The Tetrel Bond and Tetrel Halide Perovskite Semiconductors

The ion pairs [Cs+•TtX3−] (Tt = Pb, Sn, Ge; X = I, Br, Cl) are the building blocks of all-inorganic cesium tetrel halide perovskites in 3D, CsTtX3, that are widely regarded as blockbuster materials for optoelectronic applications such as in solar cells. The 3D structures consist of an anionic inorganic tetrel halide framework stabilized by the cesium cations (Cs+). We use computational methods to show that the geometrical connectivity between the inorganic monoanions, [TtX3−]∞, that leads to the formation of the TtX64− octahedra and the 3D inorganic perovskite architecture is the result of the joint effect of polarization and coulombic forces driven by alkali and tetrel bonds. Depending on the nature and temperature phase of these perovskite systems, the Tt···X tetrel bonds are either indistinguishable or somehow distinguishable from Tt–X coordinate bonds. The calculation of the potential on the electrostatic surface of the Tt atom in molecular [Cs+•TtX3−] provides physical insight into why the negative anions [TtX3−] attract each other when in close proximity, leading to the formation of the CsTtX3 tetrel halide perovskites in the solid state. The inter-molecular (and inter-ionic) geometries, binding energies, and charge density-based topological properties of sixteen [Cs+•TtX3−] ion pairs, as well as some selected oligomers [Cs+•PbI3−]n (n = 2, 3, 4), are discussed.


(a) Cesium Tetrel Iodide Perovskites
The unit-cell (top) and cage-like (bottom) structures of cesium tetrel iodide perovskites, CsTtI 3 (Tt = Pb, Sn and Ge), are shown in Figure 2a-c. The unit cell, an ion pair, [Cs + •TtI 3 − ], when periodically expanded, reveals a cage-like structure. The formation of the latter is driven by Tt···I tetrel bonds, which are equivalent and linear when Tt = Pb and Sn and quasi-linear and non-equivalent when Tt = Ge. For instance, the distances associated with the Tt-I coordinate bonds within the [TtI 3 − ] fragment and Tt···I tetrel bonds between a [TtI 3 − ] pair are 3.145 and 3.110 Å in cubic (Pm-3m) CsPbI 3 ( Figure 2a) and CsSnI 3 (Figure 2b), respectively. Because of the small size of the Ge cation, CsGeI 3 perovskite is not cubic (Figure 2c) and crystalizes in the rhombohedral space group R3m. Each of the three equivalent Ge-I coordinate bonds within the [GeI 3 − ] fragment is 2.747 Å, and each of the three equivalent Ge···I tetrel bonds between four [GeI 3 − ] fragments linked with each other in the CsGeI 3 is 3.262 Å. The latter are directional but quasi-linear, ∠I-Ge···I = 169.3 • , which is equivalent to the tilting angle, ∠Ge···I-Ge = 169.3 • , of the octahedra along each of the three principal axes. There is no structure of CsSiI 3 deposited in the ICSD.
In order to shed some light on the why the entirely negative anions [TtI 3 − ] attract each other, thus forming tetrel bonds between them (Figure 2, bottom), we performed both QTAIM and MESP analyses. The molecular graphs of cesium tetrel iodide perovskite ion pairs, [Cs + •TtI 3 − ] (Tt = Si, Ge, Sn and Pb), are shown in Figure 3a-d. The topology of the bond paths suggests that Tt-I coordinate bonds, characterized by solid lines, are shorter and stronger than the Cs···I close contacts, described by dotted lines; this is in line with the charge density values at their corresponding bcps, meaning that the Cs···I close contacts are weaker than the Tt-I coordinate bonds. Focusing on CsPbI 3 as a representative example of this series, each PbI 3 − unit, which is a face of the PbI 6 4− octahedron, is involved in an attractive coulombic interaction with Cs + , thereby forming the ion pair [Cs + •PbI 3 − ]. As shown in Figure 2a  faces of eight PbI 6 4− octahedra, forming a cage-like structure, with Cs + trapped inside the cage formed by the eight PbI 6 4− octahedra. In other words, the arrangement between the anion and the cation in CsPbI 3 is such that each face of the corner-shared PbI 6 4− octahedra hosts a Cs + cation (cf. Figure 2a). In order to shed some light on the why the entirely negative anions [TtI3 − ] attract each other, thus forming tetrel bonds between them (Figure 2, bottom), we performed both QTAIM and MESP analyses. The molecular graphs of cesium tetrel iodide perovskite ion pairs, [Cs + •TtI3 − ] (Tt = Si, Ge, Sn and Pb), are shown in Figure 3a-d. The topology of the bond paths suggests that Tt-I coordinate bonds, characterized by solid lines, are shorter and stronger than the Cs···I close contacts, described by dotted lines; this is in line with the charge density values at their corresponding bcps, meaning that the Cs···I close contacts are weaker than the Tt-I coordinate bonds. Focusing on CsPbI3 as a representative example of this series, each PbI3 − unit, which is a face of the PbI6 4− octahedron, is involved in an attractive coulombic interaction with Cs + , thereby forming the ion pair [Cs + •PbI3 − ]. As shown in Figure 2a (bottom), a single cation simultaneously interacts with eight PbI3 − faces of eight PbI6 4− octahedra, forming a cage-like structure, with Cs + trapped inside the cage formed by the eight PbI6 4− octahedra. In other words, the arrangement between the anion and the cation in CsPbI3 is such that each face of the corner-shared PbI6 4− octahedra hosts a Cs + cation (cf. Figure 2a).   