DFT Simulations as Valuable Tool to Support NMR Characterization of Halide Perovskites: the Case of Pure and Mixed Halide Perovskites

Solid state NMR spectroscopy is swiftly emerging as useful tool to characterize the structure, composition and dynamic properties of lead halide perovskites. On the other hand, interpretation of solid state NMR signatures is often challenging, because of the potential presence of many overlapping signals in small range of chemical shifts, hence complicating the extraction of detailed structural features. Here, we demonstrate the reliability of periodic Density Functional Theory in providing theoretical support for the NMR characterization of halide perovskite compounds, considering nuclei with spin I=1/2. For light 1H and 13C nuclei, we predict NMR chemical shifts in good agreement with experiment, further highlighting the effects of motional narrowing. Accurate prediction of the NMR response of 207Pb nuclei is comparably more challenging, but we successfully reproduce the downshift in frequency when changing the halide composition from pure iodine to pure bromine. Furthermore, we confirm NMR as ideal tool to study mixed halide perovskite compounds, currently at the limelight for tandem solar cells and color-tunable light emission.


Introduction
Metal halide perovskites are currently under intense scrutiny for opto-electronic applications, in particular photovoltaics and light-emission. [1][2][3] By combining inexpensive processability from room temperature solution techniques together with excellent semiconducting properties, including large optical absorption coefficient, [4] decent charge mobility [5] and hindered electron-hole recombination, [6] these systems have recently established 25.5 % photovoltaic efficiency record, [7] challenging long-time established photovoltaic technologies in the direction for low cost renewable solar energy. This class of materials is characterized by AMX 3 chemical structure, where a continuous network of MX 6 octahedra (M being a divalent metal, Pb, Sn or Ge and X being a halide) in corner shared connection is alternated by a lattice of A-cations, consisting in a small organic molecule (CH 3 NH 3 + or CH(NH 3 ) 2 + ) and/or an inorganic atomic cation (Cs + ). However, in spite of their impressive opto-electronic performances, our knowledge of the basic properties of these systems remains partly elusive. Halide perovskites are characterized by complex structural properties, as related to the rich chemical tailorability and to the existence of competing allotropic crystalline forms and phase transitions. For instance, the reference methylammonium lead halide (MAPbX 3 ) compounds show two common phase transitions, [8,9] one at low temperature, involving transition from orthorhombic to tetragonal crystal structure, and the other (taking place slightly above room temperature for the case of MAPbI 3 ) involving a transition from tetragonal to pseudocubic crystal structure. The increased mobility of the A-cation in the high temperature tetragonal and pseudocubic lattices leads to dynamic disorder, [10 -14] which impacts the optical and electronic properties of the material. [15] The characterization of this dynamics as function of the composition of the A-site is therefore an important point, to further optimize the performances of perovskite-based devices. Mixed halide perovskites also present several open questions, in terms of their structural characterization. Currently under investigation for wavelength-tunable light-emitting diodes [16] and for tandem solar cells, [17] these materials often show unstable response under operation, because of the formation of halide segregated phases. The possibility to tackle the halide composition and homogeneity from the mesoscale, down to the atomic scale is therefore important for technological applications. Another open question is then related to the socalled hollow [18] and deficient halide perovskites. [19] In these systems, the intentional incorporation of bulky organic cations leads to the formation of defective sites in the lattice. These however do not negatively affect the optical and electronic properties of the material, namely, the optical absorption onset, at least for defect concentration limited to 10 %, as reported in Ref. [19] for the specific case of the deficient MAPbI 3 . Still, quantifying the concentration of defective sites incorporated in the perovskite lattice, establishing whether they are randomly distributed (hollow), [18] or tend to self-organize (deficient) [19] is crucial for the exploitation of this promising class of materials.
In this frame, it clearly appears that in order to achieve a detailed structural characterization of halide perovskites, a smart combination of several techniques is required, so to probe the material structure both on a global scale, as in the case of the XRD, and on a very local scale. Solid state Nuclear Magnetic Resonance (NMR) spectroscopy represents the ideal method for the investigation of the local structure in halide perovskites, as the NMR response of the atoms constituting the material is strongly sensitive to their chemical environment, the nature of the chemical bonds and fine structural details (bond lengths and bond angles). NMR faced an initial success during the 80ies for the characterization of all-inorganic halide perovskite materials, as a complementary technique with respect to XRD. Seminal work by Roos and coworkers investigated the phase transition in allinorganic CsPbCl 3 and RbCdCl 3 perovskite structures, [20] followed by other NMR-based investiga-tions on the phase transitions of lead halide perovskite systems. [21,22] Wasylishen et al. instead performed seminal structural investigations of hybrid methylammonium lead halide perovskite structures by combining NMR measurements with several characterization techniques. Their results nicely highlighted the rotational dynamics associated to the methylammonium cations in the cubo-octahedral cage and further explored the related effects of temperature and phase transitions. [23,24] The interest in the NMR response of metal halide perovskite has rekindled more recently subsequent to their initial breakthrough for photovoltaic (PV) applications in 2012. [25 -27] These achievements first prompted additional structural characterization for methylammonium lead iodide perovskite reported by Baikie et al., [28] followed by NMR characterizations of pure bromide and chloride analogues, as well as alloys concomitantly by Rosales et al. [29] and Roiland et al. [12] These latter studies demonstrated the capability of NMR to characterize phase homogeneity and segregation in mixed halide compounds, through the NMR response of 207 Pb nuclei. On the strength of these findings, the last few years witnessed a major increase of the number of NMR studies on halide perovskite materials, both related to structure and composition, as well as, on their structural and compositional evolution upon device operation. Recent studies include the characterization of A-mixed halide perovskites, [30][31][32][33][34] demonstration of halide (iodine) diffusion in working devices, [35] characterization of thermal and water-induced structural decomposition, [36] and characterization of dimensionally confined perovskite systems, as layered 2D systems, [37,38] nanoparticles, [39,40] and deficient-halide perovskites. [19,41] A remarkable overview of results related to NMR characterization of metal halide perovskites can be found in Refs. [42,43] .
