercell upwards), where the overall orientation of FAs preserves the 3-fold symmetry resulting into a null total dipole moment of the system (Supplementary Fig. 1). In all cases, k-point sampling was increased until convergence was achieved. For both the relax and vc-relax calculations, the band gap converges with increasing k-point grid but to different final values, indicating that the relaxation of all the cell degrees of freedom is necessary. For all the simulation cells with dipole-aligned FAs the relax band gap converges to a value around 1.48 eV. At first glance, this is in close agreement with the experimentally measured value for the α -FAPbI 3 band gap and can be justified by the fact that, by keeping the simulation cell cubic, we are, in a sense, simulating the average (pseudocubic) α -FAPbI 3 ; however, with a closer look at the structure, we show that this is actually an artifact. Indeed, in addition to the correct band gap, it is necessary to verify that the cell distortions from the α -FAPbI 3 structure upon vc-relax and the total dipole moment of the supercell have also physical meaning. The vc-relax band gap of the 96-atom cell converges to a slightly higher value with respect to the 12-atom one. This is related to the significant distortion of the crystal structure from the cubic symmetry after vc-relax, decreasing the overlap of the electronic orbitals and consequently opening the band gap. 38 In principle, this finding aligns with experimental observations that indicate the orthorhombic perovskite phase as the most stable at low temperatures. 39 To quantify the deviations from cubic symmetry, we have analyzed the mean squared error (MSE) between the perfect α -FAPbI 3 structure and the vc-relax one as well as the distribution of the octahedra tilting angles after vc-relax for the two cases - with and without pseudo-random FA orientation (Table 2 ). In the pseudo-random FA orientation case the initial total dipole moment is smaller then 10 − 3 Debye. The pseudo-random orientation of the FAs in the starting configuration allows cubic symmetry to be maintained almost perfectly, as the MSE is reduced by two orders of magnitude for all supercells compared to the case with fully aligned FAs. The distributions of the octahedral tilting angles of the structures optimized from the pseudo-randomly oriented FA configurations average 0 degrees as expected (Supplementary Fig. 2) and the maximum value for the tilting angle converge with increasing system size (Table 2 ). The 96-atoms cell has a wide spread distribution because of the significant lattice distortions after vc-relax. In contrast, the structures optimized with all-aligned FAs do not present a octahedral tilting pattern. Local distortions of the octahedra due to a collective upward or downward motion of I ions compensate for the strongly directional dipole due to the dipole-aligned FAs (Supplementary Fig. 3). A comparison of all-aligned and pseudo-random configurations reveals that the vc-relax band gap is only improved for the 2592-atoms (3×3×3) cell. This phenomenon may be attributed to the fact that only this particular supercell allows for the complete relaxation of all degrees of freedom. This is because both the 3-fold symmetry for FAs and the octahedral tilting pattern are satisfied. It has been observed that, for all supercells, the potential energy with pseudo-randomly oriented FAs is consistently lower than that of the all-aligned case. This finding suggests that the pseudo-random configuration is a more favorable option for the system. Indeed, all-aligned configuration has a net total dipole moment for the FA that induces structural distortions in the Pb-I cages to compensate the overall dipole moment that cost energy. By adding SOC, the band gaps of the 12- and 96-atoms cells converge to slightly different values − 0.45 eV and 0.54 eV again indicating a potential problem in the description of the electronic structure related to the cell distortions for the 96-atoms cell. The band gap of the 768-atoms cell with SOC, also converges to the same value as the 12-atoms cell, and this may be related to the conservation of cubic symmetry by the 768-atoms cell. Remarkable is what is obtained with the PBE0 functional. It is known that for elements such as Pb, SOC and PBE0 contributions should cancel each other out. 25 The results of our 0 K calculations show that this effect only comes into play at k-point convergence for the 96-atoms cell, whereas it is entirely absent for the 12-atoms cell; by moving from the 12-atoms cell to the 96-atoms cell, the electron charge localization decreases as the band gap correction due to the inclusion of PBE0 decreases by 0.65 eV, allowing for SOC compensation and convergence to a band gap of 1.60 eV. It is evident that the size of the system plays a crucial role in reproducing the empirical findings of canceling out SOC-PBE0 contributions in the band gap and ensuring the structural properties of the system converge well. Characterization of α -FAPbI 3 at 300 K The thermally-averaged FAPbI 3 band gap was also calculated at 300 K by performing AIMD in the isothermal-isobaric ensemble with flexible cell option (NPT-F) at the PBE and PBE0 levels of theory for a minimum of 7 ps and up to 11 ps (Table 1 ). The band gap was computed as the average along the equilibrated AIMD trajectory (Fig. 1 -a). We run in NPT-F to avoid any kind of symmetry restriction to the system. Since we are operating at 300 K, the initial orientation of the FAs is no longer important, as the kinetic energy is sufficient to randomize the FA orientation after a few AIMD steps. The finite temperature PBE band gap for the 6144-atoms cell is 1.47 ± 0.08 eV that matches very well the experimentally measured gap, illustrating again the efficient compensation of many-body and SOC effects. Indeed, for all simulation cells, PBE0 (without SOC) consistently leads to a strong overestimation of the band gap. We also exploit the larger system sizes studied by AIMD together with its statistical approach to study the instantaneous band gap dependence from the local lattice distortions. We have subdivided the 6144-atoms and 768-atoms cells into 64 and 8 96-atoms cells, respectively, and computed the local band gap by projecting the density of states locally along the NPT-F trajectory (Fig. 1 -b, 1 -c). The 3D arrangement of the supercells was projected into a 2D map for better visualization (Supplementary Fig. 4). There is a local band gap variation for both supercells of about 1 eV, but in the 6144-atoms cell there are more defined band gap domains, while for the 768-atoms cell the band gap is more homogeneous throughout the supercell. This again indicates the importance of the system size, in order to detect the possible formation of band gap domains. It further indicates that this is a phenomenon that has to be related to the process of long-range relaxation, stressing once again the importance of correctly describing the mechanisms that we have previously shown to be related to the size of the system, such as the octahedral tilting. 31 , 35 To quantify the connection between band gap fluctuations and octahedral tilting, we have calculated the time correlation function of the band gap oscillations and the octahedral tilting for the 6144-atoms cell (Fig. 2 ), getting correlation times of about 25 fs and 30 fs, respectively, i.e. very similar time scales suggesting that the band gap variations are indeed closely connected to changes in octahedral distortions. The influence of the tilting angles on the band gap directly aligns with established literature, as the tilting angles impact the antibonding overlap of the orbitals involved in the band edges. 38 Furthermore, the FA cations motion can be characterized by two characteristic rotational correlation times of 80 fs and 250 fs for the C-H and N-N vectors, respectively (Fig. 2 ), that are farer from the value calculated for the band gap oscillations. We have also computed the time correlation functions of the bottom of the conduction band and top of the valence band eigenvalues obtaining a trend comparable with that of the band gap (Supplementary Fig. 5). Local variations in the band gap might indicate that FAPbI 3 can potentially absorb photons with different wavelengths in different regions of the sample, which may be a further explanation of why this material performs as well as AL. The electronic charge distribution due to lattice motions in halide perovskites have implications on thermal fluctuations in the electronic structure. It has been shown that off-centering of Sn 2+ and Br − widens the band gap of CsSnBr 3 . 40 Additionally, the top of the valence band and the bottom of the conduction band of FAPbI 3 are dominated by I − and Pb 2+ contributions, respectively. 23 For these reasons, we can expect that the fluctuations in the band gap occur on time scales similar to those characterizing the octahedral tilting fluctuations. The selection of an appropriate system size is imperative for the accurate description of the octahedral tilting pattern. Our findings demonstrate that PbI 6 tilting converges for a 768-atoms supercell. The last point needed to verify the proper description of FAPbI 3 is the dipole moment. Table 1 FAPbI 3 Kohn-Sham band gap for different system sizes at several theory levels and system optimization schemes. The initial configurations for the 0 K simulations were chosen with all FAs aligned except the ones indicated with a star superscript. AIMD was performed at 300 K in NPT-F and the band gaps were computed as averages along the trajectories (4–7 ps) after equilibration. Values in brackets are additive estimates from SOC and PBE0 calculations since direct calculations require too much memory for available resources. Simulation cell NPT-F PBE 300 K (eV) NPT-F PBE0 300 K (eV) relax PBE 0 K (eV) vc-relax PBE 0 K (eV) vc-relax PBE + SOC 0 K (eV) vc-relax PBE0 0 K (eV) vc-relax PBE0 + SOC 0 K (eV) \(\:n\times\:n\times\:n\) k-point grid 12-atoms \(\:(1\times\:1\times\:1)\) 3.62 ± 0.21 5.08 ± 0.21 3.68 3.72 3.46 5.82 5.54 1 - - 2.21 2.31 1.78 3.90 3.33 2 - - 1.48 1.97 1.25 3.47 2.72 4 - - 1.49 1.63 0.49 3.44 2.28 6 - - 1.49 1.56 0.45 3.38 2.25 8 - - 1.48 1.55 0.45 3.37 2.24 10 96-atoms \(\:(2\times\:2\times\:2)\) 2.16 ± 0.17 2.80 ± 0.15 2.24 2.26 1.78 3.36 2.85 1 - - 1.58 1.69 0.60 2.84 (1.75) 2 - - 1.49 1.58 0.46 2.73 (1.61) 4 - - 1.48 1.61 0.54 2.75 (1.68) 6 - - 1.61* 1.60* 0.48* 2.76* (1.64)* 6 768-atoms \(\:(4\times\:4\times\:4)\) 1.76 ± 0.04 2.28 ± 0.03 1.54 1.60 0.47 - - 1 - - 1.74* 1.74* 0.63* - - 1 - - 1.48 1.56 0.44 - - 2 - - 1.69* 1.69* 0.56* - - 2 2592-atoms \(\:(6\times\:6\times\:6)\) 1.59 ± 0.07 2.17 ± 0.03 1.51 1.71 - - - 1 - - 1.50* 1.50* - - - 1 6144-atoms \(\:(8\times\:8\times\:8)\) 1.47 ± 0.08 2.09 ± 0.05 1.47 1.67 - - - 1 - - 1.68* 1.69* - - - 1 * FA pseudo-randomly oriented Dipole moment The dipole moment was computed at 0 K and at 300 K (Table 2 ). In principle, the total dipole moment should be zero; however, for small-size simulation systems, it is not, due to the presence of residual, not fully compensated dipoles. Given the substantial decrease in dipole moment observed with increasing system size, it can be deduced that this phenomenon is associated with a long-range collective interaction, such as the long-range dipole-dipole correlations present between FA-FA and octahedral cage dipoles. 