Octahedral tilting and B-site off-centering in halide perovskites are not coupled

Domain: biology
Published: 2026-03-23
Editorial outcome: minor_revision

Abstract
Metal halide perovskites show exceptional potential for solar energy, thermoelectrics, catalysis, and other photochemical technologies, with performance rooted in electronic structure-driven properties. In ABX3 halide perovskites, localized and often aspherical local electron densities from B-site lone pairs or polarizable X− anions can distort the lattice. However, the links among electronic structure fluctuations and distortions like tilting of the BX6 octahedra and off-centering of the B-site from the center of its octahedron are not fully understood. Using group theory and ab initio molecular dynamics, we quantify how lone pairs, halide polarization, off-centering, and octahedral tilting interact in the cubic phase CsBBr3, with B = Pb, Sn, and Ge. We find that lone pair-induced off-centering and octahedral tilting are symmetry-decoupled. Instead, stereochemical lone pair expression of the B-site ion is correlated to octahedral tilting through the propensity of the B-site to form a transient, partial covalent bond with the surrounding halide ions that stiffens octahedral tilting modes. These results link local electronic asymmetry to structural fluctuations and suggest that dynamic modulation of electronic symmetry offers a pathway to control functional properties in halide perovskites.

Introduction
Lattice dynamics and their coupling to electronic structure control many important processes in metal halide perovskites. Halide perovskites exhibit soft, anharmonic phonon modes that have attracted significant interest because of their critical role in governing properties such as low thermal conductivity, dynamic disorder, carrier scattering, exciton binding and fine structure, ionic transport, and phase stability [1–10]. In cubic halide perovskites of the form ABX3, two lattice distortions that have garnered much attention are octahedral tilting [11], which occurs when alternating BX6 octahedra rotate around an axis, and off-centering of the B-site from the center of its octahedron [12].

Octahedral tilting increases the electronic gap, such that dynamic tilting increases fluctuations in the gap and alters absorption properties like the Urbach energy [11, 13–15]. Local structural changes from octahedral tilting modify orbital overlap and symmetry relevant to carrier effective masses and transport [4, 16–18]. If the B-site has a lone pair, the lone pair will localize to one side of the ion (due to s−p mixing, for example), and interactions with the surrounding halide ions produce an instability in which the B-site is displaced away from the center of the metal-halide octahedron [12, 19–24]. This off-centering, or emphanisis, significantly affects the optical properties of perovskites, such as photoluminescence [12], and can suppress their thermal conductivity [25].

Despite the importance of octahedral tilting and off-centering modes in the photophysics and thermoelectric behavior of halide perovskites, their physical origin is not fully understood, even at the level of whether or not these two modes are linked. Previous work has postulated that off-centering is coupled to octahedral tilting [26, 27]. However, a B-site lone pair and the resulting emphanisis are not a prerequisite for halide perovskites to exhibit octahedral tilting [28, 29]. Moreover, DFT calculations suggest that off-centering and octahedral tilting might even compete [30]. As a result, the question remains: how is B-site lone pair expression linked to octahedral tilting, if at all?

In this communication, we answer this question for cubic CsBBr3 perovskites using a combination of molecular dynamics simulations and group theory-based analysis of coupled electronic and nuclear fluctuations. We tune the amount of lone pair expression by varying the B-site in cubic CsBBr3 from Pb to Sn to Ge and quantify how lone pair expression couples to off-centering and octahedral tilting. By quantifying the local electronic symmetry of the B-site, we show that off-centering and octahedral tilting exhibit incommensurate symmetries, such that lone pair expression and off-centering are not directly coupled to octahedral tilting. Using the insights uncovered here by our group theory-based analysis, we suggest strategies for tuning these uncoupled low-energy instabilities in cubic halide perovskites through the local electronic structure of the component ions.

Results
At high temperatures, ABX3 perovskites typically adopt a cubic crystal structure, Fig. 1a. However, this cubic structure is only obtained upon averaging; the instantaneous structure fluctuates away from the average cubic structure. The unstable nature of the cubic structure is evidenced by imaginary frequencies in the 0 K phonon dispersion curves [11, 31–33]. The presence of imaginary frequencies indicates mechanical instability, which, when allowed to relax, causes the perovskite to fluctuate between locally distorted states. At finite temperature, the phonon dispersion curves are renormalized by their anharmonicity, resulting in nearly flat, dispersionless bands between the M and R points in the Brillouin zone, Fig. 1b, which are attributed to octahedral tilting. Their dispersionless nature indicates localized, weakly correlated phonons [34], such that the average cubic structure emerges from the dynamical averaging of transient, locally-tilted low energy domains. This behavior is observed for CsPbBr3, CsSnBr3, and CsGeBr3, albeit with different frequencies. The frequencies of the three acoustic branches shown in Fig. 1b for all three systems are of similar frequencies for CsPbBr3 and CsSnBr3, while the frequencies of CsGeBr3 are generally higher, reflecting a stiffer crystal.

The differences among the three perovskites can in part be attributed to variations in the electronic structure of the B-site cations due to increased s−p mixing, in addition to their different sizes. To characterize the local electronic structure and its fluctuations, we use maximally localized Wannier function centers (MLWFCs) as a point representation of the electron density, which we have found to be a reasonable approximation for localized charge densities [15, 35–41]. We quantify the amount of lone pair expression through the ion-MLWFC pair distribution function, g(rdip), where rdip is the distance between the center of the ion and the MLWFC (the length of the instantaneous B-MLWFC dipole). It is clear that the width of the peak in g(rdip) increases from Pb to Sn to Ge. Additionally, the maximum in g(rdip) shifts to a non-zero value in the case of Ge, indicating substantial lone pair expression. The variation in lone pair expression among the three perovskites considered here enables us to systematically identify correlations (or lack thereof) among lone pair expression, B-site off-centering, and octahedral tilting.

The amount of lone pair expression is not the only difference between the three systems studied here; they differ in unit cell length (L) and temperature. To isolate the effects of B-site lone pair expression, we model CsPbBr3, CsSnBr3, and CsGeBr3 all at the same unit cell length, corresponding to that of cubic CsPbBr3, as well as at the same temperature of 550 K. In effect, this corresponds to modeling all three systems at the same Goldschmidt tolerance factor, t=(rCs+rBr)/(sqrt(2)(rB+rBr)), equivalent to that of CsPbBr3; ri is the radius of ion i at the state point under consideration, which is under strain for two of the three systems. The amount of lone pair expression in these model systems follows expectations, Pb < Sn < Ge, as evidenced by g(rdip), (Fig. 1c, dashed lines). In fact, the larger unit cell significantly increases the amount of lone pair expression for Sn and Ge. This is expected as lattice expansion is known to stabilize and enhance the stereochemical expression of ns2 lone pairs [5]. The qualitative trends discussed below are also exhibited by the systems at the experimentally determined unit cell size, as discussed further in the SI and shown in Fig. S4-S7, and we focus on the model systems with the same L throughout the rest of the main text unless otherwise noted.