In order to shed some light on the why the entirely negative anions [TtI3 − ] attract each other, thus forming tetrel bonds between them ( Figure 2, bottom), we performed both QTAIM and MESP analyses. The molecular graphs of cesium tetrel iodide perovskite ion pairs, [Cs + •TtI3 − ] (Tt = Si, Ge, Sn and Pb), are shown in Figure 3a-d. The topology of the bond paths suggests that Tt-I coordinate bonds, characterized by solid lines, are shorter and stronger than the Cs···I close contacts, described by dotted lines; this is in line with the charge density values at their corresponding bcps, meaning that the Cs···I close contacts are weaker than the Tt-I coordinate bonds. Focusing on CsPbI3 as a representative example of this series, each PbI3 − unit, which is a face of the PbI6 4− octahedron, is involved in an attractive coulombic interaction with Cs + , thereby forming the ion pair [Cs + •PbI3 − ]. As shown in Figure 2a (bottom), a single cation simultaneously interacts with eight PbI3 − faces of eight PbI6 4− octahedra, forming a cage-like structure, with Cs + trapped inside the cage formed by the eight PbI6 4− octahedra. In other words, the arrangement between the anion and the cation in CsPbI3 is such that each face of the corner-shared PbI6 4− octahedra hosts a Cs + cation (cf. Figure 2a).  The question that immediately arises as to why the cesium tetrel halide perovskites with Tt = Pb, Sn, and Ge have been synthesized, but not with Tt = Si. This is answered below. In short, it is the result of the nucleophilic nature of the electrostatic potential on the surface of Si in molecular [Cs + •SiI 3 − ] that prevents self-assembly of these recurring units. Two distinct features can be readily seen from Figure 4. First, the tetrel atom in the ion pairs carries either a negative or a positive potential (V S,max < 0 or V S,max > 0) along the outermost extension of each I-Tt bond. In specific, V S,max > 0 in [ − ] is accompanied by an appreciable transfer of charge from the iodide to Cs + cation, varying between 0.085 and 0.116 e. This causes the charge density on the surface of Tt to be rearranged in such a manner as to produce depleted regions of charge density on its surface. The charge density-depleted regions appear along the outermost extensions of the I-Tt bonds when Tt = Pb, Sn, or Ge, but not when Tt = Si. These charge density regions on the tetrel atoms along the extension of the three I-Tt bonds are characteristic of σ-holes since they appear opposite to the I-Tt σ coordinate bonds; thus, σ-holes on Tt are electrophilic in [Cs + •PbI below. In short, it is the result of the nucleophilic nature of the electrostatic potential on the surface of Si in molecular [Cs + •SiI3 − ] that prevents self-assembly of these recurring units.
Two distinct features can be readily seen from Figure 4. First, the tetrel atom in the ion pairs carries either a negative or a positive potential (VS,max < 0 or VS,max > 0) along the outermost extension of each I-Tt bond. In specific, VS,max > 0 in [Cs + •PbI3 − ], [Cs + •SnI3 − ], and [Cs + •GeI3 − ], becomes progressively less positive, and switches to VS,max < 0 in [Cs + •SiI3 − ]. This is the result of the polarizing field induced by Cs + when it is placed close to the [TtI3 − ] anion. The polarization of the surface charge density of Tt in [TtI3 − ] is accompanied by an appreciable transfer of charge from the iodide to Cs + cation, varying between 0.085 and 0.116 e. This causes the charge density on the surface of Tt to be rearranged in such a manner as to produce depleted regions of charge density on its surface. The charge density-depleted regions appear along the outermost extensions of the I-Tt bonds when Tt = Pb, Sn, or Ge, but not when Tt = Si. These charge density regions on the tetrel atoms along the extension of the three I-Tt bonds are characteristic of σ-holes since they appear oppo-  . This indicates that Pb and Sn are entirely electrophilic, unlike Ge and Si. The MESP graphs also suggest that the stereochemically active lone pair of the Tt sites are squeezed onto the surfaces of the iodides when Tt = Pb and Sn, but not when Tt = Ge and Si. This conclusion is in accordance with a previous study where it was suggested that "a stereochemically active lone pair of electrons of the Pb atom may lie between the two I atoms in the plane" [55]. It was recently argued that . This indicates that Pb and Sn are entirely electrophilic, unlike Ge and Si. The MESP graphs also suggest that the stereochemically active lone pair of the Tt sites are squeezed onto the surfaces of the iodides when Tt = Pb and Sn, but not when Tt = Ge and Si. This conclusion is in accordance with a previous study where it was suggested that "a stereochemically active lone pair of electrons of the Pb atom may lie between the two I atoms in the plane" [55]. It was recently argued that the s 2 lone pair on heavy main-group elements in their lower oxidation states is responsible for the emergence of polar ground states in some ferroic materials and causes a crystallographically hidden, locally distorted state that appears upon warming, a phenomenon referred to as emphanisis [56]. Others have argued that PbO and PbS in both the rocksalt and litharge structures, which have distorted Pb 2+ octahedra, are not the result of chemically inert, stereochemically active lone pairs, but instead are the result of asymmetric electron densities that rely on direct electronic interaction with the coordinated anions [57]. Further discussion on the importance of stereochemically active lone pairs on Pb in Pb(II) halide compounds can be found elsewhere [58].