35 FA has a non-zero permanent dipole moment, while the octahedral distortions, i.e. contraction or elongation along different I-Pb-I axes, might induce a significant dipole moment. Table 3 reports the dipole moment contributions obtained by subtracting the dipole moment of the FA cations in the initial configuration from the total dipole moment of the system. The result is similar for all configurations (Table 3 ), and comparable with the values in Table 2 , concluding that the initial FA configuration does not affect the overall dipole moment of the cell. To estimate the actual contributions of FA and octahedral cage distortions to the dipole moment, two NPT-F simulations were performed with FA respectively PbI 6 octahedra frozen. The octahedral distortion compensates for the dipole moment of the system in cases the FA cation orientations create a non-zero dipole moment (Supplementary Fig. 6). The time correlation function associated to the dipole moment fluctuations shows that the correlation time (∼ 30 fs) is not affected by the system size (Supplementary Fig. 8) at contrast to the absolute value of the dipole moment (Table 2 ). Except for the 12-atoms and 96-atoms cells, which have already shown large size effects for structural as well as electronic properties, there is no major change in dipole moment when using PBE0. Finally, the values of the dipole moment obtained with PBE and PBE0 are statistically equivalent for the 6144-atoms cell (Table 2 ). This finding suggests that the residual dipole moment may not be directly associated with the degree of charge localization imposed by the functional. Table 2 Mean squared error (MSE) of the lattice vectors with respect to the perfect cubic FAPbI 3 α-phase (column 1) and absolute maximum value of the octahedral tilting angle (column 2) at 0 K. For the structure optimized in the presence of dipole-aligned FAs, the octahedral tilting angle refers to the global deformation of the octahedra, while for the structure optimized from pseudo-randomly oriented FAs it refers to the average absolute value of the octahedral tilting angle (Supplementary Fig. 3). Total dipole moment per stoichiometric unit (ABX3) for different simulation cells; the dipole moment was computed at 0 K (column 3) and after equilibration as average along the AIMD trajectories performed at 300 K in the NPT-F ensemble (columns 4–5). The initial configurations for the 0 K simulations were chosen with all FAs aligned except the ones indicated with a star superscript. Simulation cell MSE 0 K Octahedra tilting amplitude 0 K (deg) Total Dipole PBE 0 K (Debye/ABX 3 ) Total Dipole PBE 3000 K (Debye/ABX 3 ) Total Dipole PBE0 3000 K (Debye/ABX 3 ) 12-atoms \(\:(1\times\:1\times\:1)\) 0.08 1.37 13.17 2.32 ± 0.98 1.80 ± 0.70 96-atoms \(\:(2\times\:2\times\:2)\) 0.78 18.03 3.36 2.24 ± 1.02 4.57 ± 0.71 0.09* 17.36* 3.25* 768-atoms \(\:(4\times\:4\times\:4)\) 0.30 5.65 1.18 0.92 ± 0.26 0.89 ± 0.25 \(\:\:\:\:\:\:\:\:\:\:\:\:\:4\bullet\:{10}^{-3}\) * 6.46* 1.10* 2592-atoms \(\:(6\times\:6\times\:6)\) 0.23 5.18 0.39 0.39 ± 0.12 0.31 ± 0.09 \(\:\:\:\:\:\:\:\:\:\:\:\:\:\:8\bullet\:{10}^{-4}\) * 5.67* 0.36* 6144-atoms \(\:(8\times\:8\times\:8)\) 0.25 4.11 0.13 0.24 ± 0.07 0.24 ± 0.07 - \(\:\:\:\:\:1\bullet\:{10}^{-4}\) * 5.52 0.18* - * FA pseudo-randomly oriented Table 3 Magnitude of the total dipole moment of the system separated into FA contribution, and difference between the dipole moment and the FA contribution to the total dipole moment of the system. The values were computed for a 768-atoms cell on equilibrated AIMD snapshots starting from an initial configuration with different FA orientations. The all-aligned, random and random_best, smart_100 and smart_quasi systems are initiated from an initial configuration of FAs that are all aligned, randomly oriented, and pseudo-randomly oriented, respectively (Supplementary Fig. 7). The random and smart configurations are built with a null initial total dipole moment. FA orientation Total dipole (Debye/ABX 3 ) FA dipole (Debye/ABX 3 ) Total dipole – FA dipole (Debye/ABX 3 ) all-aligned 0.82 0.30 1.02 random 1.00 0.12 1.08 random_best 0.94 0.06 1.02 smart_100 1.05 0.00 1.05 smart_quasi 1.16 0.00 1.16 - - Discussion In conclusion, by means of large-scale first-principles calculations and ab initio molecular dynamics simulations at 300 K, we demonstrated that in order to get an accurate description of the structural and electronic properties of the α-phase of FAPbI3 the size of the simulated system needs to approach the nanoscale. In particular, we showed that three conditions have to be met simultaneously, namely a proper description of the band gap, minimization of structural distortions, and the zeroing out of the total dipole moment. For first-principles calculations, it is essential to start from an initial configuration where the FAs are pseudo-randomly oriented by preserving the 3-fold symmetry and minimizing the dipole moment. At 300 K, because of the finite temperature dynamics, the initial configuration of the FAs is not stringent and, from a 2592-atoms cell upwards, the PBE approximation is already able to describe the electronic band gap of α-FAPbI3. For the 6144-atoms cell, we have computed a band gap of 1.47 ± 0.08 eV which is in excellent agreement with the experimental values of 1.45–1.51 eV reported in literature (highlighting that PBE0 and SOC corrections only cancel out for this system size range). The same cell minimizes structural distortions with respect to the perfect α-FAPbI3 structure and has the lowest dipole moment among all the systems studied. A significant correlation was discovered between PbI6 octahedral tilting, band gap oscillations, and dipole moment. In particular, the dipole moment goes to zero only if the system size is large enough to properly relax the tilting pattern of the octahedra. Overall, an adequate size of the system (at least 6144-atoms cell) is needed to correctly describe its physics, as we have demonstrated with the identification of band gap domains related to a correct description of the octahedral tilting. Our work provides a detailed insight into the connection between structural and electronic properties of α-FAPbI3 - and MHPs in general - making an important contribution to the field of ab initio simulations dedicated to understanding fundamental physical principles, such as hole-electron transport, which is of paramount importance in the development of increasingly high-performance PSC devices. Methods First-principles calculations DFT simulations at 0 K were performed with the Quantum ESPRESSO (QE) 41 suite of codes. All the calculations were run with DOJO fully relativistic norm-conserving PBE pseudopotentials 42 , 43 and well-converged basis sets corresponding to an energy cutoff of 150 Ry for the wave functions and 600 Ry for the charge density. Different k-point Monkhorst-Pack grids 44 were used, all centered on Γ-point. Semiempirical corrections accounting for the van der Waals interactions were included with the DFT-D3 approach. 45 Different simulation cells were used, starting from a 1×1×1 α -FAPbI 3 (12-atoms) up to a 8×8×8 (6144-atoms). The electronic structure of fully relaxed structures (vc-relax) was also computed including spin-orbit coupling (SOC) and PBE0. The supercell distortion with respect to the perfect cubic α -FAPbI 3 has been estimated by the mean squared error (MSE) between the lattice vectors of the two systems. Because CP2K 46 allows to run also DFT at 0 K, Γ-point simulations were performed with both the QE and CP2K software, obtaining equal band gap values to three decimal places, which means that the results achieved with the two software packages are comparable. Ab initio molecular dynamics AIMD simulations were run in the DFT framework as implemented in the CP2K software. The PBE and PBE0 functional and the D3 dispersion correction 45 were adopted together with Goedecker-Teter-Hutter pseudopotentials 47 and a polarized double- ζ Gaussian basis set (DZVP) 48 for valence electrons. The energy cut off for the expansion of the electron density was set to 400 Ry. Simulations were run with a time step of 0.5 fs in the NPT flexible ensemble using Born-Oppenheimer dynamics for 7–12 ps (PBE) and 2–5 ps (PBE0), while the temperature was controlled by the Bussi thermostat 49 and the pressure by the Martyna barostat. 