Meanwhile, despite its successes and possibilities, a detailed interpretation of NMR spectroscopy data is usually a delicate task. Solid state NMR signatures of real samples are not-unfrequently characterized by complex lineshapes, arising from the overlap of several signatures in a small range of chemical shifts and, in the specific case of static measurements, by broadening coming from Chemical Shielding Anisotropy (CSA), hence making reliable assignments challenging. Additional support is therefore greatly welcomed to help the interpretation of experimental data, as nowadays available from periodic Density Functional Theory (DFT) simulations. [44] The development of dedicated approaches from (parameter-free) first-principles theories [44][45][46][47] indeed paves the way to provide complementary information to the experimental NMR characterization, as widely demonstrated in the literature. [48 -50] In spite of the number of important studies devoted to halide perovskites, however, the number of DFT simulations targeting NMR parameters is much more scarce. Just in the last years, few NMR investigations included estimates of the magnetic shielding tensor via DFT simulations, mainly exploiting the molecular approach from quantumchemistry. [31,34,[51][52][53][54] Here, reliable models for the halide perovskite material are obtained by cutting out a finite cluster. [55] In the present work, we will instead rely on complementary picture from periodic DFT calculations, where a representative crystal cell is replicated in space, a perspective that is more natural to discuss systems in condensed phase. Surprisingly, the periodic approach has been comparably less exploited than the molecular one, for the case of halide perovskites, [12,56] hence leaving a question mark on its reliability and accuracy in predicting the NMR properties of these interesting materials. Here, we perform extensive periodic DFT simulations aimed to unveil advantages and limitations of periodic DFT methods in simulating the NMR response of halide perovskites, with special attention to nuclei with spin quantum number I = 1/2, namely 1 H and 13 C and 207 Pb. Simulations will consider reference models for the three phases (orthorhombic, tetragonal and cubic) reported for methylammonium lead iodide and bromide perovskites, hence addressing potential influence of the reference crystallographic model. Finally, we will also simulate the 207 Pb response of mixed iodine/bromine compounds, to assess the reliability of DFT simulations in describing these systems, which are attracting large interest for tandem solar cells and tunable-color light emitting devices. [57 -60]

Methods
Periodic DFT simulations performed here are based on a series of models for methylammonium lead bromide and lead iodide perovskites, thereafter, named MAPbBr 3 and MAPbI 3 , respectively. Additional models are considered for mixed halide systems, as explained below. Pure halide phases are simulated adopting available reference models from XRD measurements, for the orthorhombic, tetragonal and cubic phases. The model employed to simulate the low temperature orthorhombic phase obeys the Pnma space group symmetry, with its MA cations in antiferroelectric ordering reported in Figure 1, as proposed by Baikie et al. on the basis of XRD measurements. [61] The tetragonal phase nominally belongs to the I4mcm [62] or I4cm, [9] which results in orientational disorder for the MA cations. For practical calculations, we here consider the specific orientation of the MA cations shown in Figure 1. The high temperature pseudocubic phase is instead modeled through 2 × 2 × 2 supercell of the nominal Pm-3m crystalline cell. To account for the orientational disorder characteristic of this phase, [13][14][15]23] we have considered four models, labeled from 1 to 4, with initial random orientation of the MA cations, as shown in Figure 1.