Our quantification of octahedral tilting and off-centering is rooted in group theory, mainly using rotor functions as order parameters to quantify atomic and electronic symmetry fluctuations [15, 42, 43]. A rotor function, M_l,m = (1/N) sum_i S_l,m(Omega_i), (1) is a function of the orientation of a collection of vectors, usually around a central atom, whose value indicates how well the collective orientation overlaps with the appropriate symmetry-adapted function, S_l,m(Omega_i), which is defined by angular momentum (l) and index m, bounded by 0 <= m <= 2l, xi indicates the basis of the harmonics which here is the cubic harmonics (K), and Omega = (phi, theta) is the solid angle of each vector. The symmetry-adapted function belongs to the irreducible representation (irrep) of the site symmetry of the central atom of interest and can be used to understand group-to-subgroup transformations straightforwardly. We use these functions to quantify fluctuations in local nuclear and electronic structures accompanying octahedral tilting and B-site off-centering.

In the cubic structure, the six bromides of the BBr6 octahedron are equivalent, and distortions away from this structure reduce the number of symmetry equivalent bromides. Octahedral tilting separates the bromides into four in-plane and two axial bromides, where only the in-plane bromides participate in the rotation that results in tilting, Fig. 2a. Tilting couples to changes in the collective orientation of the bromide electron density, with the four equivalent bromides orienting an MLWFC toward the B-site. These collective fluctuations away from the average cubic structure result from the incommensurate symmetries of the site and the local electron density [15]. Octahedral tilting transforms the environment of the B-site cation from Oh to C4h, such that changes in symmetry can be monitored with an order parameter that belongs to the T1g irrep. Here, we use the rotor function M4,4 with B-Br vectors as inputs; M4,4 = 0 in the cubic structure and |M4,4| > 0 if there is octahedral tilting. The rotor function M4,4 belongs to the T1g irrep within which the B-site environment transforms upon tilting and has been shown to be a good order parameter to quantify octahedral tilting [15]. The probability distribution of this rotor function, P(M4,4), narrows when changing the B-site from Pb to Sn to Ge, which indicates decreasing amounts of octahedral tilting from CsPbBr3 to CsSnBr3 to CsGeBr3, Fig. 2b. Therefore, increasing lone pair expression from Pb to Ge appears to decrease the propensity of halide perovskites to exhibit octahedral tilting, yet the precise nature of this coupling is not known.

Stereochemical lone pair expression has been linked to off-centering of the B-site. The predominant off-centering mode is along the [111] direction, which lowers the symmetry of the B-site from Oh to C3v as it moves toward one of the faces of the Br6 octahedron [12]. This reduction in site-symmetry creates two sets of three equivalent bromides around the B-site, Fig. 2c. To quantify the amount of off-centering in each system, we computed the radial distribution function (RDF) between the position of the B-site and the center of the octahedron, g(roct), where roct = |rB - r_oct|, (2) rB is the position of the B-site cation, and r_oct is the position of the center of the octahedron. This off-centering RDF, g(roct), indicates that Pb and Sn off-center similarly, though <roct> is slightly larger with Sn, while Ge off-centers significantly further, Fig. 2d. Therefore, increasing lone pair expression, from Pb to Ge, increases the system’s propensity to undergo B-site off-centering.

Off-centering of the B-site can occur through distortions with two different symmetries. The first distortion, discussed above, lowers the B-site symmetry from Oh to C3v and corresponds to a polarization along a [111] (off-diagonal) axis. This off-diagonal mode separates the six Br ions into two groups of three equivalent Br ions that displace in opposing off-centering directions. The second distortion lowers the B-site symmetry from Oh to C4v and corresponds to polarization along a [100] (Cartesian) axis. The change in symmetry accompanying this Cartesian distortion separates the six Br ions of the octahedron into a set of four equivalent in-plane Br ions and two axial Br ions that displace in opposing directions. Based on symmetry, the Cartesian, [100] distortion is allowed to couple to octahedral tilting while the off-diagonal, [111] distortion cannot.

To quantify which polarization mode accompanies off-centering, we computed the probability as a function of the B-MLWFC vector orientation, P(Omega). To highlight the directional preferences, we show lnP(Omega) in Fig. 3, where the value of lnP(Omega) is indicated by color, and the distortion of the surface away from a sphere is proportional to P(Omega).

The lone pair of the B-site should point toward one of the faces of the BBr6 octahedron to minimize electrostatic repulsion with the bromides, and therefore we anticipate that off-centering induces polarization along the off-diagonal [111] axis. The orientational probability distributions, P(Omega), indicate that all three systems orient their B-site lone pair along the [111] axis. All probabilities exhibit minima (white) along the [100] directions and maxima (darker colors) along the [111] directions, such that orienting the B-site lone pair along the [111] direction is preferred while orienting the lone pair along [100] is unfavorable. Because we are quantifying the orientation of the B-MLWFC vector, the probability distributions also indicate that B-site off-centering must be occurring along the [111] direction and not along another axis. This preferential orientation along the [111] direction is in agreement with experimental work that attributed the asymmetry of X-ray pair distribution functions to the transient presence of rhombohedral (R3m) distortions in the average cubic perovskite structure [12]. The relatively large lone pair expression of Ge manifests in significantly larger differences in the probability distribution, as compared to the Pb and Sn systems, such that differences between the minima and maxima in P(Omega) are an order of magnitude larger in the Ge system. Overall, larger lone pair expression favors more off-centering and an increasing preference to point the lone pair toward the face of the Br6 octahedron, consistent with expectations due to minimizing electrostatic repulsions.

The incommensurate B-site symmetries of octahedral tilting and off-centering along the [111] direction suggest that lone pair expression and octahedral tilting are not correlated. To further quantify any coupling among B-site off-centering, lone pair expression, and octahedral tilting, we computed combinations of conditional probability distributions, shown in Fig. S1-S3. To illustrate the main findings, we plot the fractional deviation of various averages in Fig. 3, which are defined as chi(y) = (<x(y)>/<x>) - 1, (3) where <x(y)> = integral dx x P(x|y) and <x> = integral dy <x(y)> = integral dy integral dx x P(x|y).

We first examine chi_rdip(roct), which quantifies how the average distance between the B-site ion and its lone pair MLWFC changes with the displacement of the B-site away from its ideal position in the cubic crystal structure. We find clear correlations between rdip and roct for all three materials. However, the precise nature of the correlation is different for each, as illustrated by the quadratic fits to each data set (solid lines). When B = Pb, rdip is roughly quadratic with roct and exhibits positive curvature. When B = Ge, rdip is also quadratic with roct but exhibits negative curvature. In contrast, rdip is approximately linear with roct when B = Sn. We attribute the different trends to the different coordination environments of the B-site. For Pb, tilting dominates over off-centering, such that the coordination shell of Pb mainly consists of 4 equivalent bromides, while off-centering dominates over tilting for Ge and its coordination shell mainly has two sets of 3 equivalent bromides. Sn displays similar amounts of tilting and off-centering, such that <rdip(roct)> behaves in a manner intermediate between Pb and Ge.