The local most maximum potential on the surface of Cs + in the ion pairs is positive, V S,max > 0, and Cs + becomes increasingly more electrophilic down the series [Cs + •PbI . This is expected, as Si is relatively more electronegative in the tetrel series, hence, the ability of Cs + to polarize electron density of small-sized Tt decreases. In all cases, both the lateral and axial portions of the halide atoms in [TtI 3 − ] are entirely negative, with V S,min < 0, as seen in Figure 4a-d. Our QTAM results in Figure 3 and Table 1 suggest that the Si-I bonds have appreciable covalency, whereas the Tt-I (Tt = Ge, Sn, and Pb) in [Cs + •TtI 3 − ] have a more ionic character. The Si-I bonds are characterized by negative values of both ∇ 2 ρ b and H b at the Si-I bcps.
Although the Tt-I (Tt = Ge, Sn, and Pb) bonds possess ionic character, the negative H b values at the Tt-I bcps are indicative of some measure of covalency. These coordinate bonds clearly have mixed bonding character. On the other hand, the cesium-centered charge-assisted alkali bonds, Cs···I, possess closed-shell character (∇ 2 ρ b > 0 and H b > 0) and the charge density at the Cs···I bcps is significantly smaller than that at the Tt-I bcps.

(b) Cesium Tetrel Bromide Perovskites
The cesium tetrel bromide perovskites, [CsTtBr 3 ] (Tt = Pb, Sn, Ge), have been reported in different temperature crystalline phases, except for [CsSiBr 3 ]; the structures of the hightemperature cubic phase are shown in Figure 5. This is probably because the surface of the Si atom along the Br-Si bond extensions is entirely negative, so the Si atom in the [Cs + •SiBr 3 − ] ion pair is unable to coulombically attract the nucleophilic bromide in a neighboring unit. This is supported by the MESP plots of [Cs + •TtBr  Table 2) were found to be very similar to those observed for the Tt-I and Cs···I bcps in [Cs + •TtI 3 − ] (Table 1) Table 2) were found to be very similar to those observed for the Tt-I and Cs···I bcps in [Cs + •TtI3 − ] (Table 1). However, the charge densities at the Tt-Br and Cs···Br bcps in [Cs + •TtBr3 − ] were slightly larger; hence the strength of the Tt-Br and Cs···Br bonds are marginally stronger than the Tt-I and Cs···I bonds in [Cs + •TtI3 -]. The values of ∇ 2 ρb are negative at the Si-Br bcps and positive at the Tt-Br (Tt = Ge, Sn, Pb) bcps, as seen in Figure 7 and listed in Table 2. The extent of charge transfer from the anion to the cation lies between 0.092 and 0.095 e.   Table 2) were found to be very similar to those observed for the Tt-I and Cs···I bcps in [Cs + •TtI3 − ] (Table 1). However, the charge densities at the Tt-Br and Cs···Br bcps in [Cs + •TtBr3 − ] were slightly larger; hence the strength of the Tt-Br and Cs···Br bonds are marginally stronger than the Tt-I and Cs···I bonds in [Cs + •TtI3 -]. The values of ∇ 2 ρb are negative at the Si-Br bcps and positive at the Tt-Br (Tt = Ge, Sn, Pb) bcps, as seen in Figure 7 and listed in Table 2. The extent of charge transfer from the anion to the cation lies between 0.092 and 0.095 e.         We, and others, have shown on several occasions that caution needs to be exercised when the potential of a σ-hole on an atom in a molecular entity is near neutral. In such a case, a higher isoelectronic density envelope may be required for mapping the potential since the choice of isoelectronic density envelope is arbitrary. Indeed, this is the case with [Cs + •GeCl 3 − ]. When the 0.001 a.u. isoelectronic density was used for mapping, the potential associated with each of the three σ-hole holes on Ge was weakly negative (V S,max = −0.6 kcal mol −1 ). However, when a 0.0015 a.u. isoelectron density was used, the V S,max of the σ-holes on the same atom was positive, V S,max = 4.4 kcal mol −1 . The positive nature of the σ-hole on Ge explains why Ge in [Cs + •GeCl 3 − ] is capable of attracting the negative portion on the Cl atoms in a neighboring interacting ion-pair, thus leading to the formation of CsGeCl 3 perovskite crystals in the crystalline phase [45]. By contrast, changing the value of the isodensity envelope did not change the negative character of the σ-holes on the Si atom along the Cl-Si bond extensions; thus, CsSiCl 3 structures should not be formed when the ion pairs are repeated periodically.