50 Different simulation cells were used, starting from a 1×1×1 α -phase FAPbI 3 (12-atoms) up to a 8×8×8 (6144-atoms). The finite temperature band gap was computed as an average of different band gaps calculated from the projected density of states (PDOS) on several AIMD snapshots after the system equilibrated (∼ 2 ps). The spatial variation of the band gap within a supercell was calculated by grouping the PDOS of the atoms of interest. Real space positions of top of the valence and bottom of the conduction bands in FAPbI 3 were computed after quenching an equilibrated AIMD snapshot to 0 K. Different initial FA orientation were tested - completely ordered, random oriented, smart oriented (total FA dipole equal to zero) - to avoid any bias on the simulations. The total dipole moment was calculated at the quantum level as in the CP2K framework, while the contribution of FAs to the dipole in the initial configurations was estimated classically, assigning a + 1 charge to each FA. Time correlation function analysis The rotational dynamics of FA and PbI 6 octahedra were characterized by the correlation function $$\:{C}_{rot}\left(t\right)=\frac{⟨\overrightarrow{\mu\:}\left(t\right)\cdot\:\overrightarrow{\mu\:}\left(0\right)⟩}{\left|\overrightarrow{\mu\:}\left(0\right)\right|}$$ Where \(\:\overrightarrow{\mu\:}\left(t\right)\) is the C-H(N-N) vector for FA or the octahedra tilting function appropriate for the PbI 6 tilting. The timescale oscillations for the band gap were quantified by the correlation function $$\:{C}_{gap}\left(t\right)=\frac{⟨{\Delta\:}{ϵ}_{cv}\left(t\right)\cdot\:{\Delta\:}{ϵ}_{cv}\left(0\right)⟩}{\left|{\Delta\:}{ϵ}_{cv}\left(0\right)\right|}$$ where ∆ ε cv ( t ) is the difference between the eigenvalues of the bottom of the conduction band and the top of the valence band. The same time correlation function has been used to compute the correlations of the dipole moment fluctuations, where the quantity correlated in time was the value of the dipole moment. Declarations Author contributions statement V.C. and L.A. conceived the idea. V.C, L.A, and V.S. performed the theoretical simulations and wrote the paper under the supervision of U.R.. All authors contributed to the discussion and writing of the paper. Competing interests The authors declare no competing interests. Additional information The online version contains supplementary material available at Data and analysis scripts are available on Zenodo at 10.5281/zenodo.13712682 . Acknowledgements U.R. acknowledges the Swiss National Foundation (grant N. 200020_219440) and computational resources from the Swiss National Computing Centre CSCS (project s1151). V.C. acknowledges computational resources from the Swiss National Computing Centre CSCS (project s1253). References Zhang, M. et al. High-Efficiency Rubidium-Incorporated Perovskite Solar Cells by Gas Quenching. ACS Energy Lett. 2, 438–444, DOI: 10.1021/acsenergylett.6b00697 (2017). Jeong, J. et al. Pseudo-halide anion engineering for α -FAPbI3 perovskite solar cells. Nature 592, 381–385, DOI: 10.1038/s41586-021-03406-5 (2021). Publisher: Nature Publishing Group. Liu, Z. et al. A holistic approach to interface stabilization for efficient perovskite solar modules with over 2,000-hour operational stability. 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Electron–phonon coupling in hybrid lead halide perovskites. Commun. 7, 11755, DOI: 10.1038/ncomms11755 (2016). Publisher: Nature Publishing Group. Maheshwari, S., Fridriksson, M. B., Seal, S., Meyer, J. & Grozema, F. C. The Relation between Rotational Dynamics of the Organic Cation and Phase Transitions in Hybrid Halide Perovskites. The J. Phys. Chem. C 123, 14652–14661, DOI: 10.1021/acs.jpcc.9b02736 (2019). Publisher: American Chemical Society. Woodward, P. M. Octahedral Tilting in Perovskites. I. Geometrical Considerations. Acta Crystallogr. Sect. B Struct. Sci. 53, 32–43, DOI: 10.1107/S0108768196010713 (1997). Jain, A. et al. Commentary: The Materials Project: A materials genome approach to accelerating materials innovation. APL Mater. 1, 011002, DOI: 10.1063/1.4812323 (2013). Meloni, S., Palermo, G., Ashari-Astani, N., Grätzel, M. & Rothlisberger, U. Valence and conduction band tuning in halide perovskites for solar cell applications. Mater. Chem. A 4, 15997–16002, DOI: 10.1039/C6TA04949D (2016). Jiang, S. et al. Phase Transitions of Formamidinium Lead Iodide Perovskite under Pressure. Am. Chem. Soc. 140, 13952–13957, DOI: 10.1021/jacs.8b09316 (2018). Publisher: American Chemical Society. Fabini, D. H. et al. Dynamic Stereochemical Activity of the Sn(2+) Lone Pair in Perovskite CsSnBr3. Am. Chem. Soc. 138, 11820–11832, DOI: 10.1021/jacs.6b06287 (2016). Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. Physics: Condens. Matter 21, 395502, DOI: 10.1088/0953-8984/21/39/395502 (2009). van Setten, M. J. et al. The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table. Phys. Commun. 226, 39–54, DOI: 10.1016/j.cpc.2018.01.012 (2018). Hamann, D. R. Optimized norm-conserving Vanderbilt pseudopotentials. Rev. B 88, 085117, DOI: 10.1103/ PhysRevB.88.085117 (2013). Publisher: American Physical Society. Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Rev. B 13, 5188–5192, DOI: 10.1103/PhysRevB.13.5188 (1976). Grimme, S. Density functional theory with London dispersion corrections. WIREs Comput. Mol. Sci. 1, 211–228, DOI: 10.1002/wcms.30 (2011). Kühne, T. D. et al. CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations. The J. Chem. Phys. 152, 194103, DOI: 10.1063/5.0007045 (2020). Perdew, J. P. et al. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Rev. Lett. 100, 136406, DOI: 10.1103/PhysRevLett.100.136406 (2008). Publisher: American Physical Society. VandeVondele, J. & Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. The J. Chem. Phys. 127, 114105, DOI: 10.1063/1.2770708 (2007). Bussi, G., Donadio, D. & Parrinello, M. Canonical sampling through velocity rescaling. The J. Chem. Phys. 126, 014101, DOI: 10.1063/1.2408420 (2007). Publisher: American Institute of Physics. Martyna, G. J., Tobias, D. J. & Klein, M. L. Constant pressure molecular dynamics algorithms. The J. Chem. Phys. 101, 4177–4189, DOI: 10.1063/1.467468 (1994). Additional Declarations There is NO Competing Interest. Supplementary Files FAPbIbandgapSI.docx Nanoscale size effects in α-FAPbI3 evinced by large-scale ab initio simulations Cite Share Download PDF Status: Published Journal Publication published 24 Jul, 2025 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5730287","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":398347881,"identity":"df3425e3-da05-44d4-8534-54d0f2178eff","order_by":0,"name":"Ursula Rothlisberger","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0002-1704-8591","institution":"Ecole Polytechnique Fédérale de Lausanne","correspondingAuthor":true,"prefix":"","firstName":"Ursula","middleName":"","lastName":"Rothlisberger","suffix":""},{"id":398347882,"identity":"2698e30a-fbb2-42d7-bb1e-37a04581b876","order_by":1,"name":"Virginia Carnevali","email":"","orcid":"https://orcid.org/0000-0002-8905-2928","institution":"École Polytechnique Fédérale de Lausanne (EPFL)","correspondingAuthor":false,"prefix":"","firstName":"Virginia","middleName":"","lastName":"Carnevali","suffix":""},{"id":398347883,"identity":"3755218d-1d8a-41ce-b91c-90fb0ce15e87","order_by":2,"name":"Lorenzo Agosta","email":"","orcid":"","institution":"Uppsala University","correspondingAuthor":false,"prefix":"","firstName":"Lorenzo","middleName":"","lastName":"Agosta","suffix":""},{"id":398347884,"identity":"fdc1b780-dcf7-495f-bae0-03709e88e773","order_by":3,"name":"Vladislav Slama","email":"","orcid":"https://orcid.org/0000-0001-7339-5523","institution":"EPFL","correspondingAuthor":false,"prefix":"","firstName":"Vladislav","middleName":"","lastName":"Slama","suffix":""},{"id":398347885,"identity":"18e27e95-05d9-45ab-b890-ee5a41f6269f","order_by":4,"name":"Nikolaos Lempesis","email":"","orcid":"https://orcid.org/0000-0002-4104-9666","institution":"EPFL","correspondingAuthor":false,"prefix":"","firstName":"Nikolaos","middleName":"","lastName":"Lempesis","suffix":""},{"id":398347886,"identity":"9260a667-b517-4eaa-983d-fb670512f1bb","order_by":5,"name":"Andrea Vezzosi","email":"","orcid":"","institution":"Ecole Polytechnique Fédérale de Lausanne","correspondingAuthor":false,"prefix":"","firstName":"Andrea","middleName":"","lastName":"Vezzosi","suffix":""}],"badges":[],"createdAt":"2024-12-29 14:05:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5730287/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5730287/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41467-025-61351-7","type":"published","date":"2025-07-24T04:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":73489032,"identity":"7a50dbbe-c20b-4e93-9a55-a032a4a879ba","added_by":"auto","created_at":"2025-01-10 12:59:37","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":322270,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Time variation of the band gap along the PBE NPT-F trajectory at 300 K for different supercells. Spatial band gap variation on 64x 96-atoms subcells contained in the 6144-atoms FAPbI3 supercell (b) and on 8x 96-atoms subcells contained in a 768-atoms supercell (c).\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5730287/v1/18e49174f267a500a416b66b.png"},{"id":73489480,"identity":"a67ff63e-234e-4202-9f45-e572f7816e5c","added_by":"auto","created_at":"2025-01-10 13:07:37","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":126743,"visible":true,"origin":"","legend":"\u003cp\u003eTime correlation function characterizing the rotational dynamics of FA cations along the N-N (blue curve) and C-H (orange curve) axes, as well as the octahedral tilting (red curve), and band gap oscillations (green curve). The analysis was done on the 6144-atoms supercell over an equilibrated MD trajectory of 5 ps at 300 K.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5730287/v1/67bf2d4878d744e3eb377eb2.png"},{"id":87556277,"identity":"71636be3-a055-46c3-8752-75c096edfa1b","added_by":"auto","created_at":"2025-07-25 07:09:23","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1424413,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5730287/v1/c393a1c0-f088-4b95-949f-f2e032f56150.pdf"},{"id":73489035,"identity":"ef09d30a-45d5-45e4-82ea-d0269a152ae3","added_by":"auto","created_at":"2025-01-10 12:59:37","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":1983577,"visible":true,"origin":"","legend":"Nanoscale size effects in \u0026#x03B1;-FAPbI3 evinced by large-scale ab initio simulations","description":"","filename":"FAPbIbandgapSI.docx","url":"https://assets-eu.researchsquare.com/files/rs-5730287/v1/ca27815f798ebd6f92981894.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Nanoscale size effects in α-FAPbI3 evinced by large-scale ab initio simulations","fulltext":[{"header":"Introduction","content":"\u003cp\u003eMetal halide perovskites (MHP) are one of the most promising classes for the photoactive layer (AL) of photovoltaic (PV) materials.