Conditional averages of rdip and roct as a function of the octahedral tilting order parameter M4,4 show little to no correlation, as evidenced by chi_rdip(M4,4) and chi_roct(M4,4) in Fig. 3b, c. As a result, both stereochemical lone pair expression and off-centering are not correlated to octahedral tilting. This lack of correlation is further supported by probability distributions of the tilting order parameter conditioned on rdip or roct, which are essentially independent of lone pair expression or off-centering, Fig. S2-S3. This lack of correlation is expected from the symmetry-based arguments discussed above; tilting and ([111]) off-centering are transformations within commensurate symmetries. Because off-centering and lone pair expression are coupled, the same symmetry arguments imply that lone pair expression is also not directly coupled to octahedral tilting.

If lone pair expression and off-centering are not correlated to octahedral tilting, why does the amount of octahedral tilting decrease from Pb to Sn to Ge? The decrease in tilting results from changes in Br–B-site bonding. The differences in Br–B-site bonding can be qualitatively observed in the Br MLWF closest to the B-site. The MLWF in the Pb system exhibits minimal directionality, consistent with the small propensity for the MLWFC to align with the z-axis. In contrast, the MLWF in the Ge system exhibits a ‘teardrop’ shape consistent with partial covalent bonding between Br and Ge. The Sn system is in between these two limits.

We can quantify the changes in the local Br− electron densities through the orientational distribution functions of the bromide MLWFCs, Fig. 5d-f. For each distribution, there are two major contributions: (1) off-diagonal peaks with four-fold symmetry that are consistent with the site symmetry when tilting [15], and (2) peaks along the ±z-axis that correspond to a Br MLWFC pointing directly toward the B-site, consistent with a partial Br–B bond. From Pb to Sn to Ge, the strength of the Br–B bond increases, evidenced by the increase in the peaks along the z-direction. The high propensity for the MLWFC to align with the local z-axis and point toward a bromide in CsGeBr3 is a signature of the partial covalent bonding observed qualitatively in the MLWF isosurfaces.

The observed evolution of MLWF shape and tilt stiffness from Pb to Sn to Ge is consistent with the electronic-structure picture developed by Papoian and Hoffman: increasing s − p mixing promotes partial covalent (directional) bonding of the anion-cation pair and converts nonbonding lone pair density into bonding character, stiffening the relevant lattice modes [44–47]. The MLWF shapes we observe correspond to a continuous transition between localized lone pair character and partial multi-center bonding. This electronic transition explains the simultaneous increase in rdip and increased Br–B bonding directionality seen for Ge. This framework rationalizes why enhanced lone pair expression (larger rdip) in Ge coexists with reduced octahedral tilting; the stronger directional Br–B bonding raises the energetic cost of octahedral rotations. The observed trend in B-Br bond stiffness can also be rationalized through the lens of Hard and Soft Acid-Base (HSAB) theory [48]. The Br− anion is a borderline base while the B-site cations, Pb2+ and Sn2+, are classified as borderline soft acids due to their large size and polarizable ns2 lone pair, while the smaller Ge2+ cation is comparatively harder. The increasing B-Br bond stiffness from Pb to Ge reflects a transition from a weaker, more polarizable soft-borderline interaction to a stronger, more iono-covalent hard-borderline interaction. This provides chemical intuition for the increase in s−p mixing and partial B-Br bonding observed in Fig. 5.

Octahedral fluctuations in halide perovskites are dictated not by lone pair expression but by the strength or stiffness of the Br–B-site bond. Octahedral tilting and B-site off-centering occur in the presence of this bond, and therefore the relative propensity for either distortion is tied to the strength of the Br–B-site bond. Fluctuations between off-centering configurations require breaking this bond, and the nature and extent of octahedral tilting fluctuations are dependent on the bond strength. As a result, a dynamical competition should exist between octahedral tilting and B-site off-centering. When Br–B-site bonds are strong, as in CsGeBr3, octahedral tilting modes will be stiff. In contrast, Br–B-site bonds are weakest in CsPbBr3, and the corresponding tilting modes will be soft, while CsSnBr3 will behave in a manner between the two limits but closer to CsPbBr3.

The qualitative differences expected for the dynamics of octahedral tilting can be quantified by time correlation functions (TCFs) of a tilting order parameter [15]. We characterize fluctuations in octahedral tilting through rotation of the B-site–Br octahedron, which can be quantified by a rotor function that belongs to the T1g irrep and takes the B-Br vectors as input; M4,m with m=3,4,5. The appropriate TCF is C_tilt(t) = (<sum_m delta M4,m(0)M4,m(t)>) / (<sum_m delta M4,m(0)^2>), (4) where Gamma_tilde(l,m) is a function that returns the irrep label, Gamma(j), corresponding to each specific combination of l=4 and m, and delta_alpha,beta is the Kronecker delta function. The octahedral tilting TCF, C_tilt(t), for CsGeBr3 exhibits a rapid oscillatory decay with a deep minimum at small times. The unnormalized TCFs are also consistent with less tilting; the GeBr6 octahedra do not tilt as far as the octahedra in the other two systems, as expected from the strong Br–Ge bonds, Fig. S6.

Fluctuations in off-centering modes correspond to exchanging the identity of the (two sets of three) equivalent Br atoms around the B-site, breaking Br–B-site bonds. The large Br–Ge bond strength makes this exchange rare and slow. We characterize fluctuations in off-centering through rotation of the B-site–MLWFC dipole, which can be quantified by a rotor function that belongs to the T1u irrep and takes the B-site–MLWFC vector as input; M1,m with m = 0,1,2. We quantify the dynamics in B-site dipole rotation through a TCF analogous to that for tilting, C_dip(t) = (<sum_m delta M1,m(0)M1,m(t)>) / (<sum_m delta M1,m(0)^2>), (5) The rotational dynamics of the Ge lone pair are extremely slow, with C_dip(t) decaying on timescales of the order of tens of picoseconds, consistent with expectations. This extremely slow decay is in part due to the deep energy minima caused by the larger unit cell. Using the experimental CsGeBr3 unit cell results in a faster decay of C_dip(t), on the order of 0.5-1 ps, Fig. S5, though the dynamic competition between tilting and off-centering remains.

Methods
Simulation Details

We performed density functional theory (DFT)-based Born-Oppenheimer ab initio molecular dynamics (AIMD) simulations of CsBBr3 (B = Pb, Sn, Ge). The simulations used the QUICKSTEP electronic structure module within the CP2K ab initio code [75]. QUICKSTEP utilizes a dual atom-centered Gaussian and Plane wave (GPW) basis approach to represent wavefunctions and electron density, efficiently and precisely implementing DFT. The molecularly optimized (MOLOPT) Goedecker-Teter-Hutter (GTH) double-zeta single polarization short-ranged (DZVP-MOLOPT-SR-GTH) Gaussian basis was selected for the expansion of orbital functions. Our simulations employed a plane-wave basis with a cutoff (Ry) and a REL CUTOFF (Ry) of (400, 55), (350, 50), and (400, 50) for B = Pb, Sn, and Ge, respectively, to represent the electron density [76]. Core electrons were represented using Goedecker-Teter-Hutter (GTH) pseudopotentials [77]. Exchange correlation interactions were approximated using the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation for the exchange-correlation functional [78], as implemented in CP2K. Grimme’s D3 van der Waals correction was applied to account for the long-range dispersion interactions in the simulations [79].