There is a potential maximum on the Cs atom in the ion-pair that appears along the extension of the C 3v axis. Its origin could be due to a weak Tt···Cs interaction in [Cs + •TtCl 3 − ] (Tt = Si, Ge, Sn, Pb), as well as the formation of three equivalent Cs···Cl alkali bonds. The surface of Cs is strongly positive relative to that of Tt in each [Cs + •TtCl 3 − ] (Tt = Si, Ge, Sn, Pb), which rationalizes why the cation lies at the center of the inorganic tetrel halide cage, thus interacting simultaneously with the lone-pair dominant regions of coordinate halides on each of the eight faces of eight octahedra (each sitting at the corner of a cage, Figure 8).
The formation of the alkali bonds in each ion pair, which is expected to mimic what occurs in the crystal (vide infra), is evident in the molecular graphs shown in Figure 10a-d and Table 3. The accumulation of charge density at the Cs···Cl bcps is weaker than that at the Tt-Cl bcps. For the latter, it trends as Pb-Cl < Sn-Cl < Ge-Cl < Si-Cl, and, with ∇ 2 ρ b > 0 and H b < 0 (see values in Table 3), these bonds have mixed bonding character. This feature is clearly distinguishable from that of the alkali bonds that are largely electrostatic in character (∇ 2 ρ b > 0 and H b > 0). thus interacting simultaneously with the lone-pair dominant regions of coordinate halides on each of the eight faces of eight octahedra (each sitting at the corner of a cage, Figure 8).
The formation of the alkali bonds in each ion pair, which is expected to mimic what occurs in the crystal (vide infra), is evident in the molecular graphs shown in Figure 10a-d and Table 3. The accumulation of charge density at the Cs···Cl bcps is weaker than that at the Tt-Cl bcps. For the latter, it trends as Pb-Cl < Sn-Cl < Ge-Cl < Si-Cl, and, with ∇ 2 ρb > 0 and Hb < 0 (see values in Table 3), these bonds have mixed bonding character. This feature is clearly distinguishable from that of the alkali bonds that are largely electrostatic in character (∇ 2 ρb > 0 and Hb > 0).    The structures of [Cs + •TtF 3 − ] are similar to the other cesium halide perovskite ion pairs discussed above. While the formation of these ion-pair systems is likely in the gas phase, they are not all stable in the crystalline phase. The instability of these perovskite structures is arguably due to the mismatch between the cavity of the fluoride-based inorganic perovskite cage formed from the repeating units of [TtF 3 − ] and the radial size of Cs + . This is not the case for CsPbF 3 , as seen in Figure 11a, the structure of which was reported in 1956 (cubic, Pm3m, ICSD ref: 30739 [59]) and 2001 (ICSD refs: 93438-93439). Smith et al. [60] have suggested that CsPbF 3 is the only experimentally synthesized AMF 3 fluoride perovskite with a polar ground state. Our search of the ICSD showed that CsSnF 3 is not cubic (space group: P12 1 /n1(14) [61]) and hence is a non-perovskite (Figure 11b). The authors of that study suggested that this system exhibits a 'zero-dimensional' crystal structure with isolated SnF 3 − anions separated by Cs + cations; again, this is not surprising since the size of the cage formed by the repeating units of the SnF 3 − anion is too small to accommodate the guest Cs + . The ICSD does not contain structures of CsTtF 3 (Tt = Ge, Sn), but it catalogues crystals such as Cs 2 GeF 6 and Cs 3 GeF 7 , suggesting that the small size of fluoride, its low polarizability, and its high electronegativity lead it to form other types of crystal structures.