NMR response of halide perovskites is computed with periodic DFT simulations, within the plane-wave/ pseudopotential formalism, as implemented in the CASTEP suite code. [44,46,63] This makes use of a perturbative approach to compute the chemical shielding tensor of the atomistic model considered. In addition, the present method takes advantage of the GIPAW (Gauge Including Projector Augmented Wave) formalism to reconnect the non-local information from the plane wave basis set to the local information associated to the electron shielding close to the nuclei, meanwhile solving the Gauge problem, typical of finiteness of the basis set. [45,46] Convergence tests have been performed to evaluate the effect of the two main computational parameters, namely, 1) the kinetic energy cutoff for plane-wave expansion of the electronic wave functions and 2) the sampling of the first Brillouin zone. We anticipate that, although accounting for the contraction of the orbitals due to relativistic effects, the present formalism for the simulation of the shielding tensor does not include Spin-Orbit-Coupling (SOC), [44] which on the other hand turns out to be important for heavy atoms. [55,[64][65][66][67] Test calculations are performed on the orthorhombic phase of MAPbBr 3 (see Figure 1), as well as for tetramethylsilane and Pb(NO 3 ) 2 , which represent the typical reference compounds for the calculation of NMR chemical shift of 13 C and 1 H nuclei, and 207 Pb, respectively. A 600 eV kinetic energy cutoff and 0.03 Å À 1 k-point mesh density result in converged values for the magnetic shielding values of 13 C nuclei, while providing discrepancy of less than 1 ppm for 1 H nuclei, as compared to more expensive computational set-ups (see Supporting Information). The chemical shielding of 207 Pb shows instead larger deviations, on the order of few tens of ppm (see Supporting Information). Still, the larger dependency of this nucleus to the computational approach, as compared to 1 H and 13 C, should be put in perspective of its very broad range of chemical shifts, on the order of thousands of ppm. Consequently, 600 eV and 0.03 Å À 1 were chosen through all this work as the best trade-off between computational cost and accuracy. Unless stated otherwise, all the models in Figure 1 are optimized with cell parameters kept fixed at their corresponding experimental values. Optimizations were performed adopting the same level of theory used for the calculation of the NMR response (600 eV and 0.03 Å À 1 ).
To compare the results from our simulations with experiments, meanwhile addressing the accuracy of our periodic DFT simulations, we resort to widespread and statistically meaningful approach for the appropriate referencing of the computed NMR parameters. This consists in computing theoretical NMR chemical shielding (σ iso ) for a set of compounds with known experimental chemical shift (δ iso ) and fitting the two datasets with a linear function. [44,47,55,64] The intercept of the line determines the reference for the definition of the chemical shift, while the slope provides indication on the accuracy of the method, with slopes close to À 1 corresponding to good quantitative matching between theory and experiment. The results of this procedure for the 1 H, 13 C and 207 Pb nuclei with respect to a set of reference compounds are illustrated in Figure 2 (details in the chosen datasets are extensively discussed in Supporting Information).
Both slope and R 2 coefficient close to À 1 for 1 H and 13 C nuclei, as shown in Figure 2, suggest good quantitative accuracy associated to our computational set-up, consistently with previous works adopting the similar periodic DFT approach. [68,69] Results for 207 Pb instead suggest lower degree of accuracy for this chemical species, however paralleling previous results reported by Alkan and Dybowski, adopting nonperiodic DFT calculations without accounting for SOC (reported slope of the linear fit and R 2 coefficient are 0.54 and 0.61, respectively). [55] The same authors also demonstrated that inclusion of SOC through Zero-Order Regular Approximation (ZORA) largely improves the results (reported slope and R 2 coefficient of 0.87 and 0.98, respectively), [55] hence demonstrating the necessity of SOC to achieve quantitative accuracy for the prediction of NMR properties of lead-based compounds.
Finally, in the effort of establishing how the detailed atomistic structure influences the predicted  [66] adopting a molecular description and focusing on the simulation of isolated PbX 6 octahedra (X = halide). Here, we still rely on frozen models of the cubic MAPbI 3 model 1, with distorted structures generated from short Born-Oppenheimer Molecular Dynamic simulation, hence used here as a simple structure generator, which automatically explores the conformations accessible by the system around the considered equilibrium structure, at a given temperature. This simulation was also performed with CASTEP, with the same computational parameters as for the previous calculations. 10 ps trajectory were generated, with time step of 1 fs, and employing an NVT ensemble, with target temperature fixed at 300 K, using Nose-Hoover thermostat. [70] Consistently, a set of 50 snapshots have been extracted each 200 fs for additional NMR simulations, as we will discuss later.
The spectra have been simulated from the NMR chemical shielding tensors computed from DFT, adopting the SIMPSON program. [71] In fact, SIMPSON is known to incorporate some aspects of the experimental spectral acquisition process. However, it does not consider the effects of rapid longitudinal (T1) relaxation of halide/halide chemical exchange nor homogeneous 207 Pb transverse (T2) relaxation. [72][73][74] Second, limited excitations brought about by the finite RF pulses used for excitation and the limited receiving bandwidth of the probe may cause intensity deviations. Additionally, our simulations do not account for scalar/dipolar couplings to halogens and the effects of halogen nuclei spin relaxation on the 207 Pb-NMR spectra.