Our simulations use a 4x4x4 supercell for all perovskites based on the experimentally determined unit cells, resulting in a cubic simulation cell with lengths L = 23.80 A (Pb), 23.18 A (Sn) [80], and 22.12 A (Ge). The supercell was taken to be 4x4x4, because any odd multiple unit cells restrict octahedral tilting [81] and 2x2x2 would result in the octahedron sharing two vertexes with the same octahedron through periodic boundaries, possibly coupling these interactions more strongly. We equilibrated the system for at least 5 ps before performing production runs. These equilibration simulations propagated the equations of motion with the canonical velocity rescaling (CSVR) thermostat to maintain a constant temperature of 300 K, 450 K, 550 K for B = Sn, Pb, and Ge, respectively, for the first set of simulations in the canonical (NVT) ensemble. A second set of AIMD simulations was run so that each B-site has the unit cell of Pb and a temperature of 550 K. For the calculation of dynamic properties, we performed AIMD simulations in the microcanonical (NVE) ensemble with a timestep of 0.5 fs for a total trajectory length of at least 260 ps and 60 ps for the first and second set of simulations, respectively. Maximally localized Wannier functions (MLWFs) [82] and their centers (MLWFCs) were computed on the fly using CP2K. The spreads of the MLWFs were minimized according to established methods [83].

Phonon Analysis

To examine collective lattice dynamics by incorporating anharmonic effects at finite temperatures, we computed phonon dispersions directly from AIMD simulations. We used dump2phonon [84] to compute the dynamical matrix directly from our AIMD simulation trajectories. The elements of the dynamical matrix contain the force constants for the vibrational modes of the system. In practice, this matrix is computed within dump2phonon through Green’s functions, which involves Fourier transforms of atomic positions. The eigenvalues of the dynamical matrix give the squares of the phonon frequencies for a given wave vector. These eigenvalues were computed and the corresponding phonon dispersion curves were constructed using the phana post-processing tool [84].