thors of that study suggested that this system exhibits a 'zero-dimensional' crystal structure with isolated SnF3 − anions separated by Cs + cations; again, this is not surprising since the size of the cage formed by the repeating units of the SnF3 − anion is too small to accommodate the guest Cs + . The ICSD does not contain structures of CsTtF3 (Tt = Ge, Sn), but it catalogues crystals such as Cs2GeF6 and Cs3GeF7, suggesting that the small size of fluoride, its low polarizability, and its high electronegativity lead it to form other types of crystal structures.  The results of our MESP calculations, shown in Figure 12, are in accordance with these rationalizations. They suggest the feasibility of the formation of [CsTtF 3 ] (Tt = Pb and Sn) structures in the solid state since the surfaces of the Tt site in the [Cs + •TtF 3 − ] ion-pair systems are highly electrophilic, with the former more so than the latter. Specifically, the surface of Pb in [Cs + •PbF 3 − ] is entirely positive along and around the F-Pb bond extensions (Figure 12a), whereas that of Sn is positive only along the F-Sn bond extensions, while the region around the outer extension of the C 3v axis is highly nucleophilic (Figure 12b). These positive sites are able to engage in a coulombic attraction with the negative site on the halogen of a neighboring unit to form structures of the types shown in Figure 11a (Table 4); Hb < 0 indicates that the bonds possess some covalency. The character of these coordinate interactions deduced from ∇ 2 ρb and Hb values are not the same as that found for the Cs···F bcps. For the latter, the sign of both ∇ 2 ρb and Hb are positive  (Table 4); H b < 0 indicates that the bonds possess some covalency. The character of these coordinate interactions deduced from ∇ 2 ρ b and H b values are not the same as that found for the Cs···F bcps. For the latter, the sign of both ∇ 2 ρ b and H b are positive ( Table 4), indicative of closed-shell (non-covalent) interactions.
[Cs•TtF3] (Tt = Pb, Sn, Ge, Si), mapped on their corresponding 0.001 a.u. isoelectronic density envelopes. The Tt atom in Cs•TtF3 faces the reader. Selected local most maxima and minima of potential (VS,max and VS,min) represented by tiny circles in red and blue, respectively, are depicted. Values are given in kcal mol −1 .
The formation of [Cs + •TtF3 − ] (Tt = Si, Ge, Sn, and Pb) ion pairs is also evident in the QTAIM-based molecular graphs shown in Figure 13a- (Table 4); Hb < 0 indicates that the bonds possess some covalency. The character of these coordinate interactions deduced from ∇ 2 ρb and Hb values are not the same as that found for the Cs···F bcps. For the latter, the sign of both ∇ 2 ρb and Hb are positive ( Table 4), indicative of closed-shell (non-covalent) interactions.

Oligomers of the [Cs + •PbI 3 − ] Ion Pair
We have sectioned the supercell structure of cubic CsPbI 3 (Figure 2a, bottom), and extracted the binary, trinary, and tertiary clusters in 1D. These were fully energy minimized at the same level of theory, [ωB97XD/def2-TZVPPD]. The geometries of the resulting [Cs + •PbI 3 − ] 2 dimer and [Cs + •PbI 3 − ] 3 trimer are shown in Figure 14a,b, respectively, together with their corresponding QTAIM-based molecular graphs in Figure 14c,d, respectively. The Pb-I bonds found in the crystal (top) are significantly elongated in the gas phase (bottom) (cf. Figure 14a,b). The ∠Pb-I···Pb angles between the ion pairs in cubic CsPbI 3 are linear, but non-linear in the gas-phase structure, leading to significant deformation passing from the solid-state structure to the gas-phase dimer and trimer. This discrepancy between the gas-phase and the solid-state geometries is not very surprising given that the role of packing forces is absent in the former. Interestingly, both the gas phase structures resemble the tilting of edge-sharing Pb-I chains in 1D, observed in the case of 3D CsPbI 3 . The ∠Pb-I···Pb angles are between 150 • and 154 • (Figure 14a,b, bottom), close to that seen between the edge-shared [PbI 6 ] 4− octahedra that are tilted relative to the corner-sharing octahedra in the low-temperature orthorhombic structure of CsPbI 3 (∠Pb-I-Pb = 148.1 • along the a-axis and 156.88 • along the c-axis; ICSD ref: 17016 [49]). On the other hand, and as noted above, the tetrel bonds between the ion pairs are longer than the Pb-I coordinate bonds and are quasi-linear (∠I-Pb···I = 166.8 • in [Cs + •PbI 3 − ] 2 ( Figure 14a) (Figure 14b). The physical chemistry of 1D CsPbI 3 has been experimentally investigated [62][63][64]. It was shown that in the orthorhombic (Pnma) γ-phase, the PbI 6 4− octahedra tilted around all three pseudocubic axes, a − a − c + , which is different to the tilt observed in the tetragonal (P 4 /mbm), a 0 a 0 c + , β-phase [65], and the bandgap increases with an increase in the octahedral tilting when the temperature cools down, allowing for the emergence of β-CsPbI 3 and γ-CsPbI 3 [66].
0 An interesting feature of the [Cs + •PbI3 − ]4 tetramer is that the Pb-I bonds are no longer equivalent, as found in cubic CsPbI3 in 3D (Figure 4a), but comparable with those found in the [Cs + •PbI3 − ]2 dimer and [Cs + •PbI3 − ]3 trimer (vide supra). Again, this is the result of the gas phase, where the role of the periodic boundary condition is nullified and no packing forces act on the system. The [Cs + •PbI3 − ] ion pairs are free to interact with each other in the gas phase at 0 K, causing the linear Pb-I-Pb bonds found in the cubic structure of An interesting feature of the [Cs + •PbI 3 − ] 4 tetramer is that the Pb-I bonds are no longer equivalent, as found in cubic CsPbI 3 in 3D (Figure 4a), but comparable with those found in the [Cs + •PbI 3 − ] 2 dimer and [Cs + •PbI 3 − ] 3 trimer (vide supra). Again, this is the result of the gas phase, where the role of the periodic boundary condition is nullified and no packing forces act on the system. The [Cs + •PbI 3 − ] ion pairs are free to interact with each other in the gas phase at 0 K, causing the linear Pb-I-Pb bonds found in the cubic structure of CsPbI 3 to change appreciably in a manner so as to adopt a significantly distorted geometry very similar to that observed in the low-temperature orthorhombic phase of the system (vide supra).