NMR Signals from Light 13 C and 1 H Nuclei
NMR spectroscopy of 13 C is informative of the organic chemical species incorporated into the perovskite and has been thoroughly exploited for hybrid halide perovskites, as consequence of the continuous increase in the complexity of their chemical formulation. For instance, it has been used for characterizing mixed A-cation methylammonium/formamidinium compounds, [30,75] to unveil structural details related to the organic spacer in layered 2D halide perovskites, [76] and to determine the successful incorporation of hydroxy-ethylammonium in deficient perovskites. [19] NMR Spectra of light 13 C nuclei as computed from DFT calculations for the various perovskite structural models for MAPbBr 3 and MAPbI 3 are shown in Figure 3, in comparison with experimental data. Translation of the computed chemical shielding in chemical shift directly follows the results from Figure 2. Experimental data in Figure 3,a are obtained from Magic Angle Spinning (MAS) NMR measurements from Ref., [19] performed at room temperature. Theoretical chemical shifts are spread between 23 ppm and 31 ppm, as function of the specific model considered, and are hence slightly underestimated with respect to the experimental results but still in overall nice agreement. The difference between NMR signals for MAPbBr 3 and MAPbI 3 depends also on the structural model considered and is overall small, hence indicating limited influence of the halide on the NMR response for 13 C. This finding is consistent with other experimental evidence from the literature. [28] NMR lineshapes computed for the orthorhombic and tetragonal phases are consistent with only one signal, with the former in nice agreement with respect to MAS NMR measurement performed in Ref. [30] at 100 K, that is, within the stability range of the orthorhombic phase. The lack of dispersion in the NMR response from the four carbon atoms composing the orthorhombic (tetragonal) model should be traced back to the fact that these atoms are equivalent (quasi-equivalent) because of the space group symmetry of this system. In other words, because of the crystal symmetry, the MA cations experience the same chemical environment (computed isotropic shielding are reported in Supporting Information, for the sake of completeness). Frozen cubic models of MAPbI 3 and MAPbBr 3 instead are characterized by broader NMR signals. This is illustrated in Figure 3,d for the special case of the model cubic 2 (Figure 1), that is found to be the most stable among the four proposed cubic models, for which we computed NMR chemical shifts ranging from 24 ppm to 30 ppm. However, similar broadening can be noticed for all considered cubic models (see Supporting Information). In contrast to the orthorhombic phase, C-atoms in the cubic models in Figure 1 are symmetry-inequivalent, hence feeling slightly different chemical environments which, in turn, results in NMR chemical shifts dispersed within a few ppm (Figure 3,d). The broadening of the spectrum is therefore informative about the local environment differences, at least in principle. On the other hand, this broadening is not found experimentally at room temperature (see Figure 3,a), [19,28] because of the occurrence of motional narrowing, that is, a decrease of linewidth when increasing the temperature. This phenomenon is characteristic of NMR techniques and is due to the fact that the correlation time for MA cation dynamics, lying in the pico-second timescale, [15,23] is much shorter than the typical time-scale probed by NMR. [50,77] As a result, because of the dynamic disorder associated to the thermal re-orientation of the MA cations at finite temperature, the NMR will not result from the convolution of the individual conformations or orientations but to an average response. A striking example from literature is represented by the 17 O-NMR signals in hydrous magnesium silicates systems. [78] These systems are characterized by rapid proton transfer between different crystalline sites and Griffin et al. hence simulated the corresponding 17 O-NMR response by averaging out the signals from oxygen atoms in symmetry equivalent sites. Theoretical spectra obtained following this procedure were in much better agreement with respect to the experiment, compared to the non-averaged spectra resulting from the convolution of the various oxygen species. Motional narrowing for the high temperature cubic phase of MAPbBr 3 and MAPbI 3 was modelled here following a similar procedure. The resulting  [19] as measured at room temperature; (b-d) computed MAS NMR spectrum for the MAPbBr 3 (orange line) and MAPbI 3 (blue line) on the orthorhombic (b), tetragonal (c) and on cubic 2 (d) models (see Figure 1); (e) MAS NMR obtained for model cubic 2, by averaging over the different carbons constituting the model, to account to some extent for motional narrowing (see text). Experimental data in Figure 3,a adapted with permission from Ref. [19] ©2017 Wiley-VCH. spectrum is shown in Figure 3,e and recovers similar agreement as the low temperature orthorhombic (Figure 3,b) and tetragonal (Figure 3,c) phases, further demonstrating importance of dynamic disorder at ambient conditions. We recall however that the reduced broadening of NMR signatures in the low and high temperature phase, as evidenced from MAS measurements in Ref. [30] and, [19] respectively, should be associated to two different mechanisms. For the low temperature phase, this corresponds to very similar chemical environment experienced by the MA cations, as due to crystal symmetry, while in the high temperature phase, this should be rather associated to motional narrowing.