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper addresses the question whether octahedral tilting distortions and off-centering in halide perovskites ABX3 are\ncoupled to each other. Both these types of distortions play a role in the structural stability and are also occurring dynamically\nat high temperature. It had been suggested in previous literature, ref. 27 in particular, that for B=Pb,Sn octahedral tilting is\nfavored while for B=Ge off-centering is favored although that paper already mentions that off-centering can also occur for the\nSn and Pb cases under lattice constant expansion. That paper also pointed out that the off-centering is related to lone pair\nchemistry, the importance of which was previously pointed out in ref. 12. The present paper uses various sophisticated\nsymmetry based order parameters and statistical distribution functions to investigate the coupling or absence of coupling of\nthese distortion modes and their relation to lone pair expression, concepts which are here quantified precisely. The paper\nconcludes that these distortion modes are of different symmetry and hence uncoupled and explains the effect of lone pairs\non tilting as resulting from an indirect effect on the B-X bond stiffness resulting from the off-centering. This work clarifies the\nissue of how lone-pair expression affects distortions in this family of materials. By quantifying these concepts it\nputs previous assertions in literature on a firmer footing and dispels other incorrect vague assertions on the relation between\noctahedral tilting to lone pair physics.\nA few clarifications are required before publishing the paper and some additional suggestions below may further increase\nthe impact of the paper.\n1) The definition of the rotor functions is somewhat incomplete, e.g. the indices ξ is not defined, I guess it relates to\nirreducible representations, and Θ_i is also not defined. Relatedly, a clear explanation for why M^4,4_κ is the relevant order\nparameter would be welcome. Similar for the M^1,m_κ\n2) Most of the results in the main paper relate to a systems CsBBr3 all taken at fixed unit cell size of B=Pb, which means that\nSn and Ge cases are under tensile strain. The authors also provide the results for each system at their own optical lattice\nconstant in the supplementary information. Since expanded volume increases the lone-pair expression, as already pointed\nout in ref. 27, and the authors here have a quantitative measure of lone-pair expression, can they discuss in some more\ndetail how the latter depends on tensile of compressive strain?\n3) The authors suggest in their discussion section that replacing the B by f-electron containing elements like Eu^2+ or Yb^2+\ncould result in stiffening of the octahedral tilting. However, is there any evidence that these elements are compatible with the\nhalide perovskite structure and would have the same type of lone-pair expression? After all, the lone pairs of group IV\nelements is related to the deeper s-levels in heavier elements compared to p-levels, which is, in fact due to the Darwin term\nrelativistic effect. Other substitutions of group II elements for the group-IV have typically led to indirect band structures and\nthe properties of halide perovskites are closely tied to using divalent ions of group-IV elements. The f-electrons in these\nlanthanides are also quite localized and as such it is not clear what their effect in bonding would be. They might rather\nintroduce magnetic effects. So, this is perhaps a somewhat premature speculation, not supported by calculations.\n4) Could the authors explain why the presence of flat dispersionless phonon modes along MR are indicative of octahedral\ntilting. The relation of imaginary modes along this direction to the octahedral tilting were described in previous literature, e.g.\nref.29, but what is the role of the mode flatness in this context?\n5) In previous literature the role of the size of the A atom in ABX3 perovskites and its relation to the Goldschmidt's tolerance\nfactor t was pointed out. How does this come into play in the off-centering vs. octahedral tilting question? It has been pointed\nout that small t is related to tilting. Do the quantitative measures introduced here have any bearing on the role of the A size\natoms.\n6) Can the analysis in terms of lone pair expression be generalized or applied also to non perovskite structures of the\nhalides, in which different connectivity of the octahedrons than corner sharing occurs, such as the yellow phases or\nmonoclinic phases. See e.g https://doi.org/10.1002/aelm.201900887.\nOverall, the writing of the paper is somewhat repetitive and I think could be substantially shortened if the authors avoid\nrestating the same arguments at several places in the manuscript.", "reviewer_2": "The authors use ab initio molecular dynamics simulations of CsBBr3 systems along with maximally localized Wannier\nfunction analysis to address an issue about the dynamics of these systems - are B-site off-centering and octahedral tilts\ncoupled. The halide perovskites are a very interesting system for a variety of reasons. Many recent experimental and\ntheoretical works have investigated the structural transitions they display, their unusual electronic properties, their optical\nproperties, electronic and lattice dynamics, and the challenges to their chemical stability. Being a perovskite system, one of\nthe crucial coordinates governing these properties is halide octahedral tilting. Moreover, the B-site ion typically being group\n14 metals, there is the interesting prospect of steric lone-pairs and their influence on properties. So the question posed by\nthe authors is indeed an important one for modeling these systems and also devising design principles for experiments.\nThe most important message from the manuscript is that, the B-site off-centering fluctuations do not couple with octahedral\ntilt fluctuations since the latter are incompatible in symmetry to the (dominantly [111]) off-centering. The authors reach this\nconclusion by analyzing the symmetry characters of the two kinds of fluctuations within the point-groups of the cubic (Oh)\nand lowered-symmetry (C2v) structures. The same conclusion is also inferred by the lack of correlations between the two\nfluctuations as obtained from the AIMD simulations. However, I am not completely convinced by this latter analysis as I\nexplain below.\nThe authors also note that the tendency for octahedral tilting decreases as the lone-pair expression increases down the\ngroup from Ge to Sn to Pb. They attribute this to stronger s-p mixing in Ge compared to Sn and Pb and, hence, stronger B-Br\nbonds. In general, this is not an original observation as it is already known that the inert pair effect, which leads to the lone\npair's inertness, increases down the group. Cation-anion interactions can result in activating the inert pair even in Pb, as in\nthe case of PbO (litharge) (see Journal of Solid State Chemistry 178 (2005) 1422–1428). While the authors speculate such\nan s-p mixing citing some previous works (very relevant though), they do not provide an analysis confirming the same in the\nsystems of their focus. While tilting stiffness is shown, its connection to the involvement of valence s-orbitals from B in\nbonding (through PDOS plots, for instance) is not confirmed.\nNevertheless, the connection, if it exists, shows a coupling of tilts and the extent of s-p mixing, i.e. lone-pair expression. So it\nis a little confusing when a claim to the contrary is made in the manuscript.\nThe conditional probability analysis done on the AIMD data is used by the authors to reject the existence of a coupling\nbetween the two features and, hence, between off-centering and tilts. Given that the lone-pair expression (x) and off-\ncentering (y) are reasonably strongly correlated, this negative result is surprising. With the correlation between x and y, it\nwould be better to compute P(x|M4,4, y) and P(y| M4,4, x). The x|y distribution clearly says that B-site lone-pair is more\npolarizable when the B ion is off-centred. Therefore, it is this combined polarized state that should be conditioned rather than\njust x or y, when checking agains the tilts.\nA clear explanation (with appropriate evidence) for the coupling of the off-centering with the lone-pair expression\n(modulation of LP polarization with off-centering) has not been provided in the manuscript. The dominance of the [111] off-\ncentering is not demonstrated in the manuscript (with P(omega) like plots as done for the MLWF) but instead assumed from\na previous study on CsSnBr3. The reason for this dominance is speculated but not clearly proved by, for instance, looking at\npotential curves for different distortions.\nOverall, I feel that the main message, while interesting for researchers in the field, is not significant to appeal to a wider\naudience. The main purposes of the work could be very convincingly demonstrated by ground-state DFT calculations\ncombined with the group theory analysis. A more detailed discussion on the possible couplings and the corresponding free\nenergy/hamiltonian being considered would also improve the appeal of the work. The AIMD, though performed with rigour,\ndoes not offer more insights than such an analysis would already deliver. Thus, I do not recommend the manuscript for\npublication in Nature communications.", "reviewer_3": "In their paper Remsing et al are discussing the dependence of octahedral tilting in halide perovskites correlated with off\ncentering of the B cation and halide polarization.