\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e,\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e Thanks to their remarkable optoelectronic properties such as a high absorption coefficient, tunable bandgap, high charge carrier mobility, and low exciton binding energy, perovskite-based solar cells (PSCs) lead to high photo conversion efficiencies (PCE) and solar cell performance. Perovskites have the chemical formula ABX\u003csub\u003e3\u003c/sub\u003e, where in the case of MHPs, A is an organic or inorganic cation, B is a divalent metal cation, and X is an anion of the halogen group. Efficient and stable PSCs require uniform and defect-free perovskite thin films at large areas, which can improve charge transport, suppress non-radiative energy loss, and minimize device degradation pathways.\u003csup\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e3\u003c/span\u003e,\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e So far, among all MHPs, formamidinum-lead-iodide (FAPbI\u003csub\u003e3\u003c/sub\u003e) in the cubic symmetry phase (\u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e) yields the best PV performance. The reported experimental band gaps for this phase are in the range of 1.45\u0026ndash;1.51 eV,\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e,5\u0026ndash;7\u003c/sup\u003e which is close to the ideal single-junction Shockley-Queisser band gap of 1.31 eV.\u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e Besides, compared with other MHPs such as methylammonium lead iodide (MAPbI\u003csub\u003e3\u003c/sub\u003e), FAPbI\u003csub\u003e3\u003c/sub\u003e exhibits an improved thermal stability due to higher activation energies for thermal degradation.\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e These two factors help to establish FAPbI\u003csub\u003e3\u003c/sub\u003e as the most promising AL among the MHPs for single-junction PSCs with PCE record of 26.7%.\u003csup\u003e10\u003c/sup\u003e\u003c/p\u003e \u003cp\u003eHowever, the pursuit of enhanced stability and performance in FAPbI\u003csub\u003e3\u003c/sub\u003e has prompted experimentalists to seek theoretical support to elucidate the pivotal mechanisms involved in surface passivation and charge transport, which are critical to guide the design of new PSCs.\u003csup\u003e\u003cspan additionalcitationids=\"CR12 CR13\" citationid=\"CR12\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e In response to this need, theorists have developed surface and bulk models of FAPbI\u003csub\u003e3\u003c/sub\u003e with an appropriate level of accuracy that can mimic the electronic and structural features of FAPbI\u003csub\u003e3\u003c/sub\u003e films. Nevertheless, numerous other salient issues persist in their theoretical intricacy. These include phenomena such as nucleation from solution, crystal growth, halide segregation in mixed halide MHPs, and defect formation, which still need to be resolved at the atomistic level. To address these collective mechanisms, simulations are required that employ an adequate level of accuracy while permitting a substantial increase in the number of atoms. Because of the system size requirements, the type of simulation techniques used to simulate FAPbI\u003csub\u003e3\u003c/sub\u003e phases have been mostly in the framework of classical molecular dynamics (CMD) and Monte Carlo (MC) simulations based on empirical force fields,\u003csup\u003e\u003cspan additionalcitationids=\"CR16 CR17\" citationid=\"CR18\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e or density functional tight binding approaches.\u003csup\u003e\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e17\u003c/span\u003e,\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e In this regard, also force-matched force fields and machine-learning potentials trained on ab initio molecular dynamics (AIMD) data are promising solutions because they allow CMD to be performed with AIMD accuracy, provided the physics is correctly described in the training data set.\u003csup\u003e\u003cspan additionalcitationids=\"CR20\" citationid=\"CR22\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u003c/sup\u003e For the generation of sufficiently accurate reference data, it is evident that an adequate and appropriate description of FAPbI\u003csub\u003e3\u003c/sub\u003e at the quantum level is of paramount importance.\u003c/p\u003e \u003cp\u003eIn this context, there is an important amount of computational literature on FAPbI\u003csub\u003e3\u003c/sub\u003e - and MHPs in general - where different periodic models and levels of theory are used, sometimes leading to contradictory results for fairly fundamental properties such as band structure and band gap,\u003csup\u003e\u003cspan additionalcitationids=\"CR23\" citationid=\"CR26\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/sup\u003e suggesting a lack of predictive ability. Several computational methods have been adopted to determine the band gap of FAPbI\u003csub\u003e3\u003c/sub\u003e, spanning from density functional theory (DFT) with different generalized gradient approximations, meta functionals or (range-separated) (meta) hybrid functionals, van der Waals functionals, to the GW approximation of many-body perturbation theory, or combinations of them. In addition, spin-orbit coupling (SOC) is usually implemented in presence of Pb/Sn atoms, which leads to a significant lowering of the band gap. In the context of lead perovskites, the combination with hybrid functionals has been empirically demonstrated to effectively mitigate the impact of SOC on the band gap.\u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e A fairly comprehensive summary of the band gap calculated for FAPbI\u003csub\u003e3\u003c/sub\u003e, depending on the level of theory used, is provided in this review\u003csup\u003e\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e. Apart from the employed theoretical method, the band gap can also be influenced by the choice of system size. In fact, part of the variations in the computed FAPbI\u003csub\u003e3\u003c/sub\u003e properties with different levels of theories might also be due to different choices of system sizes, and in particular, the use of small supercells, which imposes an ordered molecular orientation of ferroelectric multidomains, which is not observed experimentally.\u003csup\u003e\u003cspan additionalcitationids=\"CR28 CR29\" citationid=\"CR32\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e There are some examples in literature in which larger systems sizes are adopted obtaining interesting results. Carignano et al.\u003csup\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e utilize fairly large FAPbI\u003csub\u003e3\u003c/sub\u003e supercells to analyze the cation dynamics with AIMD. As demonstrated by Ma et al.,\u003csup\u003e\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/sup\u003e the electronic structure of MAPbI\u003csub\u003e3\u003c/sub\u003e must be studied at the nanoscale to ensure an accurate capture of its properties. A similar approach was adopted by Wiktor et al.\u003csup\u003e\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/sup\u003e for CsMX\u003csub\u003e3\u003c/sub\u003e (M\u0026thinsp;=\u0026thinsp;Sn, Pb; X\u0026thinsp;=\u0026thinsp;Cl, Br, I). Evidence has also been found for electron-phonon coupling in FAPbI\u003csub\u003e3\u003c/sub\u003e as a cause of long-range optical phonon modes, indicating large size effects.\u003csup\u003e\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e34\u003c/span\u003e\u003c/sup\u003e. Using CMD, Maheshwari et al.\u003csup\u003e\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e demonstrated that phase transitions in hybrid halide perovskites are driven by a complex interplay between dipole\u0026ndash;dipole interactions between organic cations and the metal halide lattice, resulting in the formation of large organized domains of organic cations. This has significant implications for the electronic structures of these materials.\u003c/p\u003e \u003cp\u003eIn this work, we want to address the key questions: what are the system size and level of theory needed to properly capture the structural and electronic properties of FAPbI\u003csub\u003e3\u003c/sub\u003e and, is the static zero Kelvin description sufficient or are simulations at finite temperature required? These are crucial questions from both theoretical and experimental perspectives.\u003c/p\u003e \u003cp\u003eUsing AIMD at 300 K and first-principles calculations, we characterized the structural and electronic properties of \u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e with increasing size of the simulation cell at the DFT level with both the PBE and PBE0 functionals. SOC was also considered when allowed by memory and/or computational resources. Our simulations show that it is necessary to use sufficiently large simulation cells (at least a 768-atom cell) in which all degrees of freedom (atomic positions and simulation cell) are fully relaxed in order to obtain an non-distorted \u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e structure, a converged band gap and a small overall dipole moment. For 0 K calculations, it is also essential to set up an initial system configuration in which the FAs cations are randomly oriented according to the 3-fold symmetry (Supplementary Fig.\u0026nbsp;1). Furthermore, with large simulation cells (from 2592 atoms upwards), PBE is able to describe the electronic band gap in good agreement with the experimental value when calculated as thermal average over finite-temperature equilibrated AIMD snapshots. We also show that the net dipole moment of the system goes to zero as the cell size increases and this is related to long-range effects due to the PbI\u003csub\u003e6\u003c/sub\u003e octahedral tilting. Finally, we demonstrate that thermal band gap oscillations are mainly related to the octahedral tilting.