NMR Spectra of 1 H nuclei computed for MAPbBr 3 and MAPbI 3 perovskites are shown in Figure 4, compared to experimental results. Also, in this case, definition of the chemical shift from the computed shielding comes from the result in Figure 2. Experimental data are obtained from MAS NMR measurements performed at room temperature, from Ref. [12] and show the presence of two signals close to 4 and 7 ppm, associated to hydrogens of the methyl and ammonium groups, respectively. Corresponding theoretical data in Figure 4,b -4,d account for the motional narrowing through averaging over the chemically equivalent hydrogen atoms residing on the CH 3 and NH 3 + groups of the MA cation (see above) and result in predicted chemical shifts at 4 ppm and 8 ppm. Hence, DFT reproduces the experimental data with reasonable accuracy and is consistent with the assignment proposed in the literature. Again, the difference between the 1 H isotropic shifts computed for MAPbI 3 and MAPbBr 3 is small and comparable to the error associated to the computational parameters. We therefore conclude that the halide has limited influence on the NMR response of 1 H nuclei, as found experimentally in Figure 4,a.

NMR Signals from 207 Pb in Pure and Mixed Halide Perovskites
Signals from 207 Pb and from the halide can instead provide a direct insight into the structural and dynamic properties of the inorganic lattice, which is responsible for the main electronic properties of this class of semiconductors. [79,80] On the halogen side, both 127 I and 79 Br represent abundant isotopes that can be probed by NMR. However, these two halides possess spin quantum number larger than 1/2, hence featuring significant quadrupolar moments, which in turn makes the measurement of their NMR of the more challenging. [81] NMR measurements of the halide species composing all inorganic CsPbX 3 perovskites have been reported in the literature only very recently. [39,42] In turn, 207 Pb nuclei are widely investigated through NMR spectroscopy, although the associated broad signatures sometimes make difficult to derive detailed structural features from NMR spectroscopic measurements. [66,67,82] Static NMR measurements are reported in Figure 5,a, for pure halide MAPbI 3 , MAPbBr 3 and MAPbCl 3 perovskites, as performed at room temperature. [12] Chemical shifts from 207 Pb nuclei indeed cover a range of around 2000 ppm, from MAPbCl 3 , peaked at À 644 ppm, to MAPbI 3 , peaked at 1430 ppm. Furthermore, Figure 5,a clearly highlights the sensitivity of the 207 Pb signal to the halide composing the perovskite. , tetragonal (c) and on the cubic 2 (d) model. Signals from hydrogens belonging to the CH 3 and NH 3 + groups are averaged, to account to some extent for motional narrowing (vide supra). Experimental data in Figure 4,a adapted with permission from Ref. [12] ©2016 Royal Society of Chemistry.
Helv. Chim. Acta 2021, 104, e2000231 Results from our calculations performed on the orthorhombic, tetragonal and cubic structures of MAPbBr 3 and MAPbI 3 are given in Figure 5,b. The isotropic shift computed for MAPbBr 3 sets between 585 and 838 ppm, nicely matching the experimental datum in Figure 5,a, while that of MAPbI 3 , which lies between 1194 and 1460 ppm, is slightly underestimated, consistently with the j slope j < 1 in Figure 2 (computed NMR parameters are reported in Supporting Information). The discrepancy between experimental and DFT chemical shifts cannot be associated to nonconverged or inaccurate computational protocol, as preliminary convergence tests clearly showed minor influence (ca. 30 ppm) of plane-wave cutoff and sampling of the first Brillouin zone, on the predicted NMR response of 207 Pb nuclei (see Supporting Information). In addition, the systematic error found for the different structural models discourages the assignment of the theoretical-experimental discrepancy to a supposed poor quality of the structural models employed here. Therefore, we believe that the main limitation of the present computational protocol is due to the lack of SOC. Many theoretical calculations from the literature indeed highlighted sizable contributions of SOC on the NMR response of heavy atoms, on the order of 1000 ppm for 119 Sn [83] and 125 Te, [64] and of 3000 ppm for 207 Pb, [65 -67] hence providing at least a crude correction for the effect of SOC on the chemical shielding of 207 Pb. Most notably, by employing non-periodic calculations accounting for SOC through ZORA, Alkan and Dybowski obtained estimated isotropic shifts for a set of lead-based compounds which correlated very well with the corresponding experimental data (slope for the linear fitting was 0.87, to compare to our value of 0.56 as reported in Figure 2,c. The latter, in turn, is very close to the value of 0.54 that these authors obtained without SOC). [55] Apart from the absolute value of the chemical shift of 207 Pb computed for MAPbI 3 and MAPbBr 3 perovskites, the present DFT calculations predict the chemical shift of 207 Pb with correct order with respect to halide composition, with reported downshift from pure MAPbI 3 to pure MAPbBr 3 of~700 ppm, against 1000 ppm found experimentally (see Figure 5). This result is indeed very important, as it provides theoretical confirmation of the experimental findings and paves the way for the theoretical support to NMR characterization of lead perovskites with mixed halide composition. This is currently a hot topic in the literature, with preparation of stable mixed halide systems at the limelight for color-tunable light emission applications and tandem solar cells. [57][58][59][60] The lineshape for the tetragonal and orthorhombic phases is strongly asymmetric and does not match the experimental signal. This result consistent with just  Figure 5,a reproduced with permission from Ref. [12] ©2016 Royal Society of Chemistry.