\nUnderstanding these correlations is vital for the development of systematic structure to property relationships in these\nmaterials, as these variables, or rather, their interplay are linked to key electronic properties such as the band gap that\ndetermine their use in applications. This paper addresses this knowledge gap from a new perspective, and allows the\nauthors to make concrete, experimentally provable predictions. The paper is written in an very clear manner. The\nconclusions are of significant importance for the field and presented in a way accessible to non-experts, which is why I\nstrongly recommend accepting this work.\nThe authors study the correlations between these variables through molecular dynamics simulations that allow them to\ncompute correlation functions that indicate off-centering, lone-pair expression and octahedral tilting.\nTheir results confirm that the width of the distribution measuring octahedral tiling increases when going from Ge to Pb,\nas expected from the increase of lone pair expression in Sn and Pb.\nThey then move to calculate a radial distribution function between the B site position and then center of the octahedron,\nwhich measures the B-site off centering along the [111] direction, and agrees with results from experiments. However, these\ntwo distortions correspond to two two different distortion pathways indicated by distinct group-subgroup relationships away\nfrom ideal octahedral symmetry, suggesting that they are uncorrelated.\nThe authors probe this conclusion by calculating conditional probability distributions.\nThese calculations indeed show that these distortions are not correlated. This result is surprising, because octahedral tilting\nis usually assumed to be correlated to lone pair expression, as it increases from Ge to Pb, i.e. by increasing expression of\nthe lone pair.]\nThe authors show that this trend can be better explained by the changes in B-Br bonding, which increases from Ge to Pb.\nThese observations are in line with ideas proposed by Hoffman and coworkers, in which the nonbonding lone pair can mix\nwith p-orbitals to result in stronger directional bonding. The main result here is that fluctuations away from octahedral\nsymmetry are due to differences in bonding between the compounds considered here. These results are confirmed by\ncomputing time correlation functions.\nI have a few minor questions/comments:\n1)The plot of the log of g(r)is hard to understand for me. Is this usual to plot it like this instead of the bare function g(r) ? Do I\nunderstand correctly that g(r) for Ge peaks at 0, while for Sn and PB the peak is nonzero, while the averages are both\nnonzero? Can I understand the definition of a B-MLWFC as being equivalent to the wannier center relative to the ideal,\noctahedral position in this work?\n2) In general some figure captions are hard to read if not zoomed in (e.g. fig 4).Fig. 2 a little hard to identify details as well —\nCould the figure be made a bit larger? The colors in Fig2. might be hard to distinguish when printed black/white.\n3) The discussion about the importance of the Halide-B bond is very interesting to me. The authors discuss a plethora of\noptions and possible alternative B cations. Is it possible to connect the discussion here to the concept of hard acids and\nbases (HSAB)? For example, the very low solubility of AgI is explained by both Ag+ and I- being soft acids/bases, resulting\nin strong bonds. In contrast, AgF is highly solvable in water. This exploration would allow it to connect it to concepts more\nfamiliar to inorganic chemists.\n4) The results as they are presented are very clear and convincing, but I have trouble figuring out what implicit assumption\nthis analysis is based on. Are there scenarios where the authors have reason to belief their analysis is qualitatively wrong,\ne.g. through change of DFT functionals or different simulation parameters. In other words, how robust are the findings and\ncould one easily expand it to other materials in which lone pair expression is considered to be important, e.g. Pb-based\nthermoelectrics (see for example the work of Jeff Snyder)."}, "author_response": "Response to Reviewers for Manuscript NCOMMS-25-66229:\n“Octahedral tilting and B-site off-centering in halide perovskites\nare not coupled”\nResponse to Reviewers\nWe thank all three reviewers for their time and insightful\nfeedback on our manuscript (NCOMMS-25-66229,”Octahedral\ntilting and B-site off-centering in halide perovskitesare not\ncoupled”). Their suggestions have helped in strengtheningthe\nmanuscript, clarifying our technical arguments, and broadening\nthe work’s connection to other key areas of materials science.\nWe have addressed all points raised. This includes clarifications,\nthe addition of new contextual discussions (Goldschmidt factor,\nHSAB principle, strain effects, and thermoelectrics), and fortifi-\ncation of our f-electron hypothesis with new citations.\nWe are confident that these revisions have substantially improved\nthe rigor, clarity, and impact of our paper.\nReviewer #1\nPoint 1: The definition of the rotor functions is somewhat incom-\nplete, e.g. the indices ξ is not defined, I guess it relates to irre-\nducible representations, and Θ is also not defined. Relatedly, a\ni\n4,4\nclear explanation for why M is the relevant order parameter\nK\n1,m\nwould be welcome. Similar for the M .\nK\nOur Response: Thank you for catching this omission. We have\nadded definitions of ξ and Θ is the sentence following Equation 1.\ni\nand others, we have changed Θ to Ω , but the meaning is the same.\ni i\nIn short, Ω represents the set of relevant orientation vectors (e.g.,\ni\nthe six B-Br vectors originating from the B-site cation). The in-\nl,m\ndex ξ defines the basis for the symmetry-adapted functions, S ;\nξ\nfor this work, we use ξ = K for Kubic harmonics, which form a\nsuitable basis for the O point group of the B-site. To clarify this,\nh\nwe add the following text to the manuscript: Kubic or Cubic?\n... ξ indicates the basis of the harmonics which here is the\ncubic harmonics (K), and Ω = (φ,θ) is the solid angle of each\nvector.\nWe have also added a new sentence after the introduction of each\norder parameter describing why it is the relevant order parameter.\nOur choice of specific rotor functions is dictated by group theory.\nThe collective BX octahedral tilting distortion (Fig. 2a) trans-\n6\nforms as the T irreducible representation (irrep) of the O point\n1g h\n4,4\ngroup. The specific rotor function M (using l = 4, m = 4\nK\nKubic harmonics) is the order parameter that possesses this T\n1g\n4,4\nsymmetry. Thus, a non-zero value, |M | > 0, is a direct, quan-\nK\ntitative measure of this distortion. We added the following to the\nmanuscript to address this point:\n4,4\nThe rotor function M belongs to the T irrep within which\nK 1g\nthe B-site environment transforms upon tilting and has been\nshown to be a good order parameter to quantify octahedral\ntilting [15].\nConversely, B-site off-centering (Fig. 2c), which creates a local\ndipole moment, is a polar vector that transforms as the T irrep.\n1u\n1,m\nThe l = 1 rotor functions, M (m = 0,1,2), are the basis for\nK\nthis T irrep and are used to quantify the dynamics of this dipole\n1u\n2\n(Fig. 6b).\nPoint 2: Most of the results in the main paper relate to a systems\nCsBBr3 all taken at fixed unit cell size of B=Pb, which means that\nSn and Ge cases are under tensile strain. The authors also provide\nthe results for each system at their own optical lattice constant in\nthe supplementary information. Since expanded volume increases\nthe lone-pair expression, as already pointed\nout in ref. 27, and the authors here have a quantitative measure of lone-pair expression,\ncan they discuss in some more detail how the latter depends on\ntensile of compressive strain\nOur Response: This is a great point, and the reviewer is correct.\nSimulating Sn and Ge at the CsPbBr lattice constant places them\n3\nunder tensile strain, and this strain increases the amount of lone\npair expression. We added a sentence to clarify this point on page\n2, citing work by Lyons and coworkers where this effect was pre-\ndicted:\nThis is expected as lattice expansion is known to stabilize and\nenhance the stereochemical expression of ns2 lone pairs [5].\nThe stereochemical expression of the lone pair requires space.\nTherefore, lattice expansion induced by tensile strain enhances its\nexpression. Conversely, compressive strain disfavors lone pair ex-\npression. This is confirmed by high-pressure experimental studies.\nWe’ve added a paragraph in the discussion section elaborating on\nthese points.\nThe relationship between lattice expansion and lone pair ex-\npession also suggests that if tensile strain activates the lone\npair, as our methods and other studies confirm [5], then com-\npressive strain should have the opposite effect. Compression\nshould quench the stereochemical expression of the lone pair.\npair should reduce s − p mixing, soften the B-Br bond, and\nthen reduce the stiffness of the octahedral tilt mode. In exper-\niments that apply compressive strain to the halide perovskites,\nthe phase reduces to that accessible by cooperating octahedral\ntilts [51-53]. Since the B-Br is softer, the electron density of Br\ncan couple to octahedral tilting, lowering the energy of these\nphases.