The molecular graphs of two different orientations of the [Cs + •PbI 3 − ] 4 tetramer are shown in Figure 16a  The formation of both Pb···I tetrel bonds and Cs···I alkali bonds between four units of the [Cs + •PbI3 − ] ion pairs in the [Cs + •PbI3 − ]4 tetramer can also be understood from the MESP plots shown in Figure 16c,d. Upon assembly, the σ-hole on three Pb atoms in three ion pairs of the tetramer is annihilated upon its attractive engagement with the iodide atom of a neighboring ion pair, thus forming Pb···I tetrel bonds. The four tetrel centers are positive (see the four green regions in Figure 16c), and one of them, which is not involved in   4 tetramer can also be understood from the MESP plots shown in Figure 16c,d. Upon assembly, the σ-hole on three Pb atoms in three ion pairs of the tetramer is annihilated upon its attractive engagement with the iodide atom of a neighboring ion pair, thus forming Pb···I tetrel bonds. The four tetrel centers are positive (see the four green regions in Figure 16c), and one of them, which is not involved in the formation of the tetrel bond (Figure 16c, top left), conceives three σ-holes on its surface; these can accept nucleophiles when in close proximity to another three ion pairs. By contrast, the Cs ions are highly electrophilic. These unequivocally provide evidence of the fact that the formation of the 3D network of the cage-like structures of cesium tetrel halide perovskites are the result of σ-hole-centered tetrel-bonded interactions between [PbI 3 − ] anions in the presence of Cs + . The physical chemistry of tetrel bonds also plays a significant role in stabilizing 1D CsPbI 3 , a material suitable for stable X-ray detection (sensitivity = 2.37 mC·Gy −1 ·cm −2 , resistivity = 7.4 × 10 9 Ω·cm, and carrier mobility-lifetime product = 3.63 × 10 −3 cm 2 ·V −1 [62]).

The interaction energies and intermolecular bond distances between [Cs + ] and [TtX 3
− ] for all the sixteen [Cs + •TtX 3 − ] ion pairs are summarized in Table 5. They are very large compared to ordinary non-covalent interactions but comparable with the binding energies of anion-molecule interactions. The charge-assisted tetrel bonds reported recently had energies (in kcal mol  [67]. Large interaction energies were also reported for tetrel bonds [62], halogen bonds [68], hydrogen bonds [68], and pnictogen bonds [69] in other anion-molecule complexes. The large interaction energies, E b and E b (BSSE), summarized in Table 5 are expected since a large part of the contribution arises from coulombic interaction between two interacting charged moieties. The binding energies are also comparable with those reported for similar halide perovskite ion-pairs at the [CCSD(T)/cc-pVTZ] level of theory [70].
For a series with a given halogen derivative [Cs + •TtX 3 − ], the interaction energies increase as the atomic size of the tetrel derivative increases (Si < Ge < Sn < Pb). This trend agrees well with the increasing strength of the σ-hole on Tt across the series for a given type of halogen derivative (see  Table 6, the largest interaction energy is calculated for cesium lead halide perovskite ion pairs, probably a consequence of the high polarizability of Pb compared to the other three tetrel derivatives. Among all the ion pairs examined, the ion pair of cesium lead fluoride perovskite, [Cs + •PbF 3 − ], is the strongest. This is expected since fluorine in the inorganic moiety [PbF 3 − ] is the most electronegative and electron-withdrawing of the halogens; hence, it strongly interacts with Cs + , which, therefore, cannot create strong σ-holes on Pb in [Cs + •PbF 3 − ]. All these trends remain valid regardless of whether E b or E b (BSSE) is considered since the BSSE is calculated to be very small.