Helv. Chim. Acta 2021, 104, e2000231 www.helv.wiley.com (8 of 17) e2000231 © 2021 The Authors. Helvetica Chimica Acta Published by Wiley-VHCA AG one single signal is not surprising and is due to symmetry equivalency among the four lead atoms in the unit cell, further characterized by large CSA, on the order of 400 ppm. Static NMR spectrum computed for the four frozen-cubic models in Figure 1 instead are characterized by more symmetric shapes of the spectra, arising from the convolution of eight symmetry inequivalent lead atoms in the cell. Notably, the lineshapes computed for these models are in far better agreement with respect to the experimental data, compared to those obtained on the orthorhombic and tetragonal models and could lead to the conclusion that the broad lineshapes of 207 Pb nuclei in MAPbX 3 perovskites may be related to a distribution of signals. However, previous works discourage this interpretation. Namely, Rosales et al. addressed the broadening observed in static NMR experiments (as those in Figure 5,a) to homogeneous broadening, [29] while Aebli et al. found sizable dipolar coupling between the lead atom and the surrounding halides, as evidenced from MAS NMR measurements, at least on the cesiumbased, chlorine and bromine-based lead compounds. [82] In this frame, it is worth to recall that the theoretical simulations in Figure 5,b, whose lineshape is dictated both by different values of the isotropic shifts and by CSA, must be addressed to the use of frozen-models, that does not consider for the rapid re-orientation of the MA cation. In light of the success of DFT simulations in reproducing qualitatively the relative chemical shift of 207 Pb nuclei as function of the halide composition of the perovskite, we further tested the capability of this approach to provide information on mixed halide perovskites. Halide mixing represents in fact an effective strategy to tune the optical properties of halide perovskites, for instance in relation to the wavelength of light emission, [57,58] and to improve the performance and stability of related opto-electronic devices. [59] Experimental NMR spectrum of mixed halide perovskite with nominal composition MAPbIBr 2 is depicted in Figure 6,a, in comparison to the spectra of pure phase iodine and bromine analogues. The mixed halide compound shows two main 207 Pb peaks, the first at intermediate frequency with respect to MAPbI 3 , and MAPbBr 3 and the second in correspondence to pure bromine composition component, assigned to mixed halide phase and segregated bromine-rich phase, respectively. [12] At the same time of the result by Roiland et al. in Figure 6,a, [12] Rosales et al. investigated similar mixed halide I : Br systems, similarly demonstrating how 207 Pb-NMR response is informative of the presence of halide segregated phases and solid state halide solutions, and how synthetic approach may affect the halide distribution in the material. [29] Subsequently, two independent works by Hanrahan et al. [73] and Karmakar et al., [56] investigated mixed Cl : Br halide perovskites, both nicely evidencing the different chemical response of the 207 Pb nuclei, as function of the surrounding halide species.
Mixed halide perovskite models are derived here from the cubic 2 × 2 × 2 supercell model from Figure 1, in light of the more realistic NMR response, compared to experiment. Mixed halide models have been prepared considering I : Br ratio and halide segregation. . In relation to halide segregation, we considered four 2 × 2 × 2 models for each composition, numbered as alloy1-alloy4, going from situation where the different halide segregated in different PbX 6 octahedra (alloy1) towards more homogeneous situations, that is, where the composition of all the PbX 6 octahedra reflects the nominal composition. A more detailed analysis of the composition of these models is reported in Supporting Information. Static NMR spectra as computed from DFT simulations for all mixed halide chemical compositions and considering more (alloy1) or less halide segregated (alloy4) models are reported in Figure 6,b. Mixed halide phases show progressive global shift of the 207 Pb-NMR signal toward lower frequency, with increasing bromine content, hence in line with the experimental result in Figure 6,a. Furthermore, mixed halide models are characterized by more dispersed 207 Pb-NMR signal, as compared to pure phase models. This is easily correlated to the contributions from PbX 6 octahedra characterized by different halide composition, as perfectly epitomized by the spectrum of the alloy1 (segregated) model associated to the 75 : 25 I : Br (MAPbI 2.25 Br 0.75 ) composition, which shows a signal centered around 600 ppm, related to pure PbBr 6 octahedron. Similar observation holds for the signal of the alloy1 model associated to the 25 : 75 I : Br (MAPbI 0.75 Br 2.25 ) composition, showing a contribution around 1200 ppm clearly associated to the pure PbI 6 octahedron composing the model. In Figure 6,c, we report the isotropic 207 Pb chemical shifts calculated in all the pure and mixed halide models investigated here, computed for each PbX 6 composition. The response of each 207 Pb nuclei is also clearly dependent upon structural details such as bond lengths and bond angles (see next Section), leading therefore to significant dispersion (up to 200 ppm) of the NMR signal for a given octahedron composition. Nevertheless, a clear downshift is evident for the 207 Pb-NMR chemical shift when passing from pure iodide to pure bromide case. Similar trend in the prediction of the NMR response of 207 Pb was reported by Karmakar et al. for mixed Cl : Br compounds. [56] Adopting non-periodic DFT simulations including ZORA for the estimate of SOC, the authors found theoretical estimates of the isotropic shift of 207 Pb in very good agreement with the experimental datum. The present calculations therefore provide a theoretical support for the use of NMR as characterization tool for halide perovskites, suggesting that the NMR response of 207 Pb nuclei can be used to estimate the composition of the individual PbX 6 octahedra within mixed halide perovskite systems. In addition, the present calculations shed light on the actual length-scale probed by NMR, clearly showing that this spectroscopic technique is able to track halide segregation already on the scale of the isolated PbX 6 octahedron.