\nPoint 3: The authors suggest in their discussion section that\nreplacing the B by f-electron containing elements like Eu2+ or\nYb2+ could result in stiffening of the octahedral tilting. However,\nis there any evidence that these elements are compatible with the\nhalide perovskite structure and would have the same type of lone-\npair expression? After all, the lone pairs of group IV elements\nis related to the deeper s-levels in heavier elements compared to\np-levels, which is, in fact due to the Darwin term relativistic ef-\nfect. Other substitutions of group II elements for the group-IV\nhave typically led to indirect band structures and the properties of\nhalide perovskites are closely tied to using divalent ions of group-\nIV elements. The f-electrons in these lanthanides are also quite\nlocalized and as such it is not clear what their effect in bonding\nwould be. They might rather introduce magnetic effects. So, this\nis perhaps a somewhat premature speculation, not supported by\ncalculations.\nOur Response: Thank you for pointing this out. Our hypothesis\nregarding f-electron containing elements is supported by calcu-\nlations in Ref 52 - make sure this changes if we need to. We\nlooked deeper into the literature and found a recent paper detail-\ning a combined computational and experimental investigation that\nsupports this hypothesis by using Eu- and Yb-doping at the B-site\n4\nto reduce ion migration in halide perovskites by reducing octahe-\ndral fluctuations. We added a sentence to the Discussion section\nregarding this point:\nIndeed, Eu and Yb doped systems have been shown to reduce\nion migration by restraining octahedral oscillations [62].\nPoint 4: Could the authors explain why the presence of flat dis-\npersionless phonon modes along MR are indicative of octahedral\ntilting. The relation of imaginary modes along this direction to\nthe octahedral tilting were described in previous literature, e.g.\nref.29, but what is the role of the mode flatness in this context?\nOur Response: This is another excellent request for clarification.\nAt finite temperature (as in our Fig. 1b), these modes are renor-\nmalized by strong anharmonicity and become real, low energy, flat\nbands. The flatness (lack of dispersion) indicates that the phonons\nare localized with weak correlation between adjacent unit cells, ul-\ntimately due to the disorder induced by anharmonic tilting modes\nat finite temperature. Our AIMD simulations our performed at fi-\nnite temperature and include anharmonicity and therefore can cap-\nture this renormalization of the bands. We have added a sentence\nwith a citation clarifying this point on page 2.\nTheir dispersionless nature indicates localized, weakly corre-\nlated phonons ?, such that the average cubic structure emerges\nfrom the dynamical averaging of transient, locally-tilted low\nenergy domains.\nPoint 5: In previous literature the role of the size of the A atom in\nABX3 perovskites and its relation to the Goldschmidt’s tolerance\nfactor t was pointed out. How does this come into play in the\noff-centering vs. octahedral tilting question?\nnected more to the tolerance factor in the revised version. Our\nwork provides the underlying mechanism for the competition that\nthe tolerance factor helps describe and demonstrates why these\nmodes seem to compete despite being symmetry-decoupled.\nThe extent of B-Br s − p mixing determines the character (extent\nof covalency) of the B-Br bond, which our work suggests con-\ntrols the stiffness of the octahedral tilting modes. At the same\ntime, increased s − p mixing increases lone pair expression and\nferroelectric or rhombohedral distortions, off-centering. Tilting\nand off-centering are not directly coupled, but both are impacted\nby the extent of s − p mixing.\nIn terms of the tolerance factor, t, low t values indicate that the\nA-site cation is too small for its cage, as in CsBBr . The standard\n3\nresponse to relieving this strain is octahedral tilting. Tilting is cou-\npled to the local Br electron density, which reorients to optimize\ninteractions. The extent to which the Br electron density can re-\norient is determined by the covalency of the B-Br bond; increased\ncovalency stiffens tilting modes and reduces the extent of tilting.\nThis is true even for fixed tolerance factor. For example, by fixing\nthe lattice parameter for the systems in the main text, we are in\neffect modeling them all at the same t. Despite being at the same\nt and having the same A-site cation, the three systems studied tilt\nto different extents, consistent with the picture described above.\nWe have modified the text in two places to make this point. First,\nwe emphasize that the systems with fixed unit cell are all at the\nsame t:\nIn effect, this corresponds to modeling all three systems at the\n√\nsame Goldschmidt tolerance factor, t = (r + r )/( 2(r +\nCs Br B\n6\nr )), equivalent to that of CsPbBr ; r is the radius of ion i at\nBr 3 i\nthe state point under consideration, which is under strain for\ntwo of the three systems.\nSecond, we added a paragraph in the discussion section presenting\nthe arguments above:\nOur findings provide a fundamental explanation for the ap-\nparent competition between tilting and off-centering in halide\nperovskites. This competition is usually framed as an A-site\ndriven structural instability (quantified by the Goldschmidt\ntolerance factor, t) and a B-site driven electronic instability\n(the ns2 lone pair). A low t-factor (< 1) indicates that the\nA-site cation is too small for its cage, as is the case here with\nA=Cs, which creates a strong energetic incentive for octahe-\ndral tilting to optimize the A-site coordination ?. The standard\nstructural response to relieve this strain is octahedral tilting,\nwhich tightens the cage around the A-site. The electron den-\nsity around the halide also reorients to fill the space around the\nA-site, and as a result, any changes to B-X mixing will alter\nthe character of the halide electron density and the resulting\nA-site-driven instability through its coupling to the electron\ndensity. We also expect that changes in the A-site, and there-\nfore, a change in the tolerance factor, will affect the environ-\nment in which the halide electron density is optimized, thereby\nstrengthening or weakening the octahedral tilting mode.\nPoint 6: Can the analysis in terms of lone pair expression be\ngeneralized or applied also to non perovskite structures of the\nhalides, in which different connectivity of the octahedrons than\ncorner sharing occurs, such as the yellow phases or\nmonoclinic phases\napplied to other structures, and we have applied similar analyses\nto superionic AgI, e.g. Ref. 32 make sure this doesn’t change.\nThe only significant change to the methodology is the choice of\nrotor functions as order parameters. The rotor function must be\nconsistent with the symmetry of the local distortion. We have also\nadded a short paragraph to the discussion section to clarify these\npoints.\nWe note that our symmetry-based analysis of dynamic cou-\npling can be applied more broadly to non-perovskite poly-\nmorphs (e.g., δ-phases) or lower dimensional structures (e.g.,\n2D metal halides). While the amount of lone pair expression\nand the octahedral connectivity may be different than that\nstudied here, the symmetries of the system can be used to\napply a similar analysis to characterize the coupling between\nlocal electronic fluctuations and distortions of the lattice, as\ndone for AgI for example [32].\nPoint 1: While the authors speculate such an s-p mixing citing\nsome previous works (very relevant though), they do not provide\nan analysis confirming the same in the systems of their focus.\nWhile tilting stiffness is shown, its connection to the involvement\nof valence s-orbitals from B in bonding (through PDOS plots, for\ninstance) is not confirmed.\nOur Response: Because the B-site has a (spherically-symmetric)\nns2-type orbital, stereochemical expression of this lone pair must\narise from mixing with a p orbital. Because this has been shown\nextensively using the same level of density functional approxima-\ntion used here (such as Reference 29), we do not include repro-\nductions of those calculations in the manuscript. All of our results\nare consistent with s − p mixing driving lone pair expression, and\nwe cite additional references on the topic including reviews that\nfocus on halide perovskites or contain significant discussions of\nthese systems (References 22-24).\nPoint 2: Nevertheless, the connection, if it exists, shows a cou-\npling of tilts and the extent of s-p mixing, i.e. lone-pair expression.\nSo it is a little confusing when a claim to the contrary is made in\nthe manuscript\nOur Response: We apologize for the confusion and have tried to\nclarify our message in the manuscript.\nWe can summarize this point in the following way. Off-centering\nis uncoupled from octahedral tilting. Lone pair expression is not\ndirectly coupled to octahedral tilting. However, s − p mixing in-\nduces lone pair expression, and s−p mixing can stiffen octahedral\ntilting by changing the electron density on the halide and creating\n9\ntransient, partially covalent B-Br bonds. Though s−p mixing has\nan impact on both modes, the modes themselves are not directly\ncoupled and can fluctuate independently.\nWe have modified the text in the following manner to make this\npoint clear:\nTilting is coupled to the local Br electron density, which re-\norients to optimize interactions. The extent to which the Br\nelectron density can reorient is determined by the covalency\nof the B-Br bond, such that increased covalency stiffens tilt-\ning modes and reduces the extent of tilting.\nPoint 3: The conditional probability analysis done on the AIMD\ndata is used by the authors to reject the existence of a coupling be-\ntween the two features and, hence, between off-centering and tilts.\nGiven that the lone-pair expression (x) and off-centering (y) are\nreasonably strongly correlated, this negative result is surprising.