As a rule of thumb, it is expected that the interaction energy increases as the intermolecular distance between interacting moieties decreases. However, this is not the case with the ion pairs explored in this study, where we find the opposite trend. The interaction energy for a given halogen derivative between [Cs + ] and [TtX 3 − ] increases when increasing the intermolecular distance between them (  Table 5. The interaction energy associated with the alkali bonds as a function of bond distance for [Cs + •TtX 3 − ] is shown in Figure 17. As a rule of thumb, it is expected that the interaction energy increases as the intermolecular distance between interacting moieties decreases. However, this is not the case with the ion pairs explored in this study, where we find the opposite trend. The interaction energy for a given halogen derivative between [Cs + ] and [TtX3 − ] increases when increasing the intermolecular distance between them (  Table 5. The interaction energy associated with the alkali bonds as a function of bond distance for [Cs + •TtX3 − ] is shown in Figure 17.  Figures 15 and 16 are -20.7, -21.3, and -22.4 kcal mol −1 , respectively. These were obtained by subtracting the total electronic energy of the oligomer from two, three, and four times the total electronic energy of the [Cs + •PbX3 -] ion pair, indicating that the binding energy is nearly additive. However, these may not be solely due to the tetrel bonds since each Cs + ion in each [Cs + •PbX3 − ] ion pair also contributes to the binding of the resulting oligomer through Cs···I alkali bonds. From these results, it is apparent that the empirical relationship Eb(QTAIM) = −½Vb is not applicable to tetrel bond energies since it largely underestimates the magnitudes. The relationship might be useful for some hydrogen-bonded systems [71], but may not be generalized to other non-covalent interactions such as the tetrel bonds explored in this work.

Discussion
This study was undertaken to reveal the underlying reasons why the TtX3 − units in-  , respectively. These were obtained by subtracting the total electronic energy of the oligomer from two, three, and four times the total electronic energy of the [Cs + •PbX 3 − ] ion pair, indicating that the binding energy is nearly additive. However, these may not be solely due to the tetrel bonds since each Cs + ion in each [Cs + •PbX 3 − ] ion pair also contributes to the binding of the resulting oligomer through Cs···I alkali bonds. From these results, it is apparent that the empirical relationship E b (QTAIM) = − 1 2 V b is not applicable to tetrel bond energies since it largely underestimates the magnitudes. The relationship might be useful for some hydrogen-bonded systems [71], but may not be generalized to other non-covalent interactions such as the tetrel bonds explored in this work.

Discussion
This study was undertaken to reveal the underlying reasons why the TtX 3 − units interact to form the inorganic cage-like tetrel halide frameworks of 3D cesium halide perovskites. We have shown that the connectivity between the TtX 3 − anions that lead to the formation of the 3D infinite inorganic framework, [TtX 3 − ] ∞ , is driven by the inorganic cation, Cs + , through the effects of both electrostatic polarization and coulombic attraction. The joint involvement of electrostatic polarization and coulombic attraction causes redistribution of the charge density profile on the electrostatic surfaces of the molecular tetrel halide perovskites, resulting in the development of positive σ-holes on the Tt atom in the TtX 3 − anions in the ion pairs [Cs + •TtX 3 − ] (Tt = Ge, Sn, Pb). This is accompanied by appreciable amount of charge transfer from the halides of the TtX 3 − anions to Cs + when they are in close proximity. The σ-holes formed on the Tt atom, therefore, are able to simultaneously attract the negative halogens from three interacting TtX 3 − anions, thus leading to the formation of TtX 6 4− octahedra, the underlying framework of the 3D CsTtX 3 cesium tetrel halide perovskites. The tetrel bonds formed are hidden between the anion moieties in the solid state structures, and to demonstrate their presence requires appropriate theoretical methods such as the QTAIM and MESP models.
There has been a failure to experimentally produce 3D tetrel halide perovskites when Tt = Si. We have explained this failure using the results of the MESP model. The underlying reason is the lack of an appreciable positive potential on the Si atom along the X-Si bond extensions in the molecular ion pair, [Cs + •SiX 3 − ]. Therefore, Si is incapable of engaging with the negative halogen in a neighboring unit because of the columbic repulsion between them.
Although the fluorinated ion-pair systems displayed electrophilic regions when Tt = Pb and Sn, the latter does not form a perovskite structure because of the small size of the cavity formed by repeating [SnF 3 − ] units, a cavity that cannot accommodate the inorganic cation. This might explain why CsSnF 3 crystalizes in low-dimension. By contrast, the results of the MESP model have showed that CsGeF 3 and CsSiF 3 perovskites cannot be formed in the crystalline phase because the Tt along the F-Tt bond extensions in the [Cs + •SiF 3 − ] (Tt = Ge, Si) ion pair has negative σ-holes. These negative σ-holes would repel the entirely negative fluorine atom(s) in a neighboring interacting unit(s), preventing the formation of Tt···F (Tt = Ge, Si) tetrel bonds. This also explains why the crystal structures of CsSiF 3 (Tt = Ge, Si) are unknown.

Computational Details
The geometries of sixteen ion pairs, [Cs + •TtX 3 − ] (Tt = Pb, Sn, Ge, Si; X = I, Br, Cl, F), were fully energy-minimized, followed by frequency calculations. The most stable conformer was considered. The ωB97XD functional [72] as implemented in the Gaussian 16 code [73], together with the def2-TZVPPD basis set retrieved from the EMSL basis set library [74], was employed. ωB97XD is known as a range-separated functional and is capable of capturing both short-range and long-range interactions. Minenkov et al. have demonstrated that the ωB97XD functional outperforms other commonly used DFT functionals (PBE and TPSS, M06 and M06L) and also produces relatively small statistical errors when considering the overall structure and inter-nuclear distances [75]. All ion-pair geometries were at an energy minimum, confirmed by positive harmonic vibrational frequencies.