Detailed Structural Information from NMR Response of 207 Pb Nuclei
We conclude this perspective on the potential contribution of atomistic simulations to the interpretation of NMR spectroscopic response, by analyzing in more detail the relationship between 207 Pb-NMR computed parameters and the detailed atomic structure of the individual perovskite PbX 6 octahedron. To this aim, we performed additional calculations on pure lead iodide composition. First, we considered an ideal cubic CsPbI 3 structure with Pm-3m space group symmetry. In the present model, the only adjustable parameter is represented by the PbÀ I bond length and we therefore  207 Pb static NMR spectra computed for various cubic models of pure phases MAPbI 3 and MAPbBr 3 perovskites and for mixed halide perovskites with different iodine:bromine ratio. Mixed halide models go from more segregated (alloy1) to more homogeneous (alloy4); c) computed isotropic shift of 207 Pb nuclei, as function of the composition of the corresponding PbX 6 octahedron. Figure 6,a reproduced with permission from Ref. [12] ©2016 Royal Society of Chemistry. evaluated the NMR response of 207 Pb nuclei with respect to this, as shown in Figure 7,a. Noteworthy, the isotropic part of the chemical shift for 207 Pb is underestimated by 1000 ppm, considering Pm-3m symme-try, as compared to the orthorhombic-tetragonal and cubic model structures as in Figures 5 and 6. [82,84] It should be noted that CSA is always zero as expected, due to lead atom site symmetry in the Pm-3m space Helv. Chim. Acta 2021, 104, e2000231 group. Furthermore, the chemical shift increases with increasing the PbÀ I distance, following a trend which is well reproduced by linear fit (R 2 parameter and Pearson parameter 1 both very close to unity, as reported in Figure 7,a). Notably, such increase can be sizable (on the order of 100 ppm) even for small shrink/elongation (0.02 Å) of the metal-halide bond length compared to the nominal equilibrium value from literature (3.165 Å), as quantified by the slope for the linear fit (4547 ppm/Å).
In order to clarify how the detailed atomistic structure of the perovskite material influences the NMR response of 207 Pb atoms, we considered differently distorted MAPbI 3 structures as extracted from a short Born-Oppenheimer molecular dynamics simulation, considering model cubic 1 of MAPbI 3 depicted in Figure 1, targeting 300 K temperature. Subsequently, we extracted 50 snapshots along the trajectory, and computed the corresponding NMR response, with results summarized in the isotropic shift-CSA 2D map in Figure 7,b. As already pointed out in the literature, NMR response of 207 Pb nuclei is spread over large range of isotropic shifts, with CSA ideally influenced by the specific deformation of the PbX 6 octahedron with respect to O h symmetry. CSA is never close to zero, indicating that the frozen structures as obtained on the fs-timescale from our molecular dynamics present PbI 6 octahedra distorted with respect to ideal Oh symmetry, a fact that has been already pointed out in the literature by molecular dynamics simulations and optical spectroscopic measurements. [10,11,15] We further dig in potential structure/NMR response correlations, by considering few snapshots from the Born-Oppenheimer molecular dynamics trajectory, highlighted in Figure 7,b, focusing in particular on lead atoms having either very large or very small isotropic chemical shifts. In light of the very local character of the NMR response, as also pointed out in Figure 7, we analyzed the structure of each corresponding PbI 6 octahedron, neglecting structural information related to the organic cations or to second nearest neighbor inorganic atoms. For the four considered octahedra, depicted in Figure 7,c, we reported bond lengths, as well as the corresponding octahedral elongation (λ oct ) parameter and the octahedral angle variance (σ), widely employed to characterize structural deformations in halide perovskites: [38,85] l oct ¼ 1 6 where l 0 is the center-to-vertex distance of a regular polyhedron of the same volume. [86] Large isotropic shifts are associated to octahedra A and B, characterized by at least one very long PbÀ I bond (3.5 Å -3.7 Å, as shown in Figure 7,c) compared to the nominal PbÀ I average distance, which is reflected by longer λ oct parameter. Small 207 Pb isotropic shifts are instead associated to octahedra C and D, characterized by more homogeneous distribution of the PbÀ I distances and λ oct value closer to 1. In this sense, it is worth to mention that Tremblay et al. already proposed a relationship between the NMR response of lead with respect to the λ oct deformation parameter, for the case of 2D halide perovskites. [38] To further extend the statistics for this structural analysis, in Figure 7,d we report the octahedral elongation λ oct parameter from the snapshots extracted from the molecular dynamics simulation, as function of the corresponding isotropic shift for 207 Pb nuclei. Indeed, a general trend is found, for which more elongated octahedra are overall associated to larger chemical shifts. Still corresponding R 2 and Pearson 1 coefficients (0.40 and 0.63, respectively) are far from pointing out a univocal correlation between these two quantities, as reflected also by the large dispersion of isotropic shifts (on the order of few hundreds of ppm) for fixed value of the octahedral elongation. In other words, although a trend is found in Figure 7,d, it is possible neither to accurately predict the chemical shift of the 207 Pb nuclei solely from the elongation parameter λ oct , nor to univocally anticipate the degree of distortion of a PbI 6 octahedron simply from a measured 207 Pb isotropic shift. In Figure 7,e, we report the same analysis on the basis of the angular distortion parameter σ, but did not find any robust correlation between structural deformation and isotropic shift. Similar result is found for the CSA, which does correlate neither with λ oct , nor with σ (see Supporting Information).