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e \u003cb\u003eCharacterization of\u003c/b\u003e \u003cb\u003eα\u003c/b\u003e\u003cb\u003e-FAPbI\u003c/b\u003e\u003csub\u003e\u003cb\u003e3\u003c/b\u003e\u003c/sub\u003e \u003cb\u003eat 0 K\u003c/b\u003e\u003c/p\u003e \u003cp\u003eThe obtained band gaps for different theoretical methods and computational schemes are reported in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e as a function of system size ranging from the minimal primitive cell (12-atoms) to larger supercells containing increasing even numbers of primitive cells. The periodicity imposed by the size of the supercell has to be compatible with the periodicity in the octahedral tilting,\u003csup\u003e\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e36\u003c/span\u003e\u003c/sup\u003e while the 12-atoms cell that is not compatible with this condition (but is still an often used setup in FAPbI\u003csub\u003e3\u003c/sub\u003e calculations) was included as a reference. At 0 K, we ran two series of calculations using as the initial structure \u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e from the material project database.\u003csup\u003e\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e In the first case (relax), only the atomic positions were optimized keeping the simulation cell fixed to cubic, while in the second (vc-relax), both the atomic positions and the lattice parameters were optimized. We also set up two initial configurations: (\u003cem\u003ei\u003c/em\u003e) \u003cem\u003eall\u003c/em\u003e \u0026minus;\u0026thinsp;\u003cem\u003ealigned\u003c/em\u003e : FA molecules are kept aligned with their dipole moments pointing all in the same direction in order to have a meaningful comparison between the 12-atoms cell and all the other supercells; (\u003cem\u003eii\u003c/em\u003e) \u003cem\u003epseudo\u003c/em\u003e\u0026thinsp;\u0026minus;\u0026thinsp;\u003cem\u003erandom\u003c/em\u003e: FAs are pseudo-randomly oriented (from the 96-atom supercell upwards), where the overall orientation of FAs preserves the 3-fold symmetry resulting into a null total dipole moment of the system (Supplementary Fig.\u0026nbsp;1). In all cases, k-point sampling was increased until convergence was achieved.\u003c/p\u003e \u003cp\u003eFor both the relax and vc-relax calculations, the band gap converges with increasing k-point grid but to different final values, indicating that the relaxation of all the cell degrees of freedom is necessary. For all the simulation cells with dipole-aligned FAs the relax band gap converges to a value around 1.48 eV. At first glance, this is in close agreement with the experimentally measured value for the \u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e band gap and can be justified by the fact that, by keeping the simulation cell cubic, we are, in a sense, simulating the average (pseudocubic) \u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e; however, with a closer look at the structure, we show that this is actually an artifact. Indeed, in addition to the correct band gap, it is necessary to verify that the cell distortions from the \u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e structure upon vc-relax and the total dipole moment of the supercell have also physical meaning. The vc-relax band gap of the 96-atom cell converges to a slightly higher value with respect to the 12-atom one. This is related to the significant distortion of the crystal structure from the cubic symmetry after vc-relax, decreasing the overlap of the electronic orbitals and consequently opening the band gap.\u003csup\u003e\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e In principle, this finding aligns with experimental observations that indicate the orthorhombic perovskite phase as the most stable at low temperatures.\u003csup\u003e\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e39\u003c/span\u003e\u003c/sup\u003e To quantify the deviations from cubic symmetry, we have analyzed the mean squared error (MSE) between the perfect \u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e structure and the vc-relax one as well as the distribution of the octahedra tilting angles after vc-relax for the two cases - with and without pseudo-random FA orientation (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). In the pseudo-random FA orientation case the initial total dipole moment is smaller then 10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e Debye. The pseudo-random orientation of the FAs in the starting configuration allows cubic symmetry to be maintained almost perfectly, as the MSE is reduced by two orders of magnitude for all supercells compared to the case with fully aligned FAs. The distributions of the octahedral tilting angles of the structures optimized from the pseudo-randomly oriented FA configurations average 0 degrees as expected (Supplementary Fig.\u0026nbsp;2) and the maximum value for the tilting angle converge with increasing system size (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The 96-atoms cell has a wide spread distribution because of the significant lattice distortions after vc-relax. In contrast, the structures optimized with all-aligned FAs do not present a octahedral tilting pattern. Local distortions of the octahedra due to a collective upward or downward motion of I ions compensate for the strongly directional dipole due to the dipole-aligned FAs (Supplementary Fig.\u0026nbsp;3). A comparison of all-aligned and pseudo-random configurations reveals that the vc-relax band gap is only improved for the 2592-atoms (3\u0026times;3\u0026times;3) cell. This phenomenon may be attributed to the fact that only this particular supercell allows for the complete relaxation of all degrees of freedom. This is because both the 3-fold symmetry for FAs and the octahedral tilting pattern are satisfied. It has been observed that, for all supercells, the potential energy with pseudo-randomly oriented FAs is consistently lower than that of the all-aligned case. This finding suggests that the pseudo-random configuration is a more favorable option for the system. Indeed, all-aligned configuration has a net total dipole moment for the FA that induces structural distortions in the Pb-I cages to compensate the overall dipole moment that cost energy.\u003c/p\u003e \u003cp\u003eBy adding SOC, the band gaps of the 12- and 96-atoms cells converge to slightly different values \u0026minus;\u0026thinsp;0.45 eV and 0.54 eV again indicating a potential problem in the description of the electronic structure related to the cell distortions for the 96-atoms cell. The band gap of the 768-atoms cell with SOC, also converges to the same value as the 12-atoms cell, and this may be related to the conservation of cubic symmetry by the 768-atoms cell. Remarkable is what is obtained with the PBE0 functional. It is known that for elements such as Pb, SOC and PBE0 contributions should cancel each other out.\u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e The results of our 0 K calculations show that this effect only comes into play at k-point convergence for the 96-atoms cell, whereas it is entirely absent for the 12-atoms cell; by moving from the 12-atoms cell to the 96-atoms cell, the electron charge localization decreases as the band gap correction due to the inclusion of PBE0 decreases by 0.65 eV, allowing for SOC compensation and convergence to a band gap of 1.60 eV. It is evident that the size of the system plays a crucial role in reproducing the empirical findings of canceling out SOC-PBE0 contributions in the band gap and ensuring the structural properties of the system converge well.\u003c/p\u003e \u003cp\u003e \u003cb\u003eCharacterization of\u003c/b\u003e \u003cb\u003eα\u003c/b\u003e\u003cb\u003e-FAPbI\u003c/b\u003e\u003csub\u003e\u003cb\u003e3\u003c/b\u003e\u003c/sub\u003e \u003cb\u003eat 300 K\u003c/b\u003e\u003c/p\u003e \u003cp\u003eThe thermally-averaged FAPbI\u003csub\u003e3\u003c/sub\u003e band gap was also calculated at 300 K by performing AIMD in the isothermal-isobaric ensemble with flexible cell option (NPT-F) at the PBE and PBE0 levels of theory for a minimum of 7 ps and up to 11 ps (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The band gap was computed as the average along the equilibrated AIMD trajectory (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e-a). We run in NPT-F to avoid any kind of symmetry restriction to the system. Since we are operating at 300 K, the initial orientation of the FAs is no longer important, as the kinetic energy is sufficient to randomize the FA orientation after a few AIMD steps. The finite temperature PBE band gap for the 6144-atoms cell is 1.47\u0026thinsp;\u0026plusmn;\u0026thinsp;0.08 eV that matches very well the experimentally measured gap, illustrating again the efficient compensation of many-body and SOC effects. Indeed, for all simulation cells, PBE0 (without SOC) consistently leads to a strong overestimation of the band gap.\u003c/p\u003e \u003cp\u003eWe also exploit the larger system sizes studied by AIMD together with its statistical approach to study the instantaneous band gap dependence from the local lattice distortions. We have subdivided the 6144-atoms and 768-atoms cells into 64 and 8 96-atoms cells, respectively, and computed the local band gap by projecting the density of states locally along the NPT-F trajectory (Fig.\u0026nbsp;\u0026lt;link rid=\"fig1\"\u0026gt;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u0026lt;/link\u0026gt;\u003c/span\u003e-b,\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e-c). The 3D arrangement of the supercells was projected into a 2D map for better visualization (Supplementary Fig.\u0026nbsp;4). There is a local band gap variation for both supercells of about 1 eV, but in the 6144-atoms cell there are more defined band gap domains, while for the 768-atoms cell the band gap is more homogeneous throughout the supercell. This again indicates the importance of the system size, in order to detect the possible formation of band gap domains. It further indicates that this is a phenomenon that has to be related to the process of long-range relaxation, stressing once again the importance of correctly describing the mechanisms that we have previously shown to be related to the size of the system, such as the octahedral tilting.\u003csup\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e31\u003c/span\u003e,\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e To quantify the connection between band gap fluctuations and octahedral tilting, we have calculated the time correlation function of the band gap oscillations and the octahedral tilting for the 6144-atoms cell (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), getting correlation times of about 25 fs and 30 fs, respectively, i.e. very similar time scales suggesting that the band gap variations are indeed closely connected to changes in octahedral distortions. The influence of the tilting angles on the band gap directly aligns with established literature, as the tilting angles impact the antibonding overlap of the orbitals involved in the band edges.