Default convergence criteria (viz. tight SCF convergence and ultrafine integration grid) were invoked.
Similar calculations, as above, were also performed for a dimer, a trimer, and a tetramer of [Cs + •PbI 3 − ], which we refer to as the [Cs + •PbI 3 − ] n (n = 2, 3, 4) oligomers, to demonstrate the charge density topologies of alkali and tetrel bonding interactions responsible for the formation of the 3D network of CsPbI 3 . The nature of physical chemistry revealed for these systems might be transferable to other oligomers when Pb in [Cs + •PbI 3 − ] n (n = 2, 3, 4) is replaced, for example, by Sn and Ge.
Relativistic spin-orbit coupling (SOC) is an important feature of tetrel halide perovskite semiconductors containing heavy atoms such as Pb because its inclusion affects the band structure without affecting the crystal geometry [76]. In particular, the inclusion of SOC can affect the direct character of the band gap transition between the extrema of valence and conduction band states, especially for periodic systems containing Pb [77]. This effect, called the Rashba-Dresselhaus effect [78], is observed specifically in non-centrosymmetric environments and can be directly measured by angle-resolved X-ray photoemission spectroscopy. However, because our calculations are aperiodic and do not involve the calculation of the band structures of the molecular entities considered, the effect of SOC was not taken into account, in line with several recent studies [67,[79][80][81][82].
The uncorrected and BSSE-corrected interaction energies (E b and E b (BSSE), respectively) of the ion pairs [Cs + •TtX 3 − ] were calculated using Equations (1) and (2), respectively. BSSE refers to the basis set superposition error, evaluated using the counterpoise procedure of Boys and Bernardi [83], and E T [Cs + •TtX QTAIM [84] calculations were performed at the same level of theory described above. Properties such as the charge density (ρ b ), the Laplacian of the charge density (∇ 2 ρ b ), and the total energy density (H b ) at the (3, −1) bond critical points (bcps), critical points where the gradients of ρ(r) vanish, were analyzed. The latter two properties at bcps provide insight into the closed-shell and/or open-shell nature of an interaction between a pair of atomic basins in molecular and intermolecular entities. For instance, the positive and negative signs of ∇ 2 ρ b were utilized to demonstrate the closed-and open-shell interactions, respectively, which were identified between the inorganic anion and cation and between the anions [85][86][87][88][89]. Similarly, the positive and negative signs of H b (H b = G b + V b ) were utilized to provide insight into the absence and presence of a covalent interaction between the corresponding moieties, respectively [86,90]. This relies on the fact that a positive H b indicates a prevalence of the gradient kinetic energy density G b over the potential energy density V b , which is typical of non-covalently bonded interactions [91][92][93]. We have also used the empirical formula E b (QTAIM) = − 1 2 V b [71] to calculate the binding energy associated with the various non-covalent interactions identified in the ion-pairs investigated.
The MESP calculations [94][95][96][97][98][99] were performed with [ωB97XD/def2-TZVPPD], utilizing the fully relaxed geometries of the ion pairs and oligomers. The signs and magnitudes of the potential extrema were computed using the 0.001 a.u. isoelectronic density envelope of the ion pairs. The magnitude of potential is a measure of the strength, whereas its (positive and negative) signs were used to provide insight into regions of charge density depletion and concentration on the electrostatic surfaces of the ion pairs, respectively. That is, th corresponding signs of the local most maxima and minima of potential (V S,max and V S,min , respectively) were utilized to arrive at these conclusions. For instance, the sign of both V S,max and V S,min can either be positive or negative, or sometimes even neutral. When positive, it is generally assumed that the region on the surface that accompanies this is electrophilic and, hence, may be suitable for accepting electron density from an interacting electron donor in close proximity. When it is negative, the region on the surface of the molecular entity that features this is nucleophilic and, hence, may be capable of donating electron density to an interacting electrophile when in close vicinity. However, it should be kept in mind that all negative or positive sites on the surface of the molecular entity may or may not always be capable of engaging in attractive interaction with a region that features the opposite reactive profile.
An electrophilic σ-hole on atom A lying opposite to the R-A covalent bond is characterized when the sign V S,max is positive (V S,max > 0) [97,100]. Similarly, a nucleophilic σ-hole on atom A in R-A is observed when V S,max is negative (V S,max < 0) [97,100]. For instance, the σ-hole on F in H-F and H 3 C-F is negative, whereas that on X in H 3 C-X, X 3 C-X, and F 5 CX (X = Cl, Br, I) is positive. The underlying equation details and applicability of the MESP model to understand non-covalent interactions have appeared in several studies [67,95,96,99,[101][102][103][104][105][106][107], hence we do not repeat them here.

Data Availability Statement:
This research did not report any data.