\nWith the correlation between x and y, it would be better to com-\npute P(x|M4,4, y) and P(y|M4,4, x). The x—y distribution clearly\nsays that B-site lone-pair is more polarizable when the B ion is off-\ncentred. Therefore, it is this combined polarized state that should\nbe conditioned rather than just x or y, when checking against the\ntilts.\nOur Response: Lone pair expression (x) and off-centering (y)\nare correlated in a nearly one-to-one mapping demonstrated by\nthe fits in Fig. 4a. As a result, conditioning the tilting distribution\non both x and y will not provide additional information. The po-\nlarized state is the off-centered state; large x states are also large\ny states. We give the entire conditional probability distributions\nin Figures S2 and S3, from which the requested information can\nbe obtained. As an example, consider a value of off-centering\n10\nof r = 3 A˚ in CsSnBr , which would imply a large value of\noct 3\n4,4\nr . The conditional probability distributions P(M |r ) or\ndip K oct\n4,4\nP(M |r ) in Figures S2 and S3, respectively, for CsSnBr , are\nK dip 3\nidentical within noise for all values of r or r . Similar con-\noct dip\nclusions hold for CsPbBr and CsGeBr . As a result, octahedral\n3 3\ntilting is not altered by changes in r or r .\noct dip\nWe have clarified this point in the text by including additional dis-\ncussion of the SI figures:\nThis lack of correlation is further supported by probability\ndistributions of the tilting order parameter conditioned on r\ndip\nor r , which are essentially independent of lone pair expres-\noct\nsion or off-centering, Fig. S2-S3.\nPoint 4: A clear explanation (with appropriate evidence) for the\ncoupling of the off-centering with the lone-pair expression (modu-\nlation of LP polarization with off-centering) has not been provided\nin the manuscript. The dominance of the [111] off-centering is\nnot demonstrated in the manuscript (with P(omega) like plots as\ndone for the MLWF) but instead assumed from a previous study\non CsSnBr3. The reason for this dominance is speculated but not\nclearly proved by, for instance, looking at potential curves for dif-\nferent distortions.\nOur Response: Thank you for pointing out the need for additional\nclarification. The dominance of [111] off-centering is supported\nby the data in Fig. 3, combined with Fig. 2b. Fig. 2b shows that the\nB-site off-centers by an average distance given by the data point\non each curve. Fig 3 displays the orientation of the B-MLWFC\ndipole moment, showing that the lone pair points along [111] di-\nrections. Because the B-MLWFC dipole is linear (a vector con-\nnecting to points), this necessarily implies that the off-centering\n11\nof the B-site must also occur along this direction.\nWe have modified the text surrounding Fig. 3 to clarify this dis-\ncussion:\nBecause we are quantifying the orientation of the B-MLWFC\nvector, the probability distributions also indicate that B-site\noff-centering must be occurring along the [111] direction and\nnot along another axis.\nPoint 5: Overall, I feel that the main message, while interesting\nfor researchers in the field, is not significant to appeal to a wider\naudience. The main purposes of the work could be very convinc-\ningly demonstrated by ground-state DFT calculations combined\nwith the group theory analysis. A more detailed discussion on the\npossible couplings and the corresponding free energy/hamiltonian\nbeing considered would also improve the appeal of the work. The\nAIMD, though performed with rigour, does not offer more insights\nthan such an analysis would already deliver.\nOur Response: We respectively disagree with the reviewer here.\nThere is extensive discussion and debate in the literature about the\nnature of the tilting and off-centering modes, including very recent\npapers in this journal, suggesting that the topic is of interest to the\njournal’s readership.\nRegarding the use of the alternative approach that the reviewer\nseems to be suggesting — a combination of ground-state DFT,\ngroup theory analysis, and possibly the use of effective Hamilto-\nnians — the availability of another approach to the problem does\nnot invalidate the use of our particular approach. The use of AIMD\nsimulations includes anharmonicity present in the system at finite\ntemperature, making it well suited to probe the coupled (and not\ncoupled) fluctuations of anharmonic modes. AIMD simulations\n12\nadditionally enable the calculation of dynamics, which is absent\nin static ground-state DFT calculations, and the dynamic compe-\ntition between octahedral tilting and lone pair dynamics presented\nin Figure 6 further supports the validity of our conclusions.\nPoint 1: The plot of the log of g(r) is hard to understand for me. Is\nthis usual to plot it like this instead of the bare function g(r)? Do I\nunderstand correctly that g(r) for Ge peaks at 0, while for Sn and\nPB the peak is nonzero, while the averages are both nonzero? Can\nI understand the definition of a B-MLWFC as being equivalent to\nthe wannier center relative to the ideal, octahedral position in this\nwork?\nOur Response: We show ln[g(r ] to better illustrate the range\ndip\nof distances sampled, and this way of plotting g(r) is used often,\nparticularly because the potential of mean force is −k T lng(r).\nB\nYou are correct that the peaks do not correspond to the average\ndistances. The g(r) for Pb and Sn are peaked at zero, while that for\nGe is peaked at a nonzero value, and all systems have an average\nthat is nonzero, (cid:104)r (cid:105) > 0, which can be obtained from g(r )\noct oct\nby noting that the probability is P(r ) = 4πr2 g(r )dr and\noct oct oct oct\ncomputing the first moment of this distribution.\nThe B-MLWFC distance is the instantaneous distance between\nthe B-site cation and the center of the maximally localized Wan-\nnier function for the lone pair of that B-site cation. Nonzero B-\nMLWFC distances correspond to configurations with the lone pair\nstereochemically expressed, such that this is a dipole moment on\nthe cation.\nWe have tried to clarify this definition in the text:\nWe quantify the amount of lone pair expression through the ion-\nMLWFC pair distribution function, g(r ), where r is the dis-\ndip dip\ntance between the center of the ion and the MLWFC (the length\nof the instantaneous B-MLWFC dipole).\nzoomed in (e.g. fig 4).Fig. 2 a little hard to identify details as\nwell — Could the figure be made a bit larger? The colors in Fig2.\nmight be hard to distinguish when printed black/white\nOur Response: We thank the reviewer. We increased the size of\nthe legends in Figures 2 and 4, as well as y-axis labels in Figure 4.\nIn Figure 2, we made the linewidths different for each curve, Pb\nis the thickest, Ge the thinnest, and Sn in between. We also made\nthe data points indicating average distances larger.\nPoint 3: The discussion about the importance of the Halide-B\nbond is very interesting to me. The authors discuss a plethora of\noptions and possible alternative B cations. Is it possible to connect\nthe discussion here to the concept of hard acids and bases\n(HSAB)? For example, the very low solubility of AgI is explained\nby both Ag+ and I- being soft acids/bases, resulting in strong\nbonds. In contrast, AgF is highly solvable in water. This explo-\nration would allow it to connect it to concepts more familiar to\ninorganic chemists.\nOur Response: We thank the review for this insight that provides\nan intuitive chemical framework for our orbital-level argument.\nWe agree that the trend in bond stiffness agrees with and connects\nwell to the HSAB concept. We have added a new paragraph to the\nDiscussion based on this idea:\nThis observed trend in B-Br bond stiffness can also be ratio-\nnalized through the lens of Hard and Soft Acid-Base (HSAB)\ntheory. The Br− anion is a borderline base while the B-site\ncations, Pb2+ and Sn2+, are classified as borderline soft acids\ndue to their large size and polarizable ns2 lone pair, while the\n15\nsmaller Ge2+ cation is comparatively harder. The increas-\ning B-Br bond stiffness from Pb to Ge reflects a transition\nfrom a weaker, more polarizable soft-borderline interaction\nto a stronger, more iono-covalent hard-borderline interaction.\nThis provides chemical intuition for the increase in s-p mixing\nand partial B-Br bonding observed in Fig. 5.\nPoint 4: The results as they are presented are very clear and con-\nvincing, but I have trouble figuring out what implicit assumption\nthis analysis is based on. Are there scenarios where the authors\nhave reason to belief their analysis is qualitatively wrong,\ne.g. through change of DFT functionals or different simulation param-\neters. In other words, how robust are the findings and could one\neasily expand it to other materials in which lone pair expression\nis considered to be important, e.g. Pb-based thermoelectrics (see\nfor example the work of Jeff Snyder).\nOur Response (still need to add this to the revisons): We thank\nthe reviewer for this suggestion. The connection to thermoelectrics\n(TEs) is important, and we have added it to the Discussion.\nWe also expect this analysis aid in understanding thermoelec-\ntric materials that exhibit off-centering. In these systems, em-\nphanisis driven by the s2 lone pair leads to ultralow lattice\nthermal conductivity [64,71-73], and can create a phonon glass\nby providing a robust source of phonon scattering, which our\nperspective suggests is a result of electronic rotation-nuclear\ntranslation coupling.\n16"}]

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