\u003csup\u003e\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e Furthermore, the FA cations motion can be characterized by two characteristic rotational correlation times of 80 fs and 250 fs for the C-H and N-N vectors, respectively (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), that are farer from the value calculated for the band gap oscillations. We have also computed the time correlation functions of the bottom of the conduction band and top of the valence band eigenvalues obtaining a trend comparable with that of the band gap (Supplementary Fig.\u0026nbsp;5). Local variations in the band gap might indicate that FAPbI\u003csub\u003e3\u003c/sub\u003e can potentially absorb photons with different wavelengths in different regions of the sample, which may be a further explanation of why this material performs as well as AL. The electronic charge distribution due to lattice motions in halide perovskites have implications on thermal fluctuations in the electronic structure. It has been shown that off-centering of Sn\u003csup\u003e2+\u003c/sup\u003e and Br\u003csup\u003e\u0026minus;\u003c/sup\u003e widens the band gap of CsSnBr\u003csub\u003e3\u003c/sub\u003e.\u003csup\u003e\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u003c/sup\u003e Additionally, the top of the valence band and the bottom of the conduction band of FAPbI\u003csub\u003e3\u003c/sub\u003e are dominated by I\u003csup\u003e\u0026minus;\u003c/sup\u003e and Pb\u003csup\u003e2+\u003c/sup\u003e contributions, respectively.\u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e For these reasons, we can expect that the fluctuations in the band gap occur on time scales similar to those characterizing the octahedral tilting fluctuations. The selection of an appropriate system size is imperative for the accurate description of the octahedral tilting pattern. Our findings demonstrate that PbI\u003csub\u003e6\u003c/sub\u003e tilting converges for a 768-atoms supercell. The last point needed to verify the proper description of FAPbI\u003csub\u003e3\u003c/sub\u003e is the dipole moment.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFAPbI\u003csub\u003e3\u003c/sub\u003e Kohn-Sham band gap for different system sizes at several theory levels and system optimization schemes. The initial configurations for the 0 K simulations were chosen with all FAs aligned except the ones indicated with a star superscript. AIMD was performed at 300 K in NPT-F and the band gaps were computed as averages along the trajectories (4\u0026ndash;7 ps) after equilibration. Values in brackets are additive estimates from SOC and PBE0 calculations since direct calculations require too much memory for available resources.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"9\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSimulation \u003c/p\u003e \u003cp\u003ecell\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNPT-F\u003c/p\u003e \u003cp\u003ePBE\u003c/p\u003e \u003cp\u003e300 K\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNPT-F\u003c/p\u003e \u003cp\u003ePBE0\u003c/p\u003e \u003cp\u003e300 K\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003erelax\u003c/p\u003e \u003cp\u003ePBE\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003evc-relax\u003c/p\u003e \u003cp\u003ePBE\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003evc-relax\u003c/p\u003e \u003cp\u003ePBE\u0026thinsp;+\u0026thinsp;SOC\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003evc-relax\u003c/p\u003e \u003cp\u003ePBE0\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003evc-relax\u003c/p\u003e \u003cp\u003ePBE0\u0026thinsp;+\u0026thinsp;SOC\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003cp\u003e(eV)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:n\\times\\:n\\times\\:n\\)\u003c/span\u003e\u003c/span\u003ek-point\u003c/p\u003e \u003cp\u003egrid\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"5\" rowspan=\"6\"\u003e \u003cp\u003e12-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(1\\times\\:1\\times\\:1)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3.62\u0026thinsp;\u0026plusmn;\u0026thinsp;0.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.08\u0026thinsp;\u0026plusmn;\u0026thinsp;0.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e5.82\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e5.54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.31\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3.90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e3.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e2.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e2.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3.38\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e2.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3.37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e2.24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"4\" rowspan=\"5\"\u003e \u003cp\u003e96-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(2\\times\\:2\\times\\:2)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.16\u0026thinsp;\u0026plusmn;\u0026thinsp;0.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.80\u0026thinsp;\u0026plusmn;\u0026thinsp;0.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e2.85\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.58\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.69\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2.84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e(1.75)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.58\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e(1.61)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e(1.68)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.61*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.60*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.48*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e2.76*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e(1.64)*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003e768-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(4\\times\\:4\\times\\:4)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.76\u0026thinsp;\u0026plusmn;\u0026thinsp;0.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.28\u0026thinsp;\u0026plusmn;\u0026thinsp;0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.74*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.74*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.63*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.69*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.69*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.56*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e2592-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(6\\times\\:6\\times\\:6)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.59\u0026thinsp;\u0026plusmn;\u0026thinsp;0.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.17\u0026thinsp;\u0026plusmn;\u0026thinsp;0.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.51\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.71\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.50*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.50*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e6144-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(8\\times\\:8\\times\\:8)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.47\u0026thinsp;\u0026plusmn;\u0026thinsp;0.08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.09\u0026thinsp;\u0026plusmn;\u0026thinsp;0.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.67\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.68*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.69*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"9\"\u003e\u003csup\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sup\u003e FA pseudo-randomly oriented\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eDipole moment\u003c/h2\u003e \u003cp\u003eThe dipole moment was computed at 0 K and at 300 K (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). In principle, the total dipole moment should be zero; however, for small-size simulation systems, it is not, due to the presence of residual, not fully compensated dipoles. Given the substantial decrease in dipole moment observed with increasing system size, it can be deduced that this phenomenon is associated with a long-range collective interaction, such as the long-range dipole-dipole correlations present between FA-FA and octahedral cage dipoles.\u003csup\u003e\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e FA has a non-zero permanent dipole moment, while the octahedral distortions, i.e. contraction or elongation along different I-Pb-I axes, might induce a significant dipole moment. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e reports the dipole moment contributions obtained by subtracting the dipole moment of the FA cations in the initial configuration from the total dipole moment of the system. The result is similar for all configurations (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), and comparable with the values in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, concluding that the initial FA configuration does not affect the overall dipole moment of the cell. To estimate the actual contributions of FA and octahedral cage distortions to the dipole moment, two NPT-F simulations were performed with FA respectively PbI\u003csub\u003e6\u003c/sub\u003e octahedra frozen. The octahedral distortion compensates for the dipole moment of the system in cases the FA cation orientations create a non-zero dipole moment (Supplementary Fig.\u0026nbsp;6). The time correlation function associated to the dipole moment fluctuations shows that the correlation time (\u0026sim; 30 fs) is not affected by the system size (Supplementary Fig.