Conclusions
Nuclear Magnetic Resonance (NMR) is rapidly becoming a standard tool for the structural characterization of halide perovskite materials. Indeed, either stand alone or in combination with other characterization tools, this spectroscopic technique has recently demonstrated its strength by unveiling the detailed material structure of various halide perovskite systems, as in the case of 3D mixed halide compounds, [12,29] 3D mixed A-cation systems, [30][31][32][33][34] 2D layered perovskites [37,38] and quantum dots. [39,43] Interpretation of NMR measurements is however inherently complicated by the complexity of the measured NMR features, often arising from the convolution of many structured signals. In this frame, relying on support from theory and simulation would be extremely welcomed. Advanced and accurate modelling approaches based on periodic Density Functional Theory are nowadays available, [44 -46] and widely testified by several studies of glassy materials. [48,49] Still, the application of these tools in the field of halide perovskite has been very limited, up to now, at least at the best of the authors knowledge. Here, we report an extensive periodic DFT investigation of the NMR response of lead halide perovskites, with focus on nuclei with spin I = 1/2, and provide several proofs of principle of the possibilities of DFT in supporting NMR structural characterization, for this class of compounds. First, we demonstrate the capability of GIPAW calculations to reproduce experimental NMR signatures of both light ( 1 H and 13 C) and heavy nuclei ( 207 Pb) of halide perovskite materials, with overall good accuracy. For the latter, we anticipate that inclusion of Spin-Orbit-Coupling will be mandatory to reach quantitative accuracy for the chemical shielding tensor, considering the important role of SOC on the electronic properties of lead-based halide perovskites. [55,66,67] Nevertheless, calculations neglecting SOC still allow to capture the main trends with respect to chemical composition of the PbX 6 octahedral frame, demonstrating qualitative agreement for the variation of the chemical shift of 207 Pb when going from MAPbI 3 to MAPbBr 3 . Second, we confirm NMR as useful tool to characterize mixed halide perovskites systems, providing in particular a detailed local description of the composition of the material, able to suitably probe the lead chemical environment at the individual PbX 6 octahedron level. This aspect is of primary importance for mixed halide perovskite-based devices, as halide mobility and segregation, represents one of the most important challenges for these materials, influencing their stability over time and performances. Third, the present simulations demonstrate the capability of DFT calculations to provide a unique insight in terms of structure-spectroscopic response relationship for the study of halide perovskites. This was done in first place by considering a simplified structural Pm-3m perovskite model. Although one should refrain from considering such simplified model for the accurate prediction of the NMR spectrum of halide perovskites, this clearly demonstrates the dependence of the chemical shift of the 207 Pb nuclei with respect to the metal-halide bond length. Structure/spectroscopic response relationships are also demonstrated considering more realistic models, referring to structural descriptors from the literature, namely the octahedral elongation parameter λ oct and the angular distortion σ. Overall, we find a slight correlation between the isotropic shift of 207 Pb nuclei and λ oct , with the NMR signal shifting to larger chemical shifts, when increasing the elongation. Our calculations clearly point out that such kind of correlation between the NMR of 207 Pb and λ oct is not overall able to univocally assign the structure of the material solely from the NMR measurement but can still be used as qualitative descriptor. Conversely, the isotropic shift of 207 Pb nuclei was not found to depend on the angular deformation σ parameter for realistic models. Overall, the present results demonstrate the suitability of periodic DFT-GIPAW calculations to support NMR investigations of halide perovskites, opening new avenues to reach deeper insights into the structural and dynamical features of mixed halide perovskites, hollow and deficient analogues, down to confined systems such as 2D layered perovskites or 1D nanorods, and to 0D systems, featuring non-connected octahedra. [84] cal support for the data analysis and the representation of the NMR results. All authors contributed to this work, to the discussions, read the manuscript and agree to its contents.