\u0026nbsp;8) at contrast to the absolute value of the dipole moment (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). Except for the 12-atoms and 96-atoms cells, which have already shown large size effects for structural as well as electronic properties, there is no major change in dipole moment when using PBE0. Finally, the values of the dipole moment obtained with PBE and PBE0 are statistically equivalent for the 6144-atoms cell (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). This finding suggests that the residual dipole moment may not be directly associated with the degree of charge localization imposed by the functional.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMean squared error (MSE) of the lattice vectors with respect to the perfect cubic FAPbI\u003csub\u003e3\u003c/sub\u003e α-phase (column 1) and absolute maximum value of the octahedral tilting angle (column 2) at 0 K. For the structure optimized in the presence of dipole-aligned FAs, the octahedral tilting angle refers to the global deformation of the octahedra, while for the structure optimized from pseudo-randomly oriented FAs it refers to the average absolute value of the octahedral tilting angle (Supplementary Fig.\u0026nbsp;3). Total dipole moment per stoichiometric unit (ABX3) for different simulation cells; the dipole moment was computed at 0 K (column 3) and after equilibration as average along the AIMD trajectories performed at 300 K in the NPT-F ensemble (columns 4\u0026ndash;5). The initial configurations for the 0 K simulations were chosen with all FAs aligned except the ones indicated with a star superscript.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSimulation \u003c/p\u003e \u003cp\u003ecell\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMSE\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eOctahedra\u003c/p\u003e \u003cp\u003etilting amplitude\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003cp\u003e(deg)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eTotal Dipole\u003c/p\u003e \u003cp\u003ePBE\u003c/p\u003e \u003cp\u003e0 K\u003c/p\u003e \u003cp\u003e(Debye/ABX\u003csub\u003e3\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eTotal Dipole\u003c/p\u003e \u003cp\u003ePBE\u003c/p\u003e \u003cp\u003e3000 K\u003c/p\u003e \u003cp\u003e(Debye/ABX\u003csub\u003e3\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eTotal Dipole\u003c/p\u003e \u003cp\u003ePBE0\u003c/p\u003e \u003cp\u003e3000 K\u003c/p\u003e \u003cp\u003e(Debye/ABX\u003csub\u003e3\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(1\\times\\:1\\times\\:1)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e13.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e \u003cp\u003e2.32\u0026thinsp;\u0026plusmn;\u0026thinsp;0.98\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e \u003cp\u003e1.80\u0026thinsp;\u0026plusmn;\u0026thinsp;0.70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e96-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(2\\times\\:2\\times\\:2)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.78\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e18.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e2.24\u0026thinsp;\u0026plusmn;\u0026thinsp;1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e4.57\u0026thinsp;\u0026plusmn;\u0026thinsp;0.71\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.09*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e17.36*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.25*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e768-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(4\\times\\:4\\times\\:4)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.92\u0026thinsp;\u0026plusmn;\u0026thinsp;0.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.89\u0026thinsp;\u0026plusmn;\u0026thinsp;0.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:4\\bullet\\:{10}^{-3}\\)\u003c/span\u003e\u003c/span\u003e*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e6.46*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.10*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e2592-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(6\\times\\:6\\times\\:6)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.39\u0026thinsp;\u0026plusmn;\u0026thinsp;0.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.31\u0026thinsp;\u0026plusmn;\u0026thinsp;0.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:8\\bullet\\:{10}^{-4}\\)\u003c/span\u003e\u003c/span\u003e*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.67*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.36*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c7\" namest=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e6144-atoms\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(8\\times\\:8\\times\\:8)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.24\u0026thinsp;\u0026plusmn;\u0026thinsp;0.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.24\u0026thinsp;\u0026plusmn;\u0026thinsp;0.07\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\:\\:\\:1\\bullet\\:{10}^{-4}\\)\u003c/span\u003e\u003c/span\u003e*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.52\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.18*\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003ctfoot\u003e \u003ctr\u003e\u003ctd colspan=\"7\"\u003e\u003csup\u003e\u003cem\u003e*\u003c/em\u003e\u003c/sup\u003e FA pseudo-randomly oriented\u003c/td\u003e\u003c/tr\u003e \u003c/tfoot\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMagnitude of the total dipole moment of the system separated into FA contribution, and difference between the dipole moment and the FA contribution to the total dipole moment of the system. The values were computed for a 768-atoms cell on equilibrated AIMD snapshots starting from an initial configuration with different FA orientations. The all-aligned, random and random_best, smart_100 and smart_quasi systems are initiated from an initial configuration of FAs that are all aligned, randomly oriented, and pseudo-randomly oriented, respectively (Supplementary Fig.\u0026nbsp;7). The random and smart configurations are built with a null initial total dipole moment.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFA \u003c/p\u003e \u003cp\u003eorientation\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTotal dipole\u003c/p\u003e \u003cp\u003e(Debye/ABX\u003csub\u003e3\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFA dipole\u003c/p\u003e \u003cp\u003e(Debye/ABX\u003csub\u003e3\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eTotal dipole \u0026ndash; FA dipole\u003c/p\u003e \u003cp\u003e(Debye/ABX\u003csub\u003e3\u003c/sub\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eall-aligned\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.82\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003erandom\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003erandom_best\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003esmart_100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"1\" nameend=\"c5\" namest=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003esmart_quasi\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e1.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e0.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e1.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eIn conclusion, by means of large-scale first-principles calculations and ab initio molecular dynamics simulations at 300 K, we demonstrated that in order to get an accurate description of the structural and electronic properties of the α-phase of FAPbI3 the size of the simulated system needs to approach the nanoscale. In particular, we showed that three conditions have to be met simultaneously, namely a proper description of the band gap, minimization of structural distortions, and the zeroing out of the total dipole moment. For first-principles calculations, it is essential to start from an initial configuration where the FAs are pseudo-randomly oriented by preserving the 3-fold symmetry and minimizing the dipole moment. At 300 K, because of the finite temperature dynamics, the initial configuration of the FAs is not stringent and, from a 2592-atoms cell upwards, the PBE approximation is already able to describe the electronic band gap of α-FAPbI3. For the 6144-atoms cell, we have computed a band gap of 1.47\u0026thinsp;\u0026plusmn;\u0026thinsp;0.08 eV which is in excellent agreement with the experimental values of 1.45\u0026ndash;1.51 eV reported in literature (highlighting that PBE0 and SOC corrections only cancel out for this system size range). The same cell minimizes structural distortions with respect to the perfect α-FAPbI3 structure and has the lowest dipole moment among all the systems studied. A significant correlation was discovered between PbI6 octahedral tilting, band gap oscillations, and dipole moment. In particular, the dipole moment goes to zero only if the system size is large enough to properly relax the tilting pattern of the octahedra. Overall, an adequate size of the system (at least 6144-atoms cell) is needed to correctly describe its physics, as we have demonstrated with the identification of band gap domains related to a correct description of the octahedral tilting. Our work provides a detailed insight into the connection between structural and electronic properties of α-FAPbI3 - and MHPs in general - making an important contribution to the field of ab initio simulations dedicated to understanding fundamental physical principles, such as hole-electron transport, which is of paramount importance in the development of increasingly high-performance PSC devices.\u003c/p\u003e"},{"header":"Methods","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n \u003ch2\u003eFirst-principles calculations\u003c/h2\u003e\n \u003cp\u003eDFT simulations at 0 K were performed with the Quantum ESPRESSO (QE)\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e41\u003c/span\u003e\u003c/sup\u003e suite of codes. All the calculations were run with DOJO fully relativistic norm-conserving PBE pseudopotentials\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e42\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e43\u003c/span\u003e\u003c/sup\u003e and well-converged basis sets corresponding to an energy cutoff of 150 Ry for the wave functions and 600 Ry for the charge density. Different k-point Monkhorst-Pack grids\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e44\u003c/span\u003e\u003c/sup\u003e were used, all centered on \u0026Gamma;-point. Semiempirical corrections accounting for the van der Waals interactions were included with the DFT-D3 approach.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e Different simulation cells were used, starting from a 1\u0026times;1\u0026times;1 \u003cem\u003e\u0026alpha;\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e (12-atoms) up to a 8\u0026times;8\u0026times;8 (6144-atoms). The electronic structure of fully relaxed structures (vc-relax) was also computed including spin-orbit coupling (SOC) and PBE0. The supercell distortion with respect to the perfect cubic \u003cem\u003e\u0026alpha;\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e has been estimated by the mean squared error (MSE) between the lattice vectors of the two systems. Because CP2K\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e46\u003c/span\u003e\u003c/sup\u003e allows to run also DFT at 0 K, \u0026Gamma;-point simulations were performed with both the QE and CP2K software, obtaining equal band gap values to three decimal places, which means that the results achieved with the two software packages are comparable.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003eAb initio molecular dynamics\u003c/h3\u003e\n\u003cp\u003eAIMD simulations were run in the DFT framework as implemented in the CP2K software. The PBE and PBE0 functional and the D3 dispersion correction\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e were adopted together with Goedecker-Teter-Hutter pseudopotentials\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e47\u003c/span\u003e\u003c/sup\u003e and a polarized double-\u003cem\u003e\u0026zeta;\u003c/em\u003e Gaussian basis set (DZVP)\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e48\u003c/span\u003e\u003c/sup\u003e for valence electrons. The energy cut off for the expansion of the electron density was set to 400 Ry. Simulations were run with a time step of 0.5 fs in the NPT flexible ensemble using Born-Oppenheimer dynamics for 7\u0026ndash;12 ps (PBE) and 2\u0026ndash;5 ps (PBE0), while the temperature was controlled by the Bussi thermostat\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e49\u003c/span\u003e\u003c/sup\u003e and the pressure by the Martyna barostat.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e Different simulation cells were used, starting from a 1\u0026times;1\u0026times;1 \u003cem\u003e\u0026alpha;\u003c/em\u003e-phase FAPbI\u003csub\u003e3\u003c/sub\u003e (12-atoms) up to a 8\u0026times;8\u0026times;8 (6144-atoms). The finite temperature band gap was computed as an average of different band gaps calculated from the projected density of states (PDOS) on several AIMD snapshots after the system equilibrated (\u0026sim; 2 ps). The spatial variation of the band gap within a supercell was calculated by grouping the PDOS of the atoms of interest. Real space positions of top of the valence and bottom of the conduction bands in FAPbI\u003csub\u003e3\u003c/sub\u003e were computed after quenching an equilibrated AIMD snapshot to 0 K. Different initial FA orientation were tested - completely ordered, random oriented, smart oriented (total FA dipole equal to zero) - to avoid any bias on the simulations. The total dipole moment was calculated at the quantum level as in the CP2K framework, while the contribution of FAs to the dipole in the initial configurations was estimated classically, assigning a\u0026thinsp;+\u0026thinsp;1 charge to each FA.\u003c/p\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003eTime correlation function analysis\u003c/h2\u003e\n \u003cp\u003eThe rotational dynamics of FA and PbI\u003csub\u003e6\u003c/sub\u003e octahedra were characterized by the correlation function\u003c/p\u003e\n \u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$$\\:{C}_{rot}\\left(t\\right)=\\frac{\u0026lang;\\overrightarrow{\\mu\\:}\\left(t\\right)\\cdot\\:\\overrightarrow{\\mu\\:}\\left(0\\right)\u0026rang;}{\\left|\\overrightarrow{\\mu\\:}\\left(0\\right)\\right|}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{\\mu\\:}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e is the C-H(N-N) vector for FA or the octahedra tilting function appropriate for the PbI\u003csub\u003e6\u003c/sub\u003e tilting. The timescale oscillations for the band gap were quantified by the correlation function\u003c/p\u003e\n \u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e$$\\:{C}_{gap}\\left(t\\right)=\\frac{\u0026lang;{\\Delta\\:}{ϵ}_{cv}\\left(t\\right)\\cdot\\:{\\Delta\\:}{ϵ}_{cv}\\left(0\\right)\u0026rang;}{\\left|{\\Delta\\:}{ϵ}_{cv}\\left(0\\right)\\right|}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere ∆\u003cem\u003e\u0026epsilon;\u003c/em\u003e\u003csub\u003e\u003cem\u003ecv\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) is the difference between the eigenvalues of the bottom of the conduction band and the top of the valence band. The same time correlation function has been used to compute the correlations of the dipole moment fluctuations, where the quantity correlated in time was the value of the dipole moment.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eAuthor contributions statement\u003c/h2\u003e \u003cp\u003eV.C. and L.A. conceived the idea. V.C, L.A, and V.S. performed the theoretical simulations and wrote the paper under the supervision of U.R.. All authors contributed to the discussion and writing of the paper.\u003c/p\u003e \u003c/p\u003e\u003cp\u003e \u003ch2\u003eCompeting interests\u003c/h2\u003e \u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e \u003c/p\u003e\u003cp\u003e \u003ch2\u003eAdditional information\u003c/h2\u003e \u003cp\u003eThe online version contains supplementary material available at\u003c/p\u003e \u003cp\u003eData and analysis scripts are available on Zenodo at \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.5281/zenodo.13712682\u003c/span\u003e\u003cspan address=\"10.5281/zenodo.13712682\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eU.R. acknowledges the Swiss National Foundation (grant N. 200020_219440) and computational resources from the Swiss National Computing Centre CSCS (project s1151). V.C. acknowledges computational resources from the Swiss National Computing Centre CSCS (project s1253).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eZhang, M. \u003cem\u003eet al. \u003c/em\u003eHigh-Efficiency Rubidium-Incorporated Perovskite Solar Cells by Gas Quenching. \u003cem\u003eACS Energy Lett. \u003c/em\u003e2, 438\u0026ndash;444, DOI: 10.1021/acsenergylett.6b00697 (2017).\u003c/li\u003e\n\u003cli\u003eJeong, J. \u003cem\u003eet al. \u003c/em\u003ePseudo-halide anion engineering for \u003cem\u003e\u0026alpha;\u003c/em\u003e-FAPbI3 perovskite solar cells. \u003cem\u003eNature \u003c/em\u003e592, 381\u0026ndash;385, DOI: 10.1038/s41586-021-03406-5 (2021). 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Phys. \u003c/em\u003e101, 4177\u0026ndash;4189, DOI: 10.1063/1.467468 (1994).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-5730287/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5730287/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFormamidinium-lead-iodide (FAPbI\u003csub\u003e3\u003c/sub\u003e) has established itself as the state of the art for high solar-energy conversion efficiency in perovskite-based solar cells. FAPbI\u003csub\u003e3\u003c/sub\u003e has a rich phase diagram, and it has been noted that long-range correlation between organic and lattice dipoles can influence phase transitions and, consequently, optoelectronic properties. In this regard, system size effects can play a crucial role for an appropriate theoretical description of FAPbI\u003csub\u003e3\u003c/sub\u003e. In this context, we perform a systematic study on the structural and electronic properties of the photoactive phase of FAPbI\u003csub\u003e3\u003c/sub\u003e (\u003cem\u003eα\u003c/em\u003e-FAPbI\u003csub\u003e3\u003c/sub\u003e) as a function of system size. Utilizing ab initio molecular dynamics at 300 K and first-principles calculations, we demonstrate that the selection of the computational system/setup must satisfy three criteria concurrently to ensure an accurate theoretical description: the (correct) value of the band gap, the extent (or the absence of) structural distortions, and the zeroing out of the total dipole moment. We demonstrate that the net dipole moment vanishes as the system size increases due to PbI\u003csub\u003e6\u003c/sub\u003e octahedra distortions rather than due to FA\u003csup\u003e+\u003c/sup\u003e rotations. Additionally, we show that thermal band gap fluctuations are predominantly correlated with octahedral tilting. The optimal agreement between simulation results and experimental properties for FAPbI\u003csub\u003e3\u003c/sub\u003e is only achieved by system sizes approaching the nanoscale.\u003c/p\u003e","manuscriptTitle":"Nanoscale size effects in α-FAPbI3 evinced by large-scale ab initio simulations","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-01-10 12:59:33","doi":"10.21203/rs.3.rs-5730287/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"45e70dae-e5e1-4dee-ae9a-4aeb87fcf70b","owner":[],"postedDate":"January 10th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":42434812,"name":"Physical sciences/Chemistry/Theoretical chemistry/Computational chemistry"},{"id":42434813,"name":"Physical sciences/Physics/Condensed-matter physics/Structure of solids and liquids"}],"tags":[],"updatedAt":"2025-07-25T07:09:18+00:00","versionOfRecord":{"articleIdentity":"rs-5730287","link":"https://doi.org/10.1038/s41467-025-61351-7","journal":{"identity":"nature-communications","isVorOnly":false,"title":"Nature Communications"},"publishedOn":"2025-07-24 04:00:00","publishedOnDateReadable":"July 24th, 2025"},"versionCreatedAt":"2025-01-10 12:59:33","video":"","vorDoi":"10.1038/s41467-025-61351-7","vorDoiUrl":"https://doi.org/10.1038/s41467-025-61351-7","workflowStages":[]},"version":"v1","identity":"rs-5730287","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5730287","identity